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Proposition 8.30. Let \( \Delta \) be a base for \( R \), let \( {R}^{ + } \) be the associated set of positive roots, and let \( \alpha \) be an element of \( \Delta \) . If \( \beta \in {R}^{ + } \) and \( \beta \neq \alpha \), then \( {s}_{\alpha } \cdot \beta \in {R}^{ + } \) . That is to say, if \( \alpha \in \Del... | Proof. Write \( \beta = \mathop{\sum }\limits_{{\gamma \in \Delta }}{c}_{\gamma }{\alpha }_{\gamma } \) with \( {c}_{\gamma } \geq 0 \) . Since \( \beta \neq \alpha \), there is some \( {c}_{\gamma } \) with \( \gamma \neq \alpha \) and \( {c}_{\gamma } > 0 \) . Now, since \( {s}_{\alpha } \cdot \beta = \beta - {n\alph... | Yes |
1. A root system is irreducible if and only if its Dynkin diagram is connected. | For Point 1, if a root system \( R \) decomposes as the direct sum of two root systems \( {R}_{1} \) and \( {R}_{2} \), then we can obtain a base \( \Delta \) for \( R \) as \( \Delta = {\Delta }_{1} \cup {\Delta }_{2} \), where \( {\Delta }_{1} \) and \( {\Delta }_{2} \) are bases for \( {R}_{1} \) and \( {R}_{2} \), ... | Yes |
Corollary 8.33. Let \( \mathfrak{g} = {\mathfrak{k}}_{\mathbb{C}} \) be a semisimple Lie algebra, let \( \mathfrak{h} = {\mathfrak{t}}_{\mathbb{C}} \) be a Cartan subalgebra of \( \mathfrak{g} \), and let \( R \subset i\mathfrak{t} \) be the root system of \( \mathfrak{g} \) relative to \( \mathfrak{h} \) . Then \( \ma... | Proof. According to Theorem 7.35, \( \mathfrak{g} \) is simple if and only if \( R \) is irreducible. But according to Point 1 of Proposition 8.32, \( R \) is irreducible if and only if the Dynkin diagram of \( R \) is connected. | Yes |
Proposition 8.35. If \( \mu \in E \) has the property that\n\n\[ 2\frac{\langle \mu ,\alpha \rangle }{\langle \alpha ,\alpha \rangle } \]\n\nis an integer for all \( \alpha \in \Delta \), then the same holds for all \( \alpha \in R \) and, thus, \( \mu \) is an integral element. | Proof. An element \( \mu \) is integral if and only if \( \left\langle {\mu ,{H}_{\alpha }}\right\rangle \) is an integer for all \( \alpha \in R \) . By Proposition 8.18, each \( {H}_{\alpha } \) with \( \alpha \in R \) can be expressed as a linear combination of the \( {H}_{\alpha } \) ’s with \( \alpha \in \Delta \)... | Yes |
Proposition 8.38. The element \( \delta \) is a strictly dominant integral element; indeed,\n\n\[ 2\frac{\langle \beta ,\delta \rangle }{\langle \beta ,\beta \rangle } = 1 \] \n\nfor each \( \beta \in \Delta \) . | Proof. If \( \beta \in \Delta \), then by Proposition 8.30, \( {s}_{\beta } \) permutes the elements of \( {R}^{ + } \) different from \( \beta \) . Thus, we can decompose \( {R}^{ + } \smallsetminus \{ \beta \} \) as \( {E}_{1} \cup {E}_{2} \), where elements of \( {E}_{1} \) are orthogonal to \( \beta \) and elements... | Yes |
Proposition 8.42. If \( \mu \) is dominant, then \( w \cdot \mu \preccurlyeq \mu \) for all \( w \in W \) . | Proof. Let \( O \) be the Weyl-group orbit of \( \mu \) . Since \( O \) is a finite set, it contains a maximal element \( \lambda \), i.e., one such that there is no \( {\lambda }^{\prime } \neq \lambda \) in \( O \) that is higher than \( \lambda \) . Then for all \( \alpha \in \Delta \), we must have \( \langle \alph... | Yes |
Proposition 8.43. If \( \mu \) is a strictly dominant integral element, then \( \mu \succcurlyeq \delta \), where \( \delta \) is as in Definition 8.37. | Proof. Since \( \mu \) is strictly dominant, \( \mu - \delta \) will still be dominant in light of Proposition 8.38. Thus, by Proposition \( {8.40},\mu - \delta \succcurlyeq 0 \), which is equivalent to \( \mu \succcurlyeq \delta \) . | Yes |
Lemma 8.45. Suppose \( K \) is a compact, convex subset of \( E \) and \( \lambda \) is an element of \( E \) that it is not in \( K \) . Then there is an element \( \gamma \) of \( E \) such that for all \( \eta \in K \), we have\n\n\[ \langle \gamma ,\lambda \rangle > \langle \gamma ,\eta \rangle \] | Proof. Since \( K \) is compact, we can choose an element \( {\eta }_{0} \) of \( K \) that minimizes the distance to \( \lambda \) . Set \( \gamma = \lambda - {\eta }_{0} \), so that\n\n\[ \left\langle {\gamma ,\lambda - {\eta }_{0}}\right\rangle = \left\langle {\lambda - {\eta }_{0},\lambda - {\eta }_{0}}\right\rangl... | Yes |
Lemma 8.46. If \( \mu \) and \( \lambda \) are dominant and \( \lambda \notin \operatorname{Conv}\left( {W \cdot \mu }\right) \), there exists a dominant element \( \gamma \in E \) such that\n\n\[ \langle \gamma ,\lambda \rangle > \langle \gamma, w \cdot \mu \rangle \]\n\nfor all \( w \in W \) . | Proof. By Lemma 8.45, we can find some \( \gamma \) in \( E \), not necessarily dominant, such that \( \langle \gamma ,\lambda \rangle > \langle \gamma ,\eta \rangle \) for all \( \eta \in \operatorname{Conv}\left( {W \cdot \mu }\right) \) . In particular,\n\n\[ \langle \gamma ,\lambda \rangle > \langle \gamma, w \cdot... | Yes |
Corollary 8.47. The following semisimple Lie algebras are simple: the special linear algebras \( \mathrm{{sl}}\left( {n + 1;\mathbb{C}}\right), n \geq 1 \) ; the odd orthogonal algebras \( \mathrm{{so}}\left( {{2n} + 1;\mathbb{C}}\right) \) , \( n \geq 1 \) ; the even orthogonal algebras \( \operatorname{so}\left( {{2n... | Proof. The Dynkin diagrams for \( {A}_{n},{B}_{n} \), and \( {C}_{n} \) are always connected, whereas the Dynkin diagram for \( {D}_{n} \) is connected for \( n \geq 3 \) . Thus, Corollary 8.33 shows that the claimed Lie algebras are simple. | Yes |
Theorem 8.49. Every irreducible root system is isomorphic to exactly one of the following:\n\n- \( {A}_{n}, n \geq 1 \)\n\n- \( {B}_{n}, n \geq 2 \)\n\n- \( {C}_{n}, n \geq 3 \)\n\n- \( {D}_{n}, n \geq 4 \)\n\n- One of the exceptional root systems \( {G}_{2},{F}_{4},{E}_{6},{E}_{7} \), and \( {E}_{8} \) . | The restrictions on \( n \) are to avoid the case of \( {D}_{2} \), which is not irreducible, and to avoid repetitions. The Dynkin diagram for \( {D}_{3} \), for example, is isomorphic to the Dynkin diagram for \( {A}_{3} \), which means (Proposition 8.32) that the \( {A}_{3} \) and \( {D}_{3} \) root systems are also ... | Yes |
Theorem 8.50. 1. If \( \mathfrak{g} \) is a complex semisimple Lie algebra and \( {\mathfrak{h}}_{1} \) and \( {\mathfrak{h}}_{2} \) are Cartan subalgebra of \( \mathfrak{g} \), there exists an automorphism \( \phi : \mathfrak{g} \rightarrow \mathfrak{g} \) such that \( \phi \left( {\mathfrak{h}}_{1}\right) = \phi \lef... | For Point 1 of the theorem see Section 16.4 of [Hum]. | No |
Proposition 9.2. If \( \left( {\pi, V}\right) \) is a finite-dimensional representation of \( \mathfrak{g} \), every weight of \( \pi \) is an integral element. | Proof. For each root \( \alpha \), let \( {\mathfrak{s}}^{\alpha } = \left\langle {{X}_{\alpha },{Y}_{\alpha },{H}_{\alpha }}\right\rangle \cong \mathrm{{sl}}\left( {2;\mathbb{C}}\right) \) be the subalgebra of \( \mathfrak{g} \) in Theorem 7.19. If \( v \) is a weight vector with weight \( \lambda \), then\n\n\[ \pi \... | Yes |
Theorem 9.3. If \( \left( {\pi, V}\right) \) is a finite-dimensional representation of \( \mathfrak{g} \), the weights of \( \pi \) and their multiplicities are invariant under the action of \( W \) on \( H \) . | Proof. For each \( \alpha \in R \), we may construct the operator\n\n\[ \n{S}_{\alpha } \mathrel{\text{:=}} {e}^{\pi \left( {X}_{\alpha }\right) }{e}^{-\pi \left( {Y}_{\alpha }\right) }{e}^{\pi \left( {X}_{\alpha }\right) }.\n\]\nIf \( \langle \alpha, H\rangle = 0 \), then \( H \) will commute with both \( {X}_{\alpha ... | Yes |
Every irreducible, finite-dimensional representation of \( \mathfrak{g} \) has a highest weight. | Enumerate the positive roots as \( {\alpha }_{1},\ldots ,{\alpha }_{N} \) . Choose a basis for \( \mathfrak{g} \) consisting of elements \( {X}_{1},\ldots ,{X}_{N} \) with \( {X}_{j} \in {\mathfrak{g}}_{{\alpha }_{j}} \), elements \( {Y}_{1},\ldots ,{Y}_{N} \) with \( {Y}_{j} \in {\mathfrak{g}}_{-{\alpha }_{j}} \), and... | Yes |
Let \( \mathfrak{g} = \mathrm{{sl}}\left( {2;\mathbb{C}}\right) \) with the usual basis \( \{ X, Y, H\} \) . The universal enveloping algebra of \( \mathfrak{g} \) is then the associative algebra with identity generated by three elements \( x, y \), and \( h \), subject to the relations\n\n\[ \n{hx} - {xh} = {2x} \n\]\... | Proof of Theorem 9.7. The operation of tensor product on vector spaces is associative, in the sense that \( U \otimes \left( {V \otimes W}\right) \) is canonically isomorphic to \( \left( {U \otimes V}\right) \otimes W \) , with the isomorphism taking \( u \otimes \left( {v \otimes w}\right) \) to \( \left( {u \otimes ... | Yes |
Proposition 9.9. If \( \pi : \mathfrak{g} \rightarrow \operatorname{End}\left( V\right) \) is a representation of a Lie algebra \( \mathfrak{g} \) (not necessarily finite dimensional), there is a unique algebra homomorphism \( \widetilde{\pi } : U\left( \mathfrak{g}\right) \rightarrow \operatorname{End}\left( V\right) ... | Proof. Apply Theorem 9.7 with \( \mathcal{A} = \operatorname{End}\left( V\right) \) and \( j\left( X\right) = \pi \left( X\right) \) . | No |
Theorem 9.10 (PBW Theorem). If \( \mathfrak{g} \) is a finite-dimensional Lie algebra with basis \( {X}_{1},\ldots ,{X}_{k} \), then elements of the form\n\n\[ i{\left( {X}_{1}\right) }^{{n}_{1}}i{\left( {X}_{2}\right) }^{{n}_{2}}\cdots i{\left( {X}_{k}\right) }^{{n}_{k}}, \]\n\nwhere each \( {n}_{k} \) is a non-negati... | In (9.8), we interpret \( i{\left( {X}_{j}\right) }^{{n}_{j}} \) as 1 if \( {n}_{j} = 0 \) . Since, actually, \( i \) is injective, we will henceforth identify \( \mathfrak{g} \) with its image under \( i \) and thus regard \( \mathfrak{g} \) as a subspace of \( U\left( \mathfrak{g}\right) \) . Thus, we may now write \... | Yes |
Corollary 9.11. If \( \mathfrak{g} \) is a Lie algebra and \( \mathfrak{h} \) is a subalgebra of \( \mathfrak{g} \), then there is a natural injection of \( U\left( \mathfrak{h}\right) \) into \( U\left( \mathfrak{g}\right) \) given by mapping any product \( {X}_{1}{X}_{2}\cdots {X}_{N} \) of elements of \( \mathfrak{h... | Proof. The inclusion of \( \mathfrak{h} \) into \( \mathfrak{g} \) induces an algebra homomorphism of \( \phi : U\left( \mathfrak{h}\right) \rightarrow \) \( U\left( \mathfrak{g}\right) \) . Let us choose a basis \( {X}_{1},\ldots ,{X}_{k} \) for \( \mathfrak{h} \) and extend it to a basis \( {X}_{1},\ldots ,{X}_{N} \)... | Yes |
Theorem 9.13. The vector \( {v}_{0} \mathrel{\text{:=}} \left\lbrack 1\right\rbrack \) is a nonzero element of \( {W}_{\mu } \) and \( {W}_{\mu } \) is a highest weight cyclic representation with highest weight \( \mu \) and highest weight vector \( {v}_{0} \) . | The hard part of the proof is establishing that \( {v}_{0} \) is nonzero; this amounts to showing that the element 1 of \( U\left( \mathfrak{g}\right) \) is not in \( {I}_{\mu } \) . For purposes of constructing the irreducible, finite-dimensional representations of \( \mathfrak{g} \), Theorem 9.13 is sufficient. Our m... | No |
Lemma 9.15. Let \( {J}_{\mu } \) denote the left ideal in \( U\left( \mathfrak{b}\right) \subset U\left( \mathfrak{g}\right) \) generated by elements of the form (9.18) and (9.19). Then 1 does not belong to \( {J}_{\mu } \) . | Proof. Let \( \mathfrak{b} \) be the direct sum (as a vector space) of \( {\mathfrak{n}}^{ + } \) and \( \mathfrak{h} \), which is easily seen to be a subalgebra of \( \mathfrak{g} \) . Let us define a one-dimensional representation \( {\sigma }_{\mu } \) of \( \mathfrak{b} \) , acting on \( \mathbb{C} \), by the formu... | Yes |
Proposition 9.17. The space \( {U}_{\mu } \) is an invariant subspace for the action of \( \mathfrak{g} \) . | Proof. Suppose that \( v \) is in \( {U}_{\mu } \) and that \( Z \) is some element of \( \mathfrak{g} \) . We want to show that \( {\pi }_{\mu }\left( Z\right) v \) is also in \( {U}_{\mu } \) . Thus, we consider\n\n\[ \n{\pi }_{\mu }\left( {X}^{1}\right) \cdots {\pi }_{\mu }\left( {X}^{l}\right) {\pi }_{\mu }\left( Z... | Yes |
Proposition 9.18. The quotient space \( {V}_{\mu } \mathrel{\text{:=}} {W}_{\mu }/{U}_{\mu } \) is an irreducible representation of \( \mathfrak{g} \) . | Proof. A simple argument shows that the invariant subspaces of the representation \( {W}_{\mu }/{U}_{\mu } \) are in one-to-one correspondence with the invariant subspaces of \( {W}_{\mu } \) that contain \( {U}_{\mu } \) . Thus, proving that \( {V}_{\mu } \) is irreducible is equivalent to showing that any invariant s... | Yes |
Let \( \alpha \) be an element of \( \Delta \) and let \( {\mathfrak{s}}^{\alpha } = \left\langle {{X}_{\alpha },{Y}_{\alpha },{H}_{\alpha }}\right\rangle \) be as in Theorem 7.19. If \( \left\langle {\mu ,{H}_{\alpha }}\right\rangle \) is a non-negative integer \( m \), then the vector \( v \mathrel{\text{:=}} \) \( \... | Proof. By the argument in proof of Theorem 4.32, the analog of (4.15) will hold here:\n\n\[ \pi \left( {X}_{\alpha }\right) \pi {\left( {Y}_{\alpha }\right) }^{j}{v}_{0} = j\left( {m - \left( {j - 1}\right) }\right) \pi {\left( {Y}_{\alpha }\right) }^{j - 1}{v}_{0}. \]\n\nThus,\n\n\[ \pi \left( {X}_{\alpha }\right) v =... | Yes |
Proposition 9.22. If \( \mu \) is dominant integral, the set of weights for \( {V}_{\mu } \) is invariant under the action of the Weyl group on \( \mathfrak{h} \) . | Proof. We continue the notation from the proof of Proposition 9.21. Since (Proposition 8.24) \( W \) is generated by the reflections \( {s}_{\alpha } \) with \( w \in \Delta \), it suffices to show that the weights of \( {W}_{\mu }/{U}_{\mu } \) are invariant under each such reflection. By Proposition 9.21, \( {\wideti... | Yes |
The value of \( C \) is independent of the choice of orthonormal basis for \( \mathfrak{k} \). | If \( \left\{ {X}_{j}\right\} \) and \( \left\{ {Y}_{j}\right\} \) are two different orthonormal bases for \( \mathfrak{k} \), then there is an orthogonal matrix \( R \) such that\n\n\[ \n{Y}_{j} = \mathop{\sum }\limits_{k}{R}_{kj}{X}_{k} \n\]\n\nThen\n\n\[ \n\mathop{\sum }\limits_{j}{Y}_{j}^{2} = \mathop{\sum }\limits... | Yes |
Lemma 10.7. Let \( X \in {\mathfrak{g}}_{\alpha } \) be a unit vector, so that \( {X}^{ * } \in {\mathfrak{g}}_{-\alpha } \) . Then under our usual identification of \( \mathfrak{h} \) with \( {\mathfrak{h}}^{ * } \), we have\n\n\[ \left\lbrack {X,{X}^{ * }}\right\rbrack = \alpha \] | Proof. According to Lemma 7.22, we have\n\n\[ \left\langle {\left\lbrack {X,{X}^{ * }}\right\rbrack ,{H}_{\alpha }}\right\rangle = \left\langle {\alpha ,{H}_{\alpha }}\right\rangle \left\langle {{X}^{ * },{X}^{ * }}\right\rangle = \left\langle {\alpha ,{H}_{\alpha }}\right\rangle ,\]\n\n(10.3)\n\nsince \( X \) (and thu... | Yes |
Proposition 10.8. If \( \left( {\pi, V}\right) \) is a one-dimensional representation of \( \mathfrak{g} \), then \( \pi \left( X\right) = 0 \) for all \( X \in \mathfrak{g} \) . | Proof. By Theorem 7.8, \( \mathfrak{g} \) decomposes as a Lie algebra direct sum of simple algebras \( {\mathfrak{g}}_{j} \) . Since the kernel of \( {\left. \pi \right| }_{{\mathfrak{g}}_{j}} \) is a ideal, the restriction of \( \pi \) to \( {\mathfrak{g}}_{j} \) must be either zero or injective. But since \( \dim {\m... | Yes |
Proposition 10.12. If \( \left( {\pi, V}\right) \) is a finite-dimensional representation of \( \mathfrak{g} \), we have the following results.\n\n1. The dimension of \( V \) is equal to the value of \( {\chi }_{\pi } \) at the origin:\n\n\[ \dim \left( V\right) = {\chi }_{\pi }\left( 0\right) \]\n\n2. Suppose \( V \) ... | Proof. The first point holds because \( \operatorname{trace}\left( I\right) = \dim V \) . Meanwhile, since \( \pi \left( H\right) \) acts as \( \langle \lambda, H\rangle I \) in each weight space \( {V}_{\lambda } \), the second point follows from the definition of \( {\chi }_{\pi } \) . | Yes |
Let \( \pi \) denote the irreducible representation of \( \mathrm{{sl}}\left( {2;\mathbb{C}}\right) \) of dimension \( m + 1 \) and let\n\n\[ H = \left( \begin{array}{rr} 1 & 0 \\ 0 & - 1 \end{array}\right) \]\n\nThen\n\n\[ {\chi }_{\pi }\left( {aH}\right) = {\chi }_{\pi }\left( \left( \begin{matrix} a & 0 \\ 0 & - a \... | The eigenvalues of \( {\pi }_{m}\left( H\right) \) are \( m, m - 2,\ldots , - m \), from which (10.8) follows. To obtain (10.9), we note that\n\n\[ \left( {{e}^{a} - {e}^{-a}}\right) {\chi }_{\pi }\left( {aH}\right) \]\n\n\[ = {e}^{\left( {m + 1}\right) a} + {e}^{\left( {m - 1}\right) a} + \cdots + {e}^{-\left( {m - 1}... | Yes |
Proposition 10.16. The exponential functions \( H \mapsto {e}^{\left( \lambda, H\right) } \), with \( \lambda \in \mathfrak{h} \), are linearly independent in \( {C}^{\infty }\left( \mathfrak{h}\right) \) . | Proof. We need to show that if the function \( f : \mathfrak-h \rightarrow \mathbb{C} \) given by\n\n\[ f\left( H\right) = {c}_{1}{e}^{\left\langle {\lambda }_{1}, H\right\rangle } + \cdots + {c}_{n}{e}^{\left\langle {\lambda }_{n}, H\right\rangle } \]\n\nis identically zero, where \( {\lambda }_{1},\ldots ,{\lambda }_... | Yes |
Corollary 10.17. Suppose \( f \in {C}^{\infty }\left( \mathfrak{h}\right) \) can be expressed as a finite linear combination of exponentials \( {e}^{\langle \lambda, H\rangle } \), with \( \lambda \) integral,\n\n\[ f\left( H\right) = \mathop{\sum }\limits_{\lambda }{c}_{\lambda }{e}^{\langle \lambda, H\rangle }.\]\n\n... | Proof. On the one hand,\n\n\[ f\left( {w \cdot H}\right) = \mathop{\sum }\limits_{\lambda }{c}_{\lambda }{e}^{\left\langle {w}^{-1} \cdot \lambda, H\right\rangle } = \mathop{\sum }\limits_{\eta }{c}_{w \cdot \eta }{e}^{\langle \eta, H\rangle }.\]\n\n(10.16)\n\nOn the other hand,\n\n\[ f\left( {w \cdot H}\right) = \det ... | Yes |
Theorem 10.18. If \( \left( {{\pi }_{\mu },{V}_{\mu }}\right) \) is the irreducible representation of \( \mathfrak{g} \) with highest weight \( \mu \), then the dimension of \( {V}_{\mu } \) may be computed as\n\n\[ \dim \left( {V}_{\mu }\right) = \frac{\mathop{\prod }\limits_{{\alpha \in {R}^{ + }}}\langle \alpha ,\mu... | Note that both \( \mu + \delta \) and \( \delta \) are strictly dominant elements, so that all the factors in both the numerator and the denominator are nonzero. | No |
The function \( P \) is Weyl alternating. | For any \( w \in W \), consider the collection of roots of the form \( {w}^{-1} \cdot \alpha \) for \( \alpha \in {R}^{ + } \) . Since \( {R}^{ + } \) contains exactly one element out of each pair \( \pm \alpha \) of roots, the same is true of the collection of \( {w}^{-1} \cdot \alpha \) ’s, with \( w \) fixed and \( ... | Yes |
Let \( \mathcal{A} \) denote the differential operator\n\n\[ \mathcal{A} = \mathop{\prod }\limits_{{\alpha \in {R}^{ + }}}{D}_{\alpha } \]\n\nFor any \( \lambda \in \mathfrak{h} \), let \( {f}_{\lambda } : \mathfrak{h} \rightarrow \mathbb{C} \) be the function given by\n\n\[ {f}_{\lambda }\left( H\right) = \mathop{\sum... | Proof. The proof that \( {f}_{\lambda } \) is Weyl alternating is elementary. Since \( {f}_{\lambda } \) is a sum of exponentials, it is a real-analytic function, meaning that it can be expanded in a convergent power series in the coordinates \( {x}_{1},\ldots ,{x}_{r} \) associated to any basis for \( \mathfrak{h} \).... | No |
If \( {\mu }_{1} \) and \( {\mu }_{2} \) denote the two fundamental weights for \( \mathrm{{sl}}\left( {3;\mathbb{C}}\right) \) then the dimension of the representation with highest weight \( {m}_{1}{\mu }_{1} + {m}_{2}{\mu }_{2} \) is given by\n\n\[ \n\frac{1}{2}\left( {{m}_{1} + 1}\right) \left( {{m}_{2} + 1}\right) ... | Proof. Note that scaling the inner product on \( \mathfrak{h} \) by a constant does not affect the right-hand side of the dimension formula, since the inner product occurs an equal number of times in the numerator and denominator. Let us then normalize the inner product so that all roots \( \alpha \) satisfy \( \langle... | Yes |
If \( {\alpha }_{1} \) and \( {\alpha }_{2} \) are the two positive simple roots for \( \mathrm{{sl}}\left( {3;\mathbb{C}}\right) \), then for any two non-negative integers \( {m}_{1} \) and \( {m}_{2} \), we have\n\n\[ p\left( {{m}_{1}{\alpha }_{1} + {m}_{2}{\alpha }_{2}}\right) = 1 + \min \left( {{m}_{1},{m}_{2}}\rig... | Proof. We have the two positive simple roots \( {\alpha }_{1} \) and \( {\alpha }_{2} \), together with one other positive root \( {\alpha }_{3} = {\alpha }_{1} + {\alpha }_{2} \). Thus, if \( \lambda \) can be expressed as a non-negative integer combination of \( {\alpha }_{1} + {\alpha }_{2} + {\alpha }_{3} \), we ca... | Yes |
Proposition 10.27 (Formal Reciprocal of the Weyl Denominator). At the level of formal exponential series, we have\n\n\[ \frac{1}{q\left( H\right) } = \mathop{\sum }\limits_{{\eta \succcurlyeq 0}}p\left( \eta \right) {e}^{-\langle \eta + \delta, H\rangle } \]\n\n(10.24)\n\nHere the sum is nominally over all integral ele... | The proposition means, more precisely, that the product of \( q\left( H\right) \) and the formal exponential series on the right-hand side of (10.24) is equal to 1 (i.e., to \( {e}^{0} \) ). To prove Proposition 10.24, we first rewrite \( q \) as a product. | Yes |
Lemma 10.28. The Weyl denominator may be computed as\n\n\[ q\left( H\right) = \mathop{\prod }\limits_{{\alpha \in {R}^{ + }}}\left( {{e}^{\langle \alpha, H\rangle /2} - {e}^{-\langle \alpha, H\rangle /2}}\right) . \] | Proof. Let \( \widetilde{q} \) denote the product on the right-hand side of (10.25). If we expand out the product in the definition of \( \widetilde{q} \), there will be a term equal to\n\n\[ \mathop{\prod }\limits_{{\alpha \in {R}^{ + }}}{e}^{\langle \alpha, H\rangle /2} = {e}^{\langle \delta, H\rangle } \]\n\nwhere \... | Yes |
Theorem 10.29 (Kostant’s Multiplicity Formula). Suppose \( \mu \) is a dominant integral element and \( {V}_{\mu } \) is the finite-dimensional irreducible representation with highest weight \( \mu \). Then if \( \lambda \) is a weight of \( {V}_{\mu } \), the multiplicity of \( \lambda \) is given by\n\n\[ \operatorna... | Proof. By the Weyl character formula and Proposition 10.27, we have\n\n\[ {\chi }_{\pi }\left( H\right) = \left( {\mathop{\sum }\limits_{{\eta \succcurlyeq 0}}p\left( \eta \right) {e}^{-\langle \eta + \delta, H\rangle }}\right) \left( {\mathop{\sum }\limits_{{w \in W}}\det \left( w\right) {e}^{\langle w \cdot \left( {\... | Yes |
Proposition 10.31 (Character Formula for Verma Modules). For any integral element \( \lambda \), the formal character of the Verma module is given by\n\n\[ \n{Q}_{{W}_{\lambda }}\left( H\right) = \mathop{\sum }\limits_{{\eta \succcurlyeq 0}}p\left( \eta \right) {e}^{\langle \lambda - \eta, H\rangle } \n\]\n\n(10.30)\n\... | Proof. Enumerate the positive roots as \( {\alpha }_{1},\ldots ,{\alpha }_{k} \) and let \( \left\{ {Y}_{j}\right\} \) be nonzero elements of \( {\mathfrak{g}}_{-{\alpha }_{j}}, j = 1,\ldots, k \) . By Theorem 9.14, the elements of the form\n\n\[ \n\pi {\left( {Y}_{1}\right) }^{{n}_{1}}\cdots \pi {\left( {Y}_{k}\right)... | Yes |
If an exponential \( {e}^{\langle \lambda, H\rangle } \) occurs with nonzero coefficient in the product \( q\left( H\right) {\chi }^{\mu }\left( H\right) \), then \( \lambda \) must be in the convex hull of the \( W \) -orbit of \( \mu + \delta \) and \( \lambda \) must differ from \( \mu + \delta \) by an element of t... | Proof. If \( \lambda \) is an integral element, then for each root \( \alpha \) ,\n\n\[ \n{s}_{\alpha } \cdot \lambda = \lambda - 2\frac{\langle \lambda ,\alpha \rangle }{\langle \alpha ,\alpha \rangle }\alpha \n\]\n\nwill differ from \( \lambda \) by an integer multiple of \( \alpha \) . Since roots are integral, we c... | Yes |
Proposition 10.34. Let \( \left( {\pi, V}\right) \) be a highest-weight cyclic representation of \( \mathfrak{g} \) (possibly infinite dimensional) with highest weight \( \lambda \) and let \( \widetilde{\pi } \) be the extension of \( \pi \) to \( U\left( \mathfrak{g}\right) \) . Then \( \widetilde{\pi }\left( C\right... | Proof. The same argument as in the proof of Proposition 10.6 shows if \( {v}_{0} \) is the highest weight vector for \( V \), then \( \widetilde{\pi }\left( C\right) {v}_{0} = {c}_{\lambda }{v}_{0} \) . Now let \( U \) be the space of all \( v \in V \) for which \( \widetilde{\pi }\left( C\right) v = {c}_{\lambda }v \)... | Yes |
Lemma 10.37. Let \( \left( {\pi, V}\right) \) be a representation of \( \mathfrak{g} \), possibly infinite dimensional, that decomposes as direct sum of weight spaces of finite multiplicity, and let \( U \) be a nonzero invariant subspace of \( V \) . Then both \( U \) and the quotient representation \( V/U \) decompos... | Proof. Let \( u \) be an element of \( U \) . By assumption, we can decompose \( u \) as \( u = \) \( {v}_{1} + \cdots + {v}_{j} \), where the \( {v}_{k} \) ’s belong to weight spaces in \( V \) corresponding to distinct weights \( {\lambda }_{1},\ldots ,{\lambda }_{j} \) . We wish to show that each \( {v}_{k} \) actua... | Yes |
Proposition 10.38. If we enumerate the elements of \( {S}_{\mu } \) in nondecreasing order as \( {S}_{\mu } = \left\{ {{\eta }_{1},\ldots ,{\eta }_{l}}\right\} \), then the matrix \( {A}_{jk} \mathrel{\text{:=}} {a}_{{\eta }_{k}}^{{\eta }_{j}} \) is upper triangular with ones on the diagonal. Thus, \( A \) is invertibl... | Proof. For any finite partially ordered set, it is possible (Exercise 6) to enumerate the elements in nondecreasing order. In our case, this means that we can enumerate the elements of \( {S}_{\mu } \) as \( {\eta }_{1},\ldots ,{\eta }_{l} \) in such a way that if \( {\eta }_{j} \preccurlyeq {\eta }_{k} \) then \( j \l... | No |
Proposition 11.7. If \( T \) is a maximal torus, the Lie algebra \( \mathfrak{t} \) of \( T \) is a maximal commutative subalgebra of \( \mathfrak{k} \) . Conversely, if \( \mathfrak{t} \) is a maximal commutative subalgebra of \( \mathfrak{k} \), the connected Lie subgroup \( T \) of \( \mathfrak{k} \) with Lie algebr... | Proof. If \( T \) is a maximal torus, it is commutative, which means that its Lie algebra \( \mathrm{t} \) is also commutative (Proposition 3.22). Suppose \( t \) is contained in a commutative subalgebra \( \mathfrak{s} \) . Then it is also contained in a maximal commutative subalgebra \( {\mathfrak{s}}^{\prime } \) co... | Yes |
Corollary 11.10. If \( K \) is a connected, compact matrix Lie group, the exponential map for \( K \) is surjective. | Proof. For any \( x \in K \), choose a maximal torus \( T \) containing \( x \) . Since the exponential map for \( T \cong {\left( {S}^{1}\right) }^{k} \) is surjective, \( x \) can be expressed as the exponential of an element of the Lie algebra of \( T \) . | Yes |
Corollary 11.11. Let \( K \) be a connected, compact matrix Lie group and let \( x \) an arbitrary element of \( K \) . Then \( x \) belongs to the center of \( K \) if and only if \( x \) belongs to every maximal torus in \( K \) . | Proof. Assume first that \( x \) belongs to the center \( Z\left( K\right) \) of \( K \), and let \( T \) be any maximal torus in \( K \) . By the torus theorem, \( x \) is contained in a maximal torus \( S \) , and this torus is conjugate to \( T \) . Thus, there is some \( y \in K \) such that \( S = {yT}{y}^{-1} \) ... | Yes |
Lemma 11.12. Let \( T \) be a fixed maximal torus in \( K \). Then every \( y \in K \) can be written in the form\n\n\[ y = {xt}{x}^{-1} \]\n\nfor some \( x \in K \) and \( t \in T \). | If \( K = \mathrm{{SU}}\left( n\right) \) and \( T \) is the diagonal subgroup of \( K \), Lemma 11.12 follows easily from the fact that every unitary matrix has an orthonormal basis of eigenvectors. The proof of the general case of Lemma 11.12 requires substantial preparation and is given in Sect. 11.5. | No |
Proposition 11.14. Let \( X, Y \), and \( f \) be as in Definition 11.13. If \( y \) is a regular value of \( f \), then \( y \) has only finitely many preimages under \( f \) . | Proof. If \( {f}^{-1}\left( {\{ y\} }\right) \) were infinite, the set would have to have an accumulation point \( {x}_{0} \), by the assumed compactness of \( X \) . Then by continuity, we would have \( f\left( {x}_{0}\right) = \) \( y \) . Since \( y \) is a regular value of \( f \), then \( {f}_{ * }\left( {x}_{0}\r... | Yes |
Corollary 11.17. Let \( X, Y \), and \( f \) be as in Theorem 11.16. If there exists a regular value \( y \) of \( f \) for which the signed number of preimages is nonzero, then \( f \) must map onto \( Y \) . | Proof. If there existed some \( {y}^{\prime } \) that is not in the range of \( f \), then \( {y}^{\prime } \) would be a regular value and the (signed) number of preimages of \( {y}^{\prime } \) would be zero. This would contradict Theorem 11.16. | Yes |
Lemma 11.18. Suppose \( {\Psi }_{t} \) is a continuous, piecewise-smooth family of orientation-preserving diffeomorphisms of \( Y \), with \( {\Psi }_{0} \) being the identity map. For any \( n \) -form \( \alpha \) on \( Y \), let \( {\alpha }_{t} = {\Psi }_{t}^{ * }\left( \alpha \right) \) . Then for all \( t \), we ... | Proof. Saying that \( {\Psi }_{t} \) is piecewise smooth means that it we can divide \( \left\lbrack {0, T}\right\rbrack \) into finitely many subintervals on each of which \( {\Psi }_{t}\left( x\right) \) is smooth in \( x \) and \( t \) . Since \( {\Psi }_{t} \) is continuous, it suffices to prove the result on each ... | Yes |
Proposition 11.19. Let \( X, Y \), and \( f \) be as in Theorem 11.16, and suppose \( f \) has mapping degree \( k \) . Then for every \( n \) -form \( \alpha \) on \( Y \), we have\n\n\[{\int }_{X}{f}^{ * }\left( \alpha \right) = k{\int }_{Y}\alpha\] | Proof. We have already noted in (11.3) that if \( y \) is a regular value of \( f \), there is a neighborhood \( U \) of \( y \) such that (11.8) holds for all \( \alpha \) supported in \( U \) . By the deformation argument in the proof of Theorem 11.16, the same result holds for any \( y \) in \( Y \) . Since \( Y \) ... | Yes |
Define a topology on \( G/H \) by decreeing that a set \( U \) in \( G/H \) is open if and only if \( {Q}^{-1}\left( U\right) \) is open in \( G \) . Then \( G/H \) is Hausdorff with respect to this topology. Furthermore, \( G/H \) has a countable dense subset. | We begin by noting that the quotient map \( Q : G \rightarrow G/H \) is open; that is, \( Q\left( U\right) \) is open in \( G/H \) whenever \( U \) is open in \( G \) . To establish this, we must show that \( {Q}^{-1}\left( {Q\left( U\right) }\right) \) is open in \( G \) . But\n\n\[ \n{Q}^{-1}\left( {Q\left( U\right) ... | Yes |
Lemma 11.21 (Slice Lemma). Let \( G \) be a matrix Lie group with Lie algebra \( \mathfrak{g} \) , let \( H \) be a closed subgroup of \( G \), and let \( \mathfrak{h} \) be the Lie algebra of \( H \) . Decompose \( \mathfrak{g} \) as a vector space as \( \mathfrak{g} = \mathfrak{h} \oplus \mathfrak{f} \) for some subs... | Proof. We identify the tangent spaces at the identity to both \( \mathfrak{f} \times H \) and \( G \) with \( \mathfrak{f} \oplus \mathfrak{h} \) . If \( X\left( t\right) \) is a smooth curve in \( f \) passing through 0 at \( t = 0 \), we have\n\n\[ {\left. \frac{d}{dt}\Lambda \left( X\left( t\right), I\right) \right|... | Yes |
Proposition 11.23. Suppose there exists an inner product on \( \mathfrak{g} \) that is invariant under the adjoint action of \( H \), and let \( V \) denote the orthogonal complement of \( \mathfrak{h} \) with respect to this inner product. Then we may identify the tangent space at each point \( \left\lbrack g\right\rb... | Proof. Since \( \mathfrak{g} = \mathfrak{h} \oplus V \), Point 2 of Theorem 11.22 tells us that every tangent vector \( v \) to \( G/H \) at the identity coset can be expressed uniquely as\n\n\[ v = {\left. \frac{d}{dt}\left\lbrack {e}^{tX}\right\rbrack \right| }_{t = 0},\;X \in V. \]\n\nFor any \( \left\lbrack g\right... | Yes |
Proposition 11.24. If \( G \) is a matrix Lie group and \( H \) is a connected compact subgroup of \( G \), there exists a volume form on \( G/H \) that is invariant under the left action of \( G \) . This form is unique up to multiplication by a constant. | Proof. Since \( H \) is compact, there exists an inner product on \( \mathfrak{g} \) that is invariant under the adjoint action of \( H \) . Let \( V \) denote the orthogonal complement of \( \mathfrak{h} \) in \( \mathfrak{g} \), so that \( V \) is invariant under the adjoint action of \( H \) . Since \( H \) is conne... | Yes |
Lemma 11.26. Let \( t \in T \) be such that the subgroup generated by \( t \) is dense in \( T \) (Proposition 11.4). Then \( {\Phi }^{-1}\left( {\{ t\} }\right) \) of \( t \) consists precisely of elements of the form \( \left( {{x}^{-1}{tx},\left\lbrack x\right\rbrack }\right) \) with \( \left\lbrack x\right\rbrack \... | Proof. If \( x \in N\left( T\right) \), then \( {x}^{-1}{tx} \in T \) and we can see that \( \Phi \left( {{x}^{-1}{tx},\left\lbrack x\right\rbrack }\right) = t \) . In the other direction, if \( {xs}{x}^{-1} = t \), then\n\n\[ \n{x}^{-1}{t}^{m}x = {s}^{m} \in T \n\]\n\nfor all integers \( m \), so that \( {x}^{-1}{Tx} ... | Yes |
Proposition 11.27. Let \( \left( {t,\left\lbrack x\right\rbrack }\right) \) be a fixed point in \( T \times \left( {K/T}\right) \) . If we identify the tangent spaces to \( T \times \left( {K/T}\right) \) and to \( K \) with \( \mathfrak{t} \oplus \mathfrak{f} \cong \mathfrak{k} \), then the differential of \( \Phi \) ... | Proof. For \( H \in \mathfrak{t} \), we compute that\n\n\[ \n{\left. \frac{d}{d\tau }\Phi \left( t{e}^{\tau H},\left\lbrack x\right\rbrack \right) \right| }_{\tau = 0} = {\left. \frac{d}{d\tau }xt{e}^{\tau H}{x}^{-1}\right| }_{\tau = 0} \n\]\n\n\[ \n= {xtH}{x}^{-1} \n\]\n\n\[ \n= \left( {{xt}{x}^{-1}}\right) \left( {{\... | Yes |
Lemma 11.28. For \( t \in T \), let \( {\operatorname{Ad}}_{{t}^{-1}}^{\prime } \) denote the restriction of \( {\operatorname{Ad}}_{{t}^{-1}} \) to \( \mathfrak{f} \). 1. If \( t \) generates a dense subgroup of \( T \), then \( {\mathrm{{Ad}}}_{{t}^{-1}}^{\prime } - I \) is an invertible linear transformation of \( \... | Proof. The operator \( {\mathrm{{Ad}}}_{{t}^{-1}}^{\prime } - I \) is invertible provided that the restriction of \( {\mathrm{{Ad}}}_{{t}^{-1}} \) to \( f \) does not have an eigenvalue of 1 . Suppose, then, that \( {\operatorname{Ad}}_{{t}^{-1}}\left( X\right) = X \) for some \( X \in \mathfrak{f} \). Then for every i... | Yes |
The Weyl group \( W \) is finite and the orientations on \( T \times \left( {K/T}\right) \) and \( K \) can be chosen so that the mapping degree of \( \Phi \) is \( \left| W\right| \), the order of the Weyl group. | If \( t \) generates a dense subgroup of \( T \), then by Lemma 11.26, \( {\Phi }^{-1}\left( {\{ t\} }\right) \) is in one-to-one correspondence with \( W \) . Furthermore, Point 1 of Lemma 11.28 then tells us that such a \( t \) is a regular value of \( \Phi \) . Thus, by Proposition 11.14, \( {\Phi }^{-1}\left( {\{ t... | Yes |
Corollary 11.32. If \( f \) is a continuous class function on \( K \), then\n\n\[{\int }_{K}f\left( x\right) {dx} = \frac{1}{\left| W\right| }{\int }_{T}\rho \left( t\right) f\left( t\right) {dt}.\] | Proof. If \( f \) is a class function, then \( f\left( {{yt}{y}^{-1}}\right) = f\left( t\right) \) for all \( y \in K \) and \( t \in T \) . Since the volume form on \( K/T \) is normalized, we have\n\n\[{\int }_{K}f\left( {{yt}{y}^{-1}}\right) d\left\lbrack y\right\rbrack = f\left( t\right)\]\n\nin which case, the Wey... | No |
Suppose \( K = \mathrm{{SU}}\left( 2\right) \) and \( T \) is the diagonal subgroup. Then Corollary 11.32 takes the form\n\n\[ \n{\int }_{\mathrm{{SU}}\left( 2\right) }f\left( x\right) {dx} = \frac{1}{2}{\int }_{-\pi }^{\pi }f\left( {\operatorname{diag}\left( {{e}^{i\theta },{e}^{-{i\theta }}}\right) }\right) 4{\sin }^... | Proof. If we use the Hilbert-Schmidt inner product on \( \mathrm{{su}}\left( 2\right) \), the orthogonal complement of \( \mathfrak{t} \) in \( \mathrm{{su}}\left( 2\right) \) is the space of matrices \( X \) of the form\n\n\[ \nX = \left( \begin{matrix} 0 & x + {iy} \\ - x + {iy} & 0 \end{matrix}\right) \n\] \n\nwith ... | Yes |
For each \( \alpha \in R \), there is an element \( x \) in \( N\left( T\right) \) such that\n\n\[{\operatorname{Ad}}_{x}\left( {H}_{\alpha }\right) = - {H}_{\alpha }\]\n\nand such that\n\n\[{\operatorname{Ad}}_{x}\left( H\right) = H\]\n\nfor all \( H \in \mathfrak{t} \) for which \( \langle \alpha, H\rangle = 0 \) . T... | Proof. Choose \( {X}_{\alpha } \) and \( {Y}_{\alpha } \) as in Theorem 7.19, with \( {Y}_{\alpha } = {X}_{\alpha }^{ * } \) . Then \( {\left( {X}_{\alpha } - {Y}_{\alpha }\right) }^{ * } = \) \( - \left( {{X}_{\alpha } - {Y}_{\alpha }}\right) \), from which it follows that \( {X}_{\alpha } - {Y}_{\alpha } \in \mathfra... | Yes |
Theorem 11.36. If \( T \) is a maximal torus in \( K \), the following results hold.\n\n1. \( Z\left( T\right) = T \) .\n\n2. The Weyl group acts effectively on \( \mathfrak{t} \) and this action is generated by the reflections \( {s}_{\alpha },\alpha \in R \), in Proposition 11.35. | Since \( Z\left( T\right) = T \), it follows that \( T \) is a maximal commutative subgroup of \( T \) (i.e., there is no commutative subgroup of \( K \) properly containing \( T \) ). Nevertheless, there may exist maximal commutative subgroups of \( K \) that are not maximal tori; see Exercise 5. | No |
Lemma 11.38. Suppose \( C \) is the fundamental Weyl chamber with respect to \( \Delta \) and that \( w \) is an element of \( W \) that maps \( C \) to \( C \) . Then \( w = 1 \) . | Proof. Let \( x \in K \) be a representative of the element \( w \in W \) . Take any \( {H}_{0} \) in the interior of \( C \) and average \( {H}_{0} \) over the action of the (finite) subgroup of \( W \) generated by \( w \) . The resulting vector \( H \) is still in the interior of \( C \) (which is convex) and is now... | Yes |
Theorem 11.39. If two elements \( s \) and \( t \) of \( T \) are conjugate in \( K \), then there exists an element \( w \) of \( W \) such that \( w \cdot s = t \) . | Proof. Suppose \( s \) and \( t \) are in \( T \) and \( t = {xs}{x}^{-1} \) for some \( x \in K \) . Let \( Z\left( t\right) \) be the centralizer of \( t \) . Since, \( {xu}{x}^{-1} \) commutes with \( t = {xs}{x}^{-1} \) for all \( u \in T \), we see that \( {xT}{x}^{-1} \subset Z\left( t\right) \) . Thus, both \( T... | Yes |
Corollary 11.40. If \( f \) is a continuous Weyl-invariant function on \( T \), then \( f \) extends uniquely to a continuous class function on \( K \) . | Proof. By the torus theorem, each conjugacy class in \( K \) intersects \( T \) in at least one point. By Theorem 11.39, each conjugacy class intersects \( T \) in a single orbit of \( W \) . Thus, if \( f \) is a \( W \) -invariant function on \( T \), we can unambiguously (and uniquely) extend \( f \) to a class func... | Yes |
Proposition 12.2. If \( \left( {\Pi, V}\right) \) is a finite-dimensional representation of \( K \), the real weights for \( \Pi \) and their multiplicities are invariant under the action of the Weyl group. | Proof. Following the proof of Theorem 6.22, we can show that if \( w \in W \) is represented by \( x \in N\left( T\right) \), then \( \Pi \left( x\right) \) will map the weight space with weight \( \lambda \) isomorphically onto the weight space with weight \( w \cdot \lambda \) . | Yes |
Proposition 12.5. Let \( \left( {\sum, V}\right) \) be a representation of \( K \) and let \( \sigma \) be the associated representation of \( \mathfrak{k} \) . If \( \lambda \in \mathfrak{t} \) is a real weight of \( \sigma \), then \( \lambda \) is an analytically integral element. | Proof. If \( v \) is a weight vector with weight \( \lambda \) and \( H \) is an element of \( \Gamma \), then on the one hand, \[ \sum \left( {e}^{2\pi H}\right) v = {Iv} = v \] while on the other hand, \[ \sum \left( {e}^{2\pi H}\right) v = {e}^{{2\pi \sigma }\left( H\right) }v = {e}^{{2\pi i}\langle \lambda, H\rangl... | Yes |
If \( \alpha \in \mathfrak{t} \) is a real root and \( {H}_{\alpha } = {2\alpha }/\langle \alpha ,\alpha \rangle \) is the associated real coroot, we have\n\n\[ \n{e}^{{2\pi }{H}_{\alpha }} = I \n\]\nin \( K \) . That is to say, \( {H}_{\alpha } \) belongs to the kernel \( \Gamma \) of the exponential map. | Proof. By Corollary 7.20, there is a Lie algebra homomorphism \( \phi : \mathrm{{su}}\left( 2\right) \rightarrow \ell \) such that the element \( {iH} = \operatorname{diag}\left( {i, - i}\right) \) in \( \mathrm{{su}}\left( 2\right) \) maps to the real coroot \( {H}_{\alpha } \) . (In the notation of the corollary, \( ... | Yes |
Proposition 12.9. If \( \lambda \) is an analytically integral element, there is a well-defined function \( {f}_{\lambda } : T \rightarrow \mathbb{C} \) such that\n\n\[ \n{f}_{\lambda }\left( {e}^{H}\right) = {e}^{i\langle \lambda, H\rangle }\n\]\n\nfor all \( H \in \mathfrak{t} \) . Conversely, for any \( \lambda \in ... | Proof. Replacing \( H \) by \( {2\pi H} \), we can equivalently write (12.2) as\n\n\[ \n{f}_{\lambda }\left( {e}^{2\pi H}\right) = {e}^{{2\pi i}\langle \lambda, H\rangle },\;H \in \mathfrak{t}.\n\]\n\n(12.3)\n\nNow, since \( T \) is connected and commutative, every \( t \in T \) can be written as \( t = \) \( {e}^{2\pi... | Yes |
Proposition 12.10. The exponentials \( {f}_{\lambda } \) in (12.2), as \( \lambda \) ranges over the set of analytically integral elements, are orthonormal with respect to the normalized volume form dt on \( T \) : | Proof. Let us identify \( T \) with \( {\left( {S}^{1}\right) }^{k} \) for some \( k \), so that \( \mathfrak{t} \) is identified with \( {\mathbb{R}}^{k} \) and the scaled exponential map is given by\n\n\[ \left( {{\theta }_{1},\ldots ,{\theta }_{n}}\right) \mapsto \left( {{e}^{{2\pi i}{\theta }_{1}},\ldots ,{e}^{{2\p... | Yes |
Consider the group \( \mathrm{{SO}}\left( 3\right) \) and let \( \mathrm{t} \) be the maximal commutative subalgebra spanned by the element \( {F}_{3} \) in Example 3.27. Let the unique positive root \( \alpha \) be chosen so that \( \left\langle {\alpha ,{F}_{3}}\right\rangle = 1 \). Then \( \mu \in \mathfrak{t} \) is... | Proof. Following Sect. 7.7, but adjusting for our current convention of using real roots (Definition 11.34), we identify \( \mathfrak{t} \) with \( \mathbb{R} \) by mapping \( a{F}_{3} \) to \( a \). The roots are then the numbers \( \pm 1 \), where we take 1 as our positive root \( \alpha \), so that \( \left\langle {... | Yes |
Let \( \mathfrak{t} \) be the diagonal subalgebra of \( \mathfrak{u}\left( 2\right) \), and write every element \( \lambda \in \mathfrak{t} \) as \[ \lambda = {c\alpha } + {d\beta } \] with \( \alpha \) and \( \beta \) as in (12.5). Then \( \lambda \) is analytically integral if and only if either \( c \) and \( d \) a... | Proof. If \( H = \operatorname{diag}\left( {{ia},{ib}}\right) \), then \( {e}^{2\pi iH} = I \) if and only if \( a \) and \( b \) are both integers. Thus, when we identify \( t \) with \( {\mathbb{R}}^{2} \), the kernel \( \Gamma \) of the exponential corresponds to the integer lattice \( {\mathbb{Z}}^{2} \) . The latt... | Yes |
Lemma 12.16. Suppose \( \\left( {\\Pi, V}\\right) \) is a finite-dimensional representation of \( K \), and let \( P \) be the operator on \( V \) given by\n\n\[ P = {\\int }_{K}\\Pi \\left( x\\right) {dx} \]\n\nThen \( P \) is a projection onto \( {V}^{K} \) . That is to say, \( P \) maps \( V \) into \( {V}^{K} \) an... | Proof. For any \( y \\in K \) and \( v \\in V \), we have\n\n\[ \\Pi \\left( y\\right) {Pv} = \\Pi \\left( y\\right) \\left( {{\\int }_{K}\\Pi \\left( x\\right) {dx}}\\right) v \]\n\n\[ = \\left( {{\\int }_{K}\\Pi \\left( {yx}\\right) {dx}}\\right) v \]\n\n\[ = {Pv} \]\n\nby the left-invariance of the form \( {dx} \). ... | Yes |
Lemma 12.17. For \( A : V \rightarrow V \) and \( B : W \rightarrow W \), we have\n\n\[ \n\operatorname{trace}\left( A\right) \operatorname{trace}\left( B\right) = \operatorname{trace}\left( {A \otimes B}\right) ,\n\]\n\nwhere \( A \otimes B : V \otimes W \rightarrow V \otimes W \) is as in Proposition 4.16. | Proof. If \( \left\{ {v}_{j}\right\} \) and \( \left\{ {w}_{l}\right\} \) are bases for \( V \) and \( W \), respectively, then \( \left\{ {{v}_{j} \otimes {w}_{l}}\right\} \) is a basis for \( V \otimes W \) . If \( {A}_{jk} \) and \( {B}_{lm} \) are the matrices of \( A \) and \( B \) with respect to \( \left\{ {v}_{... | Yes |
Lemma 12.20. If \( \left( {\Pi, V}\right) \) is an irreducible representation of \( K \), then for each operator \( A \) on \( V \), we have\n\n\[{\int }_{K}\Pi \left( y\right) {A\Pi }{\left( y\right) }^{-1}{dy} = {cI}\]\n\nfor some constant \( c \) . | Proof. Let \( B \) denote the operator on the left-hand side of (12.11). By the left-invariance of the integral, we have\n\n\[ \Pi \left( x\right) {B\Pi }{\left( x\right) }^{-1} = {\int }_{K}\Pi \left( {xy}\right) {A\Pi }{\left( xy\right) }^{-1}{dy} \]\n\n\[ = {\int }_{K}\Pi \left( y\right) {A\Pi }{\left( y\right) }^{-... | Yes |
Let \( K = \mathrm{{SU}}\left( 2\right) \), let \( T \) be the diagonal subgroup, and let \( {\Pi }_{m} \) be the irreducible representation for which the largest eigenvalue of \( {\pi }_{m}\left( H\right) \) is \( m \) . Then the character formula for \( {\Pi }_{m} \) takes the form \[ {\chi }_{{\Pi }_{m}}\left( \left... | Proof. If \( H = \operatorname{diag}\left( {1, - 1}\right) \), then we may choose the unique positive root \( \alpha \) to satisfy \( \langle \alpha, H\rangle = 2 \) . The highest weight \( \mu \) of the representation then satisfies \( \langle \mu, H\rangle = m \) . Note that \( \operatorname{diag}\left( {{e}^{i\theta... | Yes |
Proposition 12.25. Two irreducible representations of \( K \) with the same highest weight are isomorphic. | Proof. If \( \Pi \) and \( \sum \) both have highest weight \( \mu \), then the Weyl character formula shows that the characters \( {\chi }_{\Pi } \) and \( {\chi }_{\sum } \) must be equal. But if \( \Pi \) and \( \sum \) were not isomorphic, Theorem 12.15 would imply that \( {\chi }_{\Pi } \) and \( {\chi }_{\sum } =... | Yes |
Lemma 12.26. For each dominant, analytically integral element \( \mu \), there is a unique continuous function \( {\phi }_{\mu } : T \rightarrow \mathbb{C} \) satisfying\n\n\[{\phi }_{\mu }\left( {e}^{H}\right) = \frac{\mathop{\sum }\limits_{{w \in W}}\det \left( w\right) {e}^{i\langle w \cdot \left( {\mu + \delta }\ri... | Proof. By Assumption 12.21, the element \( \delta \) is analytically integral, which means that each of \( w \cdot \left( {\mu + \delta }\right) \) and \( w \cdot \delta, w \in W \), is also analytically integral. Thus, by Proposition 12.9, all of the exponentials involved are well-defined functions on \( T \) . It fol... | Yes |
Lemma 12.27. Let \( {\phi }_{\mu } \) be as in Lemma 12.26 and let \( {\Phi }_{\mu } : K \rightarrow \mathbb{C} \) be the unique continuous class function on \( K \) such that \( {\left. {\Phi }_{\mu }\right| }_{T} = {\phi }_{\mu } \) (Corollary 11.40). Then as \( \mu \) ranges over the set of dominant, analytically in... | Proof. Since the denominator in the definition of \( {\phi }_{\mu } \) is the Weyl denominator, Corollary 11.32 to the Weyl integral formula tells us that\n\n\[ \n{\int }_{K}\overline{{\Phi }_{\mu }\left( x\right) }{\Phi }_{{\mu }^{\prime }}\left( x\right) {dx} \n\]\n\n\[ \n= \frac{1}{\left| W\right| }{\int }_{T}{\left... | Yes |
If \( \lambda \) and \( \eta \) are half integral, then \( \lambda + \eta \) is analytically integral. If \( \lambda \) is half integral, then \( - \lambda \) is also half integral and \( w \cdot \lambda \) is half integral for all \( w \in W \) . | If \( \lambda = \delta + {\lambda }^{\prime } \) and \( \eta = \delta + {\eta }^{\prime } \) are half integral, then \( \lambda + \eta = {2\delta } + {\lambda }^{\prime } + {\eta }^{\prime } \) is analytically integral. If \( \lambda = \delta + {\lambda }^{\prime } \) is half integral, so is\n\n\[ - \lambda = - \delta ... | Yes |
Proposition 12.31. If \( \lambda \) and \( \eta \) are half integral, there is a well-defined function \( f \) on \( T \) such that\n\n\[ f\left( {e}^{H}\right) = \overline{{e}^{i\langle \lambda, H\rangle }}{e}^{i\langle \eta, H\rangle } \]\n\nand\n\n\[ {\int }_{T}\overline{{e}^{i\langle \lambda, H\rangle }}{e}^{i\lang... | Proof. If \( \lambda \) and \( \eta \) are half integral, then \( - \lambda \) is half integral, so that \( \eta - \lambda \) is analytically integral. Thus, by Proposition 12.9, there is a well-defined function \( f : T \rightarrow \mathbb{C} \) satisfying\n\n\[ f\left( {e}^{H}\right) = \overline{{e}^{i\langle \lambda... | Yes |
Corollary 13.4. Suppose that \( \left( {Y,\pi }\right) \) is a universal cover of \( X \), that \( l \) is a loop in \( X \), and that \( p \) is a lift of \( l \) to \( Y \) . Then \( l \) is null homotopic in \( X \) if and only if \( p \) is a loop in \( Y \), that is, if and only if \( p\left( 1\right) = p\left( 0\... | Proof. If the lift \( p \) of \( l \) is a loop, then since \( Y \) is simply connected, there is a homotopy \( {p}_{s} \) of \( p \) to a point with basepoint fixed. Then \( {l}_{s} \mathrel{\text{:=}} \pi \circ {p}_{s} \) is a homotopy of \( l \) to a point in \( X \) . In the other direction, if there is a homotopy ... | Yes |
Proposition 13.5. For a \( d \) -sphere \( {S}^{d},{\pi }_{k}\left( {S}^{d}\right) \) is trivial if \( k < d \) . | This result is plausible because for \( k < d \), the image of a \ | No |
Theorem 13.7. Suppose that \( X \) is a fiber bundle with base \( B \) and fiber \( F \) . If \( {\pi }_{1}\left( B\right) \) and \( {\pi }_{2}\left( B\right) \) are trivial, then \( {\pi }_{1}\left( X\right) \) is isomorphic to \( {\pi }_{1}\left( F\right) \) . | Proof. According to a standard topological result (e.g., Theorem 4.41 and Proposition 4.48 in [Hat]), there is a long exact sequence of homotopy groups for a fiber bundle. The portion of this sequence relevant to us is the following:\n\n\[ \n{\pi }_{2}\left( B\right) \underset{f}{ \rightarrow }{\pi }_{1}\left( F\right)... | Yes |
Proposition 13.8. Suppose \( G \) is a matrix Lie group and \( H \) is a closed subgroup of \( G \) . Then \( G \) has the structure of a fiber bundle with base \( G/H \) and fiber \( H \), where the projection map \( p : G \rightarrow G/H \) is given by \( p\left( x\right) = \left\lbrack x\right\rbrack \), with \( \le... | Proof. For any coset \( \left\lbrack x\right\rbrack \) in \( G/H \), the preimage of \( \left\lbrack x\right\rbrack \) under \( p \) is the set \( {xH} \subset G \) , which is clearly homeomorphic to \( H \) . Meanwhile, the required local triviality property of the bundle follows from Lemma 11.21 and Theorem 11.22. (I... | Yes |
Proposition 13.9. Consider the map \( p : \mathrm{{SO}}\left( n\right) \rightarrow {S}^{n - 1} \) given by\n\n\[ p\left( R\right) = R{e}_{n} \]\n\nwhere \( {e}_{n} = \left( {0,\ldots ,0,1}\right) \). Then \( \left( {\mathrm{{SO}}\left( n\right), p}\right) \) is a fiber bundle with base \( {S}^{n - 1} \) and fiber \( \m... | Proof. We think of \( \mathrm{{SO}}\left( {n - 1}\right) \) as the (closed) subgroup of \( \mathrm{{SO}}\left( n\right) \) consisting of block diagonal matrices of the form\n\n\[ R = \left( \begin{matrix} {R}^{\prime } & 0 \\ 0 & 1 \end{matrix}\right) \]\n\nwith \( {R}^{\prime } \in \mathrm{{SO}}\left( {n - 1}\right) \... | Yes |
Proposition 13.10. For all \( n \geq 3 \), the fundamental group of \( \mathrm{{SO}}\left( n\right) \) is isomorphic to \( \mathbb{Z}/2 \) . Meanwhile, \( {\pi }_{1}\left( {\mathrm{{SO}}\left( 2\right) }\right) \cong \mathbb{Z} \) . | Proof. Suppose that \( n \) is at least 4, so that \( n - 1 \) is at least 3 . Then, by Proposition 13.5, \( {\pi }_{1}\left( {S}^{n - 1}\right) \) and \( {\pi }_{2}\left( {S}^{n - 1}\right) \) are trivial and, so, Theorem 13.7 and Proposition 13.9 tell us that \( {\pi }_{1}\left( {\mathrm{{SO}}\left( n\right) }\right)... | Yes |
Proposition 13.11. The group \( \mathrm{{SU}}\left( n\right) \) is simply connected for all \( n \geq 2 \) . For all \( n \geq 1 \), we have that \( {\pi }_{1}\left( {\mathrm{U}\left( n\right) }\right) \cong \mathbb{Z} \) . | Proof. For all \( n \geq 3 \), the group \( \mathrm{{SU}}\left( n\right) \) acts transitively on the sphere \( {S}^{{2n} - 1} \) . By a small modification of the proof of Proposition 13.9, \( \mathrm{{SU}}\left( n\right) \) is a fiber bundle with base \( {S}^{{2n} - 1} \) and fiber \( \mathrm{{SU}}\left( {n - 1}\right)... | Yes |
For all \( n \geq 1 \), the compact symplectic group \( \operatorname{Sp}\left( n\right) \) is simply connected. | Proof. Since \( \operatorname{Sp}\left( n\right) \) is contained in \( \mathrm{U}\left( {2n}\right) \), it acts on the unit sphere \( {S}^{{4n} - 1} \subset {\mathbb{C}}^{2n} \) . If this action is transitive, then we can imitate the arguments from the cases of \( \mathrm{{SO}}\left( n\right) \) and \( \mathrm{{SU}}\le... | Yes |
Proposition 13.14. Every loop in \( T \) is homotopic in \( T \) to a unique loop of the form\n\n\[ \tau \mapsto {e}^{2\pi \tau \gamma },\;0 \leq \tau \leq 1 \]\n\nwith \( \gamma \in \Gamma \) . Furthermore, \( {\pi }_{1}\left( T\right) \) is isomorphic to \( \Gamma \) . | Proof. The main issue is to prove that the (scaled) exponential map \( H \mapsto {e}^{2\pi H} \) is a covering map. Since the kernel \( \Gamma \) of the exponential map is discrete, there is some \( \varepsilon > 0 \) such that every nonzero element \( \gamma \) of \( \Gamma \) has norm at least \( \varepsilon \) . Let... | Yes |
The fundamental group of \( K \) is isomorphic to the quotient group \( \Gamma /I \), where the quotient is of commutative groups. | By Theorem 13.15, every loop in \( K \) is homotopic to a loop of the form \( \tau \mapsto \) \( {e}^{2\pi \tau \gamma } \), with \( \gamma \in \Gamma \) . Under this correspondence, composition of loops corresponds to addition in \( \Gamma \) . By Theorem 13.17, two loops of the form \( \tau \mapsto {e}^{{2\pi \tau }{... | Yes |
Corollary 13.20. If \( K \) is simply connected, then every algebraically integral element is analytically integral. | Proof. In light of Theorem 13.17, \( K \) is simply connected if and only if \( I = \Gamma \) . If \( \lambda \in \mathfrak{t} \) is algebraically integral, then \( \left\langle {\lambda ,{H}_{\alpha }}\right\rangle \in \mathbb{Z} \) for all \( \alpha \), where \( {H}_{\alpha } = {2\alpha }/\langle \alpha ,\alpha \rang... | Yes |
Corollary 13.21. If \( K \) is simply connected, the element \( \delta \) (half the sum of the real, positive roots) is analytically integral. | Proof. According to Proposition 8.38 (translated into the language of real roots), the element \( \delta \) is algebraically integral. But by Corollary 13.20, if \( K \) is simply connected, every algebraically integral element is analytically integral. | Yes |
Proposition 13.23. Suppose \( x \in K \) has the form \( x = {yt}{y}^{-1} \) with \( t \in T \) and \( y \in K \) . If there exists some \( X \in \mathfrak{k} \) with \( X \notin \mathfrak{t} \) such that\n\n\[{\operatorname{Ad}}_{t}\left( X\right) = X\]\n\nthen \( x \) is singular. If no such \( X \) exists, \( x \) i... | Proof. The condition of being regular is clearly invariant under conjugation; that is, \( x \) is regular if and only if \( t \) is regular. Suppose now that there is some \( X \notin \mathfrak{t} \) with \( {\operatorname{Ad}}_{t}\left( X\right) = X \) . Then \( t \) commutes with \( {e}^{\tau X} \) for all \( \tau \i... | Yes |
Proposition 13.24. An element \( x = y{e}^{2\pi H}{y}^{-1} \) is singular if and only if there is some root \( \alpha \) for which\n\n\[ \langle \alpha, H\rangle \in \mathbb{Z}\text{.} \]\n\nIt follows that \( x = y{e}^{2\pi H}{y}^{-1} \) is singular if and only if \( {\Psi }_{ * } \) is singular at the point \( \left(... | Proof. By Proposition 13.23, \( x \) is singular if and only if \( {\operatorname{Ad}}_{{e}^{2\pi H}}\left( X\right) = X \) for some \( X \notin \mathfrak{t} \) . Now, the dimension of the eigenspace of \( {\operatorname{Ad}}_{{e}^{2\pi H}} \) with eigenvalue \( 1 \in \mathbb{R} \) is the same whether we work over \( \... | Yes |
Proposition 13.26. Suppose \( \left( {H,\left\lbrack x\right\rbrack }\right) \) and \( \left( {{H}^{\prime },\left\lbrack {x}^{\prime }\right\rbrack }\right) \) belong to \( {\mathrm{t}}_{\mathrm{{reg}}} \times \left( {K/T}\right) \) . Then \( \Psi \left( {H,\left\lbrack x\right\rbrack }\right) = \Psi \left( {{H}^{\pri... | Proof. If \( {H}^{\prime } \) and \( {x}^{\prime } \) are as in (13.6), then by the calculations preceding the statement of the proposition, we will have \( \Psi \left( {{H}^{\prime },\left\lbrack {x}^{\prime }\right\rbrack }\right) = \Psi \left( {H,\left\lbrack x\right\rbrack }\right) \) . In the other direction, if \... | Yes |
Theorem 13.27. The fundamental groups of \( K \) and \( {K}_{\text{reg }} \) are isomorphic. Specifically, every loop in \( K \) is homotopic to a loop in \( {K}_{\text{reg }} \) and a loop in \( {K}_{\text{reg }} \) is null homotopic in \( K \) only if it is null homotopic in \( {K}_{\text{reg }} \) . | To prove this result, we will first show that the singular set in \( K \) is \ | No |
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