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Proposition 22.9 (Brauer, Klimyk). Suppose that \( \lambda \) and \( \mu \) are in \( {X}^{ * }\left( T\right) \cap \) \( {\mathcal{C}}_{ + } \) . Decompose \( {\chi }_{\mu } \) into a sum of weights \( \nu \in {X}^{ * }\left( T\right) \) with multiplicities \( m\left( \nu \right) \) :\n\n\[ \n{\chi }_{\mu } = \mathop{... | Proof. By the Weyl character formula, we may write\n\n\[ \n{\chi }_{\lambda }{\chi }_{\mu } = {\Delta }^{-1}\mathop{\sum }\limits_{\nu }m\left( \nu \right) {\mathrm{e}}^{\nu }\mathop{\sum }\limits_{w}{\left( -1\right) }^{l\left( w\right) }{\mathrm{e}}^{w\left( {\lambda + \rho }\right) }.\n\]\n\nInterchange the order of... | Yes |
Proposition 23.1. Let \( G \) be a connected topological group and \( \Gamma \) a discrete normal subgroup. Then \( \Gamma \subset Z\left( G\right) \) . | Proof. Let \( \gamma \in \Gamma \) . Then \( g \rightarrow {g\gamma }{g}^{-1} \) is a continuous map \( G \rightarrow \Gamma \) . Since \( G \) is connected and \( \Gamma \) discrete, it is constant, so \( {g\gamma }{g}^{-1} = \gamma \) for all \( g \) . Therefore, \( \gamma \in Z\left( G\right) \) . | Yes |
Proposition 23.2. If \( G \) is a connected Lie group, then the fundamental group \( {\pi }_{1}\left( G\right) \) is Abelian. | Proof. Let \( p : \widetilde{G} \rightarrow G \) be the universal cover. We identify the kernel \( \ker \left( p\right) \) with \( {\pi }_{1}\left( G\right) \) . This is a discrete normal subgroup of \( \widetilde{G} \) and hence is central in \( \widetilde{G} \) by Proposition 23.1. In particular, it is Abelian. | Yes |
Proposition 23.3. The set \( {G}_{\text{sing }} \) is a finite union of submanifolds of \( G \), each of codimension at least 3 . | Proof. By Proposition 18.14, the singular elements of \( G \) are the conjugates of the kernels \( {T}_{\alpha } \) of the roots. We first show that the union of the set of conjugates of \( {T}_{\alpha } \) is the image of a manifold of codimension 3 under a smooth map. Let \( \alpha \in \Phi \) . The set of conjugates... | Yes |
Lemma 23.1. Let \( X \) and \( Y \) be Hausdorff topological spaces and \( f : X \rightarrow Y \) a local homeomorphism. Suppose that \( U \in X \) is a dense open set and that the restriction of \( f \) to \( U \) is injective. Then \( f \) is injective. | Proof. If \( {x}_{1} \neq {x}_{2} \) are elements of \( X \) such that \( f\left( {x}_{1}\right) = f\left( {x}_{2}\right) \), find open neighborhoods \( {V}_{1} \) and \( {V}_{2} \) of \( {x}_{1} \) and \( {x}_{2} \), respectively, that are disjoint, and such that \( f \) induces a homeomorphism \( {V}_{i} \rightarrow ... | Yes |
Proposition 23.5. Let \( p : X \rightarrow Y \) be a covering map. The map \( {\pi }_{1}\left( X\right) \rightarrow \) \( {\pi }_{1}\left( Y\right) \) induced by inclusion \( X \rightarrow Y \) is injective. | Proof. Suppose that \( {p}_{0} \) and \( {p}_{1} \) are loops in \( X \) with the same endpoints whose images in \( Y \) are path-homotopic. It is an immediate consequence of Proposition 13.2 that \( {p}_{0} \) and \( {p}_{1} \) are themselves path-homotopic. | No |
Proposition 23.6. The inclusion \( {G}_{\text{reg }} \rightarrow G \) induces an isomorphism of fundamental groups: \( {\pi }_{1}\left( {G}_{\text{reg }}\right) \cong {\pi }_{1}\left( G\right) \) . | Proof. Of course, we usually take the base point of \( G \) to be the identity, but that is not in \( {G}_{\text{reg }} \) . Since \( G \) is connected, the isomorphism class of its fundamental group does not change if we move the base point \( P \) into \( {G}_{\text{reg }} \) .\n\nIf \( p : \left\lbrack {0,1}\right\r... | Yes |
Proposition 23.7. We have \( {\pi }_{1}\left( {G/T}\right) = 1 \) . | Proof. Let \( {t}_{0} \in {T}_{\text{reg }} \) and consider the map \( {f}_{0} : G/T \rightarrow G,{f}_{0}\left( {gT}\right) = g{t}_{0}{g}^{-1} \) . We will show that the map \( {\pi }_{1}\left( {G/T}\right) \rightarrow {\pi }_{1}\left( G\right) \) induced by \( {f}_{0} \) is injective. We may factor \( {f}_{0} \) as\n... | Yes |
Theorem 23.1. The induced map \( {\pi }_{1}\left( T\right) \rightarrow {\pi }_{1}\left( G\right) \) is surjective. The group \( {\pi }_{1}\left( G\right) \) is finitely generated and Abelian. | Proof. One way to see this is to use have the exact sequence\n\n\[ \n{\pi }_{1}\left( T\right) \rightarrow {\pi }_{1}\left( G\right) \rightarrow {\pi }_{1}\left( {G/T}\right) \n\]\n\nof the fibration \( G \rightarrow G/T \) (Spanier [149, Theorem 10 on p. 377]). It follows using Proposition 23.7 that \( {\pi }_{1}\left... | Yes |
Lemma 23.2. Let \( H \in \mathfrak{t} \) . (i) Let \( \lambda \in \Lambda \) . Then \( \lambda \left( {\mathrm{e}}^{H}\right) = 1 \) if and only if \( \frac{1}{2\pi i}\mathrm{d}\lambda \left( H\right) \in \mathbb{Z} \) . | Proof. Since \( t \mapsto \lambda \left( {\mathrm{e}}^{tH}\right) \) is a character of \( \mathbb{R} \) we have \( \lambda \left( {\mathrm{e}}^{tH}\right) = {\mathrm{e}}^{2\pi i\theta t} \) for some \( \theta = \theta \left( {\lambda, H}\right) \) . Then \( {\left. \theta = \frac{1}{2\pi i}\frac{\mathrm{d}}{\mathrm{d}t... | Yes |
Proposition 23.8. Define \( \tau : \mathfrak{t} \rightarrow {\mathcal{V}}^{ * } \) by letting \( \tau \left( H\right) \in {\mathcal{V}}^{ * } \) be the linear functional that sends \( \lambda \) to \( \frac{1}{2\pi i}\mathrm{\;d}\lambda \left( H\right) \) . Then \( \tau \) is a linear isomorphism, and \( \tau \) maps t... | Proof. It is clear that \( \tau \) is a linear isomorphism. It follows from Lemma 23.2(ii) that it maps the kernel of exp onto the coweight lattice. The identity (23.2) follows from Proposition 18.13. | No |
Lemma 23.3. Let \( \psi : Y \rightarrow X \) be a covering map, and let \( p : \left\lbrack {0,1}\right\rbrack \rightarrow Y \) be a path. If \( \psi \circ p : \left\lbrack {0,1}\right\rbrack \rightarrow X \) is a loop that is contractible in \( X \), then \( p \) is a loop. | Proof. Let \( q = \phi \circ p \), and let \( x = q\left( 0\right) \) . Let \( y = p\left( 0\right) \), so \( \phi \left( y\right) = x \) . What we know is that \( q\left( 1\right) = x \) and what we need to prove is that \( p\left( 1\right) = y \) .\n\nSince \( q \) is contractible in \( X \), we may find a family \( ... | Yes |
Proposition 23.11. Let \( G \) be a compact connected Lie group. The following are equivalent.\n\n(i) The root system \( \Phi \) spans \( \mathcal{V} = \mathbb{R} \otimes {X}^{ * }\left( T\right) \) .\n\n(ii) The fundamental group \( {\pi }_{1}\left( G\right) \) is finite.\n\n(iii) The center \( Z\left( G\right) \) is ... | Proof. The root lattice spans \( \mathcal{V} \) if and only if the coroot lattice spans \( {\mathcal{V}}^{ * } \) , which we are identifying with \( \mathfrak{t} \) . Since \( {\Lambda }^{ \vee } \) is a lattice in \( \mathfrak{t} \) of rank equal to \( \dim \left( \mathcal{V}\right) = \dim \left( T\right) \), the coro... | Yes |
Theorem 23.2. Assume that \( G \) is semisimple. Then\n\n\[ \widetilde{\Lambda } \supseteq \Lambda \supseteq {\Lambda }_{\text{root }},\;{\widetilde{\Lambda }}^{ \vee } \supseteq {\Lambda }^{ \vee } \supseteq {\Lambda }_{\text{coroot }}^{ \vee }.\]\n\nRegarding these as lattices in the dual real vector spaces \( \mathc... | Proof. By Proposition 18.10 we have \( \widetilde{\Lambda } \supseteq \Lambda \) and \( \Lambda \supseteq {\Lambda }_{\text{root }} \) is clear since roots are characters of \( {X}^{ * }\left( T\right) \) . That \( \Lambda \) and \( {\Lambda }^{ \vee } \) are dual lattices is Lemma 23.2. That \( {\Lambda }_{\text{root ... | Yes |
Proposition 23.12. If \( G \) is semisimple and simply-connected, then \( \widetilde{\Lambda } = \Lambda \) . | Proof. This follows from (23.3) with \( {\pi }_{1}\left( G\right) = 1 \) . | No |
Proposition 24.1. The group \( \mathrm{{SL}}\left( {n,\mathbb{C}}\right) \) is the complexification of the Lie group \( \operatorname{SL}\left( {n,\mathbb{R}}\right) \) . | Proof. Given any complex analytic group \( H \) and any Lie group homomorphism \( f : \mathrm{{SL}}\left( {n,\mathbb{R}}\right) \rightarrow H \), the differential is a Lie algebra homomorphism \( \mathfrak{{sl}}\left( {n,\mathbb{R}}\right) \rightarrow \) \( \operatorname{Lie}\left( H\right) \) . Since \( \operatorname{... | Yes |
Proposition 24.2. Let \( G \) be a Lie group and let \( \mathfrak{h} \) be a Lie subalgebra of \( \operatorname{Lie}\left( G\right) \). Let \( H \) be a closed connected subset of \( G \) that is an integral submanifold of the involutory family associated with \( \mathfrak{h} \), and suppose that \( 1 \in H \). Then \(... | Proof. Let \( x \in H \) and let \( U = \left\{ {y \in H \mid {x}^{-1}y \in H}\right\} \). We show that \( U \) is open in \( H \). If \( y \in U = H \cap {xH} \), both \( H \) and \( {xH} \) are integral submanifolds for the involutory family associated with \( \mathfrak{h} \), since the vector fields corresponding to... | Yes |
Proposition 24.3. If \( G = \mathcal{G}\left( \mathbb{C}\right) \) is the algebraic complexification of \( K = \) \( \mathcal{G}\left( \mathbb{R}\right) \), then any algebraic complex representation of \( K \) extends uniquely to an algebraic representation of \( G \) . | Proof. This is clear since a polynomial function extends uniquely from \( \mathcal{G}\left( \mathbb{R}\right) \) to \( \mathcal{G}\left( \mathbb{C}\right) \) . | Yes |
Proposition 24.4. \( \mathrm{U}\left( n\right) \) is a real form of \( \mathrm{{GL}}\left( {n,\mathbb{R}}\right) \) . | Proof. Let \( {\mathcal{G}}_{1} \) be the algebraic group \( \mathrm{{GL}}\left( n\right) \), and let\n\n\[ \n{\mathcal{G}}_{2} = \left\{ {\left( {A, B}\right) \in {\operatorname{Mat}}_{n} \times {\operatorname{Mat}}_{n} \mid A \cdot {}^{t}A + B \cdot {}^{t}B = I, A \cdot {}^{t}B = B \cdot {}^{t}A}\right\} .\n\]\n\nThe... | No |
Proposition 24.5. Let \( \pi : \mathrm{{GL}}\left( {n,\mathbb{R}}\right) \rightarrow \mathrm{{GL}}\left( {m,\mathbb{C}}\right) \) be an algebraic representation. Then \( \pi \) is completely reducible. | Proof. Any irreducible algebraic representations of \( \mathrm{{GL}}\left( {n,\mathbb{R}}\right) \) can be extended to an algebraic representation of \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) and then restricted to \( \mathrm{U}\left( n\right) \), where it is completely reducible because \( \mathrm{U}\left( n\rig... | Yes |
Theorem 25.1. Let \( W \) be the Weyl group of the root system \( \Phi \), and let \( I = \) \( \left\{ {{s}_{1},\ldots ,{s}_{r}}\right\} \) be the simple reflections. Then \( \left( {W, I}\right) \) is a Coxeter group. | Proof. Let \( \left( {{W}^{\prime },{I}^{\prime }}\right) \) be the Coxeter group with generators \( \left\{ {{s}_{1}^{\prime },\ldots ,{s}_{j}^{\prime }}\right\} \) and the relations (25.1) where \( n\left( {i, j}\right) \) is the order of \( {s}_{i}{s}_{j} \) . Since the relations (25.1) are true in \( W \) we have a... | Yes |
Theorem 25.2 (Matsumoto, Tits). Let \( w \in W \) have length \( l\left( w\right) = r \) . Let \( {s}_{{i}_{1}}\cdots {s}_{{i}_{r}} = {s}_{{j}_{1}}\cdots {s}_{{j}_{r}} \) be two reduced decompositions of \( w \) into products of simple reflections. Then the corresponding words are equal in the braid group, that is, \( ... | What we will actually prove is that if \( w \) of length \( k \) has two reduced decompositions \( w = {s}_{{i}_{1}}\cdots {s}_{{i}_{k}} = {s}_{{j}_{1}}\cdots {s}_{{j}_{k}} \), then the word \( \left( {{s}_{{i}_{1}},\ldots ,{s}_{{i}_{k}}}\right) \) may be transformed into \( \left( {{s}_{{j}_{1}},\ldots ,{s}_{{j}_{k}}}... | Yes |
Proposition 25.1. Let \( n\left( {i, j}\right) \) be the order of \( {s}_{i}{s}_{j} \) in \( W \), where \( i \neq j \) . Then the \( {D}_{i} \) satisfy the braid relation\n\n\[ \n{D}_{i}{D}_{j}{D}_{i}\cdots = {D}_{j}{D}_{i}{D}_{j}\cdots \n\] \n\n(25.5) \n\nwhere the number of factors on both sides is \( n\left( {i, j}... | Proof. This calculation can be done separately for the four possible cases \( n\left( {i, j}\right) = 2,3,4 \) or 6 . The case \( n\left( {i, j}\right) = 2 \) is trivial so let us assume \( n\left( {i, j}\right) = 3 \) . We will show that \n\n\[ \n{D}_{i}{D}_{j}{D}_{i} = H\left( {\alpha }_{i}\right) H\left( {\alpha }_{... | Yes |
Proposition 25.2. We have\n\n\[ \n{\partial }_{i}^{2} = {\partial }_{i},\;{s}_{i}{\partial }_{i} = {\partial }_{i},\n\]\n\nLet \( f \in \mathcal{E} \) . Then \( {\partial }_{i}f \) is in \( \mathcal{E} \) and is invariant under \( {s}_{i} \), and if \( {s}_{i}f = f \) then \( {\partial }_{i}f = f \) . If \( f = {\mathr... | Proof. We have \( {s}_{i}{\partial }_{i} = {\left( 1 - {\mathrm{e}}^{{\alpha }_{i}}\right) }^{-1}\left( {s - {\mathrm{e}}^{{\alpha }_{i}}}\right) \) since \( {s}_{i}{\mathrm{e}}^{\lambda }{s}_{i}^{-1} = {\mathrm{e}}^{{s}_{i}\left( \lambda \right) } \) and in particular \( {s}_{i}{\mathrm{e}}^{-{\alpha }_{i}}{s}_{i}^{-1... | Yes |
Proposition 25.3. The Demazure operators also satisfy the braid relations\n\n\[ \n{D}_{i}{D}_{j}{D}_{i}\cdots = {D}_{j}{D}_{i}{D}_{j}\cdots \n\]\n\n(25.7)\n\nwhere the number of factors on both sides is \( n\left( {i, j}\right) \) . | Proof. Again there are different cases depending on whether \( n\left( {i, j}\right) = 2,3,4 \) or 6 , but in each case this can be reduced to the corresponding relation (25.5) by use of \( {\partial }_{i}^{2} = {\partial }_{i} \) . For example, if \( n\left( {i, j}\right) = 3 \), then expanding \( 0 = \left( {{\partia... | Yes |
Theorem 25.4. The affine Weyl group is also a Coxeter group (generated by \( \left. {{s}_{0},\ldots ,{s}_{r}}\right) \) . Moreover, the analog of the Matsumoto-Tits theorem is true for the affine Weyl group: if \( w \) of length \( k \) has two reduced decompositions \( w = \) \( {s}_{{i}_{1}}\cdots {s}_{{i}_{k}} = {s}... | Proof. This may be proved by the same method as Theorem 25.1 and 25.2 (Exercise 25.3). | No |
Proposition 25.4. This definition does not depend on the reduced decomposition \( v = {s}_{{i}_{1}}\cdots {s}_{{i}_{k}} \) . | Proof. By Theorem 25.2 it is sufficient to check that if \( \left( {{i}_{1},\ldots ,{i}_{k}}\right) \) is changed by a braid relation, then we can still find a subsequence \( \left( {{j}_{1},\ldots ,{j}_{l}}\right) \) representing \( u \) . We therefore find a subsequence of the form \( \left( {t, u, t,\ldots }\right) ... | Yes |
Proposition 26.1. With \( G = \mathrm{{GL}}\left( {n,\mathbb{C}}\right), K = U\left( n\right) \), and \( {B}_{0} \) as above, every element of \( g \in G \) can be factored uniquely as \( {bk} \) where \( b \in {B}_{0} \) and \( k \in K \), or as \( {a\nu k} \), where \( a \in A,\nu \in N \), and \( k \in K \) . The mu... | Proof. First let us consider \( N \times A \times K \rightarrow G \) . Let \( g \in G \) . Let \( {v}_{1},\ldots ,{v}_{n} \) be the rows of \( g \) . Then by the Gram-Schmidt orthogonalization algorithm, we find constants \( {\theta }_{ij}\left( {i < j}\right) \) such that \( {v}_{n},{v}_{n - 1} + {\theta }_{n - 1, n}{... | Yes |
Proposition 26.2. Let \( \mathfrak{b} \) be a Lie algebra, \( {\mathfrak{b}}^{\prime } \) an ideal of \( \mathfrak{b} \), and \( {\mathfrak{b}}^{\prime \prime } = \mathfrak{b}/{\mathfrak{b}}^{\prime } \) . Then \( \mathfrak{b} \) is solvable if and only if \( {\mathfrak{b}}^{\prime } \) and \( {\mathfrak{b}}^{\prime \p... | Proof. Given a chain of Lie subalgebras (26.1) satisfying \( \left\lbrack {{\mathfrak{b}}_{i},{\mathfrak{b}}_{i}}\right\rbrack \subset {\mathfrak{b}}_{i + 1} \) , one may intersect them with \( {\mathfrak{b}}^{\prime } \) or consider their images in \( {\mathfrak{b}}^{\prime \prime } \) and obtain corresponding chains ... | Yes |
Proposition 26.3. (Dynkin) Let \( \mathfrak{g} \subset \mathfrak{{gl}}\left( V\right) \) be a Lie algebra of linear transformations over a field \( F \) of characteristic zero, and let \( \mathfrak{h} \) be an ideal of \( \mathfrak{g} \) . Let \( \lambda : \mathfrak{h} \rightarrow F \) be a linear form. Then the space\... | Proof. If \( W = 0 \), there is nothing to prove, so assume \( 0 \neq {v}_{0} \in W \) . Fix an element \( X \in \mathfrak{g} \) . Let \( {W}_{0} \) be the linear span of \( {v}_{0}, X{v}_{0},{X}^{2}{v}_{0},\ldots \), and let \( d \) be the dimension of \( {W}_{0} \) .\n\nIf \( Z \in \mathfrak{h} \), then we will prove... | Yes |
Theorem 26.1. (Lie) Let \( \mathfrak{b} \subseteq \mathfrak{{gl}}\left( V\right) \) be a solvable Lie algebra of linear transformations over an algebraically closed field of characteristic zero. Assume that \( V \neq 0 \) .\n\n(i) There exists a vector \( v \in V \) that is a simultaneous eigenvector for all of \( \mat... | Proof. To prove (i), we may clearly assume that \( \mathfrak{b} \neq 0 \) . Let us first observe that \( \mathfrak{b} \) has an ideal \( \mathfrak{h} \) of codimension 1 . Indeed, since \( \mathfrak{b} \) is solvable, \( \left\lbrack {\mathfrak{b},\mathfrak{b}}\right\rbrack \) is a proper ideal, and the quotient Lie al... | Yes |
Proposition 26.4. The Lie algebra \( \mathfrak{n} \) defined by (26.5) is nilpotent. | Proof. Let \( {\Phi }_{k}^{ + } \) be the set of positive roots \( \alpha \) such that \( \alpha \) is expressible as the sum of at least \( k \) simple positive roots. Thus, \( {\Phi }_{1}^{ + } = \Phi ,{\Phi }_{1}^{ + } \supset {\Phi }_{2}^{ + } \supset {\Phi }_{3}^{ + } \supset \cdots \) , and eventually \( {\Phi }_... | Yes |
Proposition 26.5. Let \( G \) be the complexification of a compact connected Lie group \( K \), and let \( \mathfrak{n} \) be as in (26.5). If \( \pi : G \rightarrow \mathrm{{GL}}\left( V\right) \) is any representation and \( X \in \mathfrak{n} \), then \( \pi \left( X\right) \) is nilpotent as a linear transformation... | Proof. By Theorem 26.1, we may choose a basis of \( V \) such that all \( \pi \left( X\right) \) are upper triangular for \( X \in \mathfrak{b} \), where we are identifying \( \pi \left( X\right) \) with its matrix with respect to the chosen basis. What we must show is that if \( X \in \mathfrak{n} \), then the diagona... | Yes |
Theorem 26.2. (i) Let \( G \) be the complexification of a compact connected Lie group \( K \), let \( T \) be a maximal torus of \( K \), let \( \mathfrak{t} \) be the Lie algebra of \( T \) , and let \( {T}_{\mathbb{C}} \) be its complexification. Let \( \mathfrak{n} \) be as in (26.5), and let \( \mathfrak{b} = {\ma... | Proof. We will prove parts (i) and (ii) simultaneously.\n\nLet \( \pi : K \rightarrow \mathrm{{GL}}\left( V\right) \) be a faithful representation. We choose on \( V \) an inner product with respect to which \( \pi \left( k\right) \) is unitary for \( k \in K \) . By Theorem 24.1, we may extend \( \pi \) to a faithful ... | Yes |
Theorem 26.3. (Iwasawa decomposition) With notations as in Theorem 26.2 and \( {B}_{0} \) and \( A \) as above, each element of \( g \in G \) can be factored uniquely as \( {bk} \) where \( b \in {B}_{0} \) and \( k \in K \), or as a \( {\nu k} \) where \( a \in A,\nu \in N \) and \( k \in K \). The multiplication map ... | Proof. Let \( {G}^{\prime } = \mathrm{{GL}}\left( {n,\mathbb{C}}\right) ,{K}^{\prime } = U\left( n\right) ,{A}^{\prime } \) be the subgroup of \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) consisting of diagonal matrices with positive real eigenvalues, and \( {N}^{\prime } \) be the subgroup of upper triangular unipo... | Yes |
Theorem 26.4. Let \( K \) be a compact connected Lie group and \( T \) a maximal torus. Then \( X = K/T \) can be given the structure of a complex manifold in such a way that the translation maps \( g : {xT} \rightarrow {gxT} \) are holomorphic. This action of \( K \) can be extended to an action of the complexificatio... | Proof. By the Iwasawa decomposition, we may write \( G = {BK} \). Since \( B \cap K = \) \( T \), we have \( G/B \cong K/T \), and this diffeomorphism is \( K \) -equivariant. Now \( G \) is a complex Lie group and \( B \) is a closed analytic subgroup, so the quotient \( G/B \) has the structure of a complex analytic ... | Yes |
Lemma 26.1. The Lie algebra \( \mathfrak{n} \) is generated by the \( {X}_{\alpha } \) as \( \alpha \) runs through the simple positive roots. | Proof. Let \( {\mathfrak{n}}^{\prime } \) be the algebra generated by the \( {X}_{\alpha } \) with \( \alpha \) simple. Let us define the height of a positive root to be the number of simple roots into which it may be decomposed, counted with multiplicities. If \( \alpha \in {\Phi }^{ + } \) is not simple, we may write... | Yes |
Proposition 26.6. (Triangular decomposition) The linear map \( \mu \) : \( U\left( {\mathfrak{n}}_{ - }\right) \otimes U\left( {\mathfrak{t}}_{\mathbb{C}}\right) \otimes U\left( \mathfrak{n}\right) \rightarrow U\left( \mathfrak{g}\right) \) is surjective. | Proof. Let \( \mathfrak{R} = U\left( {\mathfrak{n}}_{ - }\right) U\left( {\mathfrak{t}}_{\mathbb{C}}\right) U\left( \mathfrak{n}\right) \) be the image of \( \mu \) . Since \( \mathfrak{R} \) contains generators of \( U\left( \mathfrak{g}\right) \), it is enough to show that it is closed under multiplication. It is obv... | Yes |
Theorem 26.5. Let \( \left( {\pi, V}\right) \) be an irreducible representation of \( K \) . Extend \( \left( {\pi, V}\right) \) to an irreducible analytic representation of \( G \) . Let \( \lambda \) be the highest weight. Then \( V\left( \lambda \right) = {V}^{N} \) is the space of \( N \) -invariants. | Proof. Clearly \( v \in V \) is \( N \) -invariant if and only if \( \pi \left( {X}_{\alpha }\right) = 0 \) for \( \alpha \in {\Phi }^{ + } \) , and as we have noted this is true if \( v \in V\left( \lambda \right) \) . We must show that \( N \) invariance implies that \( v \in V\left( \lambda \right) \) . Since \( {T}... | Yes |
Proposition 26.7. Let \( W \) be a \( G \) -module that decomposes into a direct sum of finite-dimensional irreducible representations. Let \( \lambda \) be a dominant weight, and let \( {\pi }_{\lambda } \) be the irreducible \( G \) -module with highest weight \( \lambda \) . Then the multiplicity of \( {\pi }_{\lamb... | Proof. Since \( N \) acts trivially in \( \lambda \) as a \( B \) -module, every \( B \) -submodule of \( W \) isomorphic to \( \lambda \) is contained in \( {W}^{N} \) . By Theorem 26.5, each copy of \( {\pi }_{\lambda } \) contains a unique vector in \( {W}^{N} \) . The statement is therefore clear. | No |
Proposition 26.8. Let \( \Omega \) be the space of \( n \times n \) symmetric complex matrices. Let \( \mathcal{P}\left( \Omega \right) \) be the ring of polynomials on \( \Omega \) with the \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) action \( \left( {gf}\right) \left( X\right) = \) \( f\left( {{}^{t}g \cdot X \cd... | Proof. For any module \( W \) the polynomial ring on \( W \) is isomorphic as a \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) -module to the symmetric algebra on \( {W}^{ * } \) . So it is sufficient to show that \( { \vee }^{2}{\mathbb{C}}^{n} \) and \( \Omega \) are dual modules. Indeed, if \( V = {\mathbb{C}}^{n} ... | Yes |
Theorem 26.6. The \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) -module \( \bigvee \left( {{ \vee }^{2}{\mathbb{C}}^{n}}\right) \) decomposes into a direct sum of irreducible representations, each with multiplicity one. Let \( \lambda \) be a dominant weight. The irreducible representation with highest weight \( \lam... | Proof. By Proposition 26.8 we may work with the representation \( \mathcal{P}\left( \Omega \right) \) . As we have explained, our task is to compute the \( N \) -invariants of the representation. If \( X = \left( {X}_{ij}\right) \in \Omega \), let \( {X}_{k}\left( {1 \leq k \leq n}\right) \) be the upper left \( k \tim... | Yes |
Theorem 26.7. Assume that \( G \) is an affine algebraic group over the complex numbers. Assume that it is also the complexification of a compact Lie group \( K \) . Let \( X \) be a complex affine algebraic variety on which the group \( G \) acts algebraically. Assume that the Borel subgroup \( B \) has a dense open o... | Proof. We need to prove that \( W \) decomposes into a direct sum of finite -dimensional modules.\n\nWe begin by showing that if \( f \in W \) then the \( G \) -translates of \( f \) span a finite-dimensional vector space \( W\left( f\right) \) . Since the group action \( G \times X \rightarrow X \) is algebraic, if \(... | Yes |
Lemma 27.1. Let \( G = \mathrm{{GL}}\left( {n, F}\right) \) for any field \( F \), and let other notations be as above. If \( s \) is a simple reflection, then \( B \cup \mathcal{C}\left( s\right) \) is a subgroup of \( G \) . | Proof. First, let us check this when \( n = 2 \) . In this case, there is only one simple root \( {s}_{\alpha } \) where \( \alpha = {\alpha }_{12} \) . We check easily that\n\n\[ \mathcal{C}\left( {s}_{\alpha }\right) = B{s}_{\alpha }B = \left\{ {\left. {\left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \in \... | Yes |
Lemma 27.2. Let \( G = \mathrm{{GL}}\left( {n, F}\right) \) for any field \( F \), and let other notations be as above. If \( \alpha \) is a simple root and \( w \in W \) such that \( w\left( \alpha \right) \in {\Phi }^{ + } \), then \( \mathcal{C}\left( w\right) \mathcal{C}\left( s\right) = \mathcal{C}\left( {ws}\righ... | Proof. We will show that\n\n\[ \n{wBs} \subseteq {BwsB}\text{.} \n\]\n\nIf this is known, then multiplying both left and right by \( B \) gives \( \mathcal{C}\left( w\right) \mathcal{C}\left( s\right) = \) \( {BwBsB} \subseteq {BwsB} = \mathcal{C}\left( {ws}\right) \) . The other inclusion is obvious, so this is suffic... | Yes |
Proposition 27.1. Let \( G = \mathrm{{GL}}\left( {n, F}\right) \) for any field \( F \), and let other notations be as above. If \( w,{w}^{\prime } \in W \) are such that \( l\left( {w{w}^{\prime }}\right) = l\left( w\right) + l\left( {w}^{\prime }\right) \), then \[ \mathcal{C}\left( {w{w}^{\prime }}\right) = \mathcal... | Proof. It is sufficient to show that if \( l\left( w\right) = r \), and if \( w = {s}_{1}\ldots {s}_{r} \) is a decomposition into simple reflections, then \[ \mathcal{C}\left( w\right) = \mathcal{C}\left( {s}_{1}\right) \ldots \mathcal{C}\left( {s}_{r}\right) \] Indeed, assuming we know this fact, let \( {w}^{\prime }... | Yes |
Theorem 27.1. With \( G = \mathrm{{GL}}\left( {n, F}\right) \) and \( B, N, I \) as above, \( \left( {B, N, I}\right) \) is a Tits’ system in \( G \) . | Proof. Only Axiom TS3 requires proof; the others can be safely left to the reader. Let \( \alpha \in \sum \) such that \( s = {s}_{\alpha } \) . First, suppose that \( w\left( \alpha \right) \in {\Phi }^{ + } \) . In this case, it follows from Lemma 27.2 that \( {wBs} \subset {BwsB} \) . Next suppose that \( w\left( \a... | No |
Proposition 27.2. Let \( G \) be the complexification of the compact connected Lie group \( K \), let \( \alpha \) be a simple positive root of \( G \) with respect to a fixed maximal torus \( T \) of \( K \), and let other notations be as above. Then \( {M}_{\alpha } \) normalizes \( {U}_{\alpha } \) . | Proof. It is clear that \( B \) normalizes \( {U}_{\alpha } \), so we need to show that \( {i}_{\alpha }\left( {\mathrm{{SL}}\left( {2,\mathbb{C}}\right) }\right) \) normalizes \( {U}_{\alpha } \) . If \( \gamma \in \left\{ {\beta \in {\Phi }^{ + } \mid \beta \neq \alpha }\right\} \) and \( \delta = \alpha \) or \( - \... | Yes |
Lemma 27.3. Let \( G \) be the complexification of the compact connected Lie group \( K \), and let other notations be as above. If \( s \) is a simple reflection, then \( B \cup \mathcal{C}\left( s\right) \) is a subgroup of \( G \) . | Proof. Indeed, if \( s = {s}_{\alpha } \), then \( B \cup \mathcal{C}\left( s\right) = {P}_{\alpha } \) . From Theorem 18.1, the group \( {M}_{\alpha } \) contains a representative of \( s \in N/T \), so it is clear that \( B \cup \mathcal{C}\left( s\right) \subset {P}_{\alpha } \) . As for the other inclusion, both \(... | Yes |
Theorem 27.2. Let \( G \) be the complexification of the compact connected Lie group \( K \) . With \( B, N, I \) as above, \( \left( {B, N, I}\right) \) is a Tits’ system in \( G \) . | Proof. The proof of this is identical to Theorem 27.1. The analog of Lemma 27.2 is true, and the proof is the same except that we use Lemma 27.3 instead of Lemma 27.1. All other details are the same. | No |
Proposition 27.3. The map \( u \mapsto {uwB} \) is a bijection of \( {U}_{ - }^{w} \) onto \( {Y}_{w} \) . | Proof. Clearly \( {BwB}/B = {UwB}/B \) . Moreover if \( u,{u}^{\prime } \in U \) then \( {uwB} = {u}^{\prime }{wB} \) if and only if \( {u}^{-1}{u}^{\prime } \in {U}_{ + }^{w} \) . We need to show that every coset in \( U/{U}_{ + }^{w} \) has a unique representative from \( {U}_{ - }^{ + } \) . This follows from Theore... | No |
Proposition 27.4. The minimal parabolic \( {P}_{i} = C\left( 1\right) \cup C\left( {s}_{i}\right) \) . The quotient \( {P}_{i}/B \) is diffeomorphic to the projective line \( {\mathbb{P}}^{1}\left( \mathbb{C}\right) \) . | Proof. By Lemma 27.1, \( C\left( 1\right) \cup C\left( {s}_{i}\right) \) is a group, so \( {P}_{i} = C\left( 1\right) \cup C\left( {s}_{i}\right) \) . Since \( \mathrm{{SL}}\left( {2,\mathbb{C}}\right) \) is simply-connected, the injection \( {i}_{{\alpha }_{k}} : \mathfrak{{sl}}\left( {2,\mathbb{C}}\right) \rightarrow... | Yes |
Theorem 27.4. The image of \( \tau \) is \( {X}_{w} \) . The Schubert variety \( {X}_{w} \) is the union of the \( {Y}_{u} \) for \( u \leq w \) in the Bruhat order. | Proof. Since \( C\left( {s}_{i}\right) \) is dense in \( {P}_{i} \), the set \( C\left( {s}_{{i}_{1}}\right) \times \ldots \times C\left( {s}_{{i}_{k}}\right) \) is dense in \( {P}_{{i}_{1}} \times \ldots \times {P}_{{i}_{k}} \) . Its image in \( X \) is \( C\left( {s}_{{i}_{1}}\right) \ldots C\left( {s}_{{i}_{k}}\righ... | Yes |
Theorem 27.5. (Borel-Weil) The space \( \Gamma \left( {\mathcal{L}}_{\lambda }\right) \) is zero unless \( \lambda \) is dominant. If \( \lambda \) is dominant, then \( \Gamma \left( {\mathcal{L}}_{\lambda }\right) \) is irreducible as a \( G \) -module, with highest weight \( \lambda \) . | Proof. We will follow the now-familiar strategy of identifying the \( N \) -fixed vectors in the module. We will take for granted the well-known fact that the space of sections \( \Gamma \left( {\mathcal{L}}_{\lambda }\right) \) is finite-dimensional. See Gunning and Rossi [60], Corollary 10 on page 241. Assume that \(... | Yes |
Proposition 28.1. Suppose that \( G \) is a connected Lie group with an involution \( \theta \) . Assume that the group\n\n\[ K = \{ g \in G \mid \theta \left( g\right) = g\} \]\n\n(28.1)\n\nis a compact Lie subgroup. In this setting, \( X = G/K \) is a symmetric space. | Proof. Clearly, \( G \) acts transitively on \( G/K \), and \( K \) is the stabilizer of the base point \( {x}_{0} \), that is, the coset \( K \in G/K \) . We put a positive definite inner product on the tangent space \( {T}_{{x}_{0}}\left( X\right) \) that is invariant under the compact group \( K \) and also under \(... | Yes |
Lemma 28.1. (Peirce decomposition) Let \( R \) be a ring, and let \( e \) and \( f \) be orthogonal central idempotents. Assume that \( 1 = e + f \) . Then Re and \( {Rf} \) are (two-sided) ideals of \( R \), and each is a ring with identity elements \( e \) and \( f \) , respectively. The ring \( R \) decomposes as \(... | Proof. It is straightforward to see that \( {Re} \) is closed under multiplication and is a ring with identity element \( e \) and similarly for \( {Rf} \) . Since \( 1 = e + f \), we have \( R = {Re} + {Rf} \), and \( {Re} \cap {Rf} = 0 \) because if \( x \in {Re} \cap {Rf} \) we can write \( x = {re} = {r}^{\prime }f... | Yes |
Lemma 28.2. Regard \( \mathbb{C} \otimes \mathbb{C} = \mathbb{C}{ \otimes }_{\mathbb{R}}\mathbb{C} \) as a \( \mathbb{C} \) -algebra with scalar multiplication \( a\left( {x \otimes y}\right) = {ax} \otimes y, a \in \mathbb{C} \) . Then \( \mathbb{C} \otimes \mathbb{C} \) and \( \mathbb{C} \oplus \mathbb{C} \) are isom... | Proof. Let\n\n\[ \n e = \frac{1}{2}\left( {1 \otimes 1 + i \otimes i}\right) ,\;f = \frac{1}{2}\left( {1 \otimes 1 - i \otimes i}\right) .\n\]\n\n(28.4)\n\nIt is easily checked that \( e \) and \( f \) are orthogonal central idempotents whose sum is the identity element \( 1 \otimes 1 \), and so we obtain a Peirce deco... | Yes |
Theorem 28.1. Let \( {K}_{0} \) be a compact connected Lie group. Then the compact and noncompact symmetric spaces of Examples 28.2 and 28.3 are in duality. | Proof. Let \( \mathfrak{g} \) and \( {\mathfrak{k}}_{0} \) be the Lie algebras of \( G \) and \( {K}_{0} \), respectively. We have \( \mathfrak{g} = \mathbb{C} \otimes {\mathfrak{k}}_{0} \) . The involution \( \theta : \mathfrak{g} \rightarrow \mathfrak{g} \) takes \( a \otimes X \rightarrow \bar{a} \otimes X \) . By L... | Yes |
Theorem 28.2. Let \( G \) be a noncompact, connected semisimple Lie group with an involution \( \theta \) satisfying Hypothesis 28.1. Then \( K \) is a maximal compact subgroup of \( G \). Indeed, if \( {K}^{\prime } \) is any compact subgroup of \( G \), then \( {K}^{\prime } \) is conjugate to a subgroup of \( K \). | Proof. This follows from Helgason [66], Theorem 2.1 of Chap. VI on page 246. (Note the hypothesis that \( K \) be compact in our Proposition 28.1.) The proof in [66] depends on showing that \( G/K \) is a space of constant negative curvature. A compact group of isometries of such a space has a fixed point ([66], Theore... | Yes |
Theorem 28.3. If \( \mathfrak{g} \) is a noncompact Lie algebra, then there exists a noncom-pact Lie group \( G \) with Lie algebra \( \mathfrak{g} \) and a Cartan involution \( \theta \) of \( G \) with fixed points that are a maximal subgroup \( K \) of \( G \) so that \( G/K \) is a symmetric space of noncompact typ... | Proof. It follows from Helgason [66], Chap. III, Theorem 6.4 on p. 181, that \( \mathfrak{g} \) has a compact form; that is, a compact Lie algebra \( {\mathfrak{g}}_{c} \) with an isomorphic complexification. It follows from Theorems 7.1 and 7.2 in Chap. III of [66] that we may arrange things so that \( {\mathfrak{g}}_... | Yes |
Consider \( \mathrm{{SL}}\left( {2,\mathbb{R}}\right) /\mathrm{{SO}}\left( 2\right) \) and \( \mathrm{{SU}}\left( 2\right) /\mathrm{{SO}}\left( 2\right) \). Unlike the general case of \( \mathrm{{SL}}\left( {n,\mathbb{R}}\right) /\mathrm{{SO}}\left( n\right) \) and \( \mathrm{{SU}}\left( n\right) /\mathrm{{SO}}\left( n... | Specifically, \( \mathrm{{SL}}\left( {2,\mathbb{R}}\right) \) acts transitively on the Poincaré upper half-plane \( \mathfrak{H} = \{ z = x + {iy} \mid x, y \in \mathbb{R}, y > 0\} \) by linear fractional transformations:\n\n\[ \operatorname{SL}\left( {2,\mathbb{R}}\right) \ni \left( \begin{array}{ll} a & b \\ c & d \e... | Yes |
Proposition 28.3. Let \( X = G/K \) and \( {X}_{c} = {G}_{c}/K \) be a pair of irreducible symmetric spaces in duality. If one is a Hermitian symmetric space, then they both are. This will be true if and only if the center of \( K \) is a one-dimensional central torus \( Z \) . In this case, the rank of \( {G}_{c} \) e... | Proof. See Helgason [66], Theorem 6.1 and Proposition 6.2, or Wolf [176], Corollary 8.7.10, for the first statement. The latter reference has two other very interesting conditions for the space to be symmetric. The fact that \( {G}_{c} \) and \( K \) are of equal rank is contained in Helgason [66] in the first paragrap... | No |
The Cayley transform is the element \( c \in \mathrm{{SU}}\left( 2\right) \) given by \[ c = \frac{1}{\sqrt{2i}}\left( \begin{matrix} 1 & - i \\ 1 & i \end{matrix}\right) ,\;\text{ so }\;{c}^{-1} = \frac{1}{\sqrt{2i}}\left( \begin{matrix} i & i \\ - 1 & 1 \end{matrix}\right) . \] Interpreted as a transformation of \( \... | The effect of the Cayley transform is shown in Fig. 28.1.  Fig. 28.1. The Cayley transform The significance of the Cayley transform is that it relates a bounded symmetric domain \( \mathfrak{D} \) to an unbounded one... | Yes |
Proposition 28.5. We have \( P{G}_{c} = \operatorname{Sp}\left( {{2n},\mathbb{C}}\right) \) and \( P \cap {G}_{c} = {cK}{c}^{-1} \) . | Proof. Indeed, \( P \) contains a Borel subgroup, the group \( B \) of matrices (28.12) with \( g \) upper triangular, so \( P{G}_{c} = \operatorname{Sp}\left( {{2n},\mathbb{C}}\right) \) follows from the Iwasawa decomposition (Theorem 26.3). The group \( K \) is \( \mathrm{U}\left( n\right) \) embedded via (28.7), and... | Yes |
Lemma 28.3. Suppose that\n\n\\[ \n g = \\left( \\begin{array}{ll} A & B \\\\ C & D \\end{array}\\right) ,\\;{g}^{\\prime } = \\left( \\begin{array}{ll} {A}^{\\prime } & {B}^{\\prime } \\\\ {C}^{\\prime } & {D}^{\\prime } \\end{array}\\right) ,\n\\]\n\nare elements of \\( {G}_{\\mathbb{C}} \\) . Then \\( {gP} = {g}^{\\p... | Proof. Most of this is safely left to the reader. We only point out the reason that \\( A{C}^{-1} \\) is symmetric. By (28.6), the matrix \\( {}^{t}{CA} \\) is symmetric, so \\( {}^{t}{C}^{-1} \\) . \\( {}^{t}{CA} \\cdot {C}^{-1} = A{C}^{-1} \\) is also. | No |
Proposition 28.6. If \( \sigma \left( Z\right) \) and \( g\left( {\sigma \left( Z\right) }\right) \) are both in \( {\Re }_{n}^{ \circ } \), where \( g = \left( \begin{array}{ll} A & B \\ C & D \end{array}\right) \) is an element of \( \operatorname{Sp}\left( {{2n},\mathbb{C}}\right) \), then \( {CZ} + D \) is invertib... | Proof. We have \[ g\left( {\sigma \left( Z\right) }\right) = \left( \begin{array}{ll} A & B \\ C & D \end{array}\right) \left( \begin{array}{ll} Z & - I \\ I & \end{array}\right) P = \left( \begin{array}{ll} {AZ} + B & - A \\ {CZ} + D & - C \end{array}\right) P. \] Since we are assuming this is in \( {\mathfrak{R}}_{n}... | Yes |
Proposition 28.7. The image of \( {\mathfrak{H}}_{n} \) under \( c \) is\n\n\[ \n{\mathfrak{D}}_{n} = \left\{ {W \in {\Re }_{n}^{ \circ } \mid I - \bar{W}W > 0}\right\} \n\]\n\nThe group \( c\operatorname{Sp}\left( {{2n},\mathbb{R}}\right) {c}^{-1} \), acting on \( {\mathfrak{D}}_{n} \) by linear fractional transformat... | Proof. The condition on \( W \) to be in \( c\left( \mathfrak{H}\right) \) is that the imaginary part of\n\n\[ \n{c}^{-1}\left( W\right) = - i\left( {W - I}\right) {\left( W + I\right) }^{-1} \n\]\n\nbe positive definite. This imaginary part is\n\n\[ \nY = - \frac{1}{2}\left( {\left( {W - I}\right) {\left( W + I\right)... | Yes |
Proposition 28.8. (i) The closure of \( {\mathfrak{D}}_{n} \) is contained within \( {\mathfrak{R}}_{n}^{ \circ } \). The boundary of \( {\mathfrak{D}}_{n} \) consists of all complex symmetric matrices \( W \) such that \( I - \bar{W}W \) is positive semidefinite but such that \( \det \left( {I - \bar{W}W}\right) = 0 \... | Proof. The diagonal entries in \( \bar{W}W \) are the squares of the lengths of the rows of the symmetric matrix \( W \). If \( I - \bar{W}W \) is positive definite, these must be less than 1. So \( {\mathfrak{D}}_{n} \) is a bounded domain within the set \( {\mathfrak{R}}_{n}^{ \circ } \) of symmetric complex matrices... | No |
Theorem 28.4. The domain \( {\mathfrak{D}}_{n} \) has \( {\mathcal{E}}_{n}\left( \mathbb{R}\right) \) as its Bergman-Shilov boundary. | Proof. Let \( f \) be a holomorphic function on \( {\mathfrak{D}}_{n} \) that is continuous on its closure. We will show that \( f \) takes its maximum on the set \( {\mathcal{E}}_{n}\left( \mathbb{R}\right) \) . This is sufficient because \( G \) acts transitively on \( {\mathcal{E}}_{n}\left( \mathbb{R}\right) \), so... | Yes |
The set defined by the inequality \( {x}_{0} > \sqrt{{x}_{1}^{2} + \cdots + {x}_{n}^{2}} \) in \( {\mathbb{R}}^{n + 1} \) is a self-dual cone, which we will denote \( \mathcal{P}\left( {n,1}\right) \) . The group of automorphisms is the group of similitudes for the quadratic form \( {x}_{0}^{2} - {x}_{1}^{2} - \cdots -... | The following special cases are worth noting: \( \mathcal{P}\left( {2,1}\right) \cong {\mathcal{P}}_{2}\left( \mathbb{R}\right) \) can be identified with the Poincaré upper half-plane, \( {\mathcal{P}}^{ \circ }\left( {3,1}\right) \) can be identified with \( {\mathcal{P}}_{2}\left( \mathbb{C}\right) \), and \( {\mathc... | Yes |
Proposition 29.1. Assume that the assumptions of Hypothesis 28.1 are satisfied. Then the map\n\n\[ \left( {Z, k}\right) \mapsto \exp \left( Z\right) k \]\n\n(29.1)\n\nis a diffeomorphism \( \mathfrak{p} \times K \rightarrow G \) . | Proof. Choosing a faithful representation \( \left( {\pi, V}\right) \) of the compact group \( {G}_{c} \), we may embed \( {G}_{c} \) into \( \mathrm{{GL}}\left( V\right) \) . We may find a positive definite invariant inner product on \( V \) and, on choosing an orthonormal basis, we may embed \( {G}_{c} \) into \( \ma... | Yes |
Proposition 29.2. Assume that the assumptions of Hypothesis 28.1 are satisfied. Let \( \mathfrak{a} \) be a maximal Abelian subspace of \( \mathfrak{p} \) . Then \( A = \exp \left( \mathfrak{a}\right) \) is a closed Lie subgroup of \( G \), and \( \mathfrak{a} \) is its Lie algebra. There exists a \( \theta \) -stable ... | Proof. By Proposition 15.2, A is an Abelian group. By Proposition 29.1, the restriction of exp to \( \mathfrak{p} \) is a diffeomorphism onto its image, which is closed in \( G \) , and since \( \mathfrak{a} \) is closed in \( \mathfrak{p} \) it follows that \( \exp \left( \mathfrak{a}\right) \) is closed and isomorphi... | Yes |
Lemma 29.1. Let \( Z \in \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) be a Hermitian matrix. If \( g \in \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) commutes with \( \exp \left( Z\right) \), then \( g \) commutes with \( Z \) . | Proof. Let \( {\lambda }_{1},\ldots ,{\lambda }_{h} \) be the distinct eigenvalues of \( Z \) . Let us choose a basis with respect to which \( Z \) has the matrix\n\n\[ \left( \begin{array}{lll} {\lambda }_{1}{I}_{{r}_{1}} & & \\ & \ddots & \\ & & {\lambda }_{h}{I}_{{r}_{h}} \end{array}\right) . \]\n\nThen \( \exp \lef... | Yes |
Proposition 29.4. Every rational character of \( A \) has the form\n\n\[ \left( {{t}_{1},\ldots ,{t}_{r}}\right) \mapsto {t}_{1}^{{k}_{1}}\cdots {t}_{r}^{{k}_{r}},\;{k}_{i} \in \mathbb{Z}. \] | Proof. Obviously (29.2) is a rational character. Extending any rational character of \( A \) to an analytic character of \( {A}_{\mathbb{C}} \) and then restricting it to \( {A}_{c} \) gives a homomorphism \( {X}^{ * }\left( A\right) \rightarrow {X}^{ * }\left( {A}_{c}\right) \), and since the characters of \( {X}^{ * ... | No |
Proposition 29.5. (i) In the context of Proposition 29.2, if \( \alpha \in {\Phi }_{\text{rel }} \), then \( {\mathfrak{X}}_{\alpha }^{\text{rel }} \cap \mathfrak{g} \) spans \( {\mathfrak{X}}_{\alpha }^{\text{rel }} \) . | Proof. We show that we may find a basis \( {X}_{1},\ldots ,{X}_{h} \) of the complex vector space \( {\mathfrak{X}}_{\alpha }^{\text{rel }} \) such that \( {X}_{i} \in \mathfrak{g} \) . Suppose that \( {X}_{1},\ldots ,{X}_{h} \) are a maximal linearly independent subset of \( {\mathfrak{X}}_{\alpha }^{\text{rel }} \) s... | Yes |
Proposition 29.6. Suppose that \( \beta \in \Phi \) . If the restriction of \( \beta \) to \( A \) is trivial, then \( {\mathfrak{X}}_{\beta } \) is contained in the complexification of \( \mathfrak{m} \) and \( \beta \) is a root of the compact group \( M \) with respect to \( {T}_{M} \) . | Proof. We show that \( {\mathfrak{X}}_{\beta } \) is \( \theta \) -stable. Let \( X \in {\mathfrak{X}}_{\beta } \) . Then\n\n\[ \left\lbrack {H, X}\right\rbrack = \mathrm{d}\beta \left( H\right) X,\;H \in \mathfrak{t}. \]\n\n(29.5)\n\nWe must show that \( \theta \left( X\right) \) has the same property. Applying \( \th... | Yes |
In the context of Proposition 29.2, let \( \alpha \in {\Phi }_{\text{rel }} \) . Let \( {A}_{\alpha } \subset \) \( A \) be the kernel of \( \alpha \), let \( {G}_{\alpha } \subset G \) be its centralizer, and let \( {\mathfrak{g}}_{\alpha } \subset \mathfrak{g} \) be the Lie algebra of \( {G}_{\alpha } \) . There exis... | Choose \( 0 \neq {X}_{\alpha } \in {\mathfrak{X}}_{\alpha } \) . By Proposition 29.5, we may choose \( {X}_{\alpha } \in \mathfrak{g} \), and denoting \( {X}_{-\alpha } = - \theta \left( {X}_{a}\right) \) we have \( {X}_{-\alpha } \in {\mathfrak{X}}_{-\alpha } \cap \mathfrak{g} \) and \( {H}_{\alpha } = \left\lbrack {{... | Yes |
Proposition 29.8. If \( \alpha \in {\Phi }_{\text{rel }}^{ + } \) is a simple positive root, then there exists a \( \beta \in {\Phi }^{ + } \) such that \( \beta \) is a simple positive root and \( \beta \mid \alpha \) . Moreover, if \( \beta \in {\Phi }^{ + } \) is a simple positive root with a restriction to \( A \) ... | Proof. Find a root \( \gamma \in \Phi \) whose restriction to \( A \) is \( \alpha \) . Since we have chosen the root systems compatibly, \( \gamma \) is a positive root. We write it as a sum of positive roots: \( \gamma = \sum {\beta }_{i} \) . Each of these restricts either trivially or to a relative root in \( {\Phi... | Yes |
Proposition 29.9. Let \( \beta \in {\Phi }^{ + } \) . Then \( - \theta \left( \beta \right) \in {\Phi }^{ + } \) . The roots \( \beta \) and \( - \theta \left( \beta \right) \) have the same restriction to \( A \) . If \( \beta \) is a simple positive root, then so is \( - \theta \left( \beta \right) \) , and if \( \al... | Proof. The fact that \( \beta \) and \( - \theta \left( \beta \right) \) have the same restriction follows from Proposition 29.5 (ii). It follows immediately that \( - \theta \left( \beta \right) \) is a positive root in \( \Phi \) . The map \( \beta \mapsto - \theta \left( \beta \right) \) permutes the positive roots,... | Yes |
Proposition 29.10. Let \( G,{G}_{c}, K,\mathfrak{g} \), and \( \theta \) satisfy Hypothesis 28.1. Let \( M \) and \( A \) be as in Propositions 29.2 and 29.3. Let \( \Phi \) and \( {\Phi }_{\text{rel }} \) be the absolute and relative root systems, and let \( {\Phi }^{ + } \) and \( {\Phi }_{\text{rel }}^{ + } \) be th... | Proof. As part of the definition of semisimplicity, it is assumed that the semisimple group \( G \) has a faithful complex representation. Since we may embed \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) in \( \mathrm{{GL}}\left( {{2n},\mathbb{R}}\right) \), it has a faithful real representation. We may assume that \... | Yes |
Theorem 29.2. (Iwasawa decomposition) With notations as above, each element of \( g \in G \) can be factored uniquely as \( {bk} \), where \( b \in {B}_{0} \) and \( k \in \) \( K \), or as a \( {\nu k} \) where \( a \in A,\nu \in N \), and \( k \in K \) . The multiplication map \( A \times N \times K \rightarrow G \) ... | Proof. This is nearly identical to Theorem 26.3, and we mostly leave the proof to the reader. We consider only the key point that \( \mathfrak{g} = \mathfrak{a} + \mathfrak{n} + \mathfrak{k} \) . It is sufficient to show that \( {\mathfrak{g}}_{\mathbb{C}} = \mathbb{C}\mathfrak{a} + \mathbb{C}\mathfrak{n} + \mathbb{C}\... | No |
Proposition 29.11. (i) If \( H \) is regular and \( Z \in \mathfrak{p} \) satisfies \( \left\lbrack {H, Z}\right\rbrack = 0 \), then \( Z \in \mathfrak{a} \) . | Proof. The element \( H \) is singular if and only if there is some \( Z \in \mathfrak{p} - \mathfrak{a} \) such that \( \left\lbrack {Z, H}\right\rbrack = 0 \), for if this is the case, then \( H \) is contained in at least two distinct maximal Abelian subspaces, namely \( \mathfrak{a} \) and any maximal Abelian subsp... | Yes |
Theorem 29.3. Let \( {\mathfrak{a}}_{1} \) and \( {\mathfrak{a}}_{2} \) be two maximal Abelian subspaces of \( \mathfrak{p} \) . Then there exists a \( k \in \mathfrak{k} \) such that \( \operatorname{Ad}\left( k\right) {\mathfrak{a}}_{1} = {\mathfrak{a}}_{2} \) . | Proof. By Proposition 29.11 (ii), \( {\mathfrak{a}}_{1} \) and \( {\mathfrak{a}}_{2} \) contain regular elements \( {H}_{1} \) and \( {H}_{2} \) . We will show that \( \left\lbrack {\operatorname{Ad}\left( k\right) {H}_{1},{H}_{2}}\right\rbrack = 0 \) for some \( k \in \mathfrak{k} \) . Choose an Ad-invariant inner pro... | Yes |
Theorem 29.4. With notations as above, \( G = {KAK} \) . | Proof. Let \( g \in G \) . Let \( p = {g\theta }{\left( g\right) }^{-1} = {g}^{t}g \) . We will show that \( p \in \exp \left( \mathfrak{p}\right) \) . By Proposition 29.1, we can write \( p = \exp \left( Z\right) {k}_{0} \), where \( Z \in \mathfrak{p} \) and \( {k}_{0} \in K \) , and we want to show that \( {k}_{0} =... | Yes |
Theorem 29.5. (Bruhat decomposition) We have\n\n\[ G = \mathop{\bigcup }\limits_{{w \in {W}_{\text{rel }}}}{BwB} \]\n | Proof. Omitted. See Helgason [66], p. 403. | No |
Proposition 30.1. Suppose in this setting that \( S \) is any set of roots such that if \( \alpha ,\beta \in S \) and if \( \alpha + \beta \subset \Phi \), then \( \alpha + \beta \in S \) . Then\n\n\[ \mathfrak{h} = {\mathfrak{t}}_{\mathbb{C}} \oplus {\bigoplus }_{\alpha \in S}{\mathfrak{X}}_{\alpha } \]\n\n is a Lie s... | Proof. It is immediate from Proposition 18.4 (ii) and Proposition 18.3 (ii) that this vector space is closed under the bracket. | No |
Proposition 31.1. The group \( \widetilde{T} \) is connected and is a maximal torus of \( \widetilde{G} \) . | Proof. Let \( \Pi \subset \widetilde{G} \) be the kernel of \( p \) . The connected component \( {\widetilde{T}}^{ \circ } \) of the identity in \( \widetilde{T} \) is a torus of the same dimension as \( T \), so it is a maximal torus in \( \widetilde{G} \) . Its image in \( G \) is isomorphic to \( {\widetilde{T}}^{ \... | Yes |
Proposition 31.2. The weight lattice \( \Lambda = {X}^{ * }\left( \widetilde{T}\right) \) consists of all elements of \( \mathcal{V} \) of the form\n\n\[ \n\frac{1}{2}\left( {\mathop{\sum }\limits_{{i = 1}}^{n}{c}_{i}{\mathbf{e}}_{i}}\right) \n\]\n\n(31.4)\n\nwhere \( {c}_{i} \in \mathbb{Z} \) are either all even or al... | Proof. From our determination of the simple reflections, which generate \( W \) , the \( W \) -invariant inner product on \( \mathcal{V} = \mathbb{R} \otimes \Lambda \) may be chosen so that the \( {\mathbf{e}}_{i} \) are orthonormal. By Proposition 18.10 every weight \( \lambda \) is in the lattice \( \widetilde{\Lamb... | Yes |
Theorem 31.2. (i) If \( N = {2n} + 1 \), the dimension of the spin representation \( \pi \left( {\varpi }_{n}\right) \) is \( {2}^{n} \). The weights that occur with nonzero multiplicity in this representation all occur with multiplicity one; they are\n\n\[ \frac{1}{2}\left( {\pm {\mathbf{e}}_{1} \pm {\mathbf{e}}_{2} \... | Proof. There is enough information in Proposition 22.4 to determine the weights in the spin representations.\n\nSpecifically, let \( \lambda = {\varpi }_{n} \) and \( N = {2n} + 1 \) or \( {2n} \), or \( \lambda = {\varpi }_{n - 1} \) if \( N = {2n} \). Let \( S\left( \lambda \right) \) be as in Exercise 22.1. Then it ... | Yes |
Theorem 31.3. If \( N = {2n} + 1 \), then \( Z\left( G\right) \cong \mathbb{Z}/2\mathbb{Z} \). If \( N = {2n} \), then \( Z\left( G\right) \cong \) \( \mathbb{Z}/4\mathbb{Z} \) if \( n \) is odd, while \( Z\left( G\right) \cong \left( {\mathbb{Z}/2\mathbb{Z}}\right) \times \left( {\mathbb{Z}/2\mathbb{Z}}\right) \) if \... | Proof. \( {X}^{ * }\left( \widetilde{T}\right) \) is described explicitly by Proposition 31.2, and we have also described the simple roots, which generate \( {\Lambda }_{\text{root }} \) . We leave the verification that \( {X}^{ * }\left( \widetilde{T}\right) /{\Lambda }_{\text{root }} \) is as described to the reader.... | No |
Proposition 31.3. If \( A \) is a simple \( \mathbb{Z}/2\mathbb{Z} \) -graded algebra then so is \( M\left( A\right) \) . | Proof. Let \( I \) be a nonzero ideal. If \( m = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \) is a nonzero element of \( I \) , then one of \( a, b, c, d \) is nonzero. Left and/or right multiplying by \( \left( \begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) \) we may assume that \( a \neq 0 \) . ... | Yes |
Proposition 31.4. In the graded ring \( D\left( A\right) \) we have \( D{\left( A\right) }_{0} \cong A \) as a ring. | Proof. We may identify \( D\left( A\right) \) with \( A \oplus A \) as a vector space in which \( a \otimes 1 + b \otimes \zeta \) is identified with the ordered pair \( \left( {a, b}\right) \) . In view of (31.5) the multiplication is\n\n\[ \left( {a, b}\right) \left( {c, d}\right) = \left( {{ac} - b\bar{d},{ad} + b\b... | Yes |
Proposition 31.5. (i) The Clifford algebra is a \( \mathbb{Z}/2\mathbb{Z} \) -graded algebra. | Proof. The tenor algebra \( T = T\left( V\right) \) is a graded algebra in which the homogeneous part of degree \( k \) is \( { \otimes }^{k}V \) . Let\n\n\[ \n{T}_{i} = {\bigoplus }_{k \equiv i{\;\operatorname{mod}\;2}}{ \otimes }^{k}V,\;\left( {i = 0,1}\right) .\n\]\n\nLet \( \mathfrak{R} \) be the vector space in \(... | Yes |
Proposition 31.6. If \( V \) is a hyperbolic plane then \( C\left( V\right) \cong M\left( F\right) \) as \( \mathbb{Z}/2\mathbb{Z} \) - graded algebras. | Proof. Let \( X = \left( \begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right) \) and \( Y = \left( \begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right) \) . With \( x, y \) such that \( q\left( x\right) = q\left( y\right) = 0 \) and \( B\left( {x, y}\right) = \frac{1}{2} \), we have \( {x}^{2} = {y}^{2} = 0 \) and \( {xy... | Yes |
Lemma 31.1. Assume that \( F \) is algebraically closed and \( V \) is nondegenerate. If \( \dim \left( V\right) \geq 2 \) then \( V \) may be decomposed as \( {V}_{0} \oplus {V}^{\prime } \) where \( {V}_{0} \) is a hyperbolic plane and \( {V}^{\prime } \) is its orthogonal complement. | Proof. Let \( v \) be any vector with \( q\left( v\right) \neq 0 \), and let \( w \) be any nonzero vector in the orthogonal complement of \( v \) . Then \( q\left( w\right) \neq 0 \) also since \( V \) otherwise it is in the kernel of the associated symmetric bilinear form \( B \), but \( B \) is nondegenerate. Let \(... | Yes |
Proposition 31.7. If \( F \) is algebraically closed and \( V \) is a nondegenerate quadratic space of dimension \( {2n} \) or \( {2n} + 1 \), then \( V \) contains Lagrangian subspaces \( W \) and \( {W}^{\prime } \) such that \( W \cap {W}^{\prime } = 0 \), and \( B \) induces a nondegenerate pairing \( W \times {W}^... | Proof. Using the Lemma 31.1 repeatedly, we may decompose \( V = {V}_{1} \oplus {V}_{2} \oplus \) \( \ldots \oplus {V}_{n} \oplus {V}^{\prime } \) where \( {V}_{i} \) are hyperbolic planes and \( {V}^{\prime } \) is either zero or one-dimensional. Each \( {V}_{i} \) is spanned by two isotropic vectors \( {x}_{i} \) and ... | Yes |
Theorem 31.4. Let \( V \) be a quadratic space with a nondegenerate symmetric bilinear form and a Lagrangian decomposition \( V = W \oplus {W}^{\prime } \oplus {V}_{0} \), and let \( \Omega = \bigwedge W \) as in Proposition 31.8. Assume that the ground field contains an element \( i \) such that \( {i}^{2} = - 1 \) . ... | Proof. Since by Lemma 31.1 the even-dimensional subspace \( W \oplus {W}^{\prime } \) is an orthogonal direct sum of hyperbolic planes, it follows from Proposition 31.6 that \( A = C\left( {W \oplus {W}^{\prime }}\right) \) is a simple algebra. If \( \dim \left( V\right) = {2n} \) then \( R = A \) .\n\nOn the other han... | Yes |
Proposition 31.9. Suppose that \( \pi : G \rightarrow \operatorname{PGL}\left( V\right) \) is a projective representation of \( G \) . Then there exists a central extension \( \widehat{G} \) of \( G \) by \( {\mathbb{C}}^{ \times } \) and a representation of \( \widehat{G} \) such that \( \pi \) is the projective repre... | Proof. Choose a map \( {\pi }^{\prime } : G \rightarrow \mathrm{{GL}}\left( V\right) \) such that \( P \circ {\pi }^{\prime } = \pi \) . Then \( {\pi }^{\prime }\left( {{g}_{1}{g}_{2}}\right) \) differs from \( {\pi }^{\prime }\left( {g}_{1}\right) {\pi }^{\prime }\left( {g}_{2}\right) \) by a scalar linear transformat... | Yes |
Proposition 31.10. Suppose that \( G \) is a simply-connected Lie group, and let \( \pi : G \rightarrow \operatorname{PGL}\left( V\right) \) be a projective representation. Then there exists a representation \( \widehat{\pi } : G \rightarrow \mathrm{{GL}}\left( V\right) \) such that \( \pi \) is the projective represen... | Proof. Let \( d = \dim \left( V\right) \) . The natural map \( \mathrm{{SL}}\left( V\right) \rightarrow \mathrm{{PGL}}\left( V\right) \) has kernel of order \( d \), consisting of scalar linear transformations \( \varepsilon {I}_{V} \) where \( \varepsilon \) is an \( d \) - th root of unity. Hence this is a covering m... | Yes |
Proposition 31.11. Let \( G \) be a group, and let \( R \) be a \( \mathbb{C} \) -algebra that has a unique isomorphism class of simple modules. Let \( \Omega \) be such a module, and let \( \omega : R \rightarrow {\operatorname{End}}_{\mathbb{C}}\left( \Omega \right) \) be the \( \mathbb{C} \) -algebra homomorphism de... | Proof. Given \( g \in G \), define another \( R \) -module structure on \( \Omega \) by means of the homomorphism \( {}^{g}\omega : R \rightarrow {\operatorname{End}}_{\mathbb{C}}\left( \Omega \right) \) given by \( {}^{g}\omega \left( {\rho \left( g\right) r}\right) = \omega \left( r\right), r \in R \) . Denote by \( ... | Yes |
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