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Lemma 3.1.21. Define \( \rho \in {\mathfrak{h}}^{ * } \) by\n\n\[ \langle \rho, Y\rangle = \frac{1}{2}\operatorname{tr}\left( {\left. \operatorname{ad}\left( Y\right) \right| }_{{\mathfrak{n}}^{ + }}\right) = \frac{1}{2}\mathop{\sum }\limits_{{\alpha \in {\Phi }^{ + }}}\langle \alpha, Y\rangle \]\n\nfor \( Y \in \mathf... | Proof. Let \( {s}_{i} \in W \) be the reflection in the root \( {\alpha }_{i} \) . By (3.8) we have\n\n\[ {s}_{i}\left( \rho \right) = - \frac{1}{2}{\alpha }_{i} + \frac{1}{2}\mathop{\sum }\limits_{{\beta \in {\Phi }^{ + } \smallsetminus \left\{ {\alpha }_{i}\right\} }}\beta = \rho - {\alpha }_{i}. \]\n\nBut we also ha... | Yes |
Lemma 3.2.2. Let \( V \) be a highest-weight representation of \( \mathfrak{g} \) as in Definition 3.2.1.\n\nThen\n\n\[ V = \mathbb{C}{v}_{0} \oplus {\bigoplus }_{\mu \prec \lambda }V\left( \mu \right) \]\n\n(3.21)\n\nwith \( \dim V\left( \mu \right) < \infty \) for all \( \mu \) . In particular, \( \dim V\left( \lambd... | Proof. Since \( \mathfrak{g} = {\mathfrak{n}}^{ - } \oplus \mathfrak{h} \oplus {\mathfrak{n}}^{ + } \), the Poincaré-Birkhoff-Witt theorem implies that there is a linear bijection \( U\left( {\mathfrak{n}}^{ - }\right) \otimes U\left( \mathfrak{h}\right) \otimes U\left( {\mathfrak{n}}^{ + }\right) \rightarrow U\left( \... | Yes |
Corollary 3.2.3. Let \( \\left( {\\pi, V}\\right) \) be a nonzero irreducible finite-dimensional representation of \( \\mathfrak{g} \) . There exists a unique dominant integral \( \\lambda \\in {\\mathfrak{h}}^{ * } \) such that \( \\dim V\\left( \\lambda \\right) = 1 \) . Furthermore, every \( \\mu \\in \\mathcal{X}\\... | Proof. By Theorem 3.1.16 we know that \( X\\left( V\\right) \\subset P\\left( \\mathfrak{g}\\right) \) . Let \( \\lambda \) be any maximal element (relative to the root order) in the finite set \( \\mathcal{X}\\left( V\\right) \) . If \( \\alpha \\in {\\Phi }^{ + } \) and \( x \\in {\\mathfrak{g}}_{\\alpha } \) then \(... | Yes |
Proposition 3.2.7. Let \( \left( {\pi, V}\right) \) be a finite-dimensional representation of \( \mathfrak{g} \) with weight space decomposition\n\n\[ V = {\bigoplus }_{\mu \in \mathcal{X}\left( V\right) }V\left( \mu \right) \]\n\nFor \( \alpha \in \Phi \) let \( \left\{ {{e}_{\alpha },{f}_{\alpha },{h}_{\alpha }}\righ... | Proof. (1): From Theorem 2.3.6 and Proposition 2.3.3 we know that \( E \) and \( F \) are nilpotent transformations. If \( X \) is any nilpotent linear transformation on \( V \), then \( \operatorname{ad}\left( X\right) \) is nilpotent on \( \operatorname{End}\left( V\right) \) and we have\n\n\[ \exp \left( X\right) A\... | Yes |
Lemma 3.2.9. Let \( \left( {\pi, V}\right) \) be a finite-dimensional representation of \( \mathfrak{g} \) and let \( X\left( V\right) \) be the set of weights of \( V \) . If \( \lambda \in \mathcal{X}\left( V\right) \) then \( \lambda - {k\alpha } \in \mathcal{X}\left( V\right) \) for all roots \( \alpha \in \Phi \) ... | Proof. We may suppose that the integer \( m = \left\langle {\lambda ,{h}_{\alpha }}\right\rangle \) is nonzero. Since \( {s}_{\alpha } \cdot \lambda = \) \( \lambda - {m\alpha } \), we have\n\n\[ \dim V\left( \lambda \right) = \dim V\left( {\lambda - {m\alpha }}\right) \]\n\nby Proposition 3.2.7. Take \( 0 \neq v \in V... | Yes |
Proposition 3.2.10. Let \( V \) be the finite-dimensional irreducible \( \mathfrak{g} \) -module with highest weight \( \lambda \) . 1. \( X\left( V\right) \) is the smallest \( \Phi \) -saturated subset of \( P\left( \mathfrak{g}\right) \) containing \( \lambda \) . 2. The orbit of \( \lambda \) under the Weyl group i... | Proof. (1): Let \( {\Psi }^{\prime } \subset X\left( V\right) \) be the smallest \( \Phi \) -saturated subset of \( P\left( \mathfrak{g}\right) \) containing \( \lambda \) . If \( {\Psi }^{\prime \prime } = \chi \left( V\right) \smallsetminus {\Psi }^{\prime } \) were nonempty, then it would contain a maximal element \... | Yes |
Proposition 3.2.11. Let \( V \) be any finite-dimensional representation of \( \mathfrak{g} \). Suppose \( \mu \in {P}_{+ + }\left( \mathfrak{g}\right), v \in \mathcal{X}\left( V\right) \), and \( \mu \prec v \). Then \( \mu \in \mathcal{X}\left( V\right) \). | Proof. By assumption, \( v = \mu + \beta \), where \( \beta = \mathop{\sum }\limits_{{i = 1}}^{l}{n}_{i}{\alpha }_{i} \in {Q}_{ + } \). We proceed by induction on \( \operatorname{ht}\left( \beta \right) = \sum {n}_{i} \), the result being true if \( \beta = 0 \). If \( \beta \neq 0 \) then\n\n\[ 0 < \left( {\beta ,\be... | Yes |
Corollary 3.2.12. Let \( {L}^{\lambda } \) be the finite-dimensional irreducible \( \mathfrak{g} \) -module with highest weight \( \lambda \) . Then \( \mathcal{X}\left( {L}^{\lambda }\right) \cap {P}_{+ + }\left( \mathfrak{g}\right) \) consists of all \( \mu \in {P}_{+ + }\left( \mathfrak{g}\right) \) such that \( \mu... | Corollary 3.2.12 and inequality (3.28) give an explicit algorithm for finding the weights of \( {L}^{\lambda } \) . Take all \( \beta \in {Q}_{ + } \) such that \( \parallel \lambda - \beta \parallel \leq \parallel \lambda \parallel \) (there are only finitely many) and write \( \mu = \lambda - \beta \) in terms of the... | Yes |
Theorem 3.2.13. Let \( \left( {\pi, V}\right) \) be an irreducible finite-dimensional \( \mathfrak{g} \) -module with highest weight \( \lambda \) and let \( \left( {{\pi }^{ * },{V}^{ * }}\right) \) be the dual module. Then the lowest weight of \( V \) is \( {w}_{0}\left( \lambda \right) \) . The highest weight of \( ... | Proof. The set \( \mathcal{X}\left( V\right) \) is invariant under \( W \) by Proposition 3.2.7. Since \( {w}_{0} \cdot {Q}_{ + } = \) \( - {Q}_{ + } \), we have \( {w}_{0}\left( \lambda \right) \prec \mu \) for all \( \mu \in \mathcal{X}\left( V\right) \), which implies the first assertion. To find the highest and low... | Yes |
Theorem 3.2.14. Suppose \( \left( {\pi, V}\right) \) is an irreducible finite-dimensional representation of \( \mathfrak{g} \) with highest weight \( \lambda \) . There is a nonzero \( \mathfrak{g} \) -invariant bilinear form on \( V \) if and only if \( - {w}_{0}\lambda = \lambda \) . In this case the form is nonsingu... | Proof. We can identify the invariant bilinear forms on \( V \) with \( {\operatorname{Hom}}_{\mathfrak{g}}\left( {V,{V}^{ * }}\right) \) as a \( \mathfrak{g} \) - module. In this identification a \( \mathfrak{g} \) -intertwining operator \( T : V \rightarrow {V}^{ * } \) corresponds to the bilinear form\n\n\[ \Omega \l... | Yes |
Lemma 3.2.15. Suppose \( G = \mathbf{{SL}}\left( {2,\mathbb{C}}\right) \) and \( \left( {\pi, V}\right) \) is the \( \left( {m + 1}\right) \) -dimensional irreducible representation of \( G \) . Then \( \pi \) is symplectic if \( m \) is odd, and \( \pi \) is orthogonal if \( m \) is even. | Proof. The element \( {w}_{0} \) acts on \( {\mathfrak{h}}^{ * } \) by -1, so every irreducible representation of \( G \) is self-dual. Recall from Section 2.3.2 that we can take for \( V \) the space of polynomials \( f\left( {{x}_{1},{x}_{2}}\right) \) that are homogeneous of degree \( m \), with action\n\n\[ \pi \le... | Yes |
Lemma 3.2.16. Let \( {\alpha }_{1},\ldots ,{\alpha }_{l} \) be the simple roots in \( {\Phi }^{ + } \) and let \( {H}_{i} \) be the coroot to \( {\alpha }_{i} \) . Then \( \left\langle {{\alpha }_{i},{h}^{0}}\right\rangle = 2 \) for \( i = 1,\ldots, l \) . Furthermore, there are integers \( {c}_{i} \geq 1 \) such that\... | Proof. Let \( {s}_{i} \in W \) be the reflection in the root \( {\alpha }_{i} \) and let \( {\check{\Phi }}_{i}^{ + } \) be the set of positive coroots with \( {H}_{i} \) deleted. Then \( {s}_{i} \) preserves \( {\check{\Phi }}_{i}^{ + } \) and \( {s}_{i}\left( {H}_{i}\right) = - {H}_{i} \) . Hence\n\n\[ \n{s}_{i}\left... | Yes |
Theorem 3.2.17. Let \( \left( {\pi, V}\right) \) be the irreducible representation of \( \mathfrak{g} \) with highest weight \( \lambda \neq 0 \) . Assume that \( - {w}_{0}\lambda = \lambda \) . Set \( m = \left\langle {\lambda ,{h}^{0}}\right\rangle \) . Then \( m \) is a positive integer. If \( m \) is odd, then \( \... | Proof. Write \( \lambda = {m}_{1}{\varpi }_{1} + \cdots + {m}_{l}{\varpi }_{l} \) in terms of the fundamental weights and let \( {c}_{i} \) be the integers in Lemma 3.2.16. Then\n\n\[ m = \mathop{\sum }\limits_{{i = 1}}^{l}{c}_{i}{m}_{i} \]\n\nfrom which it is clear that \( m \) is a positive integer.\n\nFix a nonzero ... | Yes |
Lemma 3.3.2. Let \( \left( {\rho, V}\right) \) be a completely reducible rational representation of the algebraic group \( G \) . Suppose \( W \subset V \) is an invariant subspace. Set \( \sigma \left( x\right) = {\left. \rho \left( x\right) \right| }_{W} \) and \( \pi \left( x\right) \left( {v + W}\right) = \rho \lef... | Proof. Write \( V = W \oplus U \) for some invariant subspace \( U \), and let \( P \) be the projection onto \( W \) with kernel \( U \) . If \( Y \subset W \) is an invariant subspace, then the subspace \( U \oplus Y \) is invariant. Hence there is an invariant subspace \( Z \subset V \) such that\n\n\[ V = \left( {U... | Yes |
Corollary 3.3.4. Suppose \( \left( {\rho, V}\right) \) and \( \left( {\sigma, W}\right) \) are completely reducible regular representations of \( G \) . Then \( \left( {\rho \oplus \sigma, V \oplus W}\right) \) is a completely reducible representation. | Proof. By Proposition 3.3.3, \( V \) and \( W \) are direct sums of irreducible invariant subspaces. Thus\n\n\[ V = {V}_{1} \oplus \cdots \oplus {V}_{m}\;\text{ and }\;W = {W}_{1} \oplus \cdots \oplus {W}_{n}. \]\n\nIt follows that \( V \oplus W \) satisfies condition (2) in Proposition 3.3.3. | Yes |
Proposition 3.3.5. Let \( G \) and \( H \) be linear algebraic groups with \( H \subset G \) . Assume that \( H \) is reductive and has finite index in \( G \) . Then \( G \) is reductive. | Proof. Let \( \left( {\rho, V}\right) \) be a rational representation of \( G \) and suppose \( W \subset V \) is a \( G \) - invariant subspace. Since \( H \) is reductive, there exists a \( H \) -invariant subspace \( Z \) such that \( V = W \oplus Z \) . Let \( P : V \rightarrow W \) be the projection along \( Z \) ... | Yes |
Lemma 3.3.7. The Casimir operator is independent of the choice of basis for \( \mathfrak{g} \) and commutes with \( \pi \left( \mathfrak{g}\right) \) . | Proof. We can choose a basis \( \left\{ {Z}_{i}\right\} \) for \( \mathfrak{g} \) such that \( B\left( {{Z}_{i},{Z}_{j}}\right) = {\delta }_{ij} \) . Write \( {X}_{i} = \) \( \mathop{\sum }\limits_{j}B\left( {{X}_{i},{Z}_{j}}\right) {Z}_{j} \) and \( {Y}_{i} = \mathop{\sum }\limits_{k}B\left( {{Y}_{i},{Z}_{k}}\right) {... | Yes |
Lemma 3.3.8. Let \( \left( {\pi, V}\right) \) be a highest-weight representation of \( \mathfrak{g} \) with highest weight \( \lambda \) and let \( \rho = \left( {1/2}\right) \mathop{\sum }\limits_{{\alpha \in {\Phi }^{ + }}}\alpha \) . Then the Casimir operator acts on \( V \) as a scalar:\n\n\[ \n{C}_{\pi }v = \left(... | Proof. Let \( {H}_{1},\ldots ,{H}_{l} \) be an orthonormal basis of \( {\mathfrak{h}}_{\mathbb{R}} \) with respect to \( B \) . Enumerate \( {\Phi }^{ + } = \left\{ {{\alpha }_{1},\ldots ,{\alpha }_{d}}\right\} \) and for \( \alpha \in {\Phi }^{ + } \) fix \( {X}_{\pm \alpha } \in {\mathfrak{g}}_{\pm \alpha } \), norma... | Yes |
Proposition 3.3.9. Let \( V \) be a finite-dimensional highest-weight \( \mathfrak{g} \) -module with highest weight \( \lambda \) . Then \( \lambda \) is dominant integral and \( V \) is irreducible. Hence \( V \) is isomorphic to \( {L}^{\lambda } \) . | Proof. The assumption of finite-dimensionality implies that \( \lambda \in P\left( \mathfrak{g}\right) \) by Theorem 3.1.16. Since \( \lambda \) is the maximal weight of \( V \), Theorem 2.3.6 applied to the subal-gebras \( \mathfrak{s}\left( \alpha \right) \) for \( \alpha \in {\Phi }^{ + } \) shows that \( \lambda \)... | Yes |
Corollary 3.3.14. Let \( V \) be a finite-dimensional \( \mathfrak{g} \) -module. Then \( V \) is irreducible if and only if \( \dim {V}^{{\mathfrak{n}}^{ + }} = 1 \) . | Proof. If \( V \) is irreducible then \( \dim {V}^{{\mathfrak{n}}^{ + }} = 1 \) by Corollary 3.2.3 and Proposition 3.3.9. Conversely, assume that \( \dim {V}^{{\mathfrak{n}}^{ + }} = 1 \) . By Theorem 3.3.12 there is a \( \mathfrak{g} \) -module decomposition \( V = {V}_{1} \oplus \cdots \oplus {V}_{r} \) with \( {V}_{... | Yes |
Theorem 3.3.15. Suppose \( G \) is a connected algebraic group that has a compact real form. Then \( G \) is reductive. | Before proving the theorem, we recall some properties of real forms. Let \( K \) be a compact real form of \( G \) . Write \( \iota : K \rightarrow G \) for the embedding map \( \left( {\iota \left( k\right) = k}\right) \) . Let \( \mathfrak{k} = \mathrm{d}\iota \left( {\operatorname{Lie}\left( K\right) }\right) \) . T... | Yes |
Lemma 3.3.16. If \( u \in K, v, w \in V \) then \( \langle \rho \left( u\right) v \mid \rho \left( u\right) w\rangle = \langle v \mid w\rangle \) . | Proof. By Lemmas D.2.11 and D.2.12 we have\n\n\[ \langle \rho \left( u\right) v \mid \rho \left( u\right) w\rangle = {\int }_{K}\left( {\rho \left( k\right) \rho \left( u\right) v,\rho \left( k\right) \rho \left( u\right) w}\right) \mathrm{d}k \]\n\n\[ = {\int }_{K}\left( {\rho \left( {ku}\right) v,\rho \left( {ku}\rig... | Yes |
Corollary 3.3.17. Let \( G \) be a classical group. Then \( G \) is reductive. | Proof. If \( G \) is connected, then Section 1.7.2 furnishes a compact real form for \( G \) , so we may apply Theorem 3.3.15. From Theorem 2.2.5 we know that the only non-connected classical groups are the groups \( \mathbf{O}\left( {n,\mathbb{C}}\right) \) for \( n \geq 3 \) . Since \( \mathbf{{SO}}\left( {n,\mathbb{... | Yes |
Lemma 4.1.4. Let \( \\left( {\\rho, V}\\right) \) and \( \\left( {\\tau, W}\\right) \) be irreducible representations of an associative algebra \( \\mathcal{A} \) . Assume that \( V \) and \( W \) have countable dimension over \( \\mathbb{C} \) . Then\n\n\[\\]\n\\dim {\\operatorname{Hom}}_{\\mathcal{A}}\\left( {V, W}\\... | Proof. Let \( T \\in {\\operatorname{Hom}}_{\\mathcal{A}}\\left( {V, W}\\right) \) . Then \( \\operatorname{Ker}\\left( T\\right) \) and \( \\operatorname{Range}\\left( T\\right) \) are invariant subspaces of \( V \) and \( W \), respectively. If \( T \\neq 0 \), then \( \\operatorname{Ker}\\left( T\\right) \\neq V \) ... | Yes |
Corollary 4.1.6. If \( X \) is a finite-dimensional subspace of \( V \) and \( f \in \operatorname{Hom}\left( {X, L}\right) \), then there exists \( r \in \mathcal{R} \) such that \( f = {\left. r\right| }_{X} \) . | Proof. Let \( \left\{ {{v}_{1},\ldots ,{v}_{n}}\right\} \) be a basis for \( X \) and set \( {w}_{j} = f\left( {v}_{j}\right) \) for \( j = 1,\ldots, n \) . By Theorem 4.1.5 there exists \( r \in \mathcal{R} \) such that \( r{v}_{j} = {w}_{j} \) for \( j = 1,\ldots, n \) . Hence by linearity \( {\left. r\right| }_{X} =... | Yes |
Lemma 4.1.9. Let \( \left( {\rho, V}\right) \) be completely reducible and suppose \( W \subset V \) is an invariant subspace. Set \( {\left. \sigma \left( x\right) = \rho \left( x\right) \right| }_{W} \) and \( \pi \left( x\right) \left( {v + W}\right) = \rho \left( x\right) v + W \) for \( x \in \mathcal{A} \) and \(... | Proof. The proof of Lemma 3.3.2 applies verbatim to this context. | No |
Proposition 4.1.11. Let \( \left( {\rho, V}\right) \) be a finite-dimensional representation of the associative algebra \( \mathcal{A} \) . The following are equivalent:\n\n1. \( \left( {\rho, V}\right) \) is completely reducible.\n\n2. \( V = {W}_{1} \oplus \cdots \oplus {W}_{s} \) with each \( {W}_{i} \) an irreducib... | Proof. The equivalence of the three conditions follows by the proof of Proposition 3.3.3. Now assume that \( V \) satisfies these conditions and that the \( {V}_{i} \) are all mutually equivalent as \( \mathcal{A} \) -modules. Let \( M \) be an \( \mathcal{A} \) -submodule of \( V \) . Since \( V \) is completely reduc... | Yes |
Corollary 4.1.12. Suppose \( \left( {\rho, V}\right) \) and \( \left( {\sigma, W}\right) \) are completely reducible representations of \( \mathcal{A} \) . Then \( \left( {\rho \oplus \sigma, V \oplus W}\right) \) is a completely reducible representation. | Proof. This follows from the equivalence between conditions (1) and (2) in Proposition 4.1.11. | No |
Theorem 4.1.13 (Double Commutant). Suppose \( \mathcal{A} \subset \operatorname{End}V \) is an associative algebra with identity \( {I}_{V} \) . Set \( \mathcal{B} = \operatorname{Comm}\left( \mathcal{A}\right) \) . If \( V \) is a completely reducible \( \mathcal{A} \) -module, then \( \operatorname{Comm}\left( \mathc... | Proof. By definition we have \( \mathcal{A} \subset \operatorname{Comm}\left( \mathcal{B}\right) \) . Let \( T \in \operatorname{Comm}\left( \mathcal{B}\right) \) and fix a basis \( \left\{ {{v}_{1},\ldots ,{v}_{n}}\right\} \) for \( V \) . It will suffice to find an element \( S \in \mathcal{A} \) such that \( S{v}_{i... | Yes |
Proposition 4.1.17. Characters satisfy \( \operatorname{ch}V\left( {ab}\right) = \operatorname{ch}V\left( {ba}\right) \) for all \( a, b \in \mathcal{A} \) and \( \operatorname{ch}V\left( 1\right) = \dim V \) . Furthermore, if \( U \subset V \) is a submodule and \( W = V/U \), then \( \operatorname{ch}V = \operatornam... | Proof. The first two properties are obvious from the definition. The third follows by picking a subspace \( Z \subset V \) complementary to \( U \) . Then the matrix of \( \rho \left( a\right), a \in \mathcal{A} \), is in block triangular form relative to the decomposition \( V = U \oplus Z \), and the diagonal blocks ... | Yes |
Lemma 4.1.18. Suppose \( \left( {{\rho }_{1},{V}_{1}}\right) ,\ldots ,\left( {{\rho }_{r},{V}_{r}}\right) \) are finite-dimensional irreducible representations of \( \mathcal{A} \) such that \( {\rho }_{i} \) is not equivalent to \( {\rho }_{j} \) when \( i \neq j \) . Then the set \( \left\{ {\operatorname{ch}{V}_{1},... | Proof. Set \( V = {V}_{1} \oplus \cdots \oplus {V}_{r} \) and \( \rho = {\rho }_{1} \oplus \cdots \oplus {\rho }_{r} \) . Then \( \left( {\rho, V}\right) \) is a completely reducible representation of \( \mathcal{A} \) by Proposition 4.1.11. Let \( \mathcal{B} \) be the commutant of \( \rho \left( \mathcal{A}\right) \)... | Yes |
Theorem 4.1.19. Let \( \left( {\rho, V}\right) \) be a finite-dimensional \( \mathcal{A} \) -module. The irreducible factors in a composition series for \( V \) are unique up to isomorphism and order of appearance. Furthermore, the module \( {V}_{ss} \) is uniquely determined by \( \operatorname{ch}V \) up to isomorphi... | Proof. Let \( \left( {{\rho }_{i},{U}_{i}}\right) \), for \( i = 1,\ldots, n \), be the pairwise inequivalent irreducible representations that occur in the composition series for \( V \), with corresponding multiplicities \( {m}_{i} \) . Then\n\n\[ \operatorname{ch}V = \mathop{\sum }\limits_{{i = 1}}^{n}{m}_{i}\operato... | Yes |
Theorem 4.1.20 (Hermite Reciprocity). Let \( {S}^{j}\left( {F}^{\left( k\right) }\right) \) be the jth symmetric power of \( {F}^{\left( k\right) }\right) \). Then for \( q \in {\mathbb{C}}^{ \times } \), \[ \operatorname{ch}{S}^{j}\left( {F}^{\left( k\right) }\right) \left( {d\left( q\right) }\right) = {\left\lbrack \... | To prove this theorem we need some further character identities. Fix \( k \) and write \( {f}_{j}\left( q\right) = \operatorname{ch}{S}^{j}\left( {F}^{\left( k\right) }\right) \left( {d\left( q\right) }\right) \) for \( q \in {\mathbb{C}}^{ \times } \). Let \( \left\{ {{x}_{0},\ldots ,{x}_{k}}\right\} \) be a basis for... | Yes |
Lemma 4.1.21. The generating function factors as\n\n\\[ \nf\\left( {t, q}\\right) = \\mathop{\\prod }\\limits_{{j = 0}}^{k}{\\left( 1 - t{q}^{k - {2j}}\\right) }^{-1}.\n\\]\n\n(4.13) | Proof. By definition \\( {\\left( 1 - t{q}^{k - {2j}}\\right) }^{-1} \\) is the formal power series\n\n\\[ \n\\mathop{\\sum }\\limits_{{m = 0}}^{\\infty }{t}^{m}{q}^{m\\left( {k - {2j}}\\right) }\n\\]\n\n(4.14)\n\nHence the right side of (4.13) is\n\n\\[ \n\\mathop{\\sum }\\limits_{{{m}_{0},\\ldots ,{m}_{k}}}}{t}^{{m}_... | Yes |
Lemma 4.1.22. One has the formal power series identity\n\n\\[ \n\\mathop{\\prod }\\limits_{{j = 0}}^{k}{\\left( 1 - t{q}^{k - {2j}}\\right) }^{-1} = \\mathop{\\sum }\\limits_{{j = 0}}^{\\infty }{t}^{j}{\\left\\lbrack \\begin{matrix} k + j \\\\ k \\end{matrix}\\right\\rbrack }_{q},\n\\]\n\nwhere the factors on the left ... | Proof. The proof proceeds by induction on \\( k \\) . The case \\( k = 0 \\) is the formal power series identity \\( {\\left( 1 - t\\right) }^{-1} = \\mathop{\\sum }\\limits_{{j = 0}}^{\\infty }{t}^{j} \\) . Now set\n\n\\[ \n{H}_{k}\\left( {t, q}\\right) = \\mathop{\\sum }\\limits_{{j = 0}}^{\\infty }{t}^{j}{\\left\\lb... | Yes |
Lemma 4.2.3. Let \( X \subset L \) be a finite-dimensional \( G \) -invariant subspace. Then \( {\left. {\mathcal{R}}^{G}\right| }_{X} = {\operatorname{Hom}}_{G}\left( {X, L}\right) \) | Proof. Let \( T \in {\operatorname{Hom}}_{G}\left( {X, L}\right) \) . Then by Corollary 4.1.6 there exists \( r \in \mathcal{R} \) such that \( {\left. r\right| }_{X} = T \) . Since \( G \) is reductive, condition (iii) and Proposition 4.1.15 furnish a projection \( r \mapsto {r}^{\natural } \) from \( \mathcal{R} \) t... | Yes |
Corollary 4.2.4. Assume \( \dim L < \infty \) . Set \( \mathcal{A} = \operatorname{Span}\rho \left( G\right) \) and \( \mathcal{B} = {\operatorname{End}}_{\mathcal{A}}\left( L\right) \) . Then \( L \) is a completely reducible \( \mathcal{B} \) -module. Furthermore, the following hold:\n\n1. Suppose that for every \( \... | Proof. Since \( L \) is the direct sum of \( \mathcal{B} \) -invariant irreducible subspaces by Theorem 4.2.1, it is a completely reducible \( \mathcal{B} \) -module by Proposition 4.1.11. We now prove the other assertions.\n\n(1): Let \( T \in \operatorname{End}\left( L\right) \) be the operator that acts by \( I \oti... | Yes |
Proposition 4.2.5. Suppose \( \left( {\sigma, V}\right) \) and \( \left( {\tau, W}\right) \) are irreducible. Then the outer tensor product \( \left( {\sigma \widehat{ \otimes }\tau, V \otimes W}\right) \) is an irreducible representation of \( H \times K \), and every irreducible regular representation of \( H \times ... | Proof. We have \( \operatorname{End}\left( {V \otimes W}\right) = \operatorname{End}\left( V\right) \otimes \operatorname{End}\left( W\right) = \operatorname{Span}\{ \sigma \left( H\right) \otimes \tau \left( K\right) \} \) by Corollary 4.1.7. Hence if \( 0 \neq u \in V \otimes W \), then \( \operatorname{Span}\{ \left... | Yes |
For \( \lambda \in \widehat{G} \) define \( {\varphi }_{\lambda }\left( {{v}^{ * } \otimes v}\right) \left( g\right) = \left\langle {{v}^{ * },{\pi }^{\lambda }\left( g\right) v}\right\rangle \) for \( g \in G,{v}^{ * } \in {V}^{{\lambda }^{ * }} \) , and \( v \in V \) . Extend \( {\varphi }_{\lambda } \) to a linear m... | Proof. Given \( v \in {F}^{\lambda } \) and \( {v}^{ * } \in {F}^{{\lambda }^{ * }} \), we set \( {f}_{{v}^{ * }, v} = {\varphi }_{\lambda }\left( {{v}^{ * } \otimes v}\right) \) . Then for \( x, y, z \in G \) we have\n\n\[ {f}_{x \cdot {v}^{ * }, y \cdot v}\left( z\right) = \left\langle {{\pi }^{{\lambda }^{ * }}\left... | Yes |
Theorem 4.2.12. The restriction map \( \varphi : {\left. T \mapsto T\right| }_{{V}^{{\mathrm{n}}^{ + }}} \) for \( T \in {\operatorname{End}}_{\mathfrak{g}}\left( V\right) \) gives an algebra isomorphism\n\n\[{\operatorname{End}}_{\mathfrak{g}}\left( V\right) \cong {\bigoplus }_{\mu \in \mathcal{S}}\operatorname{End}\l... | Proof. Since every finite-dimensional representation of \( \mathfrak{g} \) is completely reducible by Theorem 3.3.12, we can apply Proposition 4.1.15 (viewing \( V \) as a \( U\left( \mathfrak{g}\right) \) -module) to obtain the primary decomposition\n\n\[V = {\bigoplus }_{\mu \in {P}_{+ + }\left( \mathfrak{g}\right) }... | Yes |
Theorem 4.3.1. The Fourier transform\n\n\[ \n\mathcal{F} : \mathcal{A}\left\lbrack G\right\rbrack \rightarrow {\bigoplus }_{\lambda \in \widehat{G}}\operatorname{End}\left( {F}^{\lambda }\right) \]\n\nis an algebra isomorphism that preserves the \( * \) operation on each algebra. Furthermore, for \( f \in \mathcal{A}\l... | Proof. Since \( \mathcal{F}\left( {\delta }_{{g}_{1}{g}_{2}}\right) = {\pi }^{\lambda }\left( {{g}_{1}{g}_{2}}\right) = {\pi }^{\lambda }\left( {g}_{1}\right) {\pi }^{\lambda }\left( {g}_{2}\right) = \mathcal{F}\left( {\delta }_{{g}_{1}}\right) \mathcal{F}\left( {\delta }_{{g}_{2}}\right) \), the map \( \mathcal{F} \) ... | Yes |
Lemma 4.3.2. Suppose \( C \) is a \( G \) -invariant bilinear form on \( U \times V \) . Then \( C = 0 \) if \( U \) is not equivalent to \( {V}^{ * } \) as a \( G \) -module. If \( U = {V}^{ * } \) there is a constant \( \kappa \) such that \( C\left( {u, v}\right) = \kappa \langle u, v\rangle \), where \( \langle u, ... | Proof. We can write \( C \) as \( C\left( {u, v}\right) = \langle {Tu}, v\rangle \), where \( T \in \operatorname{Hom}\left( {U,{V}^{ * }}\right) \) . Since the form \( C \) and the canonical bilinear pairing of \( {V}^{ * } \) and \( V \) are both \( G \) invariant, we have\n\n\[ \left\langle {{g}^{-1}{Tgu}, v}\right\... | Yes |
Lemma 4.3.3 (Schur Orthogonality Relations). Suppose \( G \) is a finite group and \( \lambda ,\mu \in \widehat{G} \) . Let \( A \in \operatorname{End}\left( {F}^{\lambda }\right) \) and \( B \in \operatorname{End}\left( {F}^{\mu }\right) \) . Then\n\n\[\n\frac{1}{\left| G\right| }\mathop{\sum }\limits_{{g \in G}}{f}_{... | Proof. Define a bilinear form \( C \) on \( \operatorname{End}\left( {F}^{\lambda }\right) \times \operatorname{End}\left( {F}^{\mu }\right) \) by\n\n\[\nC\left( {A, B}\right) = \frac{1}{\left| G\right| }\mathop{\sum }\limits_{{g \in G}}{f}_{A}^{\lambda }\left( g\right) {f}_{B}^{\mu }\left( g\right) .\n\]\n\n(4.34)\n\n... | Yes |
Theorem 4.3.4 (Fourier Inversion Formula). Suppose \( G \) is a finite group. Let \( F = \{ F\left( \lambda \right) {\} }_{\lambda \in \widehat{G}} \) be in \( \mathcal{F}\mathcal{A}\left\lbrack G\right\rbrack \) . Define a function \( f \in \mathcal{A}\left\lbrack G\right\rbrack \) by\n\n\[ f\left( g\right) = \frac{1}... | Proof. The operator \( \mathcal{F}f\left( \lambda \right) \) is uniquely determined by \( \operatorname{tr}\left( {\mathcal{F}f\left( \lambda \right) A}\right) \), with \( A \) varying over \( \operatorname{End}\left( {V}^{\lambda }\right) \) . Replacing each representation by its dual, we write the formula for \( f \)... | Yes |
Corollary 4.3.5 (Plancherel Formula). Let \( \varphi ,\psi \in \mathcal{A}\left\lbrack G\right\rbrack \) . Then\n\n\[ \mathop{\sum }\limits_{{g \in G}}\varphi \left( g\right) \overline{\psi \left( g\right) } = \frac{1}{\left| G\right| }\mathop{\sum }\limits_{{\lambda \in \widehat{G}}}{d}_{\lambda }\operatorname{tr}\lef... | Proof. Let \( f = \varphi * {\left( \psi \right) }^{ * } \) . Then\n\n\[ f\left( 1\right) = \mathop{\sum }\limits_{{g \in G}}\varphi \left( g\right) \overline{\psi \left( g\right) }.\]\n\nWe can also express \( f\left( 1\right) \) by the Fourier inversion formula evaluated at \( g = 1 \) :\n\n\[ f\left( 1\right) = \fra... | Yes |
Proposition 4.3.8. The Fourier transform of \( f \in \mathcal{A}{\left\lbrack G\right\rbrack }^{G} \) has the expansion\n\n\[ \mathcal{F}f = \mathop{\sum }\limits_{{\lambda \in \widehat{G}}}\mathcal{F}f\left( \lambda \right) {E}_{\lambda } \]\n\n(4.43)\n\nIn particular, \( \dim \mathcal{A}{\left\lbrack G\right\rbrack }... | We return to a general finite group \( G \) . Under the inverse Fourier transform, the operator \( {E}_{\lambda } \) corresponds to convolution by a central function \( {e}_{\lambda } \) on \( G \) . To determine \( {e}_{\lambda } \), we apply the Fourier inversion formula (4.36):\n\n\[ {e}_{\lambda }\left( g\right) = ... | Yes |
Theorem 4.3.9. Let \( \varphi ,\psi \in \mathcal{A}{\left\lbrack G\right\rbrack }^{G} \) and \( g \in G \) . Then\n\n\[ \varphi \left( g\right) = \mathop{\sum }\limits_{{\lambda \in \widehat{G}}}\widehat{\varphi }\left( \lambda \right) {\chi }_{\lambda }\left( g\right) ,\text{ where }\widehat{\varphi }\left( \lambda \r... | Proof. Define a positive definite inner product on \( \mathcal{A}\left\lbrack G\right\rbrack \) by\n\n\[ \langle \varphi \mid \psi \rangle = \frac{1}{\left| G\right| }\mathop{\sum }\limits_{{g \in G}}\varphi \left( g\right) \overline{\psi \left( g\right) } \]\n\nLet \( \lambda ,\mu \in \widehat{G} \) . Then \( {\chi }_... | Yes |
Corollary 4.3.10 (Dual Orthogonality Relations). Suppose \( {C}_{1} \) and \( {C}_{2} \) are conjugacy classes in \( G \) . Then\n\n\[ \mathop{\sum }\limits_{{\lambda \in \widehat{G}}}{\chi }_{\lambda }\left( {C}_{1}\right) \overline{{\chi }_{\lambda }\left( {C}_{2}\right) } = \left\{ \begin{array}{ll} \left| G\right| ... | Proof. Let \( C \subset G \) be a conjugacy class. Then\n\n\[ \left| G\right| \widehat{{\varphi }_{C}}\left( \lambda \right) = \left| C\right| {\chi }_{{\lambda }^{ * }}\left( C\right) \]\n\n(4.51)\n\nTaking \( C = {C}_{1} \) and \( C = {C}_{2} \) in (4.51) and then using (4.49), we obtain (4.50). | No |
Corollary 4.3.11. Suppose \( \left( {\rho, V}\right) \) is any finite-dimensional representation of \( G \) . For \( \lambda \in \widehat{G} \) let \( {m}_{\rho }\left( \lambda \right) \) be the multiplicity of \( \lambda \) in \( \rho \) . Then \( {m}_{\rho }\left( \lambda \right) = \left\langle {{\chi }_{\rho } \mid ... | Proof. We have\n\n\[ {\chi }_{\rho } = \mathop{\sum }\limits_{{\lambda \in \widehat{G}}}{m}_{\rho }\left( \lambda \right) {\chi }_{\lambda } \]\n\nso the result on multiplicities follows from (4.48) and (4.49).\n\nBy Corollary 4.2.4 (2) there exists \( f \in \mathcal{A}{\left\lbrack G\right\rbrack }^{G} \) such that \(... | Yes |
Proposition 4.4.4. Suppose that \( G \) is a finite group that acts doubly transitively on a set \( X \) and \( \left| X\right| \geq 2 \) . Then \( \mathbb{C}\left\lbrack X\right\rbrack \) decomposes into two irreducible subspaces under \( G \), namely the constant functions and \[ V = \{ f \in \mathbb{C}\left\lbrack X... | Proof. Clearly \( \mathbb{C}\left\lbrack X\right\rbrack = \mathbb{C}1 \oplus V \) and each summand is invariant under \( G \) . To see that \( V \) is irreducible under \( G \), let \( T \in {\operatorname{End}}_{G}\left( {\mathbb{C}\left\lbrack X\right\rbrack }\right) \) . Then \[ {Tf}\left( x\right) = \mathop{\sum }\... | Yes |
Corollary 4.4.5. Let \( V \subset {\mathbb{C}}^{n} \) be the subspace \( \left\{ {x : \sum {x}_{i} = 0}\right\} \) . Then \( {\mathfrak{S}}_{n} \) acts irreducibly on \( V \) for \( n \geq 2 \), and \( {\mathfrak{A}}_{n} \) acts irreducibly on \( V \) for \( n \geq 4 \) . | Proof. Clearly \( {\mathfrak{S}}_{n} \) acts doubly transitively on \( X = \{ 1,2,\ldots, n\} \) . We claim that \( {\mathfrak{A}}_{n} \) also acts doubly transitively on \( X \) when \( n \geq 4 \) . Indeed, the isotropy group of \( \{ n\} \) in \( {\mathfrak{A}}_{n} \) is \( {\mathfrak{A}}_{n - 1} \), and it is easy ... | Yes |
Proposition 4.4.6. The restriction of the representation \( {\sigma }_{k} \) of \( {\mathfrak{S}}_{k} \) to the subspace \( { \otimes }^{k}{\mathbb{C}}^{n}\left( \lambda \right) \) is equivalent to the representation \( {\operatorname{Ind}}_{{\mathfrak{S}}_{\lambda }}^{{\mathfrak{S}}_{k}}\left( 1\right) \) on \( \mathb... | Proof. Let \( I \) be a \( k \) -tuple such that \( {\mu }_{I} = \lambda \) . Then there is a permutation \( s \) such that \( {\sigma }_{k}\left( s\right) u\left( \lambda \right) = {e}_{I} \) . Thus the map \( s \mapsto {\sigma }_{k}\left( s\right) u\left( \lambda \right) \) gives a bijection from \( {\mathfrak{S}}_{k... | Yes |
Theorem 5.1.1. Suppose \( G \) is a reductive linear algebraic group acting by a regular representation on a vector space \( V \) . Then the algebra \( \mathcal{P}{\left( V\right) }^{G} \) of \( G \) -invariant polynomials on \( V \) is finitely generated as a \( \mathbb{C} \) -algebra. | Proof. Write \( \mathcal{R} = \mathcal{P}\left( V\right) \) and \( \mathcal{J} = \mathcal{P}{\left( V\right) }^{G} \) . By the Hilbert basis theorem (Theorem A.1.2), every ideal \( \mathcal{B} \subset \mathcal{R} \) and every quotient \( \mathcal{R}/\mathcal{B} \) is finitely generated as an \( \mathcal{R} \) module. T... | Yes |
Theorem 5.1.4. The multiplication map \( f, g \mapsto {fg} \) extends to a linear isomorphism \( \mathcal{J} \otimes \mathcal{H} \rightarrow \mathcal{P} \) . Hence \( \mathcal{P} \) is a free \( \mathcal{J} \) -module on \( \dim \mathcal{H} \) generators. | The proof of this theorem will require some preliminary results. Let \( {\left( {\mathcal{{PJ}}}_{ + }\right) }^{k} \) be the homogeneous polynomials of degree \( k \) in \( {\mathcal{{PJ}}}_{ + } \). | No |
Lemma 5.1.5. One has \( {\mathcal{P}}^{k} = {\mathcal{H}}^{k} \oplus {\left( {\mathcal{{PJ}}}_{ + }\right) }^{k} \) for all \( k \) . | Proof. Define \( \langle f \mid g\rangle = \partial \left( f\right) {g}^{ * }\left( 0\right) \) for \( f, g \in \mathcal{P} \), where \( {g}^{ * } \) denotes the polynomial whose coefficients are the complex conjugates of those of \( g \) . Since \( \left\langle {{x}^{I} \mid {x}^{J}}\right\rangle = I \) ! if \( I = J ... | Yes |
Lemma 5.1.6. Let \( {f}_{1},\ldots ,{f}_{m} \in \mathcal{J} \) and suppose \( {f}_{1} \notin \mathop{\sum }\limits_{{j = 2}}^{m}{f}_{j}\mathcal{J} \) . If \( {g}_{1},\ldots ,{g}_{m} \in \mathcal{P} \) satisfy the relation \( \mathop{\sum }\limits_{{j = 1}}^{m}{f}_{j}{g}_{j} = 0 \) and \( {g}_{1} \) is homogeneous, then... | Proof. Suppose \( \deg {g}_{1} = 0 \) . Then \( {g}_{1} = c \) is constant. If \( c \) were not zero, then we could write \( {f}_{1} = - \left( {1/c}\right) \mathop{\sum }\limits_{{j = 2}}^{m}{f}_{j}{g}_{j} \), which would be a contradiction. Hence \( {g}_{1} = 0 \) and the lemma is true in this case.\n\nNow assume tha... | Yes |
Corollary 5.1.7. The series \( {p}_{\mathcal{H}}\left( t\right) = \mathop{\sum }\limits_{{j \geq 0}}\left( {\dim {\mathcal{H}}^{j}}\right) {t}^{j} \) is a polynomial and has the factorization\n\n\[ \n{p}_{\mathcal{H}}\left( t\right) = \mathop{\prod }\limits_{{k = 1}}^{n}\left( {1 + t + \cdots + {t}^{k - 1}}\right) .\n\... | Proof. Define \( f\left( t\right) = \mathop{\sum }\limits_{{j = 0}}^{\infty }\left( {\dim {\mathcal{J}}_{j}}\right) {t}^{j} \) and \( g\left( t\right) = \mathop{\sum }\limits_{{j = 0}}^{\infty }\left( {\dim {\mathcal{P}}^{j}}\right) {t}^{j} \) . Then for \( \left| t\right| < 1 \) ,\n\n\[ \nf\left( t\right) = \mathop{\p... | Yes |
Theorem 5.1.8. The space \( \mathcal{H} \) is spanned by the polynomial \[ \Delta \left( x\right) = \mathop{\prod }\limits_{{1 \leq i < j \leq n}}\left( {{x}_{i} - {x}_{j}}\right) \] and its partial derivatives of all orders. | As a preliminary to proving this theorem, we observe that \( \rho \left( s\right) \Delta \left( x\right) = \operatorname{sgn}\left( s\right) \Delta \left( x\right) \) for all \( s \in {\mathfrak{S}}_{n} \), so \( \Delta \left( x\right) \) is skew invariant under \( {\mathfrak{S}}_{n} \). Furthermore, if \( g\left( x\ri... | Yes |
Lemma 5.1.9. Suppose \( g \in {\mathcal{P}}^{m} \) and \( \partial \left( g\right) \Delta = 0 \) . Then \( g \in {\left( {\mathcal{{PJ}}}_{ + }\right) }^{m} \) . | Proof. Since \( \mathcal{H} \) is finite-dimensional, we have \( {\mathcal{P}}^{m} = {\left( {\mathcal{{PJ}}}_{ + }\right) }^{m} \) for \( m \) sufficiently large, by Lemma 5.1.5. Hence the lemma is true in this case. We assume by induction that the lemma holds for polynomials of degree greater than \( m \) . Take \( 1... | Yes |
Theorem 5.2.1. (Polynomial FFT for \( \mathbf{{GL}}\left( V\right) \) ) The map\n\n\[ \n{\mu }^{ * } : \mathcal{P}\left( {M}_{k, m}\right) \rightarrow \mathcal{P}{\left( {\left( {V}^{ * }\right) }^{k} \oplus {V}^{m}\right) }^{\mathbf{{GL}}\left( V\right) }\n\]\n\nis surjective. Hence \( \mathcal{P}{\left( {\left( {V}^{... | We shall prove this theorem in Section 5.4.2. | No |
Theorem 5.2.2. (Polynomial FFT for \( {\mathbf{O}}_{n} \) and \( {\mathbf{{Sp}}}_{n} \) ) Let \( V = {\mathbb{C}}^{n} \). 1. The homomorphism \( {\tau }^{ * } : \mathcal{P}\left( {S{M}_{k}}\right) \rightarrow \mathcal{P}{\left( {V}^{k}\right) }^{{\mathbf{O}}_{n}} \) is surjective. Hence the algebra of \( {\mathbf{O}}_{... | We shall prove this theorem in Section 5.4.3. | No |
1. Let \( G = {\mathbf{O}}_{n} \). Then \( \mathcal{P}{\left( {\left( {V}^{ * }\right) }^{k} \oplus {V}^{m}\right) }^{G} \) is generated by the quadratic polynomials \( \left( {{v}_{i},{v}_{j}}\right) ,\left( {{v}_{p}^{ * },{v}_{q}^{ * }}\right) \), and \( \left\langle {{v}_{p}^{ * },{v}_{i}}\right\rangle \), for \( 1 ... | Proof. The \( G \) -invariant bilinear form gives an isomorphism \( \varphi : {\left( {V}^{k}\right) }^{ * } \cong {V}^{k} \) as \( G \) - modules. In the orthogonal case \( \left( {{v}_{p}^{ * },{v}_{q}^{ * }}\right) = \left( {\varphi \left( {v}_{p}^{ * }\right) ,\varphi \left( {v}_{q}^{ * }\right) }\right) \) and \( ... | Yes |
1. The image of \( \mu \) consists of all matrices \( Z \) with \( \operatorname{rank}\left( Z\right) \leq \min \left( {k, m, n}\right) \) . | Proof. (1): Let \( Z \in {M}_{k, m} \) have rank \( r \leq \min \left( {k, m, n}\right) \) . We may assume \( k \leq m \) (otherwise replace \( Z \) by \( {Z}^{t} \) ). Then by row and column reduction of \( Z \) we can find \( u \in \mathbf{{GL}}\left( k\right) \) and \( w \in \mathbf{{GL}}\left( m\right) \) such that... | Yes |
Corollary 5.2.5. (SFT, Free Case) Let \( V = {\mathbb{C}}^{n} \). 1. If \( n \geq \min \left( {k, m}\right) \) then \( {\mu }^{ * } : \mathcal{P}\left( {M}_{k, m}\right) \rightarrow \mathcal{P}{\left( {\left( {V}^{ * }\right) }^{k} \oplus {V}^{m}\right) }^{\mathbf{{GL}}\left( V\right) } \) is bijective. Let \( {z}_{ij}... | Proof. From the FFT, the maps \( {\mu }^{ * },{\tau }^{ * } \), and \( {\gamma }^{ * } \) are surjective. By Lemma 5.2.4 the maps \( \mu ,\tau \), and \( \gamma \) are surjective when \( n \geq \min \left( {m, k}\right) \) . This implies that \( {\mu }^{ * },{\tau }^{ * } \), and \( {\gamma }^{ * } \) are also injectiv... | Yes |
Theorem 5.3.1. Let \( G = \mathbf{GL}\left( V\right) \) . The space of \( G \) invariants in \( {V}^{\otimes k} \otimes {V}^{* \otimes k} \) is spanned by the tensors \( \left\{ {C}_{s} : s \in {\mathfrak{S}}_{k}}\right\} \) . | Proof. Let \( T : {V}^{\otimes k} \otimes {V}^{* \otimes k} \rightarrow \operatorname{End}\left( {\bigotimes }^{k}V\right) \) be the natural isomorphism (see Appendix B.2.2). For \( s \in {\mathfrak{S}}_{k} \) we have\n\n\[ T\left( {C}_{s}\right) {e}_{J} = \sum_{I}\left\langle {e}_{I}^{*},{e}_{J}\right\rangle {e}_{s \c... | Yes |
Corollary 5.3.2. The space of \( \\mathbf{GL}\\left( V\\right) \) -invariant tensors in \( {V}^{* \\otimes k} \\otimes {V}^{\\otimes k} \) is spanned by the complete contractions \( \\left\\{ {\\lambda }_{s} : s \\in {\\mathfrak{S}}_{k}\\right\\} \) . | Proof. Let \( B : {V}^{\\otimes k} \\otimes {V}^{* \\otimes k} \\rightarrow {V}^{* \\otimes k} \\otimes {V}^{\\otimes k} \) be the natural duality map. It suffices to show that \( {\\lambda }_{s} = B\\left( {C}_{s}\\right) \). Let \( {v}_{1},\\ldots ,{v}_{k} \\in V \) and \( {v}_{1}^{* },\\ldots ,{v}_{k}^{* } \\in {V}^... | Yes |
Theorem 5.3.3. Let \( G \) be \( \mathbf{O}\left( V\right) \) or \( \mathbf{{Sp}}\left( V\right) \) . Then \( {\left\lbrack {V}^{\otimes m}\right\rbrack }^{G} = 0 \) if \( m \) is odd, and \[ {\left\lbrack {V}^{\otimes {2k}}\right\rbrack }^{G} = \operatorname{Span}\left\{ {{\sigma }_{2k}\left( s\right) {\theta }_{k} : ... | Before proving this theorem, we restate it to incorporate the symmetries of the tensor \( {\theta }_{k} \) . View \( {\mathfrak{S}}_{2k} \) as the permutations of the set \( \{ 1,2,\ldots ,{2k} - 1,{2k}\} \) . Define \( {\widetilde{\mathfrak{S}}}_{k} \subset {\mathfrak{S}}_{2k} \) as the subgroup that permutes the orde... | Yes |
Theorem 5.3.5. The complete contractions \( \left\{ {{\lambda }_{x} : x \in {X}_{k}}\right\} \) are a spanning set for the G-invariant \( {2k} \) -multilinear forms on \( V \) . | We now begin the proof of Theorem 5.3.3. Since \( V \cong {V}^{ * } \) as a \( G \) -module, it suffices to consider \( G \) -invariant tensors \( \lambda \in {V}^{* \otimes k} \) . The key idea is to shift the action of \( G \) from \( {V}^{* \otimes {2k}} \) to End \( V \) by introducing a polarization variable \( X ... | Yes |
Lemma 5.3.6. The function \( {\Phi }_{\lambda } \) has the following transformation properties:\n\n1. \( \left( {L\left( g\right) \otimes 1}\right) {\Phi }_{\lambda } = {\Phi }_{g \cdot \lambda } \) for \( g \in G \) (where \( g \cdot \lambda = {\rho }_{k}^{ * }\left( g\right) \lambda \) ).\n\n2. \( \pi \left( h\right)... | Proof. Let \( g \in G, X \in \operatorname{End}V \), and \( w \in {V}^{\otimes k} \) . Then\n\n\[ \n{\Phi }_{\lambda }\left( {{g}^{-1}X, w}\right) = \left\langle {\lambda ,{\rho }_{k}{\left( g\right) }^{-1}{X}^{\otimes k}w}\right\rangle = \left\langle {{\rho }_{k}^{ * }\left( g\right) \lambda ,{X}^{\otimes k}w}\right\r... | Yes |
Lemma 5.4.1. Let \( \mathbf{p} \in {\mathbb{N}}^{k} \) and \( \mathbf{q} \in {\mathbb{N}}^{m} \) . There is a linear isomorphism\n\n\[ \n{\mathcal{P}}^{\left\lbrack \mathbf{p},\mathbf{q}\right\rbrack }{\left( {V}^{k} \oplus {V}^{*m}\right) }^{G} \cong {\left\lbrack {\left( {V}^{* \otimes \left| \mathbf{p}\right| } \oti... | Proof. We have the isomorphisms\n\n\[ \n\mathcal{P}\left( {{V}^{k} \oplus {V}^{*m}}\right) \cong S\left( {{V}^{*k} \oplus {V}^{m}}\right)\n\]\n\n\[ \n\cong \underset{k\text{ factors }}{\underbrace{S\left( {V}^{ * }\right) \otimes \cdots \otimes S\left( {V}^{ * }\right) }} \otimes \underset{m\text{ factors }}{\underbrac... | Yes |
Lemma 5.5.1. The operator \( E \) commutes with \( \mathbf{{GL}}\left( V\right) \) and acts by the scalar \( k \) on \( \mathop{\bigwedge }\limits^{k}V \) . Hence \( E \) does not depend on the choice of basis for \( V \) . If \( T \in \operatorname{End}\left( {\bigwedge V}\right) \) and \( T : \mathop{\bigwedge }\limi... | Proof. Let \( g \in \mathbf{{GL}}\left( V\right) \) have matrix \( \left\lbrack {g}_{ij}\right\rbrack \) relative to the basis \( \left\{ {f}_{i}\right\} \) . Relations (5.45) imply that\n\n\[ \rho \left( g\right) {E\rho }{\left( g\right) }^{-1} = \mathop{\sum }\limits_{{i, k}}\left\{ {\mathop{\sum }\limits_{j}{g}_{ij}... | Yes |
Theorem 5.5.2. Let \( G = \mathbf{{GL}}\left( V\right) \) . Then \( {\operatorname{End}}_{G}\left( {\bigwedge V}\right) \) is generated by the skew Euler operator \( E \) . | Proof. From Theorem 4.2.10 we know that \( {\operatorname{Hom}}_{G}\left( {{V}^{\otimes l},{V}^{\otimes k}}\right) \) is zero if \( l \neq k \), and is spanned by the operators \( {\sigma }_{k}\left( s\right) \) with \( s \in {\mathfrak{S}}_{k} \) when \( l = k \) . Since \( P{\sigma }_{k}\left( s\right) P = \operatorn... | Yes |
Corollary 5.5.3. In the decomposition \( \land V = {\bigoplus }_{p = 1}^{d}\mathop{\bigwedge }\limits^{p}V \), the summands are irreducible and mutually inequivalent \( \mathbf{{GL}}\left( V\right) \) -modules. | Proof. This follows from Theorems 4.2.1 and 5.5.2. | No |
Lemma 5.5.4. Let \( G \) be \( \mathbf{O}\left( {V,\Omega }\right) \) (if \( \Omega \) is symmetric) or \( \mathbf{{Sp}}\left( {V,\Omega }\right) \) (if \( \Omega \) is skew symmetric). Then the space \( {\operatorname{Hom}}_{G}\left( {{V}^{\otimes l},{V}^{\otimes k}}\right) \) is zero if \( k + l \) is odd. If \( k + ... | Proof. Recall that there is a canonical \( \mathbf{{GL}}\left( V\right) \) -module isomorphism\n\n\[ \n{V}^{\otimes k} \otimes {V}^{* \otimes l} \cong \operatorname{Hom}\left( {{V}^{\otimes l},{V}^{\otimes k}}\right) \n\]\n\n(see Section B.2.2). Denote by \( {T}_{\xi } \in {\operatorname{Hom}}_{G}\left( {{V}^{\otimes l... | Yes |
Theorem 5.5.5. ( \( \Omega \) symmetric) Let \( G = \mathbf{O}\left( {V,\Omega }\right) \) . Then \( {\operatorname{End}}_{G}\left( {\bigwedge V}\right) \) is generated by the skew Euler operator \( E \) . | Proof. In this case the tensor \( \theta \) is symmetric. Hence \( {PC} = 0 \) and \( {C}^{ * }P = 0 \) . For \( v \in V \) and \( u \in {V}^{\otimes m} \) we have\n\n\[ \n{P\mu }\left( v\right) u = P\left( {v \otimes u}\right) = v \land {Pu} = \varepsilon \left( v\right) {Pu} \n\]\n\n(5.48)\n\n(from the definition of ... | Yes |
Corollary 5.5.6. ( \( \Omega \) symmetric) In the decomposition \( \bigwedge V = {\bigoplus }_{p = 1}^{d}\mathop{\bigwedge }\limits^{p}V \), the summands are irreducible and mutually inequivalent \( \mathbf{O}\left( {V,\Omega }\right) \) -modules. | Proof. The proof proceeds by using the same argument as in Corollary 5.5.3, but now using Theorem 5.5.5. | No |
Corollary 5.5.9. \( \left( {G = \mathbf{{Sp}}\left( {V,\Omega }\right) }\right) \) There is a canonical decomposition\n\n\[ \bigwedge V \cong {\bigoplus }_{k = 0}^{n}{F}^{\left( n - k\right) } \otimes {\mathcal{H}}^{k} \]\n\n(5.54)\n\nas a \( \left( {G,{\mathfrak{g}}^{\prime }}\right) \) -module, where \( \dim V = {2n}... | Proof. The eigenvalues of \( E - {nI} \) on \( \bigwedge V \) are \( n, n - 1,\ldots , - n + 1, - n \), so the only possible irreducible representations of \( {\mathfrak{g}}^{\prime } \) that can occur in \( \bigwedge V \) are the representations \( {F}^{\left( k\right) } \) with \( k = 0,1,\ldots, n \) . Now\n\n\[ X\l... | Yes |
Lemma 5.5.10. Let \( k + l \) be even. Then the space \( {\operatorname{Hom}}_{G}\left( {\mathop{\bigwedge }\limits^{l}V,\mathop{\bigwedge }\limits^{k}V}\right) \) is spanned by operators of the following forms:\n\n1. \( {YQ} \) with \( Q \in {\operatorname{Hom}}_{G}\left( {\mathop{\bigwedge }\limits^{l}V,\mathop{\bigw... | Proof. We know that \( {\operatorname{Hom}}_{G}\left( {\mathop{\bigwedge }\limits^{l}V,\mathop{\bigwedge }\limits^{k}V}\right) \) is spanned by operators \( P{\sigma }_{k}\left( s\right) A{\sigma }_{l}\left( t\right) P \) , with \( s \in {\mathfrak{S}}_{k}, t \in {\mathfrak{S}}_{l} \), and \( A \) given in cases (1),(2... | Yes |
Theorem 5.5.11. Let \( G = \mathbf{{SL}}\left( {n,\mathbb{C}}\right) \) . The representation \( {\sigma }_{r} \) on the \( r \) th exterior power \( \mathop{\bigwedge }\limits^{r}{\mathbb{C}}^{n} \) is irreducible and has highest weight \( {\varpi }_{r} \) for \( 1 \leq r < n \) . | Proof. From Corollary 5.5.3 we know that \( \mathop{\bigwedge }\limits^{r}{\mathbb{C}}^{n} \) is an irreducible \( \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) - module. Hence it is also irreducible for \( \mathbf{{SL}}\left( {n,\mathbb{C}}\right) \) . It remains only to determine its highest weight. Take the positive ... | Yes |
Theorem 5.5.13. Let \( {\sigma }_{r} \) denote the representation of \( G \) on \( \mathop{\bigwedge }\limits^{r}{\mathbb{C}}^{n} \) for \( 1 \leq r \leq n \) associated with the defining representation \( {\sigma }_{1} \) on \( {\mathbb{C}}^{n} \) . 1. Let \( n = {2l} + 1 \geq 3 \) be odd. If \( 1 \leq r \leq l \), th... | Proof. From Corollary 5.5.6 we know that \( \left( {{\sigma }_{r},\mathop{\bigwedge }\limits^{r}{\mathbb{C}}^{n}}\right) \) is an irreducible \( G \) -module for \( 1 \leq r \leq n \) . (1): If \( n = {2l} + 1 \) is odd, then \( G = {G}^{ \circ } \cup \left( {-I}\right) {G}^{ \circ } \) . Hence \( \mathop{\bigwedge }\l... | Yes |
1. If \( p > l \) then \( \mathcal{H}\left( {\mathop{\bigwedge }\limits^{p}{\mathbb{C}}^{2l},\Omega }\right) = 0 \) . | Proof. We already observed in the proof of Corollary 5.5.9 that the irreducible representations of \( {\mathfrak{g}}^{\prime } \) that occur in \( \bigwedge {\mathbb{C}}^{2l} \) are \( {F}^{\left( l - p\right) } \), where \( p = 0,1,\ldots, l \) . By definition of \( H \) and \( X \) the space \( \mathcal{H}\left( {\ma... | Yes |
Corollary 5.5.16. The map \( \mathbb{C}\left\lbrack t\right\rbrack \otimes \mathcal{H}\left( {\bigwedge {\mathbb{C}}^{2l},\Omega }\right) \rightarrow \bigwedge {\mathbb{C}}^{2l} \) given by \( f\left( t\right) \otimes u \mapsto \) \( f\left( \theta \right) \land u \) (exterior multiplication) is a \( G \) -module isomo... | Proof. Since \( H{u}_{p} = \left( {l - p}\right) {u}_{p} \) and \( {2Y} \) is exterior multiplication by \( \theta \), we have\n\n\[ {F}^{\left( l - p\right) } = {\bigoplus }_{k = 0}^{l - p}\mathbb{C}{\mathbf{\theta }}^{k} \land {u}_{p} \]\n\n(notation of Proposition 2.3.3). Now use (5.56). | No |
Corollary 5.5.17. The space \( \mathcal{H}\left( {\mathop{\bigwedge }\limits^{p}{\mathbb{C}}^{2l},\Omega }\right) \) has dimension \( \left( \begin{matrix} {2l} \\ p \end{matrix}\right) - \left( \begin{matrix} {2l} \\ p - 2 \end{matrix}\right) \) for \( p = \) \( 1,\ldots, l \) . | Proof. For \( p = 1 \) we have \( \mathcal{H}\left( {{\mathbb{C}}^{2l},\Omega }\right) = {\mathbb{C}}^{2l} \), so the dimension is as stated (with the usual convention that \( \left( \begin{array}{l} m \\ q \end{array}\right) = 0 \) when \( q \) is a negative integer). Now use induction on \( p \) and (5.57). | No |
Proposition 5.5.18. The space \( \mathcal{H}\left( {\mathop{\bigwedge }\limits^{p}{\mathbb{C}}^{2l},\Omega }\right) \) is spanned by the isotropic p-vectors for \( p = 1,\ldots, l \) . | Proof. Let \( {F}_{p} \) be the space spanned by the isotropic \( p \) -vectors. Clearly \( {F}_{p} \) is invariant under \( G \) . Any linearly independent set \( \left\{ {{v}_{1},\ldots ,{v}_{p}}\right\} \) of isotropic vectors in \( {\mathbb{C}}^{2l} \) can be embedded in a canonical symplectic basis, and \( G \) ac... | Yes |
Proposition 5.5.19. Let \( \mathfrak{g} \) be a semisimple Lie algebra. Let \( \left( {{\pi }^{\lambda },{V}^{\lambda }}\right) \) and \( \left( {{\pi }^{\mu },{V}^{\mu }}\right) \) be finite-dimensional irreducible representations of \( \mathfrak{g} \) with highest weights \( \lambda ,\mu \in \) \( {P}_{+ + }\left( \m... | Proof. The vector \( {v}_{\lambda } \otimes {v}_{\mu } \) is \( \mathfrak{b} \) -extreme of weight \( \lambda + \mu \). Hence \( U \) is irreducible by Proposition 3.3.9 and has highest weight \( \lambda + \mu \), which proves (1). Set \( M = {V}^{\lambda } \otimes {V}^{\mu } \). By Theorem 3.2.5 and (5.58) the weights... | Yes |
Corollary 5.5.20. Let \( G \) be the group \( \mathbf{SL}\left( V\right) \) or \( \mathbf{Sp}\left( V\right) \) with \( \dim V \geq 2 \), or \( \mathbf{SO}\left( V\right) \) with \( \dim V \geq 3 \). If \( {\pi }^{\lambda } \) and \( {\pi }^{\mu } \) are differentials of irreducible regular representations of \( G \), ... | Proof. This follows from Proposition 5.5.19 and Theorems 2.2.2 and 2.2.7. | No |
Theorem 5.5.21. Let \( G \) be the group \( \mathbf{SL}\left( V\right) \) or \( \mathbf{Sp}\left( V\right) \) with \( \dim V \geq 2 \), or \( \mathbf{SO}\left( V\right) \) with \( \dim V \geq 3 \) . For every dominant weight \( \mu \in {P}_{+ + }\left( G\right) \) there exists an integer \( k \) such that \( {V}^{\otim... | Proof. Suppose that \( G = \mathbf{SL}\left( {\mathbb{C}}^{l + 1}\right) \) or \( \mathbf{Sp}\left( {\mathbb{C}}^{2l}\right) \) and let \( n = l + 1 \) or \( n = {2l} \), respectively. From Theorems 5.5.11 and 5.5.15 we know that \( \bigwedge {\mathbb{C}}^{n} \) contains irreducible representations of \( G \) with high... | Yes |
Theorem 5.5.22. Let \( G = \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) and let \( \mu \) be given by (5.60). Then there exists a unique irreducible rational representation \( \left( {{\pi }_{n}^{\mu },{F}_{n}^{\mu }}\right) \) of \( G \) such that the following hold:\n\n1. The restriction of \( {\pi }_{n}^{\mu } \) to... | Proof. Let \( \left( {{\pi }_{0}, V}\right) \) be the irreducible regular representation of \( \mathbf{{SL}}\left( {n,\mathbb{C}}\right) \) with highest weight \( {\mu }_{0} \) whose existence follows from Theorem 5.5.21. We extend \( {\pi }_{0} \) to a representation \( \pi \) of \( \mathbf{{GL}}\left( {n,\mathbb{C}}\... | Yes |
Theorem 5.6.1. Let \( G \) be a reductive algebraic subgroup of \( \mathbf{GL}\left( V\right) \). Let \( G \) act on \( \mathcal{P}\left( V\right) \) by \( \rho \left( g\right) f\left( x\right) = f\left( g^{-1}x\right) \) for \( f \in \mathcal{P}\left( V\right) \) and \( g \in G \). There is a canonical decomposition\n... | Proof. Set \( \mathcal{R} = \mathbb{D}\left( V\right) \) and \( L = \mathcal{P}\left( V\right) \). Then \( L \) has countable dimension as a vector space and \( \rho \) is a locally regular representation of \( G \), since \( \mathcal{P}^{k}\left( V\right) \) is a regular \( G \)-module for each integer \( k \). We sha... | Yes |
Lemma 5.6.2. The Weyl symbol map gives a linear isomorphism\n\n\[ \n{\mathbb{D}}_{k}\left( V\right) \cong {\bigoplus }_{j = 0}^{k}{\mathcal{P}}^{j}\left( {V \oplus {V}^{ * }}\right) \n\]\n\nas \( \mathbf{{GL}}\left( V\right) \) -modules, for all \( k \) . | Proof. From (5.67) it is easy to show (by induction on filtration degree) that \( \sigma \left( T\right) = \) \( \sigma \left( S\right) \) if \( \operatorname{Gr}\left( T\right) = \operatorname{Gr}\left( S\right) \) . Thus \( \sigma \) gives a linear isomorphism\n\n\[ \n{\operatorname{Gr}}^{k}\left( {\mathbb{D}\left( V... | Yes |
Theorem 5.6.3. Let \( G \) be a reductive subgroup of \( \mathbf{GL}\left( V\right) \). Let \( \left\{ {\psi }_{1},\ldots ,{\psi }_{r}\right\} \) be a set of polynomials that generates the algebra \( \mathcal{P}{\left( V \oplus {V}^{ * }\right) }^{G} \). Suppose \( {T}_{j} \in \mathbb{D}{\left( V\right) }^{G} \) are su... | Proof. Let \( \mathcal{J} \subset \mathbb{D}{\left( V\right) }^{G} \) be the subalgebra generated by \( {T}_{1},\ldots ,{T}_{r} \). Then \( {\mathbb{D}}_{0}{\left( V\right) }^{G} = \mathbb{C}I \subset \mathcal{J} \). Let \( S \in {\mathbb{D}}_{k}{\left( V\right) }^{G} \) have filtration degree \( k \). We may assume by... | Yes |
Corollary 5.6.4. (Notation as in Theorem 5.6.3) Suppose \( {T}_{1},\ldots ,{T}_{r} \) can be chosen so that \( {\mathfrak{g}}^{\prime } = \operatorname{Span}\left\{ {{T}_{1},\ldots ,{T}_{r}}\right\} \) is a Lie subalgebra of \( \mathbb{D}{\left( V\right) }^{G} \) . Then in the duality decomposition (5.69) the space \( ... | Proof. The action of elements of \( {\mathfrak{g}}^{\prime } \) as differential operators on \( \mathcal{P}\left( V\right) \) extends to a representation \( {\rho }^{\prime } : U\left( {\mathfrak{g}}^{\prime }\right) \rightarrow \operatorname{End}\left( {\mathcal{P}\left( V\right) }\right) \) (see Appendix C.2.1). The ... | Yes |
Theorem 5.6.5. Let \( G = \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) acting by left multiplication on \( V = {M}_{n, k} \) . Set \( {\mathfrak{g}}^{\prime } = \operatorname{Span}\left\{ {{E}_{ij} : 1 \leq i, j \leq k}\right\} \) . Then \( {\mathfrak{g}}^{\prime } \) is a Lie algebra in \( \mathbb{D}\left( V\right) \)... | Proof. The FFT for \( G \) (Theorem 5.2.1) asserts that \( \mathcal{P}{\left( V \oplus {V}^{ * }\right) }^{G} \) is generated by the quadratic polynomials \( {z}_{ij} = \mathop{\sum }\limits_{{p = 1}}^{n}{x}_{pi}{\xi }_{pj} \) for \( 1 \leq i, j \leq k \), where \( {\xi }_{pj} \) are the coordinates on \( {V}^{ * } \) ... | Yes |
Corollary 5.6.6. In the canonical duality decomposition\n\n\\[ \mathcal{P}\left( {M}_{n, k}\right) \cong {\bigoplus }_{\lambda \in \mathcal{S}}{E}^{\lambda } \otimes {F}^{\lambda } \\]\n\n(5.76)\n\n(where \\( \mathcal{S} \subset \widehat{G} \\) ), each summand \\( {E}^{\lambda } \otimes {F}^{\lambda } \\) is contained ... | Proof. This follows from Theorem 5.6.5 and Corollary 5.6.4. | No |
Theorem 5.6.7. The space of homogeneous polynomials on \( {M}_{k, n} \) of degree \( d \) decomposes under the representation \( \rho \) of \( \mathbf{{GL}}\left( {k,\mathbb{C}}\right) \times \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) as\n\n\[ \n{\mathcal{P}}^{d}\left( {M}_{k, n}\right) \cong {\bigoplus }_{v}{\left( ... | Proof. We may assume \( k \geq n \) (otherwise, we use the map \( x \mapsto {x}^{t} \) to interchange \( k \) and \( n \) ). Let \( {x}_{ij} \) be the \( {ij} \) -entry function on \( {M}_{k, n} \) and let \( m \in \mathbb{N} \) . Then\n\n\[ \n\rho \left( {a, b}\right) {x}_{ij}^{m} = {a}_{i}^{-m}{b}_{j}^{m}{x}_{ij}^{m}... | Yes |
Corollary 5.6.8. As a module for \( \mathbf{{GL}}\left( {k,\mathbb{C}}\right) \times \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) , \[ S\left( {{\mathbb{C}}^{k} \otimes {\mathbb{C}}^{n}}\right) \cong {\bigoplus }_{\mu }{F}_{k}^{\mu } \otimes {F}_{n}^{\mu }, \] with the sum over all nonnegative dominant weights \( \mu \... | Proof. Define a representation \( \sigma \) of \( \mathbf{{GL}}\left( {k,\mathbb{C}}\right) \times \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) on \( \mathcal{P}\left( {M}_{k, n}\right) \) by \[ \sigma \left( {y, z}\right) f\left( x\right) = f\left( {{y}^{t}{xz}}\right) \;\text{ for }y \in \mathbf{{GL}}\left( {k,\mathb... | Yes |
Corollary 5.6.10. \( \left( {G = \mathbf{O}\left( {n,\mathbb{C}}\right) \text{with}n \geq 3}\right) \) In the canonical duality decomposition\n\n\[ \mathcal{P}\left( V\right) \cong {\bigoplus }_{\lambda \in \mathcal{S}}{E}^{\lambda } \otimes {F}^{\lambda } \]\n\n(5.85)\n\nunder the joint action of \( \mathbb{D}{\left( ... | Proof. Apply Theorem 5.6.9 and Corollary 5.6.4. | No |
Theorem 5.6.11. (Notation as above)\n\n1. \( {\mathcal{E}}^{k} \) is an irreducible \( {\mathfrak{g}}^{\prime } \) -module.\n\n2. There is an injective \( \left( {{\mathfrak{g}}^{\prime } \times G}\right) \) -intertwining map from \( {\mathcal{E}}^{k} \otimes {\mathcal{H}}^{k}\left( {\mathbb{C}}^{n}\right) \) to \( \ma... | Proof. Following the strategy outlined above, we begin by determining the irreducible \( {\mathfrak{g}}^{\prime } \) -modules in \( \mathcal{P}\left( {\mathbb{C}}^{n}\right) \) . For \( \mu \in \mathbb{C} \) let \( {\mathcal{M}}_{\mu } \) be the \( \mathbb{C} \) vector space with basis \( \left\{ {{v}_{j} : j \in \math... | Yes |
Corollary 5.6.12. The map \( \;\mathbb{C}\left\lbrack {r}^{2}\right\rbrack \otimes \mathcal{H}\left( {\mathbb{C}}^{n}\right) \rightarrow \mathcal{P}\left( {\mathbb{C}}^{n}\right) \; \) given by \( \;f \otimes u \mapsto {fu} \) (pointwise multiplication of functions) is a linear bijection. | \[ {\mathcal{P}}^{k}\left( {\mathbb{C}}^{n}\right) = {\bigoplus }_{p = 0}^{\left\lbrack k/2\right\rbrack }{r}^{2p}{\mathcal{H}}^{k - {2p}}\left( {\mathbb{C}}^{n}\right) . \] | No |
Proposition 5.6.13. The space \( {\mathcal{H}}^{k}\left( {\mathbb{C}}^{n}\right) \) is spanned by the polynomials \( {x}^{k} \), where \( x \) is an isotropic vector in \( {\mathbb{C}}^{n} \) . | Proof. Let \( {\mathcal{L}}_{k} = \operatorname{Span}\left\{ {x}^{k}\right. : x \) isotropic in \( \left. {\mathbb{C}}^{n}\right\} \) . Then \( 0 \neq {\mathcal{L}}_{k} \subset {\mathcal{H}}^{k}\left( {\mathbb{C}}^{n}\right) \) and \( {\mathcal{L}}_{k} \) is \( G \) -invariant. Hence \( {\mathcal{L}}_{k} = {\mathcal{H}... | Yes |
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