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Proposition 12.5. Let \( V \) be a finite-dimensional representation of \( \mathfrak{{sl}}\left( {2,\mathbb{R}}\right) \) . Let \( {v}_{k} \in V \) be an \( H \) -eigenvector with eigenvalue \( k \) maximal. Then \( k \) is a positive integer and \( {v}_{k} \) is contained in an irreducible subspace of \( V \) isomorph... | Proof. We have \( R{v}_{k} = 0 \) by Lemma 12.1 and the maximality of \( k \) . Then \( \Delta {v}_{k} = \left( {{k}^{2} + {2k}}\right) {v}_{k} \) follows from (12.5). Consider the submodule \( U \) generated by \( {v}_{k} \) . Every element of \( U \) is of the form \( \xi {v}_{k} \) where \( \xi \) is in the universa... | Yes |
Proposition 12.6. Let \( \left( {\pi, V}\right) \) be an irreducible complex representation of the Lie algebra \( \mathfrak{{sl}}\left( {2,\mathbb{R}}\right) \) . Then \( \Delta \) acts by a scalar \( \lambda \) on \( V \), and \( \lambda = {k}^{2} + {2k} \) for some nonnegative integer \( k \) . The representation \( ... | Proof. Let \( {v}_{k} \) be an eigenvector for \( H \) with eigenvalue \( k \) maximal. Then \( R{v}_{k} = 0 \) since otherwise \( R{v}_{k} \) is an eigenvector with eigenvalue \( k + 2 \) . By Proposition \( {12.5}{v}_{k} \) generates an irreducible subspace isomorphic to \( { \vee }^{k}{\mathbb{C}}^{2} \) . Since \( ... | No |
Theorem 12.1. Let \( \left( {\pi, V}\right) \) be any irreducible complex representation of \( \mathfrak{{sl}}\left( {2,\mathbb{R}}\right) ,\mathfrak{{su}}\left( 2\right) \) or \( \mathfrak{{sl}}\left( {2,\mathbb{C}}\right) \) . Then \( \pi \) is isomorphic to \( { \vee }^{k}{\mathbb{C}}^{2} \) for some \( k \) . | Proof. By Theorem 11.1, it is sufficient to show this for \( \mathfrak{{sl}}\left( {2,\mathbb{R}}\right) \), in which case the statement follows from Proposition 12.6. | No |
Proposition 12.7. Let \( \mathfrak{g} = \mathfrak{{sl}}\left( {2,\mathbb{R}}\right) ,\mathfrak{{su}}\left( 2\right) \) or \( \mathfrak{{sl}}\left( {2,\mathbb{C}}\right) \) . Let \( \left( {\pi, V}\right) \) be a finite-dimensional complex representation of \( \mathfrak{g} \) . If there exists \( k \geq 1 \) such that \... | Proof. There is nothing to do if \( V = \{ 0\} \) . Assume therefore that \( U \) is a maximal proper invariant subspace of \( U \) . By induction on \( \dim \left( V\right) ,\mathfrak{g} \) acts trivially on \( U \) . Now \( V/U \) is irreducible by the maximality of \( U \), and \( \Delta \) annihilates \( V/U \), so... | Yes |
Proposition 12.8. Let \( \mathfrak{g} = \mathfrak{{sl}}\left( {2,\mathbb{R}}\right) ,\mathfrak{{su}}\left( 2\right) \), or \( \mathfrak{{sl}}\left( {2,\mathbb{C}}\right) \) . Let \( \left( {\pi, V}\right) \) be a finite-dimensional complex representation of \( \mathfrak{g} \) . (i) If \( v \in V \) and \( {\Delta }^{2}... | Proof. Since \( \Delta \) commutes with the action of \( \mathfrak{g} \), the kernel \( W \) of \( {\Delta }^{k} \) is an invariant subspace. Now (i) follows from Proposition 12.7. It follows from (i) that \( {V}_{0} \cap {V}_{1} = \{ 0\} \) . Now for any linear endomorphism of a vector space, the dimension of the imag... | Yes |
Proposition 12.9. Let \( U, V, W, Q \) be \( \mathfrak{g} \) -modules, where \( \mathfrak{g} \) is one of \( \mathfrak{{sl}}\left( {2,\mathbb{R}}\right) \) , \( \mathfrak{{su}}\left( 2\right) \), or \( \mathfrak{{sl}}\left( {2,\mathbb{C}}\right) \), and let\n\n\[ 0 \rightarrow V \rightarrow W \rightarrow Q \rightarrow ... | Proof. Composition with these maps gives a short exact sequence:\n\n\[ 0 \rightarrow \operatorname{Hom}\left( {U, V}\right) \rightarrow \operatorname{Hom}\left( {U, W}\right) \rightarrow \operatorname{Hom}\left( {U, Q}\right) \rightarrow 0. \]\n\nHere, of course, \( \operatorname{Hom}\left( {U, V}\right) \) is just the... | Yes |
Theorem 12.2. Let \( \mathfrak{g} = \mathfrak{{sl}}\left( {2,\mathbb{R}}\right) ,\mathfrak{{su}}\left( 2\right) , or \( \mathfrak{{sl}}\left( {2,\mathbb{C}}\right) \). Any finite-dimensional complex representation of \( \mathfrak{g} \) is a direct sum of irreducible representations. | Proof. Let \( W \) be a \( \mathfrak{g} \)-module. If \( W \) is zero or irreducible, there is nothing to check. Otherwise, let \( V \) be a proper nonzero submodule and let \( Q = W/V \). We have an exact sequence\n\n\[ 0 \rightarrow V \rightarrow W \rightarrow Q \rightarrow 0 \]\n\nand by induction on \( \dim \left( ... | Yes |
Proposition 13.1. Suppose the space \( U \) is path-connected. The following are equivalent.\n\n(i) Every loop in \( U \) is path-homotopic to a trivial loop.\n\n(ii) Every loop \( p \) in \( U \) with \( p\left( 0\right) = p\left( 1\right) = {x}_{0} \) is path-homotopic to a trivial loop.\n\n(iii) Every continuous map... | Proof. Clearly,(i) implies (ii). Assuming (ii), if \( p \) is a loop in \( U \), let \( x \) be the endpoint \( p\left( 0\right) = p\left( 1\right) \) and (using path-connectedness) let \( q \) be a path from \( {x}_{0} \) to \( x \) . Then \( q \star p \star \left( {-q}\right) \) is a loop beginning and ending at \( {... | Yes |
Proposition 13.2. Let \( \pi : N \rightarrow M \) be a covering map.\n\n(i) If \( p : \left\lbrack {0,1}\right\rbrack \rightarrow M \) is a path, and if \( y \in {\pi }^{-1}\left( {p\left( 0\right) }\right) \), then there exists a unique path \( \widetilde{p} : \left\lbrack {0,1}\right\rbrack \rightarrow N \) such that... | Proof. If the cover is trivial, then we may assume that \( N = M \times F \) where \( F \) is discrete, and if \( y = \left( {x, f}\right) \), where \( x = p\left( 0\right) \) and \( f \in F \), then the unique solution to this problem is \( \widetilde{p}\left( t\right) = \left( {p\left( t\right), f}\right) \) .\n\nSin... | Yes |
Proposition 13.3. If \( M \) is simply connected, is \( N \) path-connected, and \( \pi \) : \( N \rightarrow M \) is a covering map, then \( \pi \) is a homeomorphism. | Proof. Since a covering map is always a local homeomorphism, what we need to show is that \( \pi \) is bijective. It is, of course, surjective. Suppose that \( n,{n}^{\prime } \in N \) have the same image in \( M \) . Since \( N \) is path-connected, let \( \widetilde{p} : \left\lbrack {0,1}\right\rbrack \rightarrow N ... | Yes |
Theorem 13.1. Let \( M \) be a path-connected space with base point \( {x}_{0} \) in which every point has a contractible neighborhood. Then there exists a simply connected space \( \widetilde{M} \) with a covering map \( \widetilde{\pi } : \widetilde{M} \rightarrow M \) . If \( \pi : N \rightarrow M \) is any pointed ... | Proof. To construct \( \widetilde{M} \), let \( \widetilde{M} \) as a set be the set of all paths \( p : \left\lbrack {0,1}\right\rbrack \rightarrow \) \( M \) such that \( p\left( 0\right) = {x}_{0} \) modulo the equivalence relation of path-homotopy. We define the covering map \( \widetilde{\pi } : \widetilde{M} \rig... | Yes |
Proposition 13.4. Let \( M, N \) and \( {N}^{\prime } \) be topological spaces such that every point has a contractible neighborhood. Assume that \( M \) is simply-connected. Let \( \pi : {N}^{\prime } \rightarrow N \) be a covering map, and let \( f : M \rightarrow N \) be continuous. Then there exists a continuous ma... | Proof. Let \( {x}_{0} \) be a base point for \( M \), and let \( {y}_{0}^{\prime } \) be an element of \( {N}^{\prime } \) such that \( \pi \left( {y}_{0}^{\prime }\right) = {y}_{0} \) where \( {y}_{0} = f\left( {x}_{0}\right) \) . If \( x \in M \), we may find a path \( p : \left\lbrack {0,1}\right\rbrack \rightarrow ... | Yes |
Theorem 13.2. Suppose that \( G \) is a path-connected group in which every point has a contractible neighborhood. Then the universal covering space \( \widetilde{G} \) admits a group structure in which both the natural inclusion map \( {\pi }_{1}\left( G\right) \hookrightarrow \widetilde{G} \) and the projection \( \w... | Proof. If \( p : \left\lbrack {0,1}\right\rbrack \rightarrow G \) and \( q : \left\lbrack {0,1}\right\rbrack \rightarrow G \) are paths, so is \( t \mapsto p \cdot q\left( t\right) = \) \( p\left( t\right) q\left( t\right) \) . If \( p\left( 0\right) = q\left( 0\right) = {1}_{G} \), the identity element in \( G \), the... | Yes |
Proposition 13.5. Let \( {S}^{r} \) denote the \( r \) -sphere. Then \( {\pi }_{1}\left( {S}^{1}\right) \cong \mathbb{Z} \), while \( {S}^{r} \) is simply-connected if \( r \geq 2 \) . | Proof. We may identify the circle \( {S}^{1} \) with the unit circle in \( \mathbb{C} \) . Then \( x \mapsto \) \( {\mathrm{e}}^{2\pi ix} \) is a covering map \( \mathbb{R} \rightarrow {S}^{1} \) . The space \( \mathbb{R} \) is contractible and hence simply-connected, so it is the universal covering space. If we give \... | Yes |
Proposition 13.6. The group \( \mathrm{{SU}}\left( 2\right) \) is simply-connected. The group \( \mathrm{{SO}}\left( 3\right) \) is not. In fact \( {\pi }_{1}\left( {\mathrm{{SO}}\left( 3\right) }\right) \cong \mathbb{Z}/2\mathbb{Z} \) . | Proof. Note that \( \mathrm{{SU}}\left( 2\right) = \left\{ {\left. \left( \begin{matrix} a & b \\ - \bar{b} & \bar{a} \end{matrix}\right) \right| \;{\left| a\right| }^{2} + {\left| b\right| }^{2} = 1}\right\} \) is homeomorphic to the 3 sphere in \( {\mathbb{C}}^{2} \) . As such, it is simply connected. We have a homom... | Yes |
Theorem 13.4. Let \( P \) be the space of positive definite Hermitian matrices. If \( g \in \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \), then \( g \) may be written uniquely as \( {pk} \), where \( k \in \mathrm{U}\left( n\right) \) and \( p \in P \) . Moreover, the multiplication map \( P \times \mathrm{U}\left( n\ri... | Proof. The matrix \( g \cdot {}^{t}\bar{g} \) is positive definite and Hermitian, so by the spectral theorem it can be diagonalized by a unitary matrix. This means we can write \( g \cdot {}^{t}\bar{g} = {\kappa a}{\kappa }^{-1} \), where \( \kappa \) is unitary and \( a \) is a diagonal matrix with positive real entri... | Yes |
Theorem 13.5. We have\n\n\[ \n{\pi }_{1}\left( {\mathrm{{GL}}\left( {n,\mathbb{C}}\right) }\right) \cong {\pi }_{1}\left( {\mathrm{U}\left( n\right) }\right) ,\;{\pi }_{1}\left( {\mathrm{{SL}}\left( {n,\mathbb{C}}\right) }\right) \cong {\pi }_{1}\left( {\mathrm{{SU}}\left( n\right) }\right) ,\n\]\n\nand\n\n\[ \n{\pi }_... | Proof. First, let \( G = \mathrm{{GL}}\left( {n,\mathbb{C}}\right), K = \mathrm{U}\left( n\right) \), and \( P \) be the space of positive definite Hermitian matrices. By the Cartan decomposition, multiplication \( K \times \) \( P \rightarrow G \) is a bijection, and in fact, a homeomorphism, so it will follow that \(... | Yes |
Proposition 13.8. Let \( G \) be a Lie group and \( H \) a closed subgroup. If the homogeneous space \( G/H \) is homeomorphic to a sphere \( {S}^{r} \) where \( r \geq 3 \), then \( {\pi }_{1}\left( G\right) \cong {\pi }_{1}\left( H\right) \) | Proof. The map \( G \rightarrow G/H \) is a fibration (Spanier [149], Example 4 on p. 91 and Corollary 14 on p. 96). It follows that there is an exact sequence\n\n\[ \n{\pi }_{2}\left( {G/H}\right) \rightarrow {\pi }_{1}\left( H\right) \rightarrow {\pi }_{1}\left( G\right) \rightarrow {\pi }_{1}\left( {G/H}\right) \n\]... | Yes |
Theorem 13.6. The groups \( \mathrm{{SU}}\left( n\right) \) are simply connected for all \( n \) . On the other hand,\n\n\[{\pi }_{1}\left( {\mathrm{{SO}}\left( n\right) }\right) \cong \left\{ \begin{matrix} \mathbb{Z} & \text{ if }n = 2, \\ \mathbb{Z}/2\mathbb{Z} & \text{ if }n > 2. \end{matrix}\right.\] | Proof. Since \( \mathrm{{SO}}\left( 2\right) \) is a circle, its fundamental group is \( \mathbb{Z} \) . By Proposition 13.6 \( {\pi }_{1}\left( {\mathrm{{SO}}\left( 3\right) }\right) \cong \mathbb{Z}/2\mathbb{Z} \) and \( {\pi }_{1}\left( {\mathrm{{SU}}\left( 2\right) }\right) \) is trivial. The group \( \mathrm{{SO}}... | Yes |
Lemma 14.1. If \( {X}_{1},\ldots ,{X}_{d} \) are vector fields on \( M \) such that \( \left\lbrack {{X}_{i},{X}_{j}}\right\rbrack \) lies in the \( {C}^{\infty }\left( M\right) \) span of \( {X}_{1},\ldots ,{X}_{d} \), and if for each \( x \in M \) we define \( {D}_{x} \) to be the span of \( {X}_{1x},\ldots ,{X}_{dx}... | Proof. Any vector field subordinate to \( D \) has the form (locally near \( x \) ) \( \mathop{\sum }\limits_{i}{f}_{i}{X}_{i} \), where \( {f}_{i} \) are smooth functions. To check that the commutator of two such vector fields is also of the same form amounts to using the formula\n\n\[ \left\lbrack {{fX},{gY}}\right\r... | Yes |
Proposition 14.1. Let \( G \) be a Lie group with Lie algebra \( \mathfrak{g} \), and let \( \mathfrak{k} \) be a Lie subalgebra of \( \mathfrak{g} \). Then there exists a local subgroup \( K \) of \( G \) with a tangent space at the identity that is \( \mathfrak{k} \). The exponential map sends a neighborhood of the i... | Proof. The Lie algebra \( \mathfrak{g} \) of \( G \) has two incarnations: as the tangent space to the identity of \( G \) and as the set of left-invariant vector fields. For definiteness, we identify \( \mathfrak{g} = {T}_{e}\left( G\right) \) and recall how the left-invariant vector field arises.\n\nIf \( g \in G \),... | Yes |
Proposition 14.2. Let \( G \) and \( H \) be Lie groups with Lie algebras \( \mathfrak{g} \) and \( \mathfrak{h} \) , respectively, and let \( \pi : \mathfrak{g} \rightarrow \mathfrak{h} \) be a Lie algebra homomorphism. Then there exists a neighborhood \( U \) of \( G \) and a local homomorphism \( \pi : U \rightarrow... | Proof. The tangent space to \( G \times H \) at the identity is \( \mathfrak{g} \oplus \mathfrak{h} \) . Let\n\n\[ \mathfrak{k} = \{ \left( {X,\pi \left( X\right) }\right) \mid X \in \mathfrak{g}\} \subset \mathfrak{g} \oplus \mathfrak{h}. \]\n\nIt is a Lie subalgebra, corresponding by Proposition 14.1 to a local subgr... | Yes |
Theorem 14.3. Let \( G \) and \( K \) be Lie groups with Lie algebras \( \mathfrak{g} \) and \( \mathfrak{k} \) . Assume \( K \) is compact and simply connected. Suppose that \( \mathfrak{g} \) and \( \mathfrak{k} \) have isomorphic complexifications. Then every finite-dimensional irreducible complex representation of ... | Proof. Let \( \left( {\pi, V}\right) \) be a finite-dimensional representation of \( G \), and let \( W \) be a proper nonzero invariant subspace. We will show that there is another invariant subspace \( {W}^{\prime } \) such that \( V = W \oplus {W}^{\prime } \) . By induction on \( \dim \left( V\right) \) , it will f... | Yes |
Theorem 14.4. Let \( \left( {\pi, V}\right) \) be a finite-dimensional irreducible complex representation of \( \mathfrak{g} = \mathfrak{{sl}}\left( {n,\mathbb{R}}\right) ,\mathfrak{{su}}\left( n\right) \), or \( \mathfrak{{sl}}\left( {n,\mathbb{C}}\right) \) . If \( \mathfrak{g} \) is \( \mathfrak{{sl}}\left( {n,\math... | Proof. We will prove this for \( \mathfrak{{sl}}\left( {n,\mathbb{R}}\right) \) and \( \mathfrak{{su}}\left( n\right) \) . By Theorem 13.6, \( K \) is simply-connected and the hypotheses of Theorem 14.3 are satisfied. For \( \mathfrak{{sl}}\left( {n,\mathbb{R}}\right) \), we can take \( G = \mathrm{{SL}}\left( {n,\math... | No |
Theorem 14.5. Let \( \left( {\pi, V}\right) \) be a finite-dimensional irreducible complex representation of \( \mathrm{{SL}}\left( {n,\mathbb{R}}\right) \) . Then \( \pi \) is completely reducible. | Proof. We take \( G = \mathrm{{SL}}\left( {n,\mathbb{R}}\right), K = \mathrm{{SU}}\left( n\right) \) . | No |
Proposition 15.1. The Lie algebra of an Abelian Lie group is Abelian. | Proof. The action of \( G \) on itself by conjugation is trivial, so the induced action Ad of \( G \) on its Lie algebra is trivial. By Theorem 8.2, it follows that \( \operatorname{ad} : \operatorname{Lie}\left( G\right) \rightarrow \operatorname{End}\left( {\operatorname{Lie}\left( G\right) }\right) \) is the zero ma... | Yes |
Proposition 15.2. If \( G \) is a Lie group, and \( X \) and \( Y \) are commuting elements of \( \operatorname{Lie}\left( G\right) \), then \( {\mathrm{e}}^{X + Y} = {\mathrm{e}}^{X}{\mathrm{e}}^{Y} \) . In particular, \( {\mathrm{e}}^{X}{\mathrm{e}}^{Y} = {\mathrm{e}}^{Y}{\mathrm{e}}^{X} \) . | Proof. First note that, since the differential of Ad is ad (Theorem 8.2), \( \operatorname{Ad}\left( {\mathrm{e}}^{tX}\right) Y = Y \) for all \( t \) . Recalling that \( \operatorname{Ad}\left( {\mathrm{e}}^{tX}\right) \) is the endomorphism of \( \operatorname{Lie}\left( G\right) \) induced by conjugation, this means... | Yes |
Proposition 15.3. Let \( T \) be a torus, and let \( \mathfrak{t} \) be its Lie algebra. Then \( \exp \) : \( \mathfrak{t} \rightarrow T \) is a homomorphism, and its kernel is a lattice. We have \( T \cong {\left( \mathbb{R}/\mathbb{Z}\right) }^{r} \cong \) \( {\mathbb{T}}^{r} \), where \( r \) is the dimension of \( ... | Proof. Let \( \mathfrak{t} \) be the Lie algebra of \( T \) . Since \( T \) is Abelian, so is \( \mathfrak{t} \), and by Proposition 15.2, exp is a homomorphism from the additive group \( \mathfrak{t} \) to \( T \) . The kernel \( \Lambda \subset \mathfrak{t} \) is discrete since exp is a local homeomorphism, and \( \L... | Yes |
Proposition 15.4. Every irreducible complex representation of \( {\left( \mathbb{R}/\mathbb{Z}\right) }^{r} \) coincides with (15.1) for suitable \( {k}_{i} \in \mathbb{Z} \) . | Proof. By classical Fourier analysis, these characters span \( {L}^{2}\left( {\left( \mathbb{R}/\mathbb{Z}\right) }^{r}\right) \) . Thus, the character \( \chi \) of any complex representation \( \pi \) is not orthogonal to (15.1) for some \( \left( {{k}_{1},\ldots ,{k}_{r}}\right) \in {\mathbb{Z}}^{r} \) . By Schur or... | Yes |
Proposition 15.5. Let \( T = {\left( \mathbb{Z}/\mathbb{R}\right) }^{r} \) and let \( \left( {\pi, V}\right) \) be an irreducible real representation. Then either \( \pi \) is trivial or \( \pi \) is two-dimensional and is one of the irreducible representations (15.2) with \( {k}_{i} \in \mathbb{Z} \) not all zero. In ... | Proof. It is straightforward to see that the real representation (15.2) is irreducible. The completeness of this set of irreducible real representations follows from the corresponding classification of the irreducible complex characters (Proposition 15.4). It is also easy to see that the complexified representation is ... | No |
Proposition 15.6. Let \( T \) be a compact torus. Then any linear character \( \chi \) of \( T \) extends uniquely to a rational character of \( {T}_{\mathbb{C}} \) . | Proof. Without loss of generality, we may assume that \( T = {\left( \mathbb{R}/\mathbb{Z}\right) }^{r} \) and that \( {T}_{\mathbb{C}} = {\left( {\mathbb{C}}^{ \times }\right) }^{r} \), where the embedding \( T \rightarrow {T}_{\mathbb{C}} \) is the map \( \left( {{x}_{1},\ldots ,{x}_{r}}\right) \rightarrow \) \( \lef... | Yes |
Theorem 15.1 (Kronecker). Let \( \left( {{t}_{1},\ldots ,{t}_{r}}\right) \in {\mathbb{R}}^{r} \), and let \( t \) be the image of this point in \( T = {\left( \mathbb{R}/\mathbb{Z}\right) }^{r} \) . Then \( t \) is a generator of \( T \) if and only if \( 1,{t}_{1},\ldots ,{t}_{r} \) are linearly independent over \( \m... | Proof. Let \( H \) be the closure of the group \( \langle t\rangle \) generated by \( t \) in \( T = {\left( \mathbb{R}/\mathbb{Z}\right) }^{r} \) . Then \( T/H \) is a compact Abelian group, and if it is not reduced to the identity it has a character \( \chi \) . We may regard this as a character of \( T \) that is tr... | Yes |
Corollary 15.1. Each compact torus \( T \) has a generator. Indeed, generators are dense in \( T \) . | Proof. We may assume that \( T = {\left( \mathbb{R}/\mathbb{Z}\right) }^{r} \) . By Kronecker’s Theorem 15.1, what we must show is that \( r \) -tuples \( \left( {{t}_{1},\ldots ,{t}_{r}}\right) \) such that \( 1,{t}_{1},\ldots ,{t}_{r} \) are linearly independent over \( \mathbb{Q} \) are dense in \( {\mathbb{R}}^{r} ... | Yes |
Proposition 15.7. Let \( T = {\left( \mathbb{R}/\mathbb{Z}\right) }^{r} \). (i) Each automorphism of \( T \) is of the form \( t \rightarrow {Mt}\left( {\;\operatorname{mod}\;{\mathbb{Z}}^{r}}\right) \), where \( M \in \) \( \mathrm{{GL}}\left( {r,\mathbb{Z}}\right) \). Thus, \( \operatorname{Aut}\left( T\right) \cong ... | Proof. If \( \phi : T \rightarrow T \) is an automorphism, then \( \phi \) induces an invertible linear transformation \( M \) of the Lie algebra \( \mathfrak{t} \) of \( T \) that commutes with the exponential map. Because \( T \) is Abelian, the exponential map \( \exp : \mathfrak{t} \rightarrow T \) is a group homom... | Yes |
Theorem 15.2. Let \( G \) be a Lie group and \( H \) a closed Abelian subgroup. Then \( H \) is a Lie subgroup of \( G \) . If \( G \) is compact, then the connected component of the identity in \( H \) is a torus. | Proof. Let \( \mathfrak{g} = \operatorname{Lie}\left( G\right) \) . The exponential map \( \mathfrak{g} \rightarrow G \) is a local homeomorphism near the origin. Let \( U \) be a neighborhood of \( 0 \in \mathfrak{g} \) such that exp has a smooth inverse \( \log : \exp \left( U\right) \rightarrow U \) . Let\n\n\[ \mat... | No |
Lemma 15.1. If \( X \in \mathfrak{h} \) and \( Y \in U \), and if \( {\mathrm{e}}^{Y} \in H \) then \( \left\lbrack {X, Y}\right\rbrack = 0 \) . | To prove the lemma, note that for any \( t > 0 \) both \( {\mathrm{e}}^{tX} \) and \( {\mathrm{e}}^{Y} \in H \) commute, so \( {\mathrm{e}}^{Y} = {\mathrm{e}}^{tX}{\mathrm{e}}^{Y}{\mathrm{e}}^{-{tX}} = \exp \left( {\operatorname{Ad}\left( {tX}\right) Y}\right) \) . If \( t \) is small enough, both \( Y \) and \( \opera... | Yes |
Proposition 15.8. Let \( G \) be a compact Lie group and \( T \) a maximal torus. Then \( N\left( T\right) \) is a closed subgroup of \( G \). The connected component \( N{\left( T\right) }^{ \circ } \) of the identity in \( N\left( T\right) \) is \( T \) itself. The quotient \( N\left( T\right) /T \) is a finite group... | Proof. We have a homomorphism \( N\left( T\right) \rightarrow \operatorname{Aut}\left( T\right) \) in which the action is by conjugation. By Proposition 15.7, \( \operatorname{Aut}\left( T\right) \cong \mathrm{{GL}}\left( {r,\mathbb{Z}}\right) \) is discrete, so any connected group of automorphisms must act trivially. ... | Yes |
Proposition 15.9. Let \( T \) be a maximal torus in the compact connected Lie group \( G \), and let \( \mathfrak{t},\mathfrak{g} \) be the Lie algebras of \( T \) and \( G \), respectively.\n\n(i) Any vector in \( \mathfrak{g} \) fixed by \( \operatorname{Ad}\left( T\right) \) is in \( \mathfrak{t} \).\n\n(ii) We have... | Proof. For (i), if \( X \in \mathfrak{g} \) is fixed by \( \operatorname{Ad}\left( T\right) \), then by Proposition 15.2, \( \exp \left( {tX}\right) \) is a one-parameter subgroup that is not contained in \( T \) but that commutes with \( T \), and unless \( X \in \mathfrak{t} \), the closure of the group it generates ... | Yes |
Corollary 15.2. If \( G \) is a compact connected Lie group and \( T \) a maximal torus, then \( \dim \left( G\right) - \dim \left( T\right) \) is even. | Proof. This follows since \( \dim \left( {G/T}\right) = \dim \left( \mathfrak{p}\right) \), and \( \mathfrak{p} \) decomposes as a direct sum of two-dimensional irreducible representations. | Yes |
Proposition 15.10. Let \( G \) be a connected Lie group and \( H \) a connected closed Lie subgroup. Then the quotient space \( G/H \) is a connected orientable manifold. | Proof. To make \( G/H \) a manifold, choose a subspace \( \mathfrak{p} \) of \( \mathfrak{g} = \operatorname{Lie}\left( G\right) \) complementary to \( \mathfrak{h} = \operatorname{Lie}\left( H\right) \) . Then \( X \rightarrow \exp \left( X\right) {gH} \) is a local homeomorphism of a neighborhood of the identity in \... | Yes |
Proposition 16.2. Let \( x \) be a point on the Riemannian manifold \( M \), and let \( X \in {T}_{x}\left( M\right) \) . Then, for sufficiently small \( \epsilon \), there is a unique geodesic \( p : \left( {-\epsilon ,\epsilon }\right) \rightarrow M \) such that \( p\left( 0\right) = x \) and \( {p}_{ * }\left( {\mat... | Proof. Let \( {x}_{1},\ldots ,{x}_{n} \) be coordinate functions. Let \( {y}_{1},\ldots ,{y}_{n} \) be a set of new variables, and rewrite (16.4) as a first-order system\n\n\[ \frac{\mathrm{d}{x}_{i}}{\mathrm{\;d}t} = {y}_{i} \]\n\n\[ \frac{\mathrm{d}{y}_{k}}{\mathrm{\;d}t} = - \{ {ij}, k\} {y}_{i}{y}_{j} \]\n\nThe con... | Yes |
Proposition 16.3. In geodesic coordinates, \( {g}_{1i} = 0 \) for \( 2 \leq i \leq n \) . Also \( {g}_{11} = 1 \) . | Proof. Having chosen coordinates so that the path \( t \mapsto \left( {t,{x}_{2,}\ldots ,{x}_{n}}\right) \) is a geodesic, we see that if all \( \mathrm{d}{x}_{i}/\mathrm{d}t = 0 \) in (16.4), for \( i \neq 1 \), then \( {\mathrm{d}}^{2}{x}_{k}/\mathrm{d}{t}^{2} = 0 \) for all \( k \) . This means that \( \{ {11}, k\} ... | Yes |
Theorem 16.1. Let \( M \) be a compact connected Riemannian manifold, and let \( x \) and \( y \) be points of \( M \) . Then there is a geodesic \( p : \left\lbrack {0,1}\right\rbrack \rightarrow M \) with \( p\left( 0\right) = x \) and \( p\left( 1\right) = y \) . | Proof. Let \( \left\{ {p}_{i}\right\} \) be a sequence of well-paced paths from \( x \) to \( y \) such that \( \left| {p}_{i}\right| \rightarrow d\left( {x, y}\right) \) . Because they are well-paced, if \( 0 \leq a < b \leq 1 \) we have \( \mathrm{d}\left( {{p}_{i}\left( a\right) ,{p}_{i}\left( b\right) }\right) = \l... | Yes |
Theorem 16.3. Let \( G \) be a compact Lie group and \( \mathfrak{g} \) its Lie algebra. Then the exponential map \( \mathfrak{g} \rightarrow G \) is surjective. | Proof. Put a Riemannian structure on \( G \) as in Theorem 16.2. By Theorem 16.1, given \( g \in G \), there exists a geodesic path from the identity to \( g \) . By Theorem 16.2, this path is of the form \( t \mapsto {\mathrm{e}}^{tX} \) for some \( X \in \mathfrak{g} \), so \( g = {\mathrm{e}}^{X} \) . | Yes |
Theorem 16.4. Let \( G \) be a compact connected Lie group, and let \( T \) be a maximal torus. Let \( g \in G \) . Then there exists \( k \in G \) such that \( g \in {kT}{k}^{-1} \) . | Proof. Let \( \mathfrak{g} \) and \( \mathfrak{t} \) be the Lie algebras of \( G \) and \( T \), respectively. Let \( {t}_{0} \) be a generator of \( T \) . Using Theorem 16.3, find \( X \in \mathfrak{g} \) and \( {H}_{0} \in \mathfrak{t} \) such that \( {\mathrm{e}}^{X} = g \) and \( {\mathrm{e}}^{{H}_{0}} = {t}_{0} \... | Yes |
Theorem 16.5 (E. Cartan). Let \( G \) be a compact connected Lie group, and let \( T \) be a maximal torus. Then every maximal torus is conjugate to \( T \), and every element of \( G \) is contained in a conjugate of \( T \) . | Proof. The second statement is contained in Theorem 16.4. As for the first statement, let \( {T}^{\prime } \) be another maximal torus, and let \( t \) be a generator. Then \( {t}^{\prime } \) is contained in \( {kT}{k}^{-1} \) for some \( k \), so \( {T}^{\prime } \subseteq {kT}{k}^{-1} \) . Since both are maximal tor... | No |
Proposition 16.5. Let \( G \) be a compact connected Lie group, \( S \subset G \) a torus (not necessarily maximal), and \( g \in {C}_{G}\left( S\right) \) an element of its centralizer. Let \( H \) be the closure of the group generated by \( S \) and \( g \) . Then \( H \) has a topological generator. That is, there e... | Proof. Since \( H \) is closed and Abelian, its connected component \( {H}^{ \circ } \) of the identity is a torus by Proposition 15.2. Let \( {h}_{0} \) be a topological generator.\n\nThe group \( H/{H}^{ \circ } \) is compact and discrete and hence finite. Since \( S \subseteq {H}^{ \circ } \) , and since \( S \) and... | Yes |
Proposition 16.6. If \( G \) is a Lie group and \( u \in G \), then the centralizer \( {C}_{G}\left( u\right) \) is a closed Lie subgroup, and its Lie algebra is \( \{ X \in \operatorname{Lie}\left( G\right) \mid \operatorname{Ad}\left( u\right) X = X\} \) . | Proof. To show that \( H = {C}_{G}\left( u\right) \) is a closed submanifold of \( G \), it is sufficient to show that its intersection with a small neighborhood of the identity is a closed submanifold since translation by an element \( h \) of \( H \) will give a diffeomorphism of that neighborhood onto a neighborhood... | Yes |
Theorem 16.6. Let \( G \) be a compact connected Lie group and \( S \subset G \) a torus (not necessarily maximal). Then the centralizer \( {C}_{G}\left( S\right) \) is a closed connected Lie subgroup of \( G \) . | Proof. We first prove that \( {C}_{G}\left( S\right) \) is connected. Let \( g \in {C}_{G}\left( S\right) \) . By Proposition 16.5, there exists an element \( h \) of \( {C}_{G}\left( S\right) \) that generates the closure \( H \) of the group generated by \( S \) and \( g \) . Let \( T \) be a maximal torus in \( G \)... | Yes |
Proposition 17.1. Let \( G \) be a locally compact group, and let \( H \) be a compact subgroup. Let \( \mathrm{d}{\mu }_{G} \) and \( \mathrm{d}{\mu }_{H} \) be left Haar measures on \( G \) and \( H \), respectively. Then there exists a regular Borel measure \( \mathrm{d}{\mu }_{G/H} \) on \( G/H \) which is invarian... | Proof. We may choose the normalization of \( \mathrm{d}{\mu }_{H} \) so that \( H \) has total volume 1 . We define a map \( \lambda : {C}_{c}\left( G\right) \rightarrow {C}_{c}\left( {G/H}\right) \) by\n\n\[ \n\left( {\lambda f}\right) \left( g\right) = {\int }_{H}f\left( {gh}\right) \mathrm{d}{\mu }_{H}\left( h\right... | Yes |
(i) Two elements of \( T \) are conjugate in \( G \) if and only if they are conjugate in \( N\left( T\right) \) . | Proof. Suppose that \( t, u \in T \) are conjugate in \( G \), say \( {gt}{g}^{-1} = u \) . Let \( H = \) \( {C}_{G}{\left( u\right) }^{ \circ } \) be the connected component of the identity in the centralizer of \( u \) . It is a closed Lie subgroup of \( G \) by Proposition 16.6. Both \( T \) and \( {gT}{g}^{-1} \) a... | Yes |
Proposition 17.2. The centralizer \( C\left( T\right) = T \) . | Proof. Since \( C\left( T\right) \subset N\left( T\right), T \) is of finite index in \( C\left( T\right) \) by Proposition 15.8. Thus, if \( x \in C\left( T\right) \), we have \( {x}^{n} \in T \) for some \( n \) . Let \( {t}_{0} \) be a generator of \( T \) . Since the \( n \) th power map \( T \rightarrow T \) is su... | Yes |
Proposition 17.3. There exists a dense open set \( \Omega \) of \( T \) such that the \( \left| W\right| \) elements \( {wt}{w}^{-1}\left( {w \in W}\right) \) are all distinct for \( t \in \Omega \) . | Proof. If \( w \in W \), let \( {\Omega }_{w} = \left\{ {t \in T \mid {wt}{w}^{-1} \neq t}\right\} \) . It is an open subset of \( T \) since its complement is evidently closed. If \( w \neq 1 \) and \( t \) is a generator of \( T \), then \( t \in {\Omega }_{w} \) because otherwise if \( n \in N\left( T\right) \) repr... | Yes |
Proposition 17.4. Let \( G = \mathrm{U}\left( n\right) \), and let \( T \) be the diagonal torus. Writing\n\n\[ t = \left( \begin{array}{lll} {t}_{1} & & \\ & \ddots & \\ & & {t}_{n} \end{array}\right) \in T \]\n\nand letting \( {\int }_{T}\mathrm{\;d}t \) be the Haar measure on \( T \) normalized so that its volume is... | Proof. This will follow from Theorem 17.2 once we check that\n\n\[ \det \left( {\left\lbrack {\operatorname{Ad}\left( {t}^{-1}\right) - {I}_{\mathfrak{p}}}\right\rbrack \mid \mathfrak{p}}\right) = \mathop{\prod }\limits_{{i < j}}{\left| {t}_{i} - {t}_{j}\right| }^{2}. \]\n\nTo compute this determinant, we may as well c... | Yes |
If \( k \in \mathbb{Z} \) then \( \mathrm{d}{\lambda }_{k}\left( {H}_{\alpha }\right) = k \) . | Although \( {H}_{\alpha } \) is not in \( \mathfrak{t}, i{H}_{\alpha } \) is and we find that\n\n\[ \mathrm{d}{\lambda }_{k}\left( {i{H}_{\alpha }}\right) = {\left. \frac{\mathrm{d}}{\mathrm{d}t}{\lambda }_{k}\left( \begin{array}{ll} {\mathrm{e}}^{it} & \\ & {\mathrm{e}}^{-{it}} \end{array}\right) \right| }_{t = 0} = {... | Yes |
Proposition 18.2. A maximal Abelian subalgebra \( \mathfrak{h} \) of \( \mathfrak{g} \) is the Lie algebra of a conjugate of \( T \) . Its dimension is the rank \( r \) of \( G \) . | Proof. By Proposition 15.2, \( \exp \left( \mathfrak{h}\right) \) is a commutative group that is connected since it is the continuous image of a connected space. By Theorem 15.2 its closure \( H \) is a Lie subgroup of \( G \), closed, connected and Abelian and therefore a torus. It is therefore contained in a maximal ... | Yes |
Lemma 18.1. Suppose that \( G \) is a compact Lie group with Lie algebra \( \mathfrak{g} \) , \( \pi : G \rightarrow \mathrm{{GL}}\left( V\right) \) a representation, and \( \mathrm{d}\pi : \mathfrak{g} \rightarrow \operatorname{End}\left( V\right) \) the differential. If \( v \in V \) and \( X \in \mathfrak{g} \) such... | Proof. We may put a \( G \) -invariant positive definite inner product \( \langle \) , \( \rangle {onV} \) . The inner product is then \( \mathfrak{g} \) -invariant, which means that \( \langle \mathrm{d}\pi \left( X\right) v, w\rangle = \) \( - \langle v,\mathrm{\;d}\pi \left( X\right) w\rangle \) . Thus, \( \mathrm{d... | Yes |
Proposition 18.3. Let \( \left( {\pi, V}\right) \) be any irreducible representation of \( G \), and let \( \alpha \) be a root.\n\n(i) If \( \mathrm{d}\pi : \mathfrak{g} \rightarrow \mathfrak{{gl}}\left( V\right) \) is the differential of \( \pi \), then\n\n\[ \mathrm{d}\pi \left( H\right) v = \mathrm{d}\lambda \left(... | Proof. For (i), if \( H \in \mathfrak{t} \) and \( t \in \mathbb{R} \), then for \( v \in V\left( \lambda \right) \) we have\n\n\[ \pi \left( {\mathrm{e}}^{tH}\right) v = \lambda \left( {\mathrm{e}}^{tH}\right) v = {\mathrm{e}}^{t\mathrm{\;d}\lambda \left( H\right) }v. \]\n\nTaking the derivative and setting \( t = 0 \... | Yes |
(i) We have \( c\left( {\mathfrak{X}}_{\alpha }\right) = {\mathfrak{X}}_{-\alpha } \) . | For (i), apply \( c \) to (18.9) using the complex antilinearity of \( c \), and the fact that \( \mathrm{d}\alpha \left( H\right) \) is purely imaginary to obtain, for \( H \in \mathfrak{t} \n\n\[ \left\lbrack {H, c\left( {X}_{\alpha }\right) }\right\rbrack = \left\lbrack {c\left( H\right), c\left( {X}_{\alpha }\right... | Yes |
Proposition 18.5. If \( \dim \left( T\right) = 1 \), then either \( G = T \) or \( \dim \left( G\right) = 3 \) . If \( \alpha \) is any root, then \( {\mathfrak{X}}_{\alpha } \) is one-dimensional, and \( \alpha , - \alpha \) are the only roots. | Proof. Since \( {\mathfrak{t}}_{\mathbb{C}} \) is one-dimensional, let \( H \) be a basis vector. Assuming \( G \neq T \) , \( \Phi \) is nonempty. The spaces \( {\mathfrak{X}}_{\alpha } \) are just the eigenspaces of \( H \) on \( {\mathfrak{p}}_{\mathbb{C}} \) . Since \( T \) is one-dimensional, so is \( \mathcal{V} ... | Yes |
(i) If \( \alpha \in \Phi \), then \( \dim \left( {\mathfrak{X}}_{\alpha }\right) = 1 \). (ii) If \( \alpha ,\beta \in \Phi \) and \( \alpha = {\lambda \beta },\lambda \in \mathbb{R} \), then \( \lambda = \pm 1 \). | Proof. The group \( H = {C}_{G}\left( {T}_{\alpha }\right) \) is a closed connected Lie subgroup by Theorem 16.6. It has \( {T}_{\alpha } \) as a normal subgroup. The Lie algebra of \( H \) is the centralizer \( \mathfrak{h} \) in \( \mathfrak{g} \) of \( {\mathfrak{t}}_{\alpha } \), so\n\n\[ \n{\mathfrak{h}}_{\mathbb{... | Yes |
Proposition 18.8. Let \( \alpha \in \Phi \) and let \( 0 \neq {X}_{\alpha } \in {\mathfrak{X}}_{\alpha } \) . Let \( {X}_{-\alpha } = - c\left( {X}_{\alpha }\right) \in \) \( {\mathfrak{X}}_{-\alpha } \) . Then \( {X}_{\alpha } \) and \( {X}_{-\alpha } \) generate a complex Lie subalgebra \( {\mathfrak{g}}_{\alpha ,\ma... | Proof. Let \( {H}_{\alpha } = \left\lbrack {{X}_{\alpha },{X}_{-\alpha }}\right\rbrack \) . By Proposition 18.4(iii), \( {H}_{\alpha } \) is a nonzero element of \( i \) thot in \( i{\mathfrak{t}}_{\alpha } \) . By Proposition 18.4(iii) and (18.9), we have \( \left\lbrack {{H}_{\alpha },{X}_{\alpha }}\right\rbrack = {2... | Yes |
Proposition 18.9. If \( H \in {\mathfrak{t}}_{\alpha } = \ker \left( {\mathrm{d}\alpha }\right) \), then \( \left\lbrack {H,{\mathfrak{g}}_{\alpha }}\right\rbrack = 0 \) . | Proof. \( H \) centralizes \( {X}_{\alpha } \) and \( {X}_{-\alpha } \) by (18.9); that is, \( \left\lbrack {H,{X}_{\alpha }}\right\rbrack = \left\lbrack {H,{X}_{-\alpha }}\right\rbrack = 0 \) , and it follows that \( \left\lbrack {H, X}\right\rbrack = 0 \) for all \( X \in {\mathfrak{g}}_{\alpha } \) . | Yes |
Theorem 18.1. Let \( \alpha \in \Phi \) . There exists a homomorphism \( {i}_{\alpha } : \mathrm{{SU}}\left( 2\right) \rightarrow \) \( C{\left( {T}_{\alpha }\right) }^{ \circ } \subset G \) such that the image of the differential \( d{i}_{\alpha } : \mathfrak{{su}}\left( 2\right) \rightarrow \mathfrak{g} \) is the Lie... | Proof. Since \( \mathrm{{SU}}\left( 2\right) \) is simply connected, it follows from Theorem 14.2 that the Lie algebra homomorphism \( \mathfrak{{su}}\left( 2\right) \rightarrow \mathfrak{g} \) of Proposition 18.8 is the differential of a homomorphism \( {i}_{\alpha } : \mathrm{{SU}}\left( 2\right) \rightarrow G \) . B... | Yes |
Proposition 18.10. Let \( \\left( {\\pi, V}\\right) \) be a finite-dimensional representation of \( G \) , and let \( \\lambda \\in {X}^{ * }\\left( T\\right) \) such that \( V\\left( \\lambda \\right) \\neq 0 \) . Then \( 2\\langle \\lambda ,\\alpha \\rangle /\\langle \\alpha ,\\alpha \\rangle \\in \\mathbb{Z} \) for ... | Proof. Let\n\n\\[ \nW = {\\bigoplus }_{k \\in \\mathbb{Z}}V\\left( {\\lambda + {k\\alpha }}\\right) \n\\]\n\nBy Proposition 18.4, this subspace is stable under \( \\mathrm{d}\\pi \\left( {X}_{\\alpha }\\right) \) and \( \\mathrm{d}\\pi \\left( {X}_{-\\alpha }\\right) \) . It is therefore invariant under the Lie algebra... | Yes |
Theorem 18.2. If \( \Phi \) is the set of roots associated with a compact Lie group and its maximal torus \( T \), then \( \Phi \) is a reduced root system. | Proof. Clearly, \( \Phi \) is a set of nonzero vectors in a Euclidean space \( \mathcal{V} \) . The fact that \( \Phi \) is invariant under \( {s}_{\alpha },\alpha \in \Phi \) follows from the construction of \( {w}_{\alpha } \in N\left( T\right) \), the conjugation of which induces \( {s}_{\alpha } \) in Theorem 18.1.... | Yes |
Proposition 18.11. Let \( \lambda \in {X}^{ * }\left( T\right) \) . Then there exists a finite-dimensional complex representation \( \left( {\pi, V}\right) \) of \( G \) such that \( V\left( \lambda \right) \neq 0 \) . | Proof. Consider the subspace \( L\left( \lambda \right) \) of \( {L}^{2}\left( G\right) \) of functions \( f \) satisfying\n\n\[ f\left( {tg}\right) = \lambda \left( t\right) f\left( g\right) \]\n\nfor \( t \in T \) . Let \( G \) act on \( L\left( \lambda \right) \) by right translation: \( \rho : G \rightarrow \operat... | Yes |
Proposition 18.12. If \( {H}_{\alpha } \) is as in Proposition 18.8 and \( {w}_{\alpha } \in N\left( T\right) \) is as in Theorem 18.1, then \( \operatorname{ad}\left( {w}_{\alpha }\right) {H}_{\alpha } = - {H}_{\alpha } \) . | Proof. Since \( {w}_{\alpha } \) lies in \( {i}_{\alpha }\left( {\mathrm{{SU}}\left( 2\right) }\right) \), and since by Proposition 18.8 the element \( - i{H}_{\alpha } \) lies in the image of the Lie algebra of \( \mathrm{{SU}}\left( 2\right) \) under the differential of \( {i}_{\alpha } \), we may work in \( \mathrm{... | No |
Proposition 18.13. Let \( \lambda \in \mathcal{V} \) and \( \alpha \in \Phi \) . Then \( \mathrm{d}\lambda \left( {H}_{\alpha }\right) = {\alpha }^{ \vee }\left( \lambda \right) \) . | Proof. First let us show that \( \lambda \) and \( \alpha \) are orthogonal if and only if \( \mathrm{d}\lambda \left( {H}_{\alpha }\right) = 0 \) with \( {H}_{\alpha } \) as in Proposition 18.4. It is sufficient to show that the orthogonal complement of \( \alpha \) is contained in the kernel of this functional since ... | Yes |
(i) \( \mathop{\bigcap }\limits_{{\alpha \in \Phi }}{T}_{\alpha } \) is the center \( Z\left( G\right) \) . | Proof. For (i), any element of \( G \) is conjugate to an element of \( T \) . If it is in \( Z\left( G\right) \), conjugation does not move it, so \( Z\left( G\right) \subset T \) . Now \( G \) is generated by \( T \) together with the subgroups \( {i}_{\alpha }\left( {\mathrm{{SU}}\left( 2\right) }\right) \) as \( \a... | Yes |
Theorem 18.3. The Weyl group \( W = N\left( T\right) /T \) is generated by the \( {w}_{\alpha } \) with \( \alpha \in \Phi \) . | Proof. Arguing by contradiction, choose \( w \in N\left( T\right) /T \) that is not in the subgroup generated by the \( {w}_{\alpha } \) . If \( \alpha \in \Phi \) let \( {\mathfrak{t}}_{\alpha } \) be the Lie algebra of the group \( {T}_{\alpha } \) which is the kernel of \( \alpha \) . They are hyperplanes in \( \mat... | Yes |
Proposition 18.15. Suppose that \( \alpha \in \Phi \) . Let \( \beta \neq \pm \alpha \) be another root. Let\n\n\[ W = {\bigoplus }_{\begin{matrix} {k \in \mathbb{Z}} \\ {\beta + {k\alpha } \in \Phi } \end{matrix}}{\mathfrak{X}}_{\beta + {k\alpha }} \]\n\nThen \( W \) is an irreducible module for \( {i}_{\alpha }\left(... | Proof. Denote \( {\mathfrak{g}}_{\alpha } = {i}_{\alpha }\left( {\mathfrak{{sl}}\left( {2,\mathbb{C}}\right) }\right) \) . First we note that \( W \) is an \( \mathfrak{{sl}}\left( {2,\mathbb{C}}\right) \) -module, since by Proposition 18.4 it is closed under the Lie bracket with \( {X}_{\alpha } \) and \( {X}_{-\alpha... | Yes |
Corollary 18.1. Suppose that \( \alpha ,\beta \) and \( \alpha + \beta \in \Phi \) . Let \( {X}_{\alpha } \) and \( {X}_{\beta } \) be nonzero elements of \( {\mathfrak{X}}_{\alpha } \) and \( {\mathfrak{X}}_{\beta } \) . Then \( \left\lbrack {{X}_{\alpha },{X}_{\beta }}\right\rbrack \) is a nonzero element of \( {\mat... | Proof. We may identify the decomposition (18.14) with the irreducible module described in (12.1). Now \( {X}_{\alpha } \) is \( {i}_{\alpha }\left( R\right) \) in the notation of that Proposition. Since \( \alpha + \beta \) is a root, \( {X}_{\beta } \) is \( {v}_{k - {2l}} \) with \( l > 0 \), and the nonvanishing of ... | Yes |
Proposition 20.2. Let \( s = {s}_{\alpha }\left( {\alpha \in \sum }\right) \) be a simple reflection, and let \( w \in W \) . We have \[ {l}^{\prime }\left( {sw}\right) = \left\{ \begin{array}{l} {l}^{\prime }\left( w\right) + 1\text{ if }{w}^{-1}\left( \alpha \right) \in {\Phi }^{ + }, \\ {l}^{\prime }\left( w\right) ... | Proof. Since \( s\left( {\Phi }^{ - }\right) \) is obtained from \( {\Phi }^{ - } \) by deleting \( - \alpha \) and adding \( \alpha \) , we see that \( {\left( sw\right) }^{-1}{\Phi }^{ - } = {w}^{-1}\left( {s{\Phi }^{ - }}\right) \) is obtained from \( {w}^{-1}{\Phi }^{ - } \) by deleting \( - {w}^{-1}\left( \alpha \... | Yes |
Proposition 20.5. If \( w \in W \), then \( l\left( w\right) = {l}^{\prime }\left( w\right) \) . | Proof. The inequality\n\n\[ \n{l}^{\prime }\left( w\right) \leq l\left( w\right) \n\]\n\nfollows from Proposition 20.2 because we may write \( w = s{w}_{1} \), where \( s \) is a simple reflection and \( l\left( {w}_{1}\right) = l\left( w\right) - 1 \), and by induction on \( l\left( {w}_{1}\right) \) we may assume tha... | Yes |
Proposition 20.6. If \( w\left( {\Phi }^{ + }\right) = {\Phi }^{ + } \), then \( w = 1 \) . | Proof. If \( w\left( {\Phi }^{ + }\right) = {\Phi }^{ + } \), then \( {l}^{\prime }\left( w\right) = 0 \), so \( l\left( w\right) = 0 \), that is, \( w = 1 \) . | Yes |
Proposition 20.7. If \( \alpha \in \Phi \), there exists an element \( w \in W \) such that \( w\left( \alpha \right) \in \sum \) . | Proof. First, assume that \( \alpha \in {\Phi }^{ + } \) . We will argue by induction on \( h\left( \alpha \right) \), which is defined by (20.3). In view of Proposition 20.1(iv), we know that \( h\left( \alpha \right) \) is a positive integer, and if \( \alpha \notin \sum \) (which we may as well assume), then \( h\le... | Yes |
Proposition 20.8. The group \( W \) contains \( {s}_{\alpha } \) for each \( \alpha \in \Phi \) . | Proof. Indeed, \( w\left( \alpha \right) \in \sum \) for some \( w \in W \), so \( {s}_{w\left( \alpha \right) } \in W \) and \( {s}_{\alpha } \) is conjugate in \( W \) to \( {s}_{w\left( \alpha \right) } \) by (20.7). Therefore, \( {s}_{\alpha } \in W \) . | Yes |
Proposition 20.9. The group \( W \) is finite. | Proof. By Proposition 20.6, \( w \in W \) is determined by \( w\left( {\Phi }^{ + }\right) \subset \Phi \) . Since \( \Phi \) is finite, \( W \) is finite. | Yes |
Proposition 20.10. Suppose that \( w \in W \) such that \( l\left( w\right) = k \) . Write \( w = \) \( {s}_{1}\cdots {s}_{k} \), where \( {s}_{i} = {s}_{{\alpha }_{i}},{\alpha }_{1},\ldots ,{\alpha }_{k} \in \sum \) . Then\n\n\[ \left\{ {\alpha \in {\Phi }^{ + } \mid w\left( \alpha \right) \in {\Phi }^{ - }}\right\} =... | Proof. By Proposition 20.5, the cardinality of \( \left\{ {\alpha \in {\Phi }^{ + } \mid w\left( \alpha \right) \in {\Phi }^{ - }}\right\} \) is \( k \), so the result will be established if we show that the described elements are distinct and in the set. Let \( w = {s}_{1}{w}_{1} \), where \( {w}_{1} = {s}_{2}\cdots {... | Yes |
Proposition 20.11. Suppose that \( w \in W \) such that \( l\left( w\right) = k \) . Write \( w = \) \( {s}_{1}\cdots {s}_{k} \), where \( {s}_{i} = {s}_{{\alpha }_{i}},{\alpha }_{1},\ldots ,{\alpha }_{k} \in \sum \) . Assume that \( x \in {\mathcal{C}}_{ + } \) such that \( {wx} \in {\mathcal{C}}_{ + } \) also.\n\n(i)... | Proof. If \( \alpha \in {\Phi }^{ + } \) and \( {w\alpha } \in {\Phi }^{ - } \), then we have \( \langle x,\alpha \rangle = 0 \) . Indeed, \( \langle x,\alpha \rangle \geq 0 \) since \( \alpha \in {\Phi }^{ + } \) and \( x \in {\mathcal{C}}_{ + } \), and \( \langle x,\alpha \rangle = \langle {wx},{w\alpha }\rangle \leq... | Yes |
Theorem 20.1. The positive Weyl chamber \( {\mathcal{C}}_{ + } \) is a fundamental domain for the action of \( W \) on \( \mathcal{V} \) . More precisely, let \( x \in \mathcal{V} \) .\n\n(i) There exists \( w \in W \) such that \( w\left( x\right) \in {\mathcal{C}}_{ + } \) .\n\n(ii) If \( w,{w}^{\prime } \in W \) and... | Proof. Let \( w \in W \) be chosen so that the cardinality of\n\n\[ S = \left\{ {\alpha \in {\Phi }^{ + }\mid \langle w\left( x\right) ,\alpha \rangle < 0}\right\} \]\n\nis as small as possible. We claim that \( S \) is empty. If not, then there exists an element of \( \beta \in \sum \cap S \) . We have \( \langle w\le... | Yes |
Proposition 20.12. The function \( w \mapsto {\left( -1\right) }^{l\left( w\right) } \in \{ \pm 1\} \) is a character of \( W \) . If \( \alpha \in \Phi \), then \( {\left( -1\right) }^{l\left( {s}_{\alpha }\right) } = - 1 \) . | Proof. If \( l\left( w\right) = k \) and \( l\left( {w}^{\prime }\right) = {k}^{\prime } \), write \( w = {s}_{1}\cdots {s}_{k} \) and \( {w}^{\prime } = {s}_{1}^{\prime }\cdots {s}_{{k}^{\prime }}^{\prime } \) as products of simple reflections. It follows from Proposition 20.4 that we may obtain a decomposition of \( ... | Yes |
Proposition 20.13. Let \( \widetilde{w} \) be a linear transformation of \( \mathcal{V} \) that maps \( \Phi \) to itself. Then there exists \( w \in W \) such that \( \widetilde{w}\left( {\mathcal{C}}_{ + }\right) = w{\mathcal{C}}_{ + } \) . The transformation \( {w}^{-1}\widetilde{w} \) of \( \mathcal{V} \) permutes ... | Proof. It is sufficient to show that \( {w}^{-1}\widetilde{w}\left( {\mathcal{C}}_{ + }^{ \circ }\right) = {\mathcal{C}}_{ + }^{ \circ } \) . Let \( x \in {\mathcal{C}}_{ + }^{ \circ } \) . Since the open Weyl chambers are defined to be the connected components of the complement of the set of hyperplanes perpendicular ... | Yes |
Proposition 20.14. If \( \mathcal{C} \) is any Weyl chamber then there is a unique element \( w \) of \( W \) such that \( \mathcal{C} = w{\mathcal{C}}_{ + } \). In particular, let \( {w}_{0} \) be the unique element such that \( - {\mathcal{C}}_{ + } = {w}_{0}\mathcal{C} \). Then \( {w}_{0}{\Phi }^{ + } = {\Phi }^{ - ... | Proof. It is clear that \( W \) permutes the Weyl chambers transitively. The uniqueness of \( w \) of \( W \) such that \( \mathcal{C} = w{\mathcal{C}}_{ + } \) follows from Theorem 20.1.\n\nRegarding \( {w}_{0} \), since\n\n\[{\mathcal{C}}_{ + } = \left\{ {x\mid \{ \alpha, x\} \text{ for }\alpha \in {\Phi }^{ + }}\rig... | Yes |
Proposition 20.15. If \( \alpha \) is a simple root, then\n\n\[ \n{s}_{\alpha }\left( \rho \right) = \rho - \alpha ,\;\alpha \in \sum .\n\] | Proof. This follows since \( {s}_{\alpha } \) changes the sign of \( \alpha \) and permutes the remaining positive roots. | No |
Proposition 20.16. If \( \lambda \in \Lambda \), then \( \lambda - w\left( \lambda \right) \in {\Lambda }_{\text{root }} \) . | Proof. This is true if \( w \) is a simple reflection by (18.1). The general case follows, since if \( w = {s}_{1}\cdots {s}_{r} \), where the \( {s}_{i} \) are simple reflections, we may write \( \lambda - w\left( \lambda \right) = \left( {\lambda - {s}_{r}\left( \lambda \right) }\right) + \left( {{s}_{r}\left( \lambd... | Yes |
Proposition 20.17. In the semisimple case \( \rho = {\varpi }_{1} + \cdots + {\varpi }_{h} \) . In particular, \( \rho \) is a dominant weight. It lies in \( {\mathcal{C}}_{ + }^{ \circ } \) . | Proof. Let \( \alpha = {\alpha }_{i} \in \sum \) . By (20.11), we have \( {\alpha }^{ \vee }\left( \rho ）) \alpha = \rho - {s}_{\alpha }\left( \rho ）) = \alpha \) . Thus, \( {\alpha }_{i}^{ \vee }\left( \rho ）) = 1 \) for each \( {\alpha }_{i} \in \sum \) . It follows that \( \rho \) is the sum of the fundamental domin... | Yes |
Proposition 20.18. Let \( \left( {\Phi ,\mathcal{V}}\right) \) be a root system that is not necessarily reduced. If \( \alpha \) and \( {\lambda \alpha } \in \Phi \) with \( \lambda > 0 \), then \( \lambda = 1,2 \) or \( \frac{1}{2} \) . Partition \( \Phi \) into positive and negative roots, and let \( \sum \) be the s... | Proof. If \( \alpha \) and \( \beta \) are proportional roots, say \( \beta = {\lambda \alpha } \), then \( 2\langle \beta ,\alpha \rangle /\langle \alpha ,\alpha \rangle \in \mathbb{Z} \) implies that \( {2\lambda } \) is an integer and, by symmetry, so is \( 2{\lambda }^{-1} \) . The first assertion is therefore clea... | Yes |
Proposition 22.1. We have \( w\left( \Delta \right) = {\left( -1\right) }^{l\left( w\right) }\Delta \) for all \( w \in W \) . | Proof. It is sufficient to check that \( {s}_{\beta }\left( \Delta \right) = - \Delta \) for every simple root \( \beta \) . We recall that \( {s}_{\beta } \) changes the sign of \( \beta \) and permutes the remaining simple roots. Of the factors in the first expression for \( \Delta \) in (22.2), only two are changed:... | Yes |
Proposition 22.2. If \( \xi \in \mathcal{E} \) satisfies \( w\left( \xi \right) = {\left( -1\right) }^{l\left( w\right) }\xi \) for all \( w \in W \), then \( \xi \) is divisible by \( \Delta \) in \( \mathcal{E} \) . | Proof. In the ring \( \mathcal{E} \), by Proposition 22.1, \( \Delta \) is a product of distinct irreducible elements \( 1 - {\mathrm{e}}^{\alpha } \), where \( \alpha \) runs through \( {\Phi }^{ + } \), times a unit \( {\mathrm{e}}^{-\rho } \) . It is therefore sufficient to show that \( \xi \) is divisible by each \... | Yes |
Proposition 22.3. If \( \lambda \in {\mathcal{C}}_{ + } \), then \( \lambda \succcurlyeq w\left( \lambda \right) \) for \( w \in W \) . If \( \lambda \in {\mathcal{C}}_{ + }^{ \circ } \) and \( w \neq 1 \), then \( w\left( \lambda \right) \succ \lambda \) . | Proof. It is easy to see that, for \( x \in \mathcal{V}, x \succcurlyeq 0 \) if and only if \( \langle x, v\rangle \geq 0 \) for all \( v \in {\mathcal{C}}_{ + }^{ \circ } \) . So if \( \lambda \in {\mathcal{C}}_{ + } \) and \( \lambda \nsucceq w\left( \lambda \right) \), then there exists \( v \in {\mathcal{C}}_{ + }^... | Yes |
Proposition 22.4. Let \( \lambda \in {\mathcal{C}}_{ + } \) . Then \( \lambda \in \operatorname{supp}\chi \left( \lambda \right) \) . Indeed, writing \( \chi \left( \lambda \right) = \) \( \mathop{\sum }\limits_{\mu }{n}_{\mu } \cdot {\mathrm{e}}^{\mu } \), we have \( {n}_{\lambda } = 1 \) . Moreover, if \( \mu \in \op... | Proof. We enlarge the ring \( \mathcal{E} \) as follows. Let \( \widehat{\mathcal{E}} \) be the \ | No |
Theorem 22.1. If \( f \) is a class function on \( G \), we have\n\n\[ \n{\int }_{G}f\left( g\right) \mathrm{d}g = \frac{1}{\left| W\right| }{\int }_{T}f\left( t\right) {\left| \Delta \left( t\right) \right| }^{2}\mathrm{\;d}t.\n\] | Proof. We will show that, in the notation of Theorem 17.2,\n\n\[ \n\det \left( {\left\lbrack {\operatorname{Ad}\left( {t}^{-1}\right) - {I}_{\mathfrak{p}}}\right\rbrack \mid \mathfrak{p}}\right) = \Delta \bar{\Delta }\n\]\n\nIndeed, since the complexification of \( \mathfrak{p} \) is the direct sum of the spaces \( {\m... | Yes |
Theorem 22.2. If \( \xi \) and \( \eta \) are characters of \( G \), identified with elements of \( \mathcal{E} \), then the inner product (22.8) agrees with the \( {L}^{2} \) inner product of the characters. | Proof. The \( {L}^{2} \) inner product of \( \xi \) and \( \eta \) is just the integral of \( \xi \cdot \bar{\eta } \) over the group and, using (22.6), this is just \( {W}^{-1} \) times the multiplicity of 0 in \( \left( {\xi \Delta }\right) \overline{\left( \eta \Delta \right) } \). | Yes |
Proposition 22.5. If \( \lambda \) and \( \mu \) are weights in \( {\mathcal{C}}_{ + } \), we have\n\n\[ \langle \chi \left( \lambda \right) ,\chi \left( \mu \right) \rangle = \left\{ \begin{array}{ll} 1 & \text{ if }\lambda = \mu , \\ 0 & \text{ otherwise. } \end{array}\right. \] | Proof. Using (22.8), this inner product is the multiplicity of 0 in\n\n\[ \frac{1}{\left| W\right| }\left\lbrack {\mathop{\sum }\limits_{{w \in W}}{\left( -1\right) }^{w}{\mathrm{e}}^{w\left( {\rho + \lambda }\right) }}\right\rbrack \left\lbrack {\mathop{\sum }\limits_{{{w}^{\prime } \in W}}{\left( -1\right) }^{{w}^{\p... | Yes |
Proposition 22.6. The set of \( \chi \left( \lambda \right) \) with \( \lambda \in \Lambda \cap {\mathcal{C}}_{ + } \) is a basis of the free \( \mathbb{Z} \) -module \( {\mathcal{E}}^{W} \) . | Proof. The linear independence of the \( \chi \left( \lambda \right) \) follows from their orthogonality. We must show that they span. Clearly, \( {\mathcal{E}}^{W} \) is spanned by elements of the form\n\n\[ B\left( \lambda \right) = \mathop{\sum }\limits_{{\mu \in W \cdot \lambda }}{\mathrm{e}}^{\mu },\;l \in \Lambda... | Yes |
Theorem 22.3. (Weyl) Assume that \( G \) is semisimple. If \( \lambda \in \Lambda \cap {\mathcal{C}}_{ + } \), then \( \chi \left( \lambda \right) \) is the character of an irreducible representation of \( G \), and each irreducible representation is obtained this way. | Proof. Let \( \chi \) be an irreducible representation of \( G \) . Regarding \( \chi \) as an element of \( {\mathcal{E}}^{W} \), we may expand \( \chi \) in terms of the \( \chi \left( \lambda \right) \) by Proposition 22.6. We write\n\n\[ \chi = \mathop{\sum }\limits_{{\lambda \in \Lambda \cap {\mathcal{C}}_{ + }}}{... | Yes |
Proposition 22.7. We have\n\n\[ \Delta = \mathop{\sum }\limits_{{w \in W}}{\left( -1\right) }^{l\left( w\right) }{\mathrm{e}}^{w\left( \rho \right) }. \] | Proof. The irreducible representation \( \chi \left( 0\right) \) with highest weight vector 0 is obviously the trivial representation. Therefore, \( \chi \left( 0\right) = {\mathrm{e}}^{0} = 1 \) . The formula now follows from (22.4). | No |
Proposition 22.8. Let \( \lambda \) be a dominant weight, and let \( \left( {\pi, V}\right) \) be an irreducible representation with highest weight \( \lambda \) . Let \( {w}_{0} \) be the longest Weyl group element (Proposition 20.14). Then the highest weight of the contragredient representation is \( - {w}_{0}\lambda... | Proof. We recall that the character of the contragredient is the complex conjugate of the character of \( \pi \) (Proposition 2.6). The weights that occur in the contragredient are therefore the negatives of the weights that occur in \( \pi \) . It follows that \( - \lambda \) is the lowest weight of \( \widehat{\pi } ... | Yes |
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