diff --git "a/FNFQT4oBgHgl3EQfRTbc/content/tmp_files/load_file.txt" "b/FNFQT4oBgHgl3EQfRTbc/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/FNFQT4oBgHgl3EQfRTbc/content/tmp_files/load_file.txt" @@ -0,0 +1,882 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf,len=881 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='13286v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='RT] 30 Jan 2023 THE SEMI-INFINITE COHOMOLOGY OF WEYL MODULES WITH TWO SINGULAR POINTS GIORGIA FORTUNA, DAVIDE LOMBARDO, ANDREA MAFFEI, VALERIO MELANI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In their study of spherical representations of an affine Lie algebra at the critical level and of unramified opers, Frenkel and Gaitsgory introduced what they called the Weyl module Vλ corresponding to a dominant weight λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This object plays an important role in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In [4], we introduced a possible analogue Vλ,µ 2 of the Weyl module in the setting of opers with two singular points, and in the case of sl(2) we proved that it has the ‘correct’ endomorphism ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In this paper, we compute the semi-infinite cohomology of Vλ,µ 2 and we show that it does not share some of the properties of the semi- infinite cohomology of the Weyl module of Frenkel and Gaitsgory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For this reason, we introduce a new module ˜Vλ,µ 2 which, in the case of sl(2), enjoys all the expected properties of a Weyl module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Introduction Let g be a complex simple Lie algebra and let ˆg be its affinization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Choose a Borel subalgebra and a maximal toral subalgebra, and let G be a simply connected algebraic group with Lie algebra equal to g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As a particular case of a more general conjecture, Frenkel and Gaitsgory proved in [6] that the semi-infinite cohomology gives an isomorphism between the category ˆgcrit-modJG of spherical representations of ˆg at the critical level (that is, representations of ˆg at the critical level with a compatible action of JG = G(C[[t]])) and the category of quasi-coherent sheaves on the space of unramified opers Opunr 1 over gL, the Langlands dual of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As they explain, the space of unramified opers is the disjoint union of its connected components Opλ,unr 1 , and the category of spherical representations is the product of certain subcategories ˆgcrit-modJG,λ, where in both cases λ ranges over all dominant weights of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The equivalence given by semi-infinite cohomology specialises to an equivalence between ˆgcrit-modJG,λ and the category of quasi-coherent sheaves over Opλ,unr 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The space Opλ,unr 1 is a non-reduced indscheme, and its reduced version, denoted by Opλ 1, is an affine scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In this paper we will denote by Zλ 1 its coordinate ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In this theory, an important role is played by the Weyl module Vλ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This module enjoys the following fundamental properties: Endˆg(Vλ 1) ≃ Zλ 1 and Ψ0(Vλ 1) ≃ Zλ 1 , where Ψn is the n-th semi-infinite cohomology group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover the semi-infinite cohomology groups Ψn(Vλ 1) are trivial for n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Dennis Gaitsgory suggested to Giorgia Fortuna to study the space of unramified opers and spherical representations in a more general context, see [3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' in fact, the definition of unramified opers as well as the definition of spherical representations can be generalized in the presence of more than one singularity, raising the question on whether or not certain statements remain true and what happens when these singularities collide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 1 2 FORTUNA, LOMBARDO, MAFFEI, MELANI In [4] we took some steps in this direction, by studying the case of sl(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, we introduced a version of the Weyl module Vλ,µ 2 of critical level of the affine Lie algebra with two singularities ˆg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Thinking of t as a coordinate near the first singularity and s as a coordinate near the second singularity, this is the version of the affine Lie algebra over the ring A = C[[a]], where a = (t−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As an A module is equal to K2 ⊗C ⊕AC2 where K2 = C[[a, t]][1/t(t− a)] and C2 is a central element (see [4], Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 for the complete definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We also introduced reduced scheme over A of unramified opers Opλ,µ 2 which gener- alize the schemes Opλ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Both objects depend on two integral dominant weights λ, µ of G, and we proved that Endˆg2(Vλ,µ 2 ) ≃ Zλ,µ 2 , where Zλ,µ 2 is the coordinate ring of Opλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In this article we study the semi-infinite cohomology of Vλ,µ 2 in order to un- derstand what relation it has with the ring Zλ,µ 2 in order to understand how the equivalence Ψ0(Vλ 1) ≃ Zλ 1 generalizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This is done in Section 4, where we compute the cohomology of Vλ,µ 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' in Section 5 we study the action of Z2, the center of a completion ˆU2 of the enveloping algebra of ˆg2 at the critical level on this module (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, we prove that the specialisation at a = 0 and the localization at a ̸= 0 of the semi-infinite cohomology of Vλ,µ 2 are isomorphic to the specialisation and localization of Zλ,µ 2 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' However, in contrast to our intuition, we also show the following result which says that Ψ0(Vλ,µ 2 ) doesn’t exactly generalize the equivalence Ψ0(Vλ 1) ≃ Zλ 1 as expected: Theorem A (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='9 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We have Ψn(Vλ,µ 2 ) = 0 for n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, Ψ0(Vλ,µ 2 ) is not isomorphic to Zλ,µ 2 as a Z2-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For this computation, we rely on the formalism introduced by Casarin in [1], which makes it possible to use vertex algebras also in the context of opers with two singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Once this formalism is in place, for the computation of the semi- infinite cohomology we can follow closely the approach taken by Frenkel and Ben Zvi in [5, Chapter 15] for the case of one singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In the last section, we restrict our attention to the Lie algebra sl(2) and introduce a submodule �Vλ,µ 2 of Vλ,µ 2 , which is generated by the highest weight vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We prove that this module is the correct one to consider, in the sense that it has the expected cohomology groups and endomorphism ring, as the following result shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Theorem B (Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='5 and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' If g = sl(2) then we have Ψn(�Vλ,µ 2 ) = 0 for n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, we have Endˆg2(�Vλ,µ 2 ) ≃ Zλ,µ 2 and Ψ0(�Vλ,µ 2 ) ≃ Zλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now briefly explain the connection between these results and Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 in Fortuna’s Thesis [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As a particular case the conjecture predicts an equivalence between quasi-coherent sheaves over the space of unramified opers with two singu- larities and the category of spherical representations over ˆg2: that is the space of smooth representations of ˆg2 with a compatible action of J2G = G(C[[a, t]]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The conjecture stated in [3] predicts an equivalence of similar categories not only in the presence of two singularities but in the presence of n-possible singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular for any finite set with n elements I we can define the space of opers on the formal disc with n-singularities OpI and the subspace of unramified opers Opunr I (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='5 in [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These are spaces over the product of n-copies of the formal disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These are easily seen to be factorization spaces, which means that SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 3 this spaces specialise nicely when restricted along or outside the diagonals of this product (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='5 in [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' There are not substantial differences between the treatment we do here or in [4] of Op2 and the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The only minor difference is that we fix a singularity to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These spaces are indschemes, and so we can define the categories QCoh(OpI), and QCoh(Opunr I ) of quasi-coherent sheaves on OpI and Opur I (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 in [3] for the actual definition), and the nice factorization properties which make them factorization categories (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2 in [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similarly, for a finite set I we can define a Lie algebra ˆgI and study its smooth representations at the critical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The objects constructed in this way live also on the product of n copies of the formal disc, and they also have nice factorization properties, in particular the collection of (completions of the) enveloping algebras specialized at the critical level ˆUI of the algebras ˆgI, is what is called a factorization algebra (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 in [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As a conseguence the collection of the categories of smooth representations at the critical of the Lie algebras ˆgI, denoted by ˆgI,crit-mod and their subcategories of spherical representations ˆgI,crit-modJG can be organized also in a factorization category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The semi-infinite cohomology can be defined also in this generality and defines a functor ΨI : ˆgI,crit-mod −→ D(QCoh(OpI)) compatible with the factorization properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' While in Fortuna’s thesis all these constructions are obtained somehow for free using the language of chiral algebras (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='6 in [3]), in this paper we use the language of vertex algebras and the formalism introduced by Casarin [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let us notice that, from this point of view, there are no differences in treating the case with two singular points and the case with an arbitrary finite number of singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For example, the proof of Theorem A above can be repeated verbatim in the case of n singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' More generally we believe that all the technical difficulties in the study of this problem already appear in the case of two singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' It is easy to see from the factorization properties and the analogous statement for the case of one singularty by Frenkel and Gaitsgory (see [7]) that the semi- infinite cohomology of a ˆgI-spherical module is supported on Opunr I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence semi- infinite cohomology restricts to a functor ΨI : ˆgI,crit-modJG −→ D(QCoh(Opunr I )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 in [3] states that this functor is exact and Ψ0 I : ˆgI,crit-modJG −→ QCoh(Opunr I ) is an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2 of [3], one possible strategy to prove this conjecture is sketched, using the factorization structure and the result proved in the case of one singularity to deduce the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, thanks to Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2 and Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 of [3] and Theorems A and B above, a more careful study of the modules Vλ,µ 2 or �Vλ,µ 2 might help in finding a proof of [3, Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1], in the case of g = sl(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In the first section we recall some definitions from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In Section 3 we recall the formalism introduced by Casarin [1] and we use it to define semi-infinite cohomology and prove some of its basic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In Sections 3 and 4 we compute the semi-infinite cohomology of Vλ,µ 2 and in Section 5 we compute the semi-infinite cohomology of �Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We thank Luca Casarin for many useful discussions and in particular for explain- ing to us the formalism introduced in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' It seems to us that Casarin’s approach provides a natural framework to treat questions concerning opers with several singu- larities, making the theory much more transparent than it was in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, 4 FORTUNA, LOMBARDO, MAFFEI, MELANI the results of [1] allowed us to streamline several arguments and calculations which would have been quite hard to carry out using the direct approach of [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Basic constructions In this section we recall some basic constructions from [4], to which we refer for further details, and we introduce the notion of semi-infinite cohomology in the context of affine Lie algebras with more than one singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We follow [4, Section 1], to which the reader is referred for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We introduce the rings A = C[[a]], Q = C((a)), R2 = C[[t, s]], K2 = C[[t, s]][1/ts], where a = t − s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall that we have expansion maps (given by suitable natural inclusions) and a specialisation map (which sends a to 0 and t, s to t, see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 in [4]) Et : K2[a−1] −→ Q((t)), Es : K2[a−1] −→ Q((s)), Sp : K2 −→ C((t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We also write E = Et × Es : K2[a−1] −→ Q((t)) × Q((s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall from [4, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1] that Sp induces an isomorphism K2/(a) ≃ C((t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These rings have natural topologies: with respect to these, the image of E is dense, and E(R2[a−1]) is dense in Q((t)) × Q((s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These rings are also equipped with residue maps Res2 : K2 → A Res1 : C((t)) → C, Rest : Q((t)) → Q, Ress : Q((s)) → Q, which behave nicely with respect to specialisation and expansion (see [4, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Finally, we recall Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='10 in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 ([4], Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let M, N be two A-modules and ϕ : M −→ N be a morphism of A-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Then a) if M is flat and ϕa : M[a−1] −→ N[a−1] is injective, then ϕ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' b) if N is flat, ϕa : M[a−1] −→ N[a−1] is surjective, and ϕ : M/aM −→ N/aN is injective, then ϕ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, if M and N are flat, ϕa : M[a−1] −→ N[a−1] is an isomorphism, and ϕ : M/aM −→ N/aN is injective, then ϕ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Affine Lie algebras and completion of the enveloping algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We follow [4, Section 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let g be a finite-dimensional Lie algebra over the complex numbers and denote by κ the Killing form of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall from [4, Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3] that for each of the rings of the previous section we introduce an affine Lie algebra: ˆg1 is the usual affine Lie algebra (we take for convenience the version defined by Laurent polynomial and not Laurent series), ˆgt and ˆgs are also versions of the the usual affine Lie algebra, while ˆg2 is an A-Lie algebra having as underlying A-module the space ˆg2 = C[t, s][1/ts] ⊗C g ⊕ A C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We also introduce the Lie algebra ˆgt,s = ˆgt ⊕ ˆgs/(Ct − Cs) (see [4, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For each of these Lie algebras, we introduce the corresponding universal envel- oping algebra, which we suitably complete and then specialize at the critical level by imposing that the central element acts as −1/2 (see Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 in [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular ˆU2 = lim ←− n U(ˆg2) (C2 = −1/2, tnsnC[t, s] ⊗ g)left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall from [4, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4] that the expansion maps and the specialisation maps induce morphisms at the level of Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, the specialisation map Sp : ˆU2 −→ ˆU1 induces an isomorphism between ˆU2/a ˆU2 and ˆU1, while the SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 5 expansion map induces a morphism E : ˆU2[a−1] −→ ˆUt,s which is injective and has dense image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, the natural inclusions ˆgt ֒→ ˆgt,s and ˆgs ֒→ ˆgt,s induce a morphism ˆUt ⊗ ˆUs −→ ˆUt,s which is also injective and with dense image (see [4, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Weyl modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We follow [4, Section 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We choose a Borel subalgebra and a maximal toral subalgebra of g, which we denote by b and t respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This data induces a choice of weights, integral weights and dominant weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For every integral dominant weight λ, [7] introduced the Weyl module Vλ 1 over the affine Lie algebra ˆg1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The representation V = V0 1, which has a structure of vertex algebra, will play a particularly important role for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This vertex algebra enjoys the following universal property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let U be a vertex algebra such that there exists a linear map x �→ ux from g to U such that (ux)(0)(uy) = u[x,y] (ux)(1)(uy) = −1 2κ(x, y)|0⟩U (ux)(n)(uy) = 0 for all n ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' There exists a unique morphism of vertex algebras α : V → U such that α(xt−1|0⟩V) = ux for all x ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Weyl modules Vλ t and Vλ s can also be defined for the Lie algebras ˆgt and ˆgs, without any significant change from [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In [4], we introduced a generalization of these modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Given two dominant weights λ, µ, we consider the irreducible repres- entations V λ and V µ of the Lie algebra g having highest weights λ, µ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In [4, Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2], given two dominant integral weights λ, µ we introduced the module Vλ,µ 2 = Indˆg2 ˆg+ 2 � A ⊗C V λ ⊗C V µ� , where ˆg+ 2 = C[t, s] ⊗ g ⊕ A C2 acts on A ⊗C V λ ⊗C V µ as f(t, s)x · (p(a) ⊗ u ⊗ v) = f(0, −a)p(a) ⊗ xu ⊗ v + f(a, 0)p(a) ⊗ u ⊗ xv, while C2 acts as −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In [4] we called this object the Weyl module of weights (λ, µ), although, as we will see, it does not have the the same properties as its 1-singularity analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We also define Wλ,µ 1 = Indˆg1 ˆg+ 1 � V λ ⊗C V µ� , where ˆg+ 1 = C[t] ⊗ g ⊕ C C1 acts on V λ ⊗C V µ as f(t)x · (u ⊗ v) = f(0)x · (u ⊗ v) and C1 acts as −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The specialisation and expansion maps are defined also for Weyl modules, and induce the following isomorphisms [4, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3]: Vλ,µ 2 aVλ,µ 2 ≃ Wλ,µ 1 , Vλ,µ 2 [a−1] ≃ Vλ t ⊗Q Vµ s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Clifford algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now define the Clifford algebra with two singularities, generalizing the construction of the classical case (see for example [5, Chapter 15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let n+ be the nilpotent radical of b and set X2 = K2 ⊗C n+ ⊕ K2 ⊗C n∗ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We equip X2 with the unique A-bilinear form such that K2 ⊗C n+ and K2 ⊗C n∗ + are isotropic subspaces and (f ⊗ x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' g ⊗ ϕ) = Res2(fg) ϕ(x) 6 FORTUNA, LOMBARDO, MAFFEI, MELANI for all f, g ∈ K2, x ∈ n+ and ϕ ∈ n∗ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by Cℓ2 the associated Clifford algebra over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' There are obvious variants of the same construction where we replace K2 with the ring C[t±1] or one of the rings Q[t±1], Q[s±1], Q[t±1] × Q[s±1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We obtain Clifford algebras that we denote by Cℓ1, Cℓt, Cℓs, Cℓt,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The algebra CℓU in [5, Section 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1] is a completion of Cℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These Clifford algebras have a natural grading called the charge and denoted by ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' It can be defined as follows: the elements of the base ring have charge 0, while for ψ ∈ n and ψ∗ ∈ n∗ we have ch ψ = −1, ch ψ∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2) The relations defining each Clifford algebra are homogeneous, hence the charge induces a well-defined grading on the Clifford algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now introduce completions of the tensor product ˆU2 ⊗A Cℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We define ˆU2 ˆ⊗ACℓ2 = lim ←− n ˆU2 ⊗A Cℓ2 � (ts)nR2g ⊗ 1, 1 ⊗ (ts)nR2n+, 1 ⊗ (ts)nR2n∗ + � left ideal and we notice that, as in the case of the algebra ˆU2, this A-module has a nat- ural structure of A-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We introduce the completed Clifford algebras ˆU1 ˆ⊗Cℓ1, ˆUt ˆ⊗QCℓt, ˆUs ˆ⊗QCℓs, and ˆUt,s ˆ⊗QCℓt,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The specialisation and expansion map de- termine morphisms Sp : ˆU2 ˆ⊗ACℓ2 −→ ˆU1 ˆ⊗Cℓ1 and E : ( ˆU2 ˆ⊗ACℓ2)[a−1] −→ ˆUt,s ˆ⊗QCℓt,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Arguing exacly as in [4, Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='7 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='9] we see that E is injective with dense image, while the specialisation map induces an isomorphism ˆU2 ˆ⊗ACℓ2/a( ˆU2 ˆ⊗ACℓ2) ≃ ˆU1 ˆ⊗Cℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Finally, we have an injective map I : ˆUt ˆ⊗QCℓt → ˆUt,s ˆ⊗QCℓt,s induced by the natural inclusion Kt → Kt,s = Kt × Ks given by f �→ (f, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similarly, we have an injective map J : ˆUs ˆ⊗QCℓs → ˆUt,s ˆ⊗QCℓt,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 of [4], the product of these maps I ⊗ J : ( ˆUt ˆ⊗QCℓt) ⊗Q ( ˆUs ˆ⊗QCℓs) → ˆUt,s ˆ⊗QCℓt,s is injective with dense image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Fock module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now describe the “fermionic” Fock spaces corresponding to the Clifford algebras defined in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As above, for the construction in the case of one singularity we refer to [5, Section 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4]: here we mimic this definition in the case of two singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We define Cℓ+ 2 as the A-subalgebra of Cℓ2 generated by R2 ⊗ n+ and R2 ⊗ n∗ + and we define the Fock module Λ2 = Cℓ2 ⊗Cℓ+ 2 A |0⟩Λ2 where R2 ⊗ n+ and R2 ⊗ n∗ + acts trivially on |0⟩Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The charge (see equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2)) induces a grading on the Fock space by setting ch |0⟩Λ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by Λn 2 the subspace of elements of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similar constructions can be given for all the other Clifford algebras Cℓ1, Cℓt, Cℓs, and Cℓt,s, giving Fock modules Λ1, Λt, Λs, and Λt,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Specialisation and expansion, induce maps also at the level of the Fock spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Arguing as in [4, Section 6] (where we considered the module Vλ,µ 2 ), it is easy to prove the following Lemma: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' a) The specialisation map Sp : Λ• 2 −→ Λ• 1 is homogeneous of degree zero and induces an isomorphism Λ• 2/aΛ• 2 ≃ Λ• 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' b) We have a homogeneous isomorphism of degree zero Λ• t,s ≃ Λ• t ⊗Q Λ• s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 7 c) The expansion map E : Λ• 2[a−1] −→ Λ• t ⊗Q Λ• s is a homogeneous isomorphism of degree zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall also that the Fock space Λ = Λ1 has a natural structure of vertex super- algebra with the following universal property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let U be a vertex superalgebra such that there exists a linear map x �→ ux from n∗ + ⊕ n∗ + to the space of odd elements of U such that (1) for all ϕ, ψ ∈ n and for all ϕ∗, ψ∗ ∈ n∗ + (uψ)(n)(uϕ) = (uψ∗)(n)(uϕ∗) = (uψ)(m)(uψ∗) = (uψ∗)(m)(uψ) = 0 for all n ⩾ 0 and for all m ⩾ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (2) (uψ)(0)(uψ∗) = (uψ∗)(0)(uψ) = ⟨ψ, ψ∗⟩|0⟩U for all ψ ∈ n and ψ∗ ∈ n∗ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Then there exists a unique morhism of vertex superalgebras α : Λ → U such that α(ψt−1|0⟩Λ) = uψ and α(ψ∗t−1|0⟩Λ) = uψ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For each of the objects introduced above – base rings, enveloping al- gebras, Clifford algebras, and Fock spaces – it is not hard to construct explicit bases (or topological bases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We give the details in the case of two singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The construction of a basis depends on the choice of a basis of C[t, s][1/ts] as an A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Following [4], Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 and Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1) we introduce the following bases, indexed by 1 2Z: for n ∈ Z we define � zn = tnsn zn+ 1 2 = tn+1sn � wn = tnsn wn+ 1 2 = tnsn+1 The elements zm for m ∈ 1 2Z form a basis of C[t, s][1/ts] as an A-module, and the elements wn are the dual basis with respect to the residue bilinear form: more precisely, one has Res2(znw−m− 1 2 ) = δn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This specific choice of basis is not particularly important, and several others would be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' However, some properties need to be satisfied for our approach to work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particolar with our choice, the elements zm (or wm) with m ⩾ 0 form an A-basis of C[t, s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Since K2 is an A-free module, we deduce that the enveloping algebras of g2 and Cℓ2 are A-free modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, as R2 is a direct summand of K2, we also deduce that Vλ,µ 2 and Λ2 are also A-free modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Explicit bases of these modules, as well as an explicit topological basis of the algebra ˆU2 ˆ⊗ACℓ2, can be obtained using the Poincar´e-Birkhoff-Witt theorem and its analogue for Clifford algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Vertex algebras and semi-infinite cohomology In this section, we recall some results obtained by Casarin [1] which allow us to use the formalism of vertex algebras also in the context of several singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, using this formalism we develop a notion of semi-infinite cohomology for ˆU2-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Distributions and vertex algebra morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let R be a complete topo- logical associative A-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Following [1, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4], we denote by FA(K2, R) the space of continuous A-linear morphisms from K2 to R and call it the space of 2-fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We refer to [1] for the definitions of mutually local 2-fields (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1), of the n-products X(n)Y of two 2-fields (Definitions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='7) and of the derivative ∂(X) of a 2-field (before Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The definition in [1] applies also to the other ring we are considering: K1, Kt, Ks, Kt,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 8 FORTUNA, LOMBARDO, MAFFEI, MELANI In particular to define n products it is necessary to choose what in [1] is called a global coordinate (see definition ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We choose always t as a global coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' More explicitly for the rings K2, K1, Kt and Ks we choose t = s + a as a global coordinate, and for the ring Kt,s = Kt × K2 we choose (t, t) = (t, s + a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We also use some foundational results proved in this context in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, the following result will be crucial for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 ([1], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let F be a C-linear subspace of FA(K2, R) of mutually local 2-fields closed under derivation and n-products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let 1 be a field such that 1(f) is central for every f ∈ K2, that ∂ 1 = 0 and such that 1(n)X = δn,−1X for all X ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Then the vector dpace F + C1, endowed with n-products and derivation T = ∂, is a C-vertex algebra with 1 as vacuum vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' It is straightforward to generalize the constructions and results in [1] to the case of superalgebras R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We are interested in the case where R is the superalgebra ˆU2 ˆ⊗ACℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For x ∈ g, ψ ∈ n+ and ψ∗ ∈ n∗ + we define the 2-fields x(2)(g) = (x⊗g)⊗1Cℓ2, ψ[2](g) = 1 ˆU2⊗(ψ⊗g), (ψ∗)[2](g) = 1 ˆU2⊗(ψ∗⊗g) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1) for all g ∈ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The first of these fields has even parity with respect to the superal- gebra structure, while the second and third ones are odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These fields are mutually local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We consider the minimal C-linear subspace F(2) of ˆU2 ˆ⊗ACℓ2 closed under n-products and derivation and containing the fields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, we define 12(f) = Res2(f) � 1 ˆU2 ⊗ 1Cℓ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' It is easy to check that this data satisfies the hypothesis of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Therefore, V(2) = F(2) + C12 has a structure of vertex superalgebra, and by the universal properties of the vertex algebra V (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2) and of the vertex superalgebra Λ• (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4) it follows that there exists a morphism of vertex superalgebras Φ(2) : V ⊗C Λ• −→ V(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2) This homomorphism will allow us to easily introduce many elements in V(2), hence also in ˆU2 ˆ⊗ACℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similar constructions apply if the algebra ˆU2 ˆ⊗ACℓ2 is replaced by the algebras ˆU1 ˆ⊗Cℓ1, ˆUt ˆ⊗QCℓt, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence, we construct the fields x(1), ψ[1], x(t), ψ[t], the vertex superalgebras V(1), V(t), and homomorphisms of vertex algebras Φ(1) : V⊗C Λ• −→ V(1), Φ(t) : V ⊗C Λ• −→ V(t), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Notice that we have a specialisation morphism SpF : FA(K2, ˆU2 ˆ⊗Cℓ2) −→ FC(K1, ˆU2 ˆ⊗Cℓ1) and an expansion map EF : FA(K2, ˆU2 ˆ⊗ACℓ2) −→ FQ(Kt,s, ˆU2 ˆ⊗QCℓt,s), determined by the conditions � SpF(X) � (Sp(f)) = Sp(ϕ(f)) and � EF(X) � (E(f)) = E(ϕ(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These maps commute with n-products and derivations and satisfy SpF(12) = 11 and EF(12) = 1t,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, by construction they satisfy SpF(x(2)) = x(1) and EF(x(2)) = x(t,s) for x ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similar relations hold for ψ[2] and (ψ∗)[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This implies in particular that the homomorphisms SpF and EF restrict to homomorphisms of vertex algebras Sp : V(2) −→ V(1) and E : V(2) −→ V(t,s) such that Sp ◦Φ(2) = Φ(1) E ◦ Φ(2) = Φ(t,s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We can also describe the morphism Φ(2) through the morphisms Φ(t) and Φ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall from the end of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4 the maps I, J from ˆUt ˆ⊗QCℓt and ˆUs ˆ⊗QCℓs to ˆUt,s ˆ⊗QCℓt,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These maps induce maps at the level of fields IF : FQ(Kt, ˆUt ˆ⊗QCℓt) → SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 9 FQ(Kt,s, ˆUt,s ˆ⊗QCℓt,s) and JF : FQ(Ks, ˆUs ˆ⊗QCℓs) → FQ(Kt,s, ˆUt,s ˆ⊗QCℓt,s), given by IF(X)(f, g) = I(X(f)) and JF(X)(f, g) = J(X(g)) for all (f, g) ∈ Kt×Ks = Kt,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The maps IF and JF preserve n-products, commute with derivations, and satisfy IF(1t) + JF(1s) = 1t,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover we notice that I(u) and J(v) commute for all u ∈ ˆUt ˆ⊗QCℓt and v ∈ ˆUs ˆ⊗QCℓs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By the discussion in [1, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2], this implies IF ◦ Φ(t) + JF ◦ Φ(s) = Φ(t,s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This is the only statement where it is relevant the choice of the global coordinate we have done in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Semi-infinite cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now define a notion of semi-infinite cohomo- logy for ˆU2-modules, in analogy with the analogous notion for ˆU1-modules described for example in [5, Chapter 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' To this end, we introduce some notation for elements in the vertex superalgebra V ⊗ Λ•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As in the the case of ˆU1, to describe these ele- ments we choose a basis Ja of g compatible with the decomposition g = n−⊕t⊕n+, where n+ is the nilpotent radical of b and n− is the radical of the opposite nilpotent borel subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by cb,d e the structure coefficients of the Lie bracket with respect to this basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by Φ ⊔ Γ the indexing set of the basis Ja, so that, if α ∈ Φ, then Jα = eα = f−α is a root vector of weight α and, if α ∈ Γ, then Jα ∈ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We also denote by ψ∗ α for α ∈ Φ+ the basis of n∗ + dual to the basis eα of n+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' With each element in n+ ⊗ · · · ⊗ n+ ⊗ n∗ + ⊗ · · · ⊗ n∗ + we associate an element in the vertex superalgebra Λ as follows: N(ψ1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' ψℓ ⊗ ψ∗ 1 ⊗ · · · ⊗ ψ∗ m) = (ψ1t−1) · · · (ψℓt−1) · (ψ∗ 1t−1) · · · (ψ∗ mt−1) · |0⟩Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similarly, with an element in g ⊗ n∗ + we associate an element in the vertex superalgebra V ⊗ Λ∗ by setting M(x ⊗ ψ∗) = (xt−1) · |0⟩V ⊗ (ψ∗t−1) · |0⟩Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Following [5, Chapter 15] we define q =M(I) − 1 2|0⟩V ⊗ N(B) = � α∈Φ+ (eαt−1) · |0⟩V ⊗ (ψ∗ αt−1) · |0⟩Λ − 1 2 � α,β∈Φ+ cα,β α+β |0⟩V ⊗ (eα+βt−1) · (ψ∗ αt−1) · (ψ∗ βt−1) · |0⟩Λ, where I ∈ g ⊗ n∗ + represents the inclusion of n+ in g and B ∈ n+ ⊗ n∗ + ⊗ n∗ + is the Lie bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now define the boundary operator d(2) std ∈ ˆU2 ˆ⊗ACℓ2 as follows: d(2) std := � Φ(2)(q) � (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The boundary operator that we will use to define the semi-infinite cohomology is a deformation of d(2) std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let ψ∗ pr = � α simple ψ∗ α ∈ n∗ +, and define χ(2) = 1 ˆU2 ⊗ ψ∗ pr = Φ(2)(N(ψ∗ pr))(1) ∈ ˆU2 ⊗A Cℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similar constructions yield χ(s), χ(t), χ(s), and χ(s,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Finally set d(2) = d(2) std + χ(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As we will check in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3, this is an element that squares to zero, and therefore, it can be used to define the semi-infinite cohomology of a ˆU2-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 10 FORTUNA, LOMBARDO, MAFFEI, MELANI Similarly we can define d(1) std, χ(1), d(1), d(t) std, χ(t), d(t), and so on, as elements of the corresponding superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By the discussion at the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 we have Sp(d(2)) = d(1), E(d(2)) = d(t,s), and I(d(t)) + J(d(s)) = d(t,s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let M be an ˆU2 module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Consider the ˆU2 ˆ⊗ACℓ2-graded module M ⊗AΛ• 2, where the grading is given by charge on Λ• 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The element d(2) acts on this module as a boundary operator of degree one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Define Ψn(M) as the corresponding cohomology of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similar constructions apply to modules over the algebras ˆU1, ˆUt, ˆUs or ˆUt,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let Z2 be the center of the algebra ˆU2, and similarly introduce the center Z1 of ˆU1 and the centers Zt and Zs of ˆUt and ˆUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' If M is an ˆU2-module, the action of Z2 on M ⊗A Λ• 2 commutes with the differential d(2) and preserves the charge, hence induces an action of Z2 on the semi-infinite cohomology groups of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' A similar action is defined in the case of ˆU1-modules or ˆUt-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall that a module M over a topological algebra R is said to be smooth if the action of R on M is continuous with respect to the discrete topology on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Notice that, if M is a smooth ˆU2-module, then, since the map E has dense image, the action of ˆU2 on M extends to a smooth action of ˆUt,s on M[a−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similarly, if Mt is a smooth ˆUt-module and Ms is a smooth ˆUs-module, then there is an induced action of ˆUt,s on Mt ⊗Q Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In the next section we will use the following properties of the semi-infinite cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' a) Given a short exact sequence of ˆU2-modules, there is an induced long exact sequence in semi-infinite cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' b) Let M be an ˆU1-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The semi-infinite cohomology of M as an ˆU1-module is isomorphic to the semi-infinite cohomology of M considered as an ˆU2- module through the map Sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' c) Let M be an ˆUt,s-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The semi-infinite cohomology of M as an ˆUt,s- module is isomorphic to the semi-infinite cohomology of M considered as an ˆU2-module through the map E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, this applies to the case where M = N[a−1] is the localization of a smooth ˆU2-module N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' d) Let Mt be a smooth ˆUt-module, Ms be a smooth ˆUs-module, and let M := Mt ⊗Q Ms, regarded as a ˆUt,s-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The complex computing the semi- infinite cohomology of M is the total complex associated with the double com- plex given by the tensor product of the complex computing the semi-infinite cohomology of Mt and that of Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, being the base ring Q a field, if Mt and Ms have non zero semi-infinite cohomology only in degree zero, then M considered as an ˆUt,s-module has semi-infinite cohomology only in degree zero and the cohomology in degree zero is isomorphic to the product of the tensor product of Ψ0(Mt) and Ψ0(Ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Part a) follows from the fact that Λ2 is a free module over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Part b) follows from the fact that, since a ∈ A acts trivially on M, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 a) we have M ⊗A Λ• 2 ≃ M ⊗C Λ• 2 aΛ• 2 ≃ M ⊗C Λ• 1 and moreover, by construction, d(1) = Sp(d(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Part c) follows from the fact that, since the action of a on M is invertible, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 c) we have M ⊗A Λ• 2 = M ⊗A Λ• 2[a−1] = M ⊗A Λ• t,s SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 11 and, moreover, by construction, d(t,s) = E(d(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Finally, from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 c) we have (Mt ⊗Q Λ• t) ⊗Q (Ms ⊗Q Λ• s) ≃ M ⊗Q Λ• t,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Part d) then follows from the equality d(t,s) = I(d(t)) + J(d(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Commutation relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For their computation of the semi-infinite cohomo- logy of V, Frenkel and Ben Zvi (see [5] Chapter 15) relied on the choice of a clever basis of V ⊗ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For all x ∈ g, they define ˆx = xt−1 · |0⟩V ⊗ |0⟩Λ + N(αx), where αx ∈ n∗ +⊗n∗ + represents the linear map n+ → n+ obtained as the composition of adx : n+ −→ n+, the natural projection π : g −→ g/b−, and the inverse of the isomorphism n+ ∼= g/b− induced by π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Using the map Φ(2) from Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2) we define ˆx(2) = Φ(2)(ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' To compute the semi-infinite cohomology of Vλ,µ 2 we will need some information about the commutation relations among the elements ˆx(2), ψ[2], and (ψ∗)[2], and the boundary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These are easy to compute because all these objects are constructed through the map Φ(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let us make this remark precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Given an element x in V⊗Λ, denote by x(z) the corresponding field in the vertex superalgebra and by x(2) : K2 −→ ˆU2 ˆ⊗Cℓ2 the 2-field Φ(2)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For any choice of elements x, y ∈ V ⊗ Λ, the commutator of the corresponding fields is given by [x(z), y(w)] = � n⩾0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (x(n)y)(w)∂n wδ(z − w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We have a similar Operator Product Expansion formula for 2-fields (see [1], Pro- position 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4) [x(2)(f), y(2)(g)] = � n⩾0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' � (x(2))(n)(y(2)) � (g ∂nf), where the product (x(2))(n)(y(2))) is the product of 2-fields defined in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' However, since Φ(2) is a map of vertex algebras we get (x(2))(n)(y(2)) = (x(n)y)(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence, if we know the commutator of x(z), y(w), we immediately deduce that of x(2) and y(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similar considerations apply when we want to compute [x(2)(1), y(2)(g)] assuming we know the commutator of x(0) and y(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In this case, the usual OPE formula gives [x(0), y(w)] = (x(0)y)(w), while the OPE formula for 2-fields gives [x(2)(1), y(2)(g)] = � (x(2))(0)(y(2)) � (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Using again the fact that Φ(2) is a map of vertex algebras, we get [x(2)(1), y(2)] = Φ(2) �� [x(0), y(w)](|0⟩V ⊗ |0⟩Λ) � |w=0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These formulas are enough to determine all commutation relations among the ele- ments ˆx(2), ψ[2], (ψ∗)[2] and the boundary operators from those obtained by Frenkel and Ben Zvi in [5, Chapter 15], without the need of any further computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We summarise these results in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4 below, which (in light of the above) follows from Sections 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='9 of [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In the statement, we denote by epr, hpr, fpr the sl(2)-triple such that fpr = � α simple λαfα, κ(fpr, eα) = 1 for all simple root α and hpr ∈ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 12 FORTUNA, LOMBARDO, MAFFEI, MELANI Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' for all x ∈ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' y ∈ b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' z ∈ n+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' w ∈ b−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' ψ ∈ n+ and ψ∗ ∈ n∗ + we have: a) (d(2) std)2 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' [d(2) std,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' χ(2)]+ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' b) (χ(2))2 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (d(2))2 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' c) [χ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' ψ[2]]+ = ⟨ψ∗ pr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' ψ⟩ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' [χ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (ψ∗)[2]]+ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' d) [χ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' ˆz(2)] = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' [χ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' ˆw(2)] = � α ∈Φ+ κ([fpr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' z],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' eα)ψ∗ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' e) [d(2) std,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' ψ[2]]+ = ˆψ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' [d(2) std,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (ψ∗)[2]]+ = −1 2Φ � 1 ˆU2 ⊗ N(ψ∗ ◦ B) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' f) [d(2) std,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' ˆy(2)]+ = 0 where in the second formula of e) the element ψ∗ ◦ B ∈ n∗ + ⊗ n∗ + represents the composition of the bracket with the map ψ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, if we choose a basis Ja as at the beginnin of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2, for all γ ∈ Φ+ we have [d(2) std, ˆf (2) γ ]+ = � α∈Φ+,a∈Φ−⊔Γ cα,−γ a ( ˆJa)(2) (−1)(ψ∗ α)[2] − 1 2 κ(e−γ, fγ) ∂(ψ∗ −γ)[2] − � α,β∈Φ+, a∈Φ⊔Γ cα,a β cβ,−γ a ∂(ψ∗ α)[2] By specialisation and localization we obtain that similar formulas hold also in the case of our various other superalgebras ˆUt ˆ⊗Cℓt, ˆUt,s ⊗ Cℓt,s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The semi-infinite cohomology of Vλ,µ 2 In this section we compute the semi-infinite cohomology of Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by C• 2 = C• 2(λ, µ) the complex Vλ,µ 2 ⊗A Λ• 2 and similarly we introduce the complexes C• t = C• t (λ) = Vλ t ⊗Q Λ• t and C• s = C• s (µ) = Vµ s ⊗Q Λ• s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We further introduce the complexes C• 1(ν) = Vν 1 ⊗C Λ• 1 and C• 1(λ, µ) = Wλ,µ 1 ⊗C Λ• 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence, we have C• 1(λ, µ) ≃ ⊕C• 1(ν), where the sum ranges over the irreducible factors of V λ ⊗ V µ counted with multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by Op1 the indscheme of opers on the punctured disc and, for every integral dominant weight ν, we write Opν 1 for the associated connected component of the space of unramified opers without monodromy, equipped with its reduced structure (see, for example, [7] for a more complete definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We also denote by vν a highest weight vector in the g-module V ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Feigin and Frenkel [2] constructed an isomorphism F1 : Funct(Op1) −→ Z1 between the space of functions over Op1 and the center Z1 of ˆU1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall the following result, which combines Theorem 1, Theorem 2 and the proof of Proposition 1 in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 (Frenkel and Gaitsgory [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The action of Z1 on Vν 1 and the Feigin- Frenkel isomorphism induce an isomorphism G1 : Funct(Opν 1) −→ Endˆg1(Vν 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, the element vν ⊗|0⟩Λ is a cocycle in C• 1(ν) and the map z �→ [z ·vν ⊗|0⟩Λ from Z1 to Ψ0(Vν 1) induces isomorphisms of Z1-modules Funct(Opν 1) ≃ Endˆg1(Vν 1) ≃ Ψ0(Vν 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Finally, Ψn(Vν 1) vanishes for all n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The result of Frenkel and Gaitsgory generalises easily to the case of the modules Vλ t and Vµ s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 13 By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3, as in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3, by the compatibility of boundary operators we get homomorphisms of complexes Sp : C• 2 → C• 1(λ, µ) and E : C• 2 → C• t (λ) ⊗Q C• s (µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These induce isomorphisms C• 2[a−1] ≃ C• t (λ) ⊗Q C• s (µ) and C• 2 aC• 2 ≃ C• 1(λ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1) From these isomorphisms and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 it follows easily that Ψn(Vλ,µ 2 ) is zero for n ̸= 0, 1, and we could also get information on the cohomology in degrees zero and one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' However, it is easier to compute these cohomology groups directly by adapting the strategy employed by Frenkel and Ben Zvi in [5, Chapter 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In order to do this, we now introduce certain subcomplexes of C• 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by 1V0,0 2 the element 1 ∈ A ⊗C C ⊗C C ⊂ V0,0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by E• 2 the subcomplex of C• 2(0, 0) spanned by elements of the form ˆx(2) 1 (g1) · · · ˆx(2) a (ga) · 1V0,0 2 ⊗ ψ(2) 1 (ℓ1) · · · ψ(2) b (ℓb) · |0⟩Λ2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2) where xi, ψi ∈ n+ and g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' , ga, ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' ℓb ∈ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By the commutation relations of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 we see that E• 2 is a subcomplex of C• 2(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We define also analogous complexes E• t , E• s and E• 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These complexes were de- noted by C′ in [5] and by C0 in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By construction, these subcomplexes are compat- ible with specialisation and localization, and there are isomorphisms E• 2/aE• 2 ≃ E• 1 and E• 2[a−1] ≃ E• t ⊗Q E• s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by D• 2 = D• 2(λ, µ) the subcomplex of C• 2(λ, µ) spanned by elements of the form ˆy(2) 1 (h1) · · · ˆy(2) c (hc) · w ⊗ (ψ∗ 1)(2)(k1) · · · (ψ∗ d)(2)(kd) · |0⟩Λ2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3) where w ∈ V λ ⊗ V µ, yi ∈ b− = n− + t, ψ∗ i ∈ n∗ + and h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' , hc, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' , kd ∈ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By the commutation relations of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 we see that E• 2 is a subcomplex of C• 2(λ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We define also analogous complexes D• t (λ), D• s(µ) and D• 1(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These complexes were denoted by C0 in [5] and by C′ in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Finally, we denote by D• 1(λ, µ) the analogous subcomplex of C• 1(λ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By construction, these subcomplexes are com- patible with specialisation and localization, and there are isomorphisms D• 2/aD• 2 ≃ D• 1(λ, µ) and D• 2[a−1] ≃ D• t (λ) ⊗Q D• s(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' There is an isomorphism of complexes E• 2 ⊗ D• 2 −→ C• 2 defined by � x · 1V0,0 2 ⊗ ψ · |0⟩Λ2 � ⊗ � y · w ⊗ ψ∗ · |0⟩Λ2 � �−→ x · y · w ⊗ ψ · ψ∗ · |0⟩Λ2, where x = ˆx(2) 1 (g1) · · · ˆx(2) a (ga) and ψ = ψ(2) 1 (ℓ1) · · · ψ(2) b (ℓb) are as in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2), y = ˆy(2) 1 (h1) · · · ˆy(2) c (hc) and ψ∗ = (ψ∗)(2)(k1) · · · (ψ∗)(2)(kd) are as in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3), and w is an element of V λ ⊗ V µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now compute the cohomology of the complex E• 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We will need the following result by Frenkel and Ben Zvi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4 ([5, Section 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hn(E• 1) = 0 for n ̸= 0 and Ψ0(E• 1) = C[|0⟩V ⊗ |0⟩Λ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This result generalizes easily to the case of E• t and E• s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Localizing and special- izing, we deduce the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hn(E• 2) = 0 for n ̸= 0 and H0(E• 2) = A[1V0,0 2 ⊗ |0⟩Λ2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 14 FORTUNA, LOMBARDO, MAFFEI, MELANI Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By definition, the complex E• 2 is concentrated in non-positive degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence, the long exact sequence induced by 0 � E• 2 a· � E• 2 � E• 1 � 0 implies that Hn(E• 2) is torsion free for every n, and that the specialisation of H0(E• 2) is isomorphic to H0(E• 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Since semi-infinite cohomology commutes with localization (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3), using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4 we get the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ We now compute the cohomology of D• 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The strategy is similar, but the ar- gument is less straightforward since we do not have an explicit representative for H0(D• 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Following the strategy in [5], we introduce the following bigraded struc- ture on D• 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall that the height ht(α) of a root α is equal to the sum of the coefficients of α when written as a sum of simple roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let also epr, hpr, fpr be an sl(2)-triple such that fpr = � α simple fα and hpr belongs to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We define a bidegree, with values in 1 2Z × 1 2Z and denoted by bideg, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' On elements of ˆg2, we set bideg(x ⊗ g) = (−n, n) if x ∈ g is such that [hpr, x] = 2 n x and g ∈ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We set also the bidegree of the central element C2 ∈ ˆg2 to be (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This induces a bidegree on U(ˆg2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' On the space X2 = K2 ⊗ n+ ⊕ K2 ⊗ n∗ + (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4) we define bideg eα ⊗ g = (− ht(α), −1 + ht(α)) bideg ψ∗ α ⊗ g = (ht(α), 1 − ht(α)) for α a positive root and g any element of K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This induces a bidegree on the Clifford algebra Cℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, if W is any finite-dimensional representation of g, then we set bideg w = (−n, n) if w ∈ W is such that hpr · w = 2 n w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These choices induces a bidegree on the module C• 2(λ, µ), and the element ˆx(2)(g) is homogeneous of bidegree (−n, n) if [hpr, x] = 2 n x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Finally, notice that if an element has bidegree (p, q), then it has charge p + q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, we introduce the submodule Dp,q 2 of elements of Dp+q 2 of bidegree (p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We notice also that bideg d(2) std = (0, 1) and that bideg χ(2) = (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particu- lar, D•,• 2 is a double complex and D• 2 is the associated total complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Following Frenkel and Ben Zvi [5, Chapter 15], the cohomology of the rows of this double complex is easy to describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let a be the centralizer of fpr in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall from [5, Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 and Section 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='9] that the space spanned by monomials of the form (ˆp1)n1 · · · (ˆpk)nk ·|0⟩V ⊗|0⟩Λ with pi ∈ a generates a commutative vertex subalgebra F1 of V ⊗ Λ• isomorphic to S•(a ⊗ t−1C[t−1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3, it follows that for x, y ∈ a the fields ˆx(2) and ˆy(2) commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We define F2(λ, µ) as the span of elements of the form ˆx(2) 1 (g1) · · · ˆx(2) k (gk) · (v ⊗ |0⟩Λ2) ∈ Vλ,µ 2 ⊗A Λ• 2 with x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' , xk ∈ a and v ∈ V λ ⊗ V µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Notice that all these elements have charge equal to zero, and that the space F2(λ, µ) splits as a direct sum F2(λ, µ) = � q F −q,q 2 (λ, µ) according to the bidegree introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, by Propos- ition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4 d), these elements are annihilated by the action of χ(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similarly we construct subspaces F −q,q 1 (ν) ⊂ Vν 1 ⊗C Λ• 1, F −q,q t (λ) ⊂ Vλ t ⊗Q Λ• t , F −q,q s (µ) ⊂ Vλ s ⊗Q Λ• s, and F −q,q 1 (λ, µ) ⊂ Wλ,µ 1 ⊗C Λ• 1, In particular, F −q,q 1 (λ, µ) = SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 15 � ν F −q,q 1 (ν) where the sum is over all irreducible factors of V λ ⊗C V µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By con- struction, the specialisation and localization maps induce isomorphisms F −q,q 2 (λ, µ) aF −q,q 2 (λ, µ) ≃ F −q,q 1 (λ, µ) and F −q,q 2 (λ, µ)[a−1] ≃ � b+c=q F −b,b t (λ) ⊗Q F −c,c s (µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall the following result on the cohomology of D•,q 1 with respect to the bound- ary χ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='7 ([5, Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='10] and [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let 2pν = ⟨ν, hpr⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' a) Dp,q 1 (ν) = 0 for q > pν and for p < −q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, Dp,q 1 = 0 for q > pλ+µ and for p < −q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' b) Hn(D•,q 1 (ν)) = 0 for n ̸= −q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, Hn(D•,q 1 (λ, µ)) = 0 for n ̸= −q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' c) The map v �→ [v] from F −q,q 1 (ν) to H−q(D•,q 1 (ν)) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Finally, it follows from c) that the map v �→ [v] from F −q,q 1 (λ, µ) to H−q(D•,q 1 (λ, µ)) is also an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similar results hold for the complexes D•,q t (λ) and D•,q s (µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' From this result we deduce the cohomology of the complex D•,q q with respect to the boundary operator χ(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let 2p0 = ⟨λ + µ, hpr⟩ as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' a) Dp,q 2 = 0 for q > p0 and for p < −q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' b) Hn(D•,q 2 ) = 0 for n ̸= −q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' c) The map v �→ [v] from F −q,q 2 (λ, µ) to H−q(D•,q 2 (λ, µ)) is an isomorphism of A-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Part a) is clear for the definition of Dp,q 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For parts b) and c), we start by studying the localization of the cohomology groups of D•,q 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Equivalently, we aim to compute the cohomology of the localization of the row D•,q 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This localization can be rewritten as � b+c=q D•,b t (λ) ⊗ D•,c s (µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, it follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='7 that its cohomology is concentrated in degree −q, and that its cohomology in this degree is given by � b+c=q F −b,b t (λ) ⊗ F −c,c s (µ), which is the localization of F −q,q 2 (λ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Since specialisation is compatible with bideg, we have an isomorphism D•,q 2 /aD•,q 2 ≃ D•,q 1 (λ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='7, the associated long exact sequence shows that Hn(D•,q 2 ) is torsion-free for n ̸= −q + 1, and that the map ι : H−q(D•,q 2 )/aH−q(D•,q 2 ) → H−q(D•,q 1 (λ, µ)) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now prove c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Notice that both F −q,q 2 (λ, µ) and H−q(D•,q 2 (λ, µ)) are torsion- free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We have already shown that the localization of the natural maps between them is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' To study its specialisation, we compose it with the injection ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This composition is the isomorphism of the last remark of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We conclude by applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In order to prove b), it is enough to notice that from the above discussion we know that, for n ̸= −q, the module Hn(D•,q 2 ) = 0 is torsion-free, and that its localization is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ 16 FORTUNA, LOMBARDO, MAFFEI, MELANI Let now be ϕ(q) i be an A-basis of F −q,q 2 (λ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Since the cohomology in degree −q of the complex D•,q+1 2 is zero, there exists an element ϕ(q) i,1 ∈ D−q−1,q+1 2 such that χ(2)(ϕ(q) i,1 ) = −d(2) std(ϕ(q) i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By induction, we can construct elements ϕ(q) i,0 = ϕ(q) i and ϕ(q) i,ℓ ∈ D−q−ℓ,q+ℓ 2 such that their sum ˜ϕ(q) i = p0−q � ℓ=0 ϕ(q) i,ℓ satisfies d(2)( ˜ϕ(q) i ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now prove the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' a) Ψn(Vλ,µ 2 ) = 0 for n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' b) We have an isomorphism Ψ0(Vλ,µ 2 ) aΨ0(Vλ,µ 2 ) ≃ Ψ0(Wλ,µ 1 ) ≃ � ν Ψ0(Vν 1) where the sum ranges over all irreducible components V ν of V λ ⊗V µ, counted with multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' c) The elements � ˜ϕ(q) i � are an A-basis of Ψ0(Vλ,µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='8 we deduce that the classes of the elements ˜ϕ(q) i form an A-basis of H0(D• 2), and that Hn(D• 2) = 0 for n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As the complex D• 2 is con- centrated in non-negative degrees, by a standard homological argument we deduce that Hn(Vλ,µ 2 ) is isomorphic to the n-th cohomology of the complex H0(D• 2)⊗AE• 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='5, we immediately obtain parts a) and c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The second isomorphism appearing in part b) is clear, while the first follows from a) and the long exact sequence associated with the isomorphism C• 2 aC• 2 ≃ C• 1(λ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ We will use the following Corollary in the next Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The element [vλ ⊗ vµ] ∈ Ψ0(Vλ,µ) is indivisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By the previous theorem we can choose [vλ ⊗ vµ] as an element of a basis of the free A module Ψ0(Vλ,µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The action of the center In this section we study the action of the center Z2 on the semi-infinite cohomo- logy of the module Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In this section we show that Vλ,µ 2 is not a perfect analogue of the Weyl module Vν 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Indeed, we show that, as a Z2-module, the semi-infinite cohomology of Vλ,µ 2 is not isomorphic to Endˆg2(Vλ,µ 2 ) or to Funct(Opλ,µ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We begin by observing that the module Ψ0(Vν 1) has no non-trivial Z1-equivariant automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' First we notice, that by construction, the action of Z2 commutes with localization and specialisation, as introduced before Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Concretely, we have: Et(z · x) = Et(z) · Et(x), Es(z · x) = Es(z) · Es(x), Sp(z · x) = Sp(z) · Sp(x) for all z ∈ Z2 and for all x ∈ Ψ0(Vλ,µ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' If K : Endˆgt(Vλ t )⊗QEndˆgs(Vµ s ) −→ Ψ0(Vλ t )⊗QΨ0(Vµ s ) is a (Zt⊗Zs)- equivariant isomorphism, then K(IdVλ t ⊗ IdVµ s ) = q[vλ] ⊗ [vµ] for some q ∈ Q ∖ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' It follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 that Endˆgt׈gs(Vλt ⊗Q Vµs) is isomorphic to Funct(Opλ t ×Spec Q Opµ s ) = Funct(Opλ t ) ⊗Q Funct(Opµ s ) and this is a polynomial ring in infinitely many variables over the field Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, its only invertible elements are the non-zero scalars in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 also implies that Funct(Opλ t ) is isomorphic as a Zt- module to Ψ0(Vλ t ), with an isomorphism given by z −→ Gt(z) · [vλ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ Before proving that Vλ,µ 2 does not have the “right” semi-infinite cohomology we recall some properties of the modules Vν 1 that will be needed also in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by Zν 1 the coordinate ring of the scheme Opν 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall that the schemes Opν 1 for different values of ν are disjoint, so that the map Z1 −→ Zν1 1 × · · · × Zνk 1 is surjective if the weights νi are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall also that the ring Zν 1 is a polynomial ring in infinitely many variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This implies that (1) There are no nontrivial ˆg1-morphisms between the ˆU1-modules Vν 1 and Vν′ 1 if ν ̸= ν′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (2) There are no nontrivial extensions between the ˆU1-modules Vν 1 and Vν′ 1 if ν ̸= ν′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (3) Assume that α : � Zνi −→ � Zνi is a map of Z-modules and that the weights νi are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' If 1 is in the image of α then α is an isomorphism and α(Zνi 1 ) = Zνi 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By the Feigin-Frenkel Theorem (see [4] Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2) the ring Funct(Op2) is isomorphic to Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In the sequel we will identify these rings through this isomorph- ism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular the ring Funct(Opλ,µ 2 ) is a quotient of Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We will denote Funct(Opλ,µ 2 ) by Zλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now prove that Zλ,µ 2 and Ψ0(Vλ,µ 2 ) are not isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Assume that V λ ⊗ V µ is not irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Then the two Z2- modules Endˆg2(Vλ,µ 2 ) and Ψ0(Vλ,µ 2 ) are not isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similarly the two Z2- modules Zλ,µ 2 and Ψ0(Vλ,µ 2 ) are not isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Suppose H : Endˆg2(Vλ,µ 2 ) −→ Ψ0(Vλ,µ 2 ) is a Z2-equivariant isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='28 in [4] that Z2[1/a] is dense in Zt,s, and therefore the localization of H is a (Zt ⊗Q Zs)-equivariant isomorphism Endˆgt(Vλ t ) ⊗Q Endˆgs(Vµ s ) −→ Ψ0(Vλ t ) ⊗Q Ψ0(Vµ s ), where we used the identification of the localization of Ψ0(Vλ,µ 2 ) with Ψ0(Vλ t ) ⊗Q Ψ0(Vµ s ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' From Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='10 we deduce that H(IdVλ,µ 2 ) = [q vλ ⊗ vµ], where q ∈ A and qvλ ⊗ vµ ∈ Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We set w = qvλ ⊗ vµ ∈ Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By specialisation, H gives a Z1-equivariant isomorphism H : Endˆg2(Vλ,µ 2 ) a Endˆg2(Vλ,µ 2 ) −→ Ψ0(Vλ,µ 2 ) aΨ0(Vλ,µ 2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1) This isomorphism sends IdVλ,µ 2 to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Now consider the decomposition V λ ⊗ V µ = � V ν as g-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='9, the target of the map H in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1) decomposes as � Ψ0(Vν 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The element w is a multiple of vλ ⊗ vµ hence its class belongs to Ψ0(Vλ+µ 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As H is Z1-equivariant and Vλ+µ 1 is stable by the action of ˆg1, we get that the image of H is contained in the direct summand Ψ0(Vλ+µ 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, if V λ ⊗ V µ is not irreducible, the map H cannot be surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This proves the 18 FORTUNA, LOMBARDO, MAFFEI, MELANI first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The second claim follows since the map from Zλ,µ 2 to Ψ0(Vλ,µ 2 ) factors through Endˆg2(Vλ,µ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' A Weyl module for sl(2) In this section, we propose an alternative Weyl module in the context of opers with two singularities, in the case of g = sl(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We fix the following notation: e, h, f is an sl(2)-triple such that h ∈ t and e ∈ n+, while ψ∗ ∈ n∗ + is the dual of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We identify dominant weights with natural numbers and we assume from now on that λ ⩾ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In this case, the differential of the complex computing semi-infinite cohomology takes the simpler form d(2) = ψ∗ + � ewn ⊗ ψ∗z−n−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let �Vλ,µ 2 be the ˆU2-submodule of Vλ,µ 2 generated by the highest weight vector 1 ⊗ vλ ⊗ vµ ∈ A ⊗ V λ ⊗ V µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We will prove that this module has the “correct” semi-infinite cohomology and the “correct” endomorphism ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We start by giving a more explicit description of the module �Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' If X is a subspace of U(g) and Y is a subspace of a g-module Z we denote by X · Y the subspace of Z generated by the products x · y with x ∈ X and y ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We define an increasing filtration F i of �Vλ,µ 2 as follows F i = U(g) · (C Id ⊗ Id + Id ⊗ g)i · (vλ ⊗ vµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This is an increasing filtration of V λ ⊗ V µ by g-modules and for i large enough we have F i = V λ ⊗ V µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Choose a g-stable complement Gi+1 of F i in F i+1 and set G0 = F 0, so that F i = �i j=0 Gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' If we set F i(V µ) = (CId + n−)ivµ, it is easy to check by induction on i that F i = U(g) · (Id ⊗ Id + Id ⊗ n−)i(vλ ⊗ vµ) = U(g) · � V λ ⊗ F i(V µ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In the case of g = sl(2) we have Gi ≃ V λ+µ−2i and F µ(V µ) = V µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let U − 2 ⊂ U(ˆg2) be the A-span of Poincar´e-Birkhoff-Witt monomials of the form (x1wa1) · · · (xkwak) with xi ∈ g and ai < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This is a complement of U(ˆg+ 2 ) in U(ˆg2), so that in particular we have Vλ,µ 2 = U − 2 ⊗C (V λ ⊗ V µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' If λ ⩾ µ then �Vλ,µ 2 = µ � i=0 aiU − 2 ⊗C F i = µ � i=0 aiU − 2 ⊗C Gi Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' To understand the module �Vλ,µ 2 we need to compute the ˆg+ 2 -submodule of A⊗C V λ ⊗C V µ generated by 1⊗vλ ⊗vµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Notice that every element of the form xg, with x ∈ g and g ∈ C[[t, s]] divisible by ts, acts trivially on A ⊗ V λ ⊗ V µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence we need to understand the action of elements of the form z = x1 · · · xℓ · (y1t) · · · (ymt) · (vλ ⊗ vµ), with xi, yi ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, elements of g act in the standard way on the tensor product V λ ⊗ V µ, while elements of the form xt with x ∈ g act via −a(Id ⊗ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This implies the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ We now describe the specialisation of the module �Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We introduce the fol- lowing decreasing filtration of �Vλ,µ 2 : Fi = �Vλ,µ 2 ∩ aiVλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1) SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 19 By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 we have the following description of the terms of this filtration as A-modules: Fi = aiU − 2 ⊗C F i ⊕ µ � j=i+1 ajU − 2 ⊗C Gj In particular we have F0 = �Vλ,µ 2 , Fj = ajVλ,µ 2 for j ⩾ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' a) Let ui ∈ Gi be the highest weight vector and set ˜wi = aiui ∈ �Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Then ˜wi ∈ �Vλ,µ 2 and ai−1ui /∈ �Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' b) There is an isomorphism of ˆU1-modules Fi + a�Vλ,µ 2 a�Vλ,µ 2 ≃ µ � j=i Vλ+µ−2j 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The quotient Fi+a�Vλ,µ 2 a�Vλ,µ 2 is generated as a ˆU1-module by the classes of ˜wi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' , ˜wµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular �Vλ,µ 2 /a�Vλ,µ 2 ≃ Wλ,µ 1 is generated by ˜w0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' , ˜wµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The first claim follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We prove part b) by decreasing induction on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1, for i > µ the quotient is zero and the claim is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For i ⩽ µ, consider the map U − 2 ⊗ Gi −→ Fi + ai+1Vλ,µ 2 ai+1Vλ,µ 2 + Fi ∩ a�Vλ,µ 2 ≃ (Fi + a�Vλ,µ 2 )/a�Vλ,µ 2 (Fi+1 + a�Vλ,µ 2 )/a�Vλ,µ 2 sending an element u ⊗ v to the class of aiu ⊗ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This map induces an isomorphism U − 2 aU − 2 ⊗ Gi ≃ (Fi + a�Vλ,µ 2 )/a�Vλ,µ 2 (Fi+1 + a�Vλ,µ 2 )/a�Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2) Moreover, notice that U− 2 aU− 2 ⊗ Gi ≃ U − 1 ⊗ Gi, where U − 1 = U(t−1g[t−1]) ⊂ U(ˆg1) = U1, and that U − 1 ⊗ Gi has a natural structure of U1-module, as it can be identified with Vλ+µ−2i 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' With this U1-action, the isomorphism 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2 is U1-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Now the claim follows by the inductive hypothesis, combined with the fact that there are no nontrivial extensions between modules Vν 1 and Vν′ 1 if ν ̸= ν′ and that the highest weight vector of V ν generates the module Vν 1 as an U1-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ Notice that, although the specialisations at a = 0 of Vλ,µ 2 and �Vλ,µ 2 are iso- morphic, the specialisation of �Vλ,µ 2 , is generated by vλ ⊗ vµ while in the first case this vector generates the submodule Vλ+µ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As a corollary, we get the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The following hold: a) Ψn(�Vλ,µ 2 ) = 0 for n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' b) The inclusion of �Vλ,µ 2 in Vλ,µ 2 induces isomorphisms Ψ0(�Vλ,µ 2 )[a−1] ≃ Ψ0(Vλ,µ 2 )[a−1] ≃ Ψ0(Vλ t ) ⊗Q Ψ0(Vµ s ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' c) Ψ0(�Vλ,µ 2 ) is torsion-free with respect to the action of A, and the natural pro- jection induces isomorphisms Ψ0(�Vλ,µ 2 ) aΨ0(�Vλ,µ 2 ) ≃ Ψ0 � �Vλ,µ 2 a�Vλ,µ 2 � ≃ Ψ0(Wλ,µ 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We use the filtration introduced in Equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Notice that Fi Fi+1 = aiU − 2 ⊗ F i ai+1U − 2 ⊗ F i ≃ U − 1 ⊗C F i ≃ Indˆg1 ˆg+ 1 F i, 20 FORTUNA, LOMBARDO, MAFFEI, MELANI where we consider F i as a ˆg+ 1 -module on which tg[t] acts trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Notice that Indˆg1 ˆg+ 1 F i is a sum of modules of the form Vν 1, hence in particular has trivial non- zero cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence, arguing by decreasing induction on i, starting from i = µ, it follows that Fi has trivial semi-infinite cohomology in degree different from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Indeed for i = µ we have Fµ = aµVλ,µ 2 ≃ Vλ,µ 2 and this is the content of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For i = 0 this implies claim a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Part b) follows from the fact that semi-infinite cohomology commutes with local- ization (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3) combined with the isomorphism �Vλ,µ 2 [a−1] = Vλ,µ 2 [a−1] ≃ Vλ t ⊗Q Vµ s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' To prove c), consider the exact sequence 0 � �Vλ,µ 2 a � �Vλ,µ 2 � �Vλ,µ 2 a�Vλ,µ 2 � 0 By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2, the last module in this sequence is isomorphic to Wλ,µ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In par- ticular, the semi-infinite cohomology groups Ψn of the modules appearing in this sequence are zero for n ̸= 0, and c) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ To prove that the semi-infinite cohomology of �Vλ,µ 2 is isomorphic to Zλ,µ 2 we will use the action of a particular central element in Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall from [4] the definition of the 2-Sugawara operator S(2) 1/2 = � n∈ 1 2 Z,b : (Jbwn)(Jbz−n) : (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3) where J1, J2, J3 are the basis elements e, h, f and J1, J2, J3 are the dual basis elements f, h/2, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As proved in [4], the element S(2) 1/2 is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Its specialisation is the Sugawara operator S(1) 1 = � n∈Z,b : (Jbtn) (Jbt−n) : (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4) which is an element of Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' It is straightforward to check that the action of S(1) 1 on the Weyl module Vν 1 is given by multiplication by ν(ν + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The element ˆwℓ = � et−1�ℓ ˜wℓ belongs to Z2 · (vλ ⊗ vµ) + a�Vλ,µ 2 for ℓ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' , µ, Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We notice first that the element vλ ⊗ f ℓvµ belongs to F ℓ \\ F ℓ−1 and has weight λ + µ − 2ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence, up to a non-zero constant we have vλ ⊗ f ℓvµ = uℓ + u′ ℓ, where we recall that uℓ is the highest weight vector in Gℓ ≃ V λ+µ−2ℓ ⊂ V λ ⊗ V µ and u′ ℓ ∈ F ℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, recall from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2 that aℓ−1F ℓ ⊂ �Vλ,µ 2 , hence aℓ � et−1�ℓ vλ ⊗ f ℓvµ = � et−1�ℓ ˜wℓ + � et−1�ℓ (aℓu′ ℓ) ≡ � et−1�ℓ ˜wℓ mod a�Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence, the lemma is equivalent to the fact that ˆwℓ = aℓ � et−1�ℓ vλ ⊗ f ℓvµ is in Z2 · vλ ⊗ vµ + a�Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We prove this statement by induction on ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For ℓ = 0 it is trivially true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Now assume ˆwℓ is in Z2 · vλ ⊗ vµ + a�Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We compute S(2) 1/2( ˆwℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In order to do this, we notice that the action of xtisj on �Vλ,µ 2 /a�Vλ,µ 2 is equal to the action of xti+j on the same module, and that vλ ⊗ e f ℓvµ is in F ℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We have S(2) 1/2 ˆwℓ = 2 � n>0 et−n · ftn · ˆwℓ + 2 � n>0 ft−n · etn · ˆwℓ + � n>0 ht−n · htn · ˆwℓ + e · f · ˆwℓ + e · f · ˆwℓ + 1 2h · h · ˆwℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 21 In the second infinite sum above, the element etn commutes with et−1, hence etn · ˆwℓ ∈ a�Vλ,µ 2 for all n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The summands of the third series are of the form htn · (et−1)ℓ · ˆwℓ = (et−1)ℓhtn · ˆwℓ + 2ℓ(et−1)ℓ−1etn−1 · ˆwℓ, hence they vanish for n ⩾ 3, while for n = 1, 2 they are easily checked to be elements of a�Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The summands of the first series are given by ftn·(et−1)ℓ· ˆwℓ = (et−1)ℓftn· ˆwℓ−ℓ(et−1)ℓ−1htn−1· ˆwℓ−ℓ(ℓ−1)(et−1)ℓ−2etn−2· ˆwℓ, and all terms are zero or in a�Vλ,µ 2 but for the case n = 1, for which we get (et−1) · (ft) · (et−1)ℓ · ˆwℓ = aℓ+1(et−1)ℓ+1 · (vλ ⊗ f ℓ+1vµ) − ℓ(et−1)ℓh · (vλ ⊗ f ℓvµ) − ℓ(ℓ − 1)(et−1)ℓ · ˆwℓ = ˆwℓ+1 + K1 ˆwℓ for some constant K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Finally, e·f · ˆwℓ+e·f · ˆwℓ+ 1 2h·h· ˆwℓ belongs to K2 ˆwℓ+a�Vλ,µ 2 for some constant K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence we get S(2) 1/2 ˆwℓ ≡ ˆwℓ+1 + K ˆwℓ mod a�Vλ,µ 2 for some constant K, proving our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ We now prove that the zero-th semi-infinite cohomology of the module �Vλ,µ 2 is isomorphic to Zλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For g = sl(2) the map Φ : Zλ,µ 2 −→ Ψ0��Vλ,µ 2 � given by Φ(z) = z · [vλ ⊗ vµ] is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By [4], Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4, the action of Z2 on Vλ,µ 2 , hence on �Vλ,µ 2 , factors through Zλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover vλ ⊗ vµ is a cycle, so the map Φ is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Since we know that both modules are torsion-free, to prove that Φ is an isomorphism it suffices to prove that the localization Φa and the specialisation Φ are isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The fact that Φa is an isomorphism is the content of part b) of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We need to prove that Φ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 and [4, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='13] we have Zλ,µ 2 aZλ,µ 2 ≃ µ � i=0 Zλ+µ−2i 1 and Ψ0(�Vλ,µ 2 ) aΨ0(�Vλ,µ 2 ) ≃ µ � i=0 Ψ0(Vλ+µ−2i 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 these two Z1-modules are isomorphic, but we need to prove that our specific map Φ provides an isomorphism between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2 it is enough to prove that Φ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We prove that the image of Φ contains Ψ0(Fℓ+a�Vλ,µ 2 /a�Vλ,µ 2 ) arguing by reverse induction on ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For ℓ = 0 we get our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For ℓ > µ there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Now assume ℓ ⩽ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Consider again the exact sequence 0 � Fℓ+1+a�Vλ,µ 2 a�Vλ,µ 2 � Fℓ+a�Vλ,µ 2 a�Vλ,µ 2 �aℓU − 2 ⊗C Gℓ �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We know that the last module is isomorphic to aℓU − 2 ⊗C Gℓ ≃ Vλ+µ−2ℓ 1 = Indˆg1 ˆg+ 1 (V λ+µ−2ℓ) and that it is generated by the element ˜wℓ ∈ aℓGℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Notice this sequence of Z1- modules splits by Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Taking semi-infinite cohomology we get a short exact sequence 0 �Ψ0 � Fℓ+1+a�Vλ,µ 2 a�Vλ,µ 2 � �Ψ0 � Fℓ+a�Vλ,µ 2 a�Vλ,µ 2 � �Ψ0 � aℓU − 2 ⊗C Gℓ� �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' and we know that the last Z2-module is generated by ˜wℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence it is enough to prove that this element is in the image of Zλ,µ 2 (vλ ⊗vµ) in Ψ0� �Vλ,µ 2 /Fℓ+1 +a�Vλ,µ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 22 FORTUNA, LOMBARDO, MAFFEI, MELANI By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4 we know that ˆwℓ is in this image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Now we prove that ˜wℓ and ˆwℓ define the same element in the semi-infinite cohomology of aℓU − 2 ⊗C Gℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This is a claim about the cohomology of the module Vν 1 for ν = λ + µ − 2ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For any ν we prove that � et−1�hvν + � et−1�h−1vν is a coboundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Indeed the boundary operator in the case of sl(2) is equal to d(1) = ψ∗ + � n∈Z (etn) ⊗ ψ∗t−1−n, so a simple computation shows d(1) �� et−1�h−1vν ⊗ (ψt−1)|0⟩Λ � = � et−1�h−1vν ⊗ |0⟩Λ + � et−1�hvν ⊗ |0⟩Λ, which implies our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ Recall that in [4] we computed the endomorphism ring of Vλ,µ 2 , showing that it is isomorphic to Zλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now prove the same result for the module �Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The action of the center Z2 on �Vλ,µ 2 induces an isomorphism Zλ,µ 2 ≃ Endˆg2(�Vλ,µ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We already recalled at the beginning of the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='5 that the action of Z2 on �Vλ,µ 2 factors through Zλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by α : Zλ,µ 2 −→ End(�Vλ,µ 2 ) this action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Since both modules have no A-torsion, in order to prove that α is an isomorphism it suffices to show that its localization and its specialisation are isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, since our modules are finitely generated and have no torsion we have Endˆg2 � �Vλ,µ 2 � [a−1] ≃ Endˆg2[a−1] � �Vλ,µ 2 [a−1] � ≃ Endˆgt,s � Vλ ⊗Q Vµ s � ≃ Zλ t ⊗Q Zµ t ≃ Zλ,µ 2 [a−1], hence the localization of α is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Finally, we prove that the specialisation of α is also an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We have already recalled that by [4, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='13] we have Zλ,µ 2 /aZλ,µ 2 ≃ �µ i=0 Zλ+µ−2i 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 we have the following abstract isomorphisms of Z1-modules: Zλ,µ 2 aZλ,µ 2 ≃ µ �� i=0 Zλ+µ−2i 1 ≃ µ � i=0 Endˆg1(Vλ+µ−2i 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, since �Vλ,µ 2 has no nontrivial A-torsion, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2 and Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2 part (1) we have the inclusion Endˆg1 � �Vλ,µ 2 � a Endˆg1 � �Vλ,µ 2 � ⊂ Endˆg1 � �Vλ,µ 2 a�Vλ,µ 2 � ≃ µ � i=0 Endˆg1(Vλ+µ−2i 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence, composing the specialisation of the map α with this inclusion and the iso- morphisms above we get a Z1-equivariant endomorphism of �µ i=0 Zλ+µ−2i 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, α(1) = 1, hence we conclude by Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2 (3) that the specialisation of α is also an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ References [1] Casarin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' A Feigin Frenkel theorem with n singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' preprint, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' [2] Feigin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=', and Frenkel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Affine Kac-Moody algebras at the critical level and Gelfand- Diki˘ı algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In Infinite analysis, Part A, B (Kyoto, 1991), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 16 of Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' World Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=', River Edge, NJ, 1992, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 197–215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' [3] Fortuna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The Beilinson-Bernstein Localization Theorem for the affine Grassmannian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' MIT, PhD thesis, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 23 [4] Fortuna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=', Lombardo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=', Maffei, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=', and Melani, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Local opers with two singularities: the case of sl(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=', 394 (2022), 1303–1360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' [5] Frenkel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=', and Ben-Zvi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Vertex algebras and algebraic curves, second ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 88 of Mathematical Surveys and Monographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' American Mathematical Society, Providence, RI, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' [6] Frenkel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=', and Gaitsgory, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Local geometric Langlands correspondence: the spherical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In Algebraic analysis and around, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 54 of Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Stud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Pure Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Japan, Tokyo, 2009, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 167–186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' [7] Frenkel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=', and Gaitsgory, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Weyl modules and opers without monodromy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In Arithmetic and geometry around quantization, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 279 of Progr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Birkh¨auser Boston, Boston, MA, 2010, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 101–121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' E-mail addresses: giorgiafortuna@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='com, davide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='lombardo@unipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='it, andrea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='maffei@unipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='it, valerio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='melani@unifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='it' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'}