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Proposition 8.4.6. Suppose that \( A\left( q\right) > 1 \) . Then\n\n\[{\alpha }_{q}\left( \delta \right) \geq \left( {1 - A{\left( q\right) }^{-1}}\right) - \delta\]\n\nin the interval \( 0 \leq \delta \leq 1 - A{\left( q\right) }^{-1} \) . | Proof. Let \( \delta \in \left\lbrack {0,1 - A{\left( q\right) }^{-1}}\right\rbrack \) . Choose a sequence of function fields \( {F}_{i}/{\mathbb{F}}_{q} \) of genus \( {g}_{i} \) such that\n\n\[{g}_{i} \rightarrow \infty \;\text{ and }\;{n}_{i}/{g}_{i} \rightarrow A\left( q\right) ,\]\n\n(8.11)\n\nwhere \( {n}_{i} \ma... | Yes |
Theorem 8.4.7 (Tsfasman-Vladut-Zink Bound). Let \( q = {\ell }^{2} \) be a square. Then we have for all \( \delta \) with \( 0 \leq \delta \leq 1 - {\left( {q}^{1/2} - 1\right) }^{-1} \) , \[ {\alpha }_{q}\left( \delta \right) \geq \left( {1 - \frac{1}{{q}^{1/2} - 1}}\right) - \delta \] | Proof. By Corollary 7.4.8 we have \( A\left( q\right) = {q}^{1/2} - 1 \) if \( q \) is a square. Now the assertion follows immediately from Proposition 8.4.6. | Yes |
Proposition 8.5.3. Under the above hypotheses the system (8.36) has a unique solution, namely the vector \( {\left( {e}_{\nu }\right) }_{\nu \in N\left( f\right) } \) . | Proof. As \( {h}_{\mu } \in \mathcal{L}\left( G\right) \), we have\n\n\[ \left\lbrack {a,{h}_{\mu }}\right\rbrack = \left\lbrack {c + e,{h}_{\mu }}\right\rbrack = \left\lbrack {e,{h}_{\mu }}\right\rbrack = \mathop{\sum }\limits_{{\nu = 1}}^{n}{e}_{\nu } \cdot {h}_{\mu }\left( {P}_{\nu }\right) = \mathop{\sum }\limits_{... | Yes |
Theorem 8.5.5 (Skorobogatov-Vladut). (a) Provided \( {G}_{1} \) and \( t \) satisfy (8.24), the algorithm 8.5.4 decodes all errors of weight \( \leq t \) .\n\n(b) One can choose the divisor \( {G}_{1} \) in such a way that the algorithm 8.5.4 decodes all errors \( e \) of weight\n\n\[ \operatorname{wt}\left( e\right) \... | Proof. (a) is obvious from Proposition 8.5.2 and 8.5.3, and (b) follows from Remark 8.5.1(b). | No |
Theorem 9.1.2 (Delsarte). For a code \( C \) over \( {\mathbb{F}}_{{q}^{m}} \) , \[ {\left( {\left. C\right| }_{{\mathbb{F}}_{q}}\right) }^{ \bot } = \operatorname{Tr}\left( {C}^{ \bot }\right) \] | Proof. Recall that we denote by \( \langle \) , \( \rangle {thecanonicalinnerproducton}{\mathbb{F}}_{q}^{n}({resp}. \) on \( \left. {\left( {\mathbb{F}}_{{q}^{m}}\right) }^{n}\right) \) . In order to prove \( {\left( {\left. C\right| }_{{\mathbb{F}}_{q}}\right) }^{ \bot } \supseteq \operatorname{Tr}\left( {C}^{ \bot }\... | Yes |
Lemma 9.1.3. Let \( C \) be a code of length \( n \) over \( {\mathbb{F}}_{{q}^{m}} \). Then\n\n\[ \dim C \leq \dim \operatorname{Tr}\left( C\right) \leq m \cdot \dim C \]\n\n(9.5)\n\nand\n\n\[ \dim C - \left( {m - 1}\right) \left( {n - \dim C}\right) \leq {\left. \dim C\right| }_{{\mathbb{F}}_{q}} \leq \dim C. \]\n\n(... | Proof. By Delsarte's Theorem and (9.3) we have\n\n\[ \dim \operatorname{Tr}\left( C\right) = \dim {\left( {\left. {C}^{ \bot }\right| }_{{\mathbb{F}}_{q}}\right) }^{ \bot } = n - {\left. \dim {C}^{ \bot }\right| }_{{\mathbb{F}}_{q}} \geq n - \dim {C}^{ \bot } = \dim C. \]\n\nThis proves (9.5). The lower estimate in (9.... | No |
Proposition 9.1.4. Let \( C \) be a code over \( {\mathbb{F}}_{{q}^{m}} \) and let \( U \subseteq C \) be a subcode with the additional property \( {U}^{q} \subseteq C \) . Then\n\n\[ \n\dim \operatorname{Tr}\left( C\right) \leq m \cdot \left( {\dim C - \dim U}\right) + {\left. \dim U\right| }_{{\mathbb{F}}_{q}}.\n\] | Proof. We consider the \( {\mathbb{F}}_{q} \) -linear map \( \phi : U \rightarrow C \), given by \( \phi \left( u\right) \mathrel{\text{:=}} {u}^{q} - u \) . The kernel of \( \phi \) is easily seen to be\n\n\[ \n\operatorname{Ker}\left( \phi \right) = {\left. U\right| }_{{\mathbb{F}}_{q}}\n\]\n\n(9.7)\n\nSince \( \oper... | Yes |
Corollary 9.1.5. Let \( C \) be a code of length \( n \) over \( {\mathbb{F}}_{{q}^{m}} \) and let \( V \subseteq {C}^{ \bot } \) be a subcode of \( {C}^{ \bot } \) satisfying \( {V}^{q} \subseteq {C}^{ \bot } \) . Then\n\n\[ \n{\left. \dim C\right| }_{{\mathbb{F}}_{q}} \geq \dim C - \left( {m - 1}\right) \left( {n - \... | Proof. We use Proposition 9.1.4 and Delsarte's Theorem:\n\n\[ \n{\left. \dim C\right| }_{{\mathbb{F}}_{q}} = \dim \operatorname{Tr}{\left( {C}^{ \bot }\right) }^{ \bot } = n - \dim \operatorname{Tr}\left( {C}^{ \bot }\right) \]\n\n\[ \n\geq n - \left( {m \cdot \left( {\dim {C}^{ \bot } - \dim V}\right) + \dim {\left. V... | Yes |
Theorem 9.1.6. Let \( F \) be an algebraic function field of genus \( g \) over the constant field \( {\mathbb{F}}_{{q}^{m}} \) . Consider the AG codes\n\n\[ \n{C}_{\mathcal{L}} \mathrel{\text{:=}} {C}_{\mathcal{L}}\left( {D, G}\right) \;\text{ and }\;{C}_{\Omega } \mathrel{\text{:=}} {C}_{\Omega }\left( {D, G}\right) ... | Proof. Let \( U \mathrel{\text{:=}} {C}_{\mathcal{L}}\left( {D,{G}_{1}}\right) \) . It follows from (9.10) that \( {U}^{q} \subseteq {C}_{\mathcal{L}} \) . We can apply Proposition 9.1.4 and obtain\n\n\[ \n\dim \operatorname{Tr}\left( {C}_{\mathcal{L}}\right) \leq m\left( {\ell \left( G\right) - \ell \left( {G}_{1}\rig... | Yes |
Every code \( C \subseteq {\mathbb{F}}_{q}^{n} \) over \( {\mathbb{F}}_{q} \) can be represented as \( C = \) \( {\operatorname{Tr}}_{D}\left( V\right) \) for a suitable choice of \( V \) and \( D \). | This can be seen as follows. Choose \( m \in \mathbb{N} \) sufficiently large such that \( {q}^{m} \geq n \) . Let \( F \mathrel{\text{:=}} {\mathbb{F}}_{{q}^{m}}\left( z\right) \) be the rational function field over \( {\mathbb{F}}_{{q}^{m}} \) . Choose \( n \) distinct elements \( {\alpha }_{1},\ldots ,{\alpha }_{n} ... | Yes |
Theorem 9.2.5. Let \( F = {\mathbb{F}}_{{p}^{m}}\left( z\right) \) be the rational function field over \( {\mathbb{F}}_{{p}^{m}} \) (where \( p \) is a prime number), \( D = {P}_{1} + \ldots + {P}_{n} \) with pairwise distinct places \( {P}_{i} \in {\mathbb{P}}_{F} \) of degree one, and \( s \mathrel{\text{:=}} {p}^{m}... | \[ \left| {w - \frac{p - 1}{p}n}\right| \leq \frac{p - 1}{2p}\left( {-2 + \deg A + \deg {A}^{0}}\right) \cdot \left\lbrack {2{p}^{m/2}}\right\rbrack + \frac{p - 1}{p}s, \] where the divisors \( A \) and \( {A}^{0} \) are defined by (9.26) and (9.27). | Yes |
Lemma 9.2.7. Suppose \( f \in V \) is degenerate. Then\n\n\[{\operatorname{Tr}}_{D}\left( f\right) = \left( {\alpha ,\alpha ,\ldots ,\alpha }\right) \;\text{ with }\alpha \in {\mathbb{F}}_{p}.\n\]\n\nHence the weight of \( {\operatorname{Tr}}_{D}\left( f\right) \) is 0 or \( n \) . | Proof. Write \( f = \gamma + \left( {{h}^{p} - h}\right) \) with \( \gamma \in {\mathbb{F}}_{{p}^{m}} \) and \( h \in F \) . Since \( {v}_{{P}_{i}}\left( f\right) \geq 0 \), the Triangle Inequality yields \( {v}_{{P}_{i}}\left( h\right) \geq 0 \) for \( 1 \leq i \leq n \) . Setting \( {\gamma }_{i} \mathrel{\text{:=}} ... | Yes |
Proposition 9.2.8. Suppose \( f \in V \) is non-degenerate. Then the polynomial \( \varphi \left( Y\right) \mathrel{\text{:=}} {Y}^{p} - Y - f \in F\left\lbrack Y\right\rbrack \) is irreducible over \( F \) . Let \( {E}_{f} \mathrel{\text{:=}} F\left( y\right) \) where \( y \) is a root of \( \varphi \left( Y\right) \)... | Proof. We use some facts about Artin-Schreier extensions, cf. Appendix A. The polynomial \( \varphi \left( Y\right) = {Y}^{p} - Y - f \) is either irreducible in \( F\left\lbrack Y\right\rbrack \) or it has a root in \( F \) ; i.e., \( f = {h}^{p} - h \) with \( h \in F \) . Since \( f \) is assumed to be non-degenerat... | Yes |
Lemma 9.2.9. Suppose \( f \in V \) is non-degenerate. Then\n\n\[ g\left( {E}_{f}\right) \leq \frac{p - 1}{2}\left( {-2 + \deg A + \deg {A}^{0}}\right) ,\]\n\nwhere \( A \) and \( {A}^{0} \) are defined by (9.26) and (9.27). | Proof. This lemma is an easy application of Proposition 3.7.8(d). Observe that all places \( P \notin \operatorname{supp}A \) are unramified in \( {E}_{f}/F \), and for \( P \in \operatorname{supp}A \) the integer \( {m}_{P} \) (as defined in Proposition 3.7.8) is obviously bounded by \( {v}_{P}\left( A\right) \) . | No |
Corollary 9.2.11. Notation as in Theorem 9.2.5. If \( V \neq \{ 0\} \) and \( V \neq {\mathbb{F}}_{{p}^{m}} \) , the minimum distance \( d \) of \( {\operatorname{Tr}}_{D}\left( V\right) \) is bounded from below by\n\n\[ d \geq \frac{p - 1}{p}n - \frac{s}{p} - \frac{p - 1}{2p}\left( {-2 + \deg A + \deg {A}^{0}}\right) ... | Proof. The assumption \( V \neq \{ 0\} \) and \( V \neq {\mathbb{F}}_{{p}^{m}} \) implies that \( \deg A > 0 \), hence the right-hand side of (9.34) is \( \leq n \) . Therefore it is sufficient to estimate the weight \( {w}_{f} = w\left( {{\operatorname{Tr}}_{D}\left( f\right) }\right) \) for a non-degenerate element \... | Yes |
We consider the dual \( {C}^{ \bot } \) of the BCH code \( C \) over \( {\mathbb{F}}_{p} \) of length \( n = {p}^{m} - 1 \) and designed distance \( \delta = {2t} + 1 > 1 \) . Thus\n\n\[ C = \left\{ {\left( {{c}_{0},{c}_{1},\ldots ,{c}_{n - 1}}\right) \in {\mathbb{F}}_{p}^{n} \mid \mathop{\sum }\limits_{{i = 0}}^{{n - ... | We show now how (9.35) and (9.36) follow from our previous results. Let \( F = {\mathbb{F}}_{{p}^{m}}\left( z\right) \) . For \( 1 \leq i \leq n \) let \( {P}_{i} \in {\mathbb{P}}_{F} \) be the zero of \( z - {\beta }^{i - 1} \) . Set \( D = {P}_{1} + \ldots + {P}_{n} \) and \( \left( z\right) = {P}_{0} - {P}_{\infty }... | Yes |
Proposition 9.2.13. Consider a Goppa code \( \Gamma \left( {L, g\left( z\right) }\right) \) over \( {\mathbb{F}}_{p} \), where \( L = \) \( {\mathbb{F}}_{{p}^{m}} \) and \( g\left( z\right) \in {\mathbb{F}}_{{p}^{m}}\left\lbrack z\right\rbrack \) is a monic polynomial without zeros in \( {\mathbb{F}}_{{p}^{m}} \) (see ... | Proof. By Proposition 2.3.11 and Delsarte's Theorem we can represent the dual code \( \Gamma {\left( L, g\left( z\right) \right) }^{ \bot } \) as follows:\n\n\[ \Gamma {\left( L, g\left( z\right) \right) }^{ \bot } = {\left( {\left. {C}_{\mathcal{L}}{\left( D,{G}_{0} - {P}_{\infty }\right) }^{ \bot }\right| }_{{\mathbb... | Yes |
Lemma 1.1.2. Let \( V \) be an \( n \) -dimensional vector space over \( \mathbb{F} \) and let \( B \) be a symmetric nondegenerate bilinear form over \( \mathbb{F} \) . 1. If \( \mathbb{F} = \mathbb{C} \) then there exists a basis \( \left\{ {{v}_{1},\ldots ,{v}_{n}}\right\} \) of \( V \) such that \( B\left( {{v}_{i}... | Proof. We first observe that if \( M \) is a symmetric bilinear form on \( V \) such that \( M\left( {v, v}\right) = 0 \) for all \( v \in V \), then \( M = 0 \) . Indeed, using the symmetry and bilinear-ity we have \[ {4M}\left( {v, w}\right) = M\left( {v + w, v + w}\right) - M\left( {v - w, v - w}\right) = 0\;\text{ ... | Yes |
Lemma 1.1.5. Let \( V \) be a 2n-dimensional vector space over \( \mathbb{F} \) and let \( B \) be a nondegenerate, skew-symmetric bilinear form on \( V \) . Then there exists a basis \( \left\{ {{v}_{1},\ldots ,{v}_{2n}}\right\} \) for \( V \) such that the matrix \( \left\lbrack {B\left( {{v}_{i},{v}_{j}}\right) }\ri... | Proof. Let \( v \) be a nonzero element of \( V \) . Since \( B \) is nondegenerate, there exists \( w \in V \) with \( B\left( {v, w}\right) \neq 0 \) . Replacing \( w \) with \( B{\left( v, w\right) }^{-1}w \), we may assume that \( B\left( {v, w}\right) = 1 \) . Let\n\n\[ W = \{ x \in V : B\left( {v, x}\right) = 0\t... | No |
Lemma 1.1.7. Let \( V \) be an \( n \) -dimensional vector space over \( \mathbb{C} \) and let \( B \) be a nondegenerate Hermitian form on \( V \) . Then there exist an integer \( p \), with \( n \geq p \geq 0 \), and \( \begin{matrix} a & b & a & s & i & s & \\ \left\{ {{v}_{1},\ldots ,{v}_{n}}\right\} & o & f & V & ... | The proof of Lemma 1.1.7 is almost identical to that of Lemma 1.1.2 and will be left as an exercise. | No |
Proposition 1.3.1. If \( H \) is an open subgroup of a topological group \( G \), then \( H \) is also closed in \( G \) . | Proof. We note that \( G \) is a disjoint union of left cosets. If \( g \in G \) then the left coset \( {gH} = {L}_{g}\left( H\right) \) is open, since \( {L}_{g} \) is a homeomorphism. Hence the union of all the left cosets other than \( H \) is open, and so \( H \) is closed. | Yes |
Proposition 1.3.2. Let \( G \) be a topological group. Then the identity component of \( G \) (that is, the connected component \( {G}^{ \circ } \) that contains the identity element \( e \) ) is a normal subgroup. | Proof. Let \( {G}^{ \circ } \) be the identity component of \( G \) . If \( h \in {G}^{ \circ } \) then \( h \in {L}_{h}\left( {G}^{ \circ }\right) \) because \( e \in {G}^{ \circ } \) . Since \( {L}_{h} \) is a homeomorphism and \( {G}^{ \circ } \cap {L}_{h}\left( {G}^{ \circ }\right) \) is nonempty, it follows that \... | Yes |
Lemma 1.3.3. Suppose \( g \in \mathbf{{GL}}\left( {n,\mathbb{R}}\right) \) satisfies \( \parallel g - I\parallel < \log 2/\left( {1 + \log 2}\right) \) . Then \( \parallel \log \left( g\right) \parallel < \log 2 \) and \( \exp \left( {\log \left( g\right) }\right) = g \) . Furthermore, if \( X \in {B}_{\log 2}\left( 0\... | Proof. Since \( \log 2/\left( {1 + \log 2}\right) < 1 \), the power series for \( \log \left( g\right) \) is absolutely convergent and\n\n\[ \parallel \log \left( g\right) \parallel \leq \mathop{\sum }\limits_{{m = 1}}^{\infty }\parallel g - I{\parallel }^{m} = \frac{\parallel g - I\parallel }{1 - \parallel g - I\paral... | Yes |
Proposition 1.3.7. If \( X, Y \in {M}_{n}\left( \mathbb{R}\right) \), then\n\n\[ \exp \left( {X + Y}\right) = \mathop{\lim }\limits_{{k \rightarrow \infty }}{\left( \exp \left( \frac{1}{k}X\right) \exp \left( \frac{1}{k}Y\right) \right) }^{k}, \]\n\n(1.14)\n\n\[ \exp \left( \left\lbrack {X, Y}\right\rbrack \right) = \m... | Proof. For \( k \) a sufficiently large integer, Lemma 1.3.6 implies that\n\n\[ \exp \left( {\frac{1}{k}X}\right) \exp \left( {\frac{1}{k}Y}\right) = \exp \left( {\frac{1}{k}\left( {X + Y}\right) + \mathrm{O}\left( {1/{k}^{2}}\right) }\right) ,\]\n\nwhere \( \mathrm{O}\left( r\right) \) denotes a matrix function of \( ... | Yes |
Theorem 1.3.8. If \( G \) is a closed subgroup of \( \mathbf{{GL}}\left( {n,\mathbb{R}}\right) \), then \( \operatorname{Lie}\left( G\right) \) is a Lie subalgebra of \( {M}_{n}\left( \mathbb{R}\right) \) . | Proof. If \( X \in \operatorname{Lie}\left( G\right) \), then \( {tX} \in \operatorname{Lie}\left( G\right) \) for all \( t \in \mathbb{R} \) . If \( X, Y \in \operatorname{Lie}\left( G\right) \) and \( t \in \mathbb{R} \) , then\n\n\[ \exp \left( {t\left( {X + Y}\right) }\right) = \mathop{\lim }\limits_{{k \rightarrow... | Yes |
Lemma 1.3.10. Suppose \( H \subset G \subset \mathbf{{GL}}\left( {n,\mathbb{R}}\right) \) with \( H \) a closed subgroup of \( G \) and \( G \) a closed subgroup of \( \mathbf{{GL}}\left( {n,\mathbb{R}}\right) \) . Then \( H \) is a closed subgroup of \( \mathbf{{GL}}\left( {n,\mathbb{R}}\right) \), and\n\n\[ \operator... | Proof. It is obvious that \( H \) is a closed subgroup of \( \mathbf{{GL}}\left( {n,\mathbb{R}}\right) \) . If \( X \in \operatorname{Lie}\left( H\right) \) then \( \exp \left( {tX}\right) \in H \subset G \) for all \( t \in \mathbb{R} \) . Thus \( X \in \operatorname{Lie}\left( G\right) \) . | Yes |
Proposition 1.3.14. Let \( \varphi : H \rightarrow G \) be a continuous homomorphism. There exists a unique Lie algebra homomorphism \( \mathrm{d}\varphi : \operatorname{Lie}\left( H\right) \rightarrow \operatorname{Lie}\left( G\right) \), called the differential of \( \varphi \), such that \( \varphi \left( {\exp \lef... | Proof. If \( X \in \operatorname{Lie}\left( H\right) \) then \( t \mapsto \varphi \left( {\exp {tX}}\right) \) defines a continuous homomorphism of \( \mathbb{R} \) into \( \mathbf{{GL}}\left( {n,\mathbb{R}}\right) \) . Hence Theorem 1.3.5 implies that there exists \( \mu \left( X\right) \in {M}_{n}\left( \mathbb{R}\ri... | Yes |
Corollary 1.3.15. The homomorphism \( \varphi \) is real analytic. | Proof. This follows immediately from the definition of the Lie group structures on \( G \) and \( H \) using exponential coordinates (as in the proof of Theorem 1.3.12), together with Proposition 1.3.14. | Yes |
Lemma 1.3.16. The differential of the inclusion map satisfies \( {\left( \mathrm{d}{\iota }_{G}\right) }_{I}\left( {T{\left( G\right) }_{I}}\right) = \) \( \left\{ {{\left( {X}_{A}\right) }_{I} : A \in \operatorname{Lie}\left( G\right) }\right\} \) . | Proof. For \( A \in \operatorname{Lie}\left( G\right) \) the one-parameter group \( t \mapsto \exp \left( {tA}\right) \) is a \( {C}^{\infty } \) map from \( \mathbb{R} \) to \( G \), by definition of the manifold structure of \( G \) (see Theorem 1.3.12). We define the tangent vector \( {v}_{A} \in T{\left( G\right) }... | Yes |
Proposition 1.3.17. Every left-invariant regular vector field on \( G \) is of the form \( {X}_{A}^{G} \) for a unique \( A \in \operatorname{Lie}\left( G\right) \) . Furthermore, if \( A, B \in \operatorname{Lie}\left( G\right) \) then \( \left\lbrack {{X}_{A}^{G},{X}_{B}^{G}}\right\rbrack = {X}_{\left\lbrack A, B\rig... | Proof. Since a left-invariant vector field \( X \) is uniquely determined by the tangent vector \( {X}_{I} \) at \( I \), the first statement follows from Lemma 1.26. Likewise, to prove the commutator formula it suffices to show that\n\n\[ \n{\left\lbrack {X}_{A}^{G},{X}_{B}^{G}\right\rbrack }_{I} = {\left( {X}_{\left\... | Yes |
Proposition 1.4.4. The maps \( \mu : G \times G \rightarrow G \) and \( \eta : G \rightarrow G \) given by multiplication and inversion are regular. If \( f \in \mathcal{O}\left\lbrack G\right\rbrack \) then there exist an integer \( p \) and \( {f}_{i}^{\prime },{f}_{i}^{\prime \prime } \in \mathcal{O}\left\lbrack G\r... | Proof. Cramer’s rule says that \( \eta \left( g\right) = \det {\left( g\right) }^{-1}\operatorname{adj}\left( g\right) \), where \( \operatorname{adj}\left( g\right) \) is the transposed cofactor matrix of \( g \) . Since the matrix entries of \( \operatorname{adj}\left( g\right) \) are polynomials in the matrix entrie... | Yes |
Lemma 1.4.7. Let \( G = \mathbf{GL}\left( {n,\mathbb{C}}\right) \) and let \( v \in T{\left( G\right) }_{g} \) . Set \( {a}_{ij} = v\left( {x}_{ij}\right) \) and \( A = \left\lbrack {a}_{ij}\right\rbrack \in \) \( {M}_{n}\left( \mathbb{C}\right) \) . Then \( v = {v}_{A} \) . Hence the map \( A \mapsto {v}_{A} \) is a l... | Proof. By (1.34) we have \( v\left( 1\right) = v\left( {1 \cdot 1}\right) = {2v}\left( 1\right) \) . Hence \( v\left( 1\right) = 0 \) . In particular, if \( f = \mathop{\det }\limits^{k} \) for some positive integer \( k \), then\n\n\[ 0 = v\left( {f \cdot {f}^{-1}}\right) = v\left( f\right) f{\left( g\right) }^{-1} + ... | Yes |
Corollary 1.4.8. \( \left( {G = \mathbf{{GL}}\left( {n,\mathbb{C}}\right) }\right) \) If \( X \in \operatorname{Der}\left( {\mathcal{O}\left\lbrack G\right\rbrack }\right) \) then \( X \) is given by (1.32), where \( {c}_{ij} = X\left( {x}_{ij}\right) \) | Proof. For fixed \( g \in G \), the linear functional \( f \mapsto {Xf}\left( g\right) \) is a tangent vector at \( g \) . Hence \( {Xf}\left( g\right) = {v}_{A}\left( f\right) \), where \( {a}_{ij} = X\left( {x}_{ij}\right) \left( g\right) \) . Now define \( {c}_{ij}\left( g\right) = X\left( {x}_{ij}\right) \left( g\r... | Yes |
Proposition 1.4.9. Let \( G \) be an algebraic subgroup of \( \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) . Then \( \mathfrak{g} \) is a Lie subalgebra of \( {M}_{n}\left( \mathbb{C}\right) \) (viewed as a Lie algebra over \( \mathbb{C} \) ). Furthermore, the map \( A \mapsto \) \( {X}_{A} \) is an injective complex l... | Proof. Since the map \( A \mapsto {X}_{A} \) is complex linear, it follows that \( A + {\lambda B} \in \mathfrak{g} \) if \( A, B \in \mathfrak{g} \) and \( \lambda \in \mathbb{C} \) . The differential operators \( {X}_{A}{X}_{B} \) and \( {X}_{B}{X}_{A} \) on \( \mathcal{O}\left\lbrack {\mathbf{{GL}}\left( V\right) }\... | Yes |
Proposition 1.5.1. The representations \( \\left( {L,\\mathcal{O}\\left\\lbrack G\\right\\rbrack }\\right) \) and \( \\left( {R,\\mathcal{O}\\left\\lbrack G\\right\\rbrack }\\right) \) are locally regular. | Proof. For any \( f \\in \\mathcal{O}\\left\\lbrack G\\right\\rbrack \\), equation (1.31) furnishes functions \( {f}_{i}^{\\prime },{f}_{i}^{\\prime \\prime } \\in \\mathcal{O}\\left\\lbrack G\\right\\rbrack \) such that\n\n\[ \nL\\left( x\\right) f = \\mathop{\\sum }\\limits_{{i = 1}}^{n}{f}_{i}^{\\prime }\\left( {x}^... | Yes |
Theorem 1.5.2. The differential of a rational representation \( \left( {\pi, V}\right) \) is the unique linear map \( \mathrm{d}\pi : \mathfrak{g} \rightarrow \operatorname{End}\left( V\right) \) such that\n\n\[ \n{X}_{A}\left( {{f}_{C} \circ \pi }\right) \left( I\right) = {f}_{\mathrm{d}\pi \left( A\right) C}\left( I\... | Proof. For fixed \( A \in \mathfrak{g} \), the map \( C \mapsto {X}_{A}\left( {{f}_{C} \circ \pi }\right) \left( I\right) \) is a linear functional on \( \operatorname{End}\left( V\right) \) . Hence there exists a unique \( D \in \operatorname{End}\left( V\right) \) such that\n\n\[ \n{X}_{A}\left( {{f}_{C} \circ \pi }\... | Yes |
Proposition 1.5.4. The range of \( \mathrm{d}\pi \) is contained in \( \mathfrak{h} \) . Hence \( \mathrm{d}\pi \) is a Lie algebra homomorphism from \( \mathfrak{g} \) to \( \mathfrak{h} \) . Furthermore, if \( K \subset \mathbf{{GL}}\left( W\right) \) is a linear algebraic group and \( \rho : H \rightarrow K \) is a ... | Proof. We first verify that \( \mathrm{d}\pi \left( A\right) \in \mathfrak{h} \) for all \( A \in \mathfrak{g} \) . Let \( f \in {\mathcal{J}}_{H} \) and \( h \in H \) . Then\n\n\[ \left( {{X}_{\mathrm{d}\pi \left( A\right) }f}\right) \left( h\right) = L\left( {h}^{-1}\right) \left( {{X}_{\mathrm{d}\pi \left( A\right) ... | Yes |
Lemma 1.5.6. Let \( A \in \mathfrak{g} \) and \( g \in G \) . Then \( {gA}{g}^{-1} \in \mathfrak{g} \) . | Proof. For \( A \in {M}_{n}\left( \mathbb{C}\right), g \in \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \), and \( t \in \mathbb{C} \) we have\n\n\[ g\exp \left( {tA}\right) {g}^{-1} = \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{{t}^{k}}{k!}{\left( gA{g}^{-1}\right) }^{k} = \exp \left( {{tgA}{g}^{-1}}\right) .\n\]\n\nN... | Yes |
Theorem 1.5.7. The differential of the adjoint representation of \( G \) is the representation ad : \( \mathfrak{g} \rightarrow \operatorname{End}\left( \mathfrak{g}\right) \) given by\n\n\[ \operatorname{ad}\left( A\right) \left( B\right) = \left\lbrack {A, B}\right\rbrack \;\text{ for }A, B \in \mathfrak{g}. \]\n\nFu... | Proof. Equation (1.52) is the special case of equation (1.50) with \( \pi \) the defining representation of \( G \) on \( {\mathbb{C}}^{n} \) and \( \mathrm{d}\pi \left( A\right) = A \) . The derivation property follows from the Jacobi identity. | No |
Lemma 1.6.1 (Taylor’s formula). Suppose \( A \in {M}_{n}\left( \mathbb{C}\right) \) is nilpotent and \( f \) is a regular function on \( \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) . Then there exists an integer \( k \) such that \( {\left( {X}_{A}\right) }^{k}f = 0 \) and\n\n\[ f\left( {\exp A}\right) = \mathop{\sum ... | Proof. Since \( \det \left( {\exp {zA}}\right) = 1 \), the function \( z \mapsto \varphi \left( z\right) = f\left( {\exp {zA}}\right) \) is a polynomial in \( z \in \mathbb{C} \) . Hence there exists a positive integer \( k \) such that \( {\left( d/dz\right) }^{k}\varphi \left( z\right) = 0 \) . Furthermore,\n\n\[ {\v... | Yes |
Theorem 1.6.2. Let \( G \subset \mathbf{GL}\left( {n,\mathbb{C}}\right) \) be a linear algebraic group with Lie algebra \( \mathfrak{g} \). If \( A \in {M}_{n}\left( \mathbb{C}\right) \) is nilpotent, then \( A \in \mathfrak{g} \) if and only if \( \exp A \in G \). Furthermore, if \( A \in \mathfrak{g} \) is a nilpoten... | Proof. Take \( f \in {\mathcal{J}}_{G} \). If \( A \in \mathfrak{g} \), then \( {X}_{A}^{m}f \in {\mathcal{J}}_{G} \) for all integers \( m \geq 0 \). Hence \( {\left( {X}_{A}\right) }^{m}f\left( I\right) = 0 \) for all \( m \), and so by Taylor’s formula (1.53) we have \( f\left( {\exp A}\right) = 0 \). Thus \( \exp A... | Yes |
Lemma 1.6.4. Let \( \left( {\varphi ,{\mathbb{C}}^{n}}\right) \) be a regular representation of \( {\mathbb{C}}^{ \times } \) . For \( p \in \mathbb{Z} \) define \( {E}_{p} = \left\{ {v \in {\mathbb{C}}^{n} : \varphi \left( z\right) v = {z}^{p}v}\right. \) for all \( \left. {z \in {\mathbb{C}}^{ \times }}\right\} \) . ... | Proof. Since \( \mathcal{O}\left\lbrack {\mathbb{C}}^{ \times }\right\rbrack = \mathbb{C}\left\lbrack {z,{z}^{-1}}\right\rbrack \), the entries in the matrix \( \varphi \left( z\right) \) are Laurent polynomials. Thus there is an expansion\n\n\[ \varphi \left( z\right) = \mathop{\sum }\limits_{{p \in \mathbb{Z}}}{z}^{p... | Yes |
Theorem 1.6.6. Let \( G \) be an algebraic group with Lie algebra \( \mathfrak{g} \) . Suppose \( \left( {\rho, V}\right) \) is a regular representation of \( G \) .\n\n1. If \( A \in \mathfrak{g} \) and \( A = S + N \) is its additive Jordan decomposition, then \( \mathrm{d}\rho \left( S\right) \) is semisimple, \( \m... | Proof. We know from Theorem 1.6.2 that \( \mathrm{d}\rho \left( N\right) \) is nilpotent and \( \rho \left( u\right) \) is unipotent, and since \( \mathrm{d}\rho \) is a Lie algebra homomorphism, we have\n\n\[ \left\lbrack {\mathrm{d}\rho \left( N\right) ,\mathrm{d}\rho \left( S\right) }\right\rbrack = \mathrm{d}\rho \... | Yes |
Corollary 1.6.7. Suppose \( G \) and \( H \) are algebraic groups with Lie algebras \( \mathfrak{g} \) and \( \mathfrak{h} \) . Let \( \rho : G \rightarrow H \) be a regular homomorphism such that \( {d\rho } : \mathfrak{g} \rightarrow \mathfrak{h} \) is surjective. Then \( \rho \left( {G}_{u}\right) = {H}_{u} \) . | Proof. By Theorem 1.6.2 the map \( N \mapsto \exp \left( N\right) \) from \( {\mathfrak{g}}_{n} \) to \( {G}_{u} \) is a bijection, and by Theorem 1.6.6 we have \( {H}_{u} = \exp \left( {\mathfrak{h}}_{n}\right) = \exp \left( {{d\rho }\left( {\mathfrak{g}}_{n}\right) }\right) = \rho \left( {G}_{u}\right) \) . | Yes |
Lemma 1.7.2. Let \( G \subset \mathbf{GL}\left( {n,\mathbb{C}}\right) \) be an algebraic subgroup. Then \( G \) is defined over \( \mathbb{R} \) if and only if \( {\mathcal{J}}_{G} \) is invariant under \( f \mapsto \bar{f} \) . | Proof. Assume that \( G \) is defined over \( \mathbb{R} \) . If \( {f}_{1} \in {\mathcal{J}}_{\mathbb{R}, G} \) then \( {f}_{1} = {\bar{f}}_{1} \) . Hence \( {f}_{1}\left( {\sigma \left( g\right) }\right) = \) \( \overline{{f}_{1}\left( g\right) } = 0 \) for \( g \in G \) . Since \( {\mathcal{J}}_{\mathbb{R}, G} \) is... | Yes |
Proposition 1.7.7. Suppose \( \left( {\rho, V}\right) \) is a regular representation of \( G \) . Then a subspace \( W \subset V \) is invariant under \( \mathrm{d}\rho \left( \mathfrak{k}\right) \) if and only if it is invariant under \( {G}^{ \circ } \) . In particular, \( V \) is irreducible under \( \mathfrak{k} \)... | Proof. Assume \( W \) is invariant under \( \mathfrak{k} \) . Since the map \( X \mapsto \mathrm{d}\rho \left( X\right) \) from \( \mathfrak{g} \) to \( \operatorname{End}\left( V\right) \) is complex linear, it follows from (1.63) that \( W \) is invariant under \( \mathfrak{g} \) . Let \( {W}^{ \bot } \subset {V}^{ *... | Yes |
Lemma 2.1.2. Let \( T \) be an algebraic torus of rank \( l \) . The group \( \mathcal{X}\left( T\right) \) is isomorphic to \( {\mathbb{Z}}^{l} \) . Furthermore, \( \mathcal{X}\left( T\right) \) is a basis for \( \mathcal{O}\left\lbrack T\right\rbrack \) as a vector space over \( \mathbb{C} \) . | Proof. We may assume that \( T = {\left( {\mathbb{C}}^{ \times }\right) }^{l} \) with coordinate functions \( {x}_{1},\ldots ,{x}_{l} \) . Thus \( \mathcal{O}\left\lbrack T\right\rbrack = \mathbb{C}\left\lbrack {{x}_{1},\ldots ,{x}_{l},{x}_{l}^{-1},\ldots ,{x}_{l}^{-1}}\right\rbrack \) . For \( t = \left\lbrack {{x}_{1... | Yes |
Proposition 2.1.3. Let \( T \) be an algebraic torus. Suppose \( \left( {\rho, V}\right) \) is a regular representation of \( T \) . Then there exists a finite subset \( \Psi \subset X\left( T\right) \) such that\n\n\[ V = {\bigoplus }_{\chi \in \Psi }V\left( \chi \right) \]\n\nwhere \( V\left( \chi \right) = \{ v \in ... | Proof. Since \( {\left( {\mathbb{C}}^{ \times }\right) }^{l} \cong {\mathbb{C}}^{ \times } \times {\left( {\mathbb{C}}^{ \times }\right) }^{l - 1} \), the existence of the decomposition (2.2) follows from Lemma 1.6.4 by induction on \( l \) . The last statement is clear from the definition of \( V\left( \chi \right) \)... | Yes |
Lemma 2.1.4. Let \( T \) be an algebraic torus. Then there exists an element \( t \in T \) with the following property: If \( f \in \mathcal{O}\left\lbrack T\right\rbrack \) and \( f\left( {t}^{n}\right) = 0 \) for all \( n \in \mathbb{Z} \), then \( f = 0 \) . | Proof. We may assume \( T = {\left( {\mathbb{C}}^{ \times }\right) }^{l} \) . Choose \( t \in T \) such that its coordinates \( {t}_{i} = {x}_{i}\left( t\right) \) satisfy \[ {t}_{1}^{{p}_{1}}\cdots {t}_{l}^{{p}_{l}} \neq 1\;\text{ for all }\left( {{p}_{1},\ldots ,{p}_{l}}\right) \in {\mathbb{Z}}^{l} \smallsetminus \{ ... | Yes |
Theorem 2.1.5. Let \( G \) be \( \mathbf{{GL}}\left( {n,\mathbb{C}}\right) ,\mathbf{{SL}}\left( {n,\mathbb{C}}\right) ,\mathbf{{SO}}\left( {{\mathbb{C}}^{n}, B}\right) \) or \( \mathbf{{Sp}}\left( {{\mathbb{C}}^{2l},\Omega }\right) \) in the form given above, where \( H \) is the diagonal subgroup in \( G \) . Suppose ... | Proof. We have \( G \subset \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) . An element \( h \in H \) acts on the standard basis \( \left\{ {{e}_{1},\ldots ,{e}_{n}}\right\} \) for \( {\mathbb{C}}^{n} \) by \( h{e}_{i} = {\theta }_{i}\left( h\right) {e}_{i} \) . Here the characters \( {\theta }_{i} \) are given as follow... | Yes |
Theorem 2.1.7. (Notation as in Theorem 2.1.5) If \( g \in G \) is semisimple then there exists \( \gamma \in G \) such that \( {\gamma g}{\gamma }^{-1} \in H \) . | Proof. When \( G \) is \( \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) or \( \mathbf{{SL}}\left( {n,\mathbb{C}}\right) \), let \( \left\{ {{v}_{1},\ldots ,{v}_{n}}\right\} \) be a basis of eigenvectors for \( g \) and define \( \gamma {v}_{i} = {e}_{i} \), where \( \left\{ {e}_{i}\right\} \) is the standard basis for \... | Yes |
Corollary 2.1.8. If \( T \) is any torus in \( G \), then there exists \( \gamma \in G \) such that \( {\gamma T}{\gamma }^{-1} \subset H \) . In particular, if \( T \) is a maximal torus in \( G \), then \( {\gamma T}{\gamma }^{-1} = H \) . | Proof. Choose \( t \in T \) satisfying the condition of Lemma 2.1.4. By Theorem 2.1.7 there exists \( \gamma \in G \) such that \( {\gamma t}{\gamma }^{-1} \in H \) . We want to show that \( {\gamma x}{\gamma }^{-1} \in H \) for all \( x \in T \) . To prove this, take any function \( \varphi \in {\mathcal{J}}_{H} \) an... | Yes |
Lemma 2.2.1. The group \( \mathbf{{SL}}\left( {2,\mathbb{C}}\right) \) is generated by \( {N}^{ + } \cup {N}^{ - } \) . | Proof. Let \( g = \left\lbrack \begin{array}{ll} a & b \\ c & d \end{array}\right\rbrack \) with \( {ad} - {bc} = 1 \) . If \( a \neq 0 \) we can use elementary row and column operations to factor\n\n\[ g = \left\lbrack \begin{matrix} 1 & 0 \\ {a}^{-1}c & 1 \end{matrix}\right\rbrack \left\lbrack \begin{matrix} a & 0 \\... | Yes |
Theorem 2.2.2. Suppse that \( G \) is \( \mathbf{{SL}}\left( {l + 1,\mathbb{C}}\right) ,\mathbf{{SO}}\left( {{2l} + 1,\mathbb{C}}\right) \), or \( \mathbf{{Sp}}\left( {l,\mathbb{C}}\right) \) with \( l \geq 1 \) , or that \( G \) is \( \mathbf{{SO}}\left( {{2l},\mathbb{C}}\right) \) with \( l \geq 2 \) . Then \( G \) i... | Proof. We have \( G \subset \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) (where \( n = l + 1,{2l} \), or \( {2l} + 1 \) ). Let \( {G}^{\prime } \) be the subgroup generated by the unipotent elements of \( G \) . Since the conjugate of a unipotent element is unipotent, we see that \( {G}^{\prime } \) is a normal subgrou... | Yes |
Theorem 2.2.4. Let \( G \) be a linear algebraic group that is generated by unipotent elements. Then \( G \) is connected as an algebraic group and as a Lie group. | Proof. Suppose \( {f}_{1},{f}_{2} \in \mathcal{O}\left\lbrack G\right\rbrack ,{f}_{1} \neq 0 \), and \( {f}_{1}{f}_{2} = 0 \) . We must show that \( {f}_{2} = 0 \) . Translating \( {f}_{1} \) and \( {f}_{2} \) by an element of \( G \) if necessary, we may assume that \( {f}_{1}\left( I\right) \neq \) 0 . Let \( g \in G... | Yes |
Theorem 2.2.5. The groups \( \mathbf{GL}\left( {n,\mathbb{C}}\right) ,\mathbf{SL}\left( {n,\mathbb{C}}\right) ,\mathbf{SO}\left( {n,\mathbb{C}}\right) \), and \( \mathbf{Sp}\left( {n,\mathbb{C}}\right) \) are connected (as linear algebraic groups and Lie groups) for all \( n \geq 1 \) . | Proof. The homomorphism \( \lambda, g \mapsto {\lambda g} \) from \( {\mathbb{C}}^{ \times } \times \mathbf{SL}\left( {n,\mathbb{C}}\right) \) to \( \mathbf{GL}\left( {n,\mathbb{C}}\right) \) is surjective. Hence the connectedness of \( \mathbf{GL}\left( {n,\mathbb{C}}\right) \) will follow from the connectedness of \(... | Yes |
Theorem 2.2.7. Suppose \( G \) is a linear algebraic group with Lie algebra \( \mathfrak{g} \) . Let \( \left( {\pi, V}\right) \) be a regular representation of \( G \) and \( W \subset V \) a subspace.\n\n1. If \( \pi \left( g\right) W \subset W \) for all \( g \in G \) then \( \mathrm{d}\pi \left( A\right) W \subset ... | Proof. This follows by the same argument as in Proposition 1.7.7, using the exponentials of nilpotent elements of \( \mathfrak{g} \) to generate \( G \) in part (2). | Yes |
Lemma 2.3.1. Let \( V \) be a \( \mathfrak{g} \) -module (possibly infinite-dimensional) and let \( {v}_{0} \in V \) be such that \( x{v}_{0} = 0 \) and \( h{v}_{0} = \lambda {v}_{0} \) for some \( \lambda \in \mathbb{C} \) . Set \( {v}_{j} = {y}^{j}{v}_{0} \) for \( j \in \mathbb{N} \) and \( {v}_{j} = 0 \) for \( j <... | Proof. The equation for \( y{v}_{j} \) follows by definition, and the equation for \( h{v}_{j} \) follows from the commutation relation (proved by induction on \( j \) ) \[ h{y}^{j}v = {y}^{j}{hv} - {2jv}\;\text{ for all }v \in V\text{ and }j \in \mathbb{N}. \] From (2.17) and the relation \( {xyv} = {yxv} + {hv} \) on... | Yes |
Lemma 2.3.2. Suppose \( V \) is a finite-dimensional \( \mathfrak{g} \) -module and \( 0 \neq {v}_{0} \in V \) satisfies \( h{v}_{0} = \lambda {v}_{0} \) and \( x{v}_{0} = 0 \) . Let \( k \) be the smallest nonnegative integer such that \( {y}^{k}{v}_{0} \neq 0 \) and \( {y}^{k + 1}{v}_{0} = 0 \) . Then \( \lambda = k ... | Proof. Such an integer \( k \) exists by (2.19), since \( V \) is finite-dimensional and the weight spaces are linearly independent. Lemma 2.3.1 implies that \( W \) is invariant under \( x, y \), and \( h \) . Furthermore, \( {v}_{0}, y{v}_{0},\ldots ,{y}^{k}{v}_{0} \) are eigenvectors for \( h \) with respective eige... | Yes |
Proposition 2.3.3. Let \( k \geq 0 \) be an integer: The representation \( \left( {{\rho }_{k},{F}^{\left( k\right) }}\right) \) of \( \mathfrak{g} \) on \( {\mathbb{C}}^{k + 1} \) defined by\n\n\[{\rho }_{k}\left( x\right) = {X}_{k},\;{\rho }_{k}\left( h\right) = {H}_{k},\;\text{ and }\;{\rho }_{k}\left( y\right) = {Y... | Proof. Suppose that \( W \subset {F}^{\left( k\right) } \) is a nonzero invariant subspace. Since \( {xW}\left( \lambda \right) \subset \) \( W\left( {\lambda + 2}\right) \), there must be \( \lambda \) with \( W\left( \lambda \right) \neq 0 \) and \( {xW}\left( \lambda \right) = 0 \) . But from the echelon form of \( ... | Yes |
Corollary 2.3.4. The weights of a finite-dimensional \( \mathfrak{g} \) -module \( V \) are integers. | Proof. There are \( \mathfrak{g} \) -invariant subspaces \( 0 = {V}_{0} \subset {V}_{1} \subset \cdots \subset {V}_{k} = V \) such that the quotient modules \( {W}_{j} = {V}_{j}/{V}_{j - 1} \) are irreducible for \( j = 1,\ldots, k - 1 \) . The weights are the eigenvalues of \( h \) on \( V \), and this set is the unio... | Yes |
Theorem 2.3.6. Let \( V \) be a finite-dimensional \( \mathfrak{g} \) -module with \( \dim V > 0 \) . Then there exist integers \( {k}_{1},\ldots ,{k}_{r} \) (not necessarily distinct) such that \( V \) is equivalent to \( {F}^{\left( {k}_{1}\right) } \oplus {F}^{\left( {k}_{2}\right) } \oplus \cdots \oplus {F}^{\left(... | The key step in the proof of Theorem 2.3.6 is the following result:\n\nLemma 2.3.7. Suppose | No |
Corollary 2.3.8. Let \( \\left( {\\rho, V}\\right) \) be a finite-dimensional representation of \( \\mathfrak{{sl}}\\left( {2,\\mathbb{C}}\\right) \) . There exists a regular representation \( \\left( {\\pi, W}\\right) \) of \( \\mathbf{{SL}}\\left( {2,\\mathbb{C}}\\right) \) such that \( \\left( {\\mathrm{d}\\pi, W}\\... | Proof. By Theorem 2.3.6 we may assume that \( V = {F}^{\\left( {k}_{1}\\right) } \\oplus {F}^{\\left( {k}_{2}\\right) } \\oplus \\cdots \\oplus {F}^{\\left( {k}_{r}\\right) } \) . Each of the summands is the differential of a representation of \( \\mathbf{{SL}}\\left( {2,\\mathbb{C}}\\right) \) by Proposition 2.3.5. | Yes |
1. \( \dim {\mathfrak{g}}_{\alpha } = 1 \) for all \( \alpha \in \Phi \) . | Proof of (1): We shall calculate the roots and root vectors for each type of classical group. We take the Lie algebras in the matrix form of Section 2.1.2. In this realization the algebras are invariant under the transpose. For \( A \in \mathfrak{h} \) and \( X \in \mathfrak{g} \) we have \( {\left\lbrack A, X\right\rb... | Yes |
Lemma 2.4.2. (Notation as in Theorem 2.4.1) For each \( \alpha \in \Phi \) there exist \( {e}_{\alpha } \in {\mathfrak{g}}_{\alpha } \) and \( {f}_{\alpha } \in {\mathfrak{g}}_{-\alpha } \) such that the element \( {h}_{\alpha } = \left\lbrack {{e}_{\alpha },{f}_{\alpha }}\right\rbrack \in \mathfrak{h} \) satisfies \( ... | Proof. By Theorem 2.4.1 we can pick \( X \in {\mathfrak{g}}_{\alpha } \) and \( Y \in {\mathfrak{g}}_{-\alpha } \) such that \( \left( {X, Y}\right) \neq 0 \) . Set \( A = \left\lbrack {X, Y}\right\rbrack \in \mathfrak{h} \) . Then\n\n\[ \left\lbrack {A, X}\right\rbrack = \langle \alpha, A\rangle X,\;\left\lbrack {A, Y... | Yes |
Lemma 2.4.3. For every \( \alpha ,\beta \in \Phi \) with \( \alpha \neq \pm \beta \), the space \( {V}_{\alpha ,\beta } \) is invariant and irreducible under \( \operatorname{ad}\left( {\mathfrak{s}\left( \alpha \right) }\right) \) . | Proof. From (2.25) we have \( \left\lbrack {{\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{\beta + {k\alpha }}}\right\rbrack \subset {\mathfrak{g}}_{\beta + \left( {k + 1}\right) \alpha } \) and \( \left\lbrack {{\mathfrak{g}}_{-\alpha },{\mathfrak{g}}_{\beta + {k\alpha }}}\right\rbrack \subset {\mathfrak{g}}_{\beta + \left(... | Yes |
Corollary 2.4.4. If \( \alpha ,\beta \in \Phi \) and \( \alpha + \beta \in \Phi \), then \( \left\lbrack {{\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{\beta }}\right\rbrack = {\mathfrak{g}}_{\alpha + \beta } \) . | Proof. Since \( \alpha + \beta \in \Phi \), we have \( \alpha \neq \pm \beta \) . Thus \( {V}_{\alpha ,\beta } \) is irreducible under \( {\mathfrak{s}}_{\alpha } \) and contains \( {\mathfrak{g}}_{\alpha + \beta } \) . Hence by (2.16) the operator \( E = \pi \left( {e}_{\alpha }\right) \) maps \( {\mathfrak{g}}_{\beta... | No |
Corollary 2.4.5. Let \( \alpha ,\beta \in \Phi \) with \( \beta \neq \pm \alpha \) . Let \( p \) be the largest integer \( j \geq 0 \) such that \( \beta + {j\alpha } \in \Phi \) and let \( q \) be the largest integer \( k \geq 0 \) such that \( \beta - {k\alpha } \in \Phi \) . Then\n\n\[ \left\langle {\beta ,{h}_{\alp... | Proof. The largest eigenvalue of \( \pi \left( {h}_{\alpha }\right) \) is the positive integer \( n = \left\langle {\beta ,{h}_{\alpha }}\right\rangle + {2p} \) . Since \( \pi \) is irreducible, Proposition 2.3.3 implies that the eigenspaces of \( \pi \left( {h}_{\alpha }\right) \) are \( {\mathfrak{g}}_{\beta + {r\alp... | Yes |
Lemma 2.4.10. Let \( \Phi \) be the root system for a classical Lie algebra \( \mathfrak{g} \) of rank \( l \) and type \( A, B, C \), or \( D \) (in the case of type \( D \) assume that \( l \geq 3 \) ). Let the system of simple roots \( \Delta \subset \Phi \) be chosen as above. Let \( {\Phi }^{ + } \) be the positiv... | Proof. Property (1) is clear from the definition of a system of simple roots. Properties (2)-(5) follow on a case-by-case basis from the calculations made above. We leave the details as an exercise. | No |
Theorem 2.4.11. Let \( \mathfrak{g} \) be the Lie algebra of \( \mathbf{{SL}}\left( {l + 1,\mathbb{C}}\right) ,\mathbf{{Sp}}\left( {{\mathbb{C}}^{2l},\Omega }\right) \), or \( \mathbf{{SO}}\left( {{\mathbb{C}}^{{2l} + 1}, B}\right) \) with \( l \geq 1 \), or the Lie algebra of \( \mathbf{{SO}}\left( {{\mathbb{C}}^{2l},... | Proof. The fact that \( {\mathfrak{n}}^{ + } \) and \( {\mathfrak{n}}^{ - } \) are subalgebras follows from property (1) in Lemma 2.4.10. Equation (2.37) is clear from Theorem 2.4.1 and the decomposition\n\n\[ \n\Phi = {\Phi }^{ + } \cup \left( {-{\Phi }^{ + }}\right) \n\]\n\nFor \( \alpha \in \Phi \) let \( {h}_{\alph... | Yes |
Lemma 2.5.1. Let \( V \) be a finite-dimensional complex vector space and let \( A \in \) \( \operatorname{End}\left( V\right) \) . Suppose there exist \( {X}_{i},{Y}_{i} \in \operatorname{End}\left( V\right) \) such that \( A = \mathop{\sum }\limits_{{i = 1}}^{k}\left\lbrack {{X}_{i},{Y}_{i}}\right\rbrack \) and \( \l... | Proof. Let \( \sum \) be the spectrum of \( A \), and let \( {\left\{ {P}_{\lambda }\right\} }_{\lambda \in \sum } \) be the resolution of the identity for \( A \) (see Lemma B.1.1). Then \( {P}_{\lambda }{X}_{i} = {X}_{i}{P}_{\lambda } = {P}_{\lambda }{X}_{i}{P}_{\lambda } \) for all \( i \), so\n\n\[ \n{P}_{\lambda }... | Yes |
Corollary 2.5.6. Suppose \( \mathfrak{g} \subset \operatorname{End}\left( V\right) \) is a solvable Lie algebra and that \( V \) is completely reducible as a \( \mathfrak{g} \) -module. Then \( \mathfrak{g} \) is abelian. In particular, if \( V \) is an irreducible \( \mathfrak{g} \) -module, then \( \dim V = 1 \) . | Proof. Let \( \mathfrak{z} \) be the center of \( \mathfrak{g} \) . If \( \mathfrak{z} \neq \mathfrak{g} \), then \( \mathfrak{g}/\mathfrak{z} \) would be a nonzero solvable Lie algebra and hence would contain a nonzero abelian ideal. But this would contradict part (3) of Theorem 2.5.3, so we must have \( \mathfrak{z} ... | Yes |
Corollary 2.5.8. Let \( \mathfrak{g} \) be a Lie subalgebra of \( \operatorname{End}\left( V\right) \) that has no nonzero abelian ideals. Then the bilinear form \( \operatorname{tr}\left( {XY}\right) \) on \( \mathfrak{g} \) is nondegenerate, and \( \mathfrak{g} = {\mathfrak{g}}_{1} \oplus \cdots \oplus {\mathfrak{g}}... | Proof. Let \( \mathfrak{r} = \{ X \in \mathfrak{g} : \operatorname{tr}\left( {XY}\right) = 0\; \) for all \( Y \in \mathfrak{g}\} \) be the radical of the trace form. Then \( \mathfrak{r} \) is an ideal in \( \mathfrak{g} \), and by Cartan’s criterion \( \mathfrak{r} \) is a solvable Lie algebra. Suppose \( \mathfrak{r... | Yes |
Corollary 2.5.9. Let \( V \) be a finite-dimensional complex vector space. Suppose \( \mathfrak{g} \) is a Lie subalgebra of \( \operatorname{End}\left( V\right) \) such that \( V \) is completely reducible as a representation of \( \mathfrak{g} \) . Let \( \mathfrak{z} = \{ X \in \mathfrak{g} : \left\lbrack {X, Y}\rig... | Proof. Theorem 2.5.3 implies that \( \mathfrak{g}/\mathfrak{z} \) has no nonzero abelian ideals; hence \( \mathfrak{g}/\mathfrak{z} \) is semisimple (Corollary 2.5.8). Since \( \mathfrak{g}/\mathfrak{z} \) is a direct sum of simple algebras, it satisfies \( \left\lbrack {\mathfrak{g}/\mathfrak{z},\mathfrak{g}/\mathfrak... | Yes |
Theorem 2.5.11. The Lie algebra \( \mathfrak{g} \) is semisimple if and only if its Killing form is nondegenerate. | Proof. Assume that \( \mathfrak{g} \) is semisimple. Since the adjoint representation of a simple Lie algebra is faithful, the same is true for a semisimple Lie algebra. Hence a semisimple Lie algebra \( \mathfrak{g} \) is isomorphic to a Lie subalgebra of \( \operatorname{End}\left( \mathfrak{g}\right) \) . Let\n\n\[ ... | Yes |
Corollary 2.5.12. Suppose \( \mathfrak{g} \) is a semisimple Lie algebra and \( D \in \operatorname{Der}\left( \mathfrak{g}\right) \) . Then there exists \( X \in \mathfrak{g} \) such that \( D = \operatorname{ad}X \) . | Proof. The derivation property \( D\left( \left\lbrack {Y, Z}\right\rbrack \right) = \left\lbrack {D\left( Y\right), Z}\right\rbrack + \left\lbrack {Y, D\left( Z\right) }\right\rbrack \) can be expressed as the commutation relation\n\n\[ \left\lbrack {D,\operatorname{ad}Y}\right\rbrack = \operatorname{ad}D\left( Y\righ... | Yes |
Corollary 2.5.13. Let \( \mathfrak{g} \) be a semisimple Lie algebra. If \( X \in \mathfrak{g} \) and \( \operatorname{ad}X = S + N \) is the additive Jordan decomposition in \( \operatorname{End}\left( \mathfrak{g}\right) \) (with \( S \) semisimple, \( N \) nilpotent, and \( \left\lbrack {S, N}\right\rbrack = 0) \), ... | Proof. Let \( \lambda \in \mathbb{C} \) and set\n\n\[ \n{\mathfrak{g}}_{\lambda }\left( X\right) = \mathop{\bigcup }\limits_{{k \geq 1}}\operatorname{Ker}{\left( \operatorname{ad}X - \lambda \right) }^{k} \n\]\n\n(the generalized \( \lambda \) eigenspace of \( \operatorname{ad}X \) ). The Jordan decomposition of \( \op... | Yes |
Theorem 2.5.14 (Engel). Let \( V \) be a nonzero finite-dimensional vector space and let \( \mathfrak{g} \subset \operatorname{End}\left( V\right) \) be a Lie algebra. Assume that every \( X \in \mathfrak{g} \) is a nilpotent linear transformation. Then there exists a nonzero vector \( {v}_{0} \in V \) such that \( X{v... | Proof. For \( X \in \operatorname{End}\left( V\right) \) write \( {L}_{X} \) and \( {R}_{X} \) for the linear transformations of \( \operatorname{End}\left( V\right) \) given by left and right multiplication by \( X \), respectively. Then ad \( X = {L}_{X} - {R}_{X} \) and \( {L}_{X} \) commutes with \( {R}_{X} \) . He... | Yes |
Corollary 2.5.15. There exists a basis for \( V \) in which the elements of \( \mathfrak{g} \) are represented by strictly upper-triangular matrices. | Proof. This follows by repeated application of Theorem 2.5.14, replacing \( V \) by \( V/\mathbb{C}{v}_{0} \) at each step. | No |
Corollary 2.5.16. Suppose \( \mathfrak{g} \) is a semisimple Lie algebra. Then there exists a nonzero element \( X \in \mathfrak{g} \) such that \( \operatorname{ad}X \) is semisimple. | Proof. We argue by contradiction. If \( \mathfrak{g} \) contained no nonzero elements \( X \) with ad \( X \) semisimple, then Corollary 2.5.13 would imply that \( \operatorname{ad}X \) is nilpotent for all \( X \in \mathfrak{g} \) . Hence Corollary 2.5.15 would furnish a basis for \( \mathfrak{g} \) such that \( \oper... | Yes |
Lemma 2.5.17. Let \( \mathfrak{h} \) be a toral subalgebra. Then \( \left\lbrack {\mathfrak{h},\mathfrak{h}}\right\rbrack = 0 \) . | Proof. Let \( X \in \mathfrak{h} \) . Then \( \mathfrak{h} \) is an invariant subspace for the semisimple transformation ad \( X \) . If \( \left\lbrack {X,\mathfrak{h}}\right\rbrack \neq 0 \) then there would exist an eigenvalue \( \lambda \neq 0 \) and an eigenvector \( Y \in \mathfrak{h} \) such that \( \left\lbrack... | Yes |
Theorem 2.5.20. The roots and root spaces satisfy the following properties:\n\n1. \( \Phi \) spans \( {\mathfrak{h}}^{ * } \) .\n\n2. If \( \alpha \in \Phi \) then \( \dim \left\lbrack {{\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{-\alpha }}\right\rbrack = 1 \) and there is a unique element \( {h}_{\alpha } \in \left\lbrac... | Proof. (1): If \( h \in \mathfrak{h} \) and \( \langle \alpha, h\rangle = 0 \) for all \( \alpha \in \Phi \), then \( \left\lbrack {h,{\mathfrak{g}}_{\alpha }}\right\rbrack = 0 \) and hence \( \left\lbrack {h,\mathfrak{g}}\right\rbrack = 0 \) .\n\nThe center of \( \mathfrak{g} \) is trivial, since \( \mathfrak{g} \) ha... | Yes |
Lemma 2.5.21. Each root \( \alpha \in \Phi \) is in \( {\mathfrak{h}}_{\mathbb{Q}}^{ * } \), and the element \( {h}_{\alpha } \) is in \( {\mathfrak{h}}_{\mathbb{Q}} \) . Let \( a, b \in {\mathfrak{h}}_{\mathbb{Q}} \) . Then \( B\left( {a, b}\right) \in \mathbb{Q} \) and \( B\left( {a, a}\right) > 0 \) if \( a \neq 0 \... | Proof. Set \( {a}_{ij} = \left\langle {{\alpha }_{j},{H}_{i}}\right\rangle \) and let \( A = \left\lbrack {a}_{ij}\right\rbrack \) be the corresponding \( l \times l \) matrix. The entries of \( A \) are integers by Theorem 2.5.20 (4), and the columns of \( A \) are linearly independent. Hence \( A \) is invertible. Fo... | Yes |
Let \( {\mathfrak{h}}_{\mathbb{R}} \) be the real span of \( \left\{ {{h}_{\alpha } : \alpha \in \Phi }\right\} \) and let \( {\mathfrak{h}}_{\mathbb{R}}^{ * } \) be the real span of the roots. Then the Killing form is real-valued and positive definite on \( {\mathfrak{h}}_{\mathbb{R}} \) . Furthermore, \( {\mathfrak{h... | This follows immediately from (2.49) and Lemma 2.5.21. | No |
Theorem 2.5.24. The simple root vectors \( \left\{ {{E}_{1},\ldots ,{E}_{l},{F}_{1},\ldots ,{F}_{l}}\right\} \) generate \( \mathfrak{g} \) . They satisfy the relations \( \left\lbrack {{E}_{i},{F}_{j}}\right\rbrack = 0 \) for \( i \neq j \) and \( \left\lbrack {{H}_{i},{H}_{j}}\right\rbrack = 0 \), where \( {H}_{i} = ... | Proof. Let \( {\mathfrak{g}}^{\prime } \) be the Lie subalgebra generated by the \( {E}_{i} \) and \( {F}_{j} \) . Since \( \left\{ {{H}_{1},\ldots ,{H}_{l}}\right\} \) is a basis for \( \mathfrak{h} \), we have \( \mathfrak{h} \subset {\mathfrak{g}}^{\prime } \) . We show that \( {\mathfrak{g}}_{\beta } \in {\mathfrak... | Yes |
Theorem 2.5.27. The semisimple Lie algebra \( \mathfrak{g} \) is simple if and only if \( \Delta \) is indecomposable. | Proof. Assume that \( \Delta = {\Delta }_{1} \cup {\Delta }_{2} \) is decomposable. Let \( \alpha \in {\Delta }_{1} \) and \( \beta \in {\Delta }_{2} \) . Then \( p = 0 \) in (2.50), since \( \left( {\alpha ,\beta }\right) = 0 \) . Hence \( \beta + \alpha \) is not a root, and we already know that \( \beta - \alpha \) ... | Yes |
Theorem 2.5.28. Let \( \mathfrak{g} \) be a semisimple Lie algebra over \( \mathbb{C} \) and let \( {\mathfrak{h}}_{1} \) and \( {\mathfrak{h}}_{2} \) be Cartan subalgebras of \( \mathfrak{g} \) . Then there exists an automorphism \( \varphi \in \operatorname{Int}\left( \mathfrak{g}\right) \) such that \( \varphi \left... | To prove this theorem, we need some preliminary results. Let \( \mathfrak{g} = {\mathfrak{n}}^{ - } + \mathfrak{h} + {\mathfrak{n}}^{ + } \) be the triangular decomposition of \( \mathfrak{g} \) from Corollary 2.5.25 and let \( \mathfrak{b} = \mathfrak{h} + {\mathfrak{n}}^{ + } \) be the corresponding Borel subalgebra.... | No |
Lemma 2.5.29. Suppose \( Z \in \mathfrak{b} \) is semisimple. Write \( Z = H + Y \), where \( H \in \mathfrak{h} \) and \( Y \in {\mathfrak{n}}^{ + } \) . Then \( \dim \operatorname{Ker}\left( {\operatorname{ad}Z}\right) = \dim \operatorname{Ker}\left( {\operatorname{ad}H}\right) \geq \dim \mathfrak{h} \), with equalit... | Proof. Enumerate the positive roots in order of nondecreasing height as \( \left\{ {{\beta }_{1},\ldots ,{\beta }_{n}}\right\} \) and take an ordered basis for \( \mathfrak{g} \) as\n\n\[ \left\{ {{X}_{-{\beta }_{n}},\ldots ,{X}_{-{\beta }_{1}},{H}_{1},\ldots ,{H}_{l},{X}_{{\beta }_{1}},\ldots ,{X}_{{\beta }_{n}}}\righ... | Yes |
Lemma 2.5.30. Let \( H \in \mathfrak{h} \) be regular. Define \( f\left( X\right) = \exp \left( {\operatorname{ad}X}\right) H - H \) for \( X \in {\mathfrak{n}}^{ + } \) . Then \( f \) is a polynomial map of \( {\mathfrak{n}}^{ + } \) onto \( {\mathfrak{n}}^{ + } \) . | Proof. Write the elements of \( {\mathfrak{n}}^{ + } \) as \( X = \mathop{\sum }\limits_{{\alpha \in {\Phi }^{ + }}}{X}_{\alpha } \) with \( {X}_{\alpha } \in {\mathfrak{g}}_{\alpha } \) . Then\n\n\[ f\left( X\right) = \mathop{\sum }\limits_{{k \geq 1}}\frac{1}{k!}{\left( \operatorname{ad}X\right) }^{k}H = - \mathop{\s... | Yes |
Corollary 2.5.31. Suppose \( Z \in \mathfrak{b} \) is semisimple and \( \dim \operatorname{Ker}\left( {\operatorname{ad}Z}\right) = \dim \mathfrak{h} \) . Then there exist \( X \in {\mathfrak{n}}^{ + } \) and a regular element \( H \in \mathfrak{h} \) such that \( \exp \left( {\operatorname{ad}X}\right) H = Z \) . | Proof. Write \( Z = H + Y \) with \( H \in \mathfrak{h} \) and \( Y \in {\mathfrak{n}}^{ + } \) . By Lemma 2.5.29, \( H \) is regular, so by Lemma 2.5.30 there exists \( X \in {\mathfrak{n}}^{ + } \) with \( \exp \left( {\operatorname{ad}X}\right) H = H + Y = Z \) . | Yes |
Lemma 2.5.32. Suppose \( {\mathfrak{b}}_{i} = {\mathfrak{h}}_{i} + {\mathfrak{n}}_{i} \) are Borel subalgebras of \( \mathfrak{g} \), for \( i = 1,2 \) . Then\n\n\[{\mathfrak{b}}_{1} = {\mathfrak{b}}_{1} \cap {\mathfrak{b}}_{2} + {\mathfrak{n}}_{1}\]\n\n(2.58) | Proof. The right side of (2.58) is contained in the left side, so it suffices to show that both sides have the same dimension. For any subspace \( V \subset \mathfrak{g} \) let \( {V}^{ \bot } \) be the orthogonal of \( V \) relative to the Killing form on \( \mathfrak{g} \) . Then \( \dim {V}^{ \bot } = \dim \mathfrak... | Yes |
Theorem 3.1.1. \( {W}_{G} \) is a finite group and the representation of \( {W}_{G} \) on \( {\mathfrak{h}}^{ * } \) is faithful. | Proof. Let \( s \in {\operatorname{Norm}}_{G}\left( H\right) \) . Suppose \( s \cdot \theta = \theta \) for all \( \theta \in \mathfrak{X}\left( H\right) \) . Then \( {s}^{-1}{hs} = h \) for all \( h \in H \), and hence \( s \in H \) by Theorem 2.1.5. This proves that the representation of \( {W}_{G} \) on \( {\mathfra... | Yes |
For \( G = \mathbf{Sp}\left( {{\mathbb{C}}^{2l},\Omega }\right) \), the subgroup \( {T}_{l} \subset {W}_{G} \) is normal, and \( {W}_{G} \) is the semidirect product of \( {T}_{l} \) and \( \bar{\pi }\left( {\mathfrak{S}}_{l}\right) \). The action of \( {W}_{G} \) on the coordinate functions in \( \mathcal{O}\left\lbra... | Recall that a group \( K \) is a semidirect product of subgroups \( L \) and \( M \) if \( M \) is a normal subgroup of \( K, L \cap M = 1 \), and \( K = L \cdot M \). By (3.3) we see that it suffices to prove that \( {W}_{G} = {T}_{l}\pi \left( {\mathfrak{S}}_{l}\right) \). Suppose \( s \in \) \( {\operatorname{Norm}}... | Yes |
Lemma 3.1.3. Let \( G = \mathbf{SO}\left( {\mathbb{C}}^{2l + 1}, B\right) \). The subgroup \( T_l \subset W_G \) is normal, and \( W_G \) is the semidirect product of \( T_l \) and \( \bar{\varphi}\left( \mathfrak{S}_l\right) \). The action of \( W_G \) on the coordinate functions in \( \mathcal{O}\left\lbrack H\right\... | Proof. Suppose \( s \in \operatorname{Norm}_G\left( H\right) \). Then there exists \( \sigma \in \mathfrak{S}_{2l + 1} \) such that \( s \) is given by (3.2), with \( n = 2l + 1 \). The action of \( s \) as an automorphism of \( H \) is\n\n\[ s \cdot \operatorname{diag}\left\lbrack a_1,\ldots, a_n\right\rbrack s^{-1} =... | No |
Lemma 3.1.4. Let \( G = \mathbf{{SO}}\left( {{\mathbb{C}}^{2l}, B}\right) \). The subgroup \( {R}_{l} \subset {W}_{G} \) is normal, and \( {W}_{G} \) is the semidirect product of \( {R}_{l} \) and \( \bar{\pi }\left( {\mathfrak{S}}_{l}\right) \). The action of \( {W}_{G} \) on the coordinate functions in \( \mathcal{O}... | Proof. By the same argument as in the proof of Lemma 3.1.2 we see that the normalizer of \( H \) in \( \mathbf{O}\left( {{\mathbb{C}}^{2l}, B}\right) \) is given by the \( H \) cosets of the elements \( {\beta }_{F}\pi \left( \sigma \right) \) as \( \sigma \) ranges over \( {\mathfrak{S}}_{l} \) and \( F \) ranges over... | Yes |
For every \( \alpha \in \Phi \) there exists \( w \in W \) such that \( w \) acts on \( {\mathfrak{h}}^{ * } \) by the reflection \( {s}_{\alpha } \) . | Type \( \mathbf{A}\left( {G = \mathbf{{SL}}\left( {l + 1,\mathbb{C}}\right) }\right) \) : Here \( W \cong {\mathfrak{S}}_{l + 1} \) acts on \( {\mathfrak{h}}^{ * } \) by permutations of \( {\varepsilon }_{1},\ldots ,{\varepsilon }_{l + 1} \) . Let \( \alpha = {\varepsilon }_{i} - {\varepsilon }_{j} \) . Then\n\n\[ \lef... | Yes |
Theorem 3.1.16. Let \( \left( {\pi, V}\right) \) be a finite-dimensional representation of \( \mathfrak{g} \) . For \( \mu \in {\mathfrak{h}}^{ * } \) set\n\n\[ \nV\left( \mu \right) = \{ v \in V : \pi \left( Y\right) v = \langle \mu, Y\rangle v\;\text{ for all }Y \in \mathfrak{h}\} .\n\]\n\nCall \( \mu \in {\mathfrak{... | Proof. Take the three-dimensional simple algebra \( \mathfrak{s}\left( \alpha \right) \) containing \( {h}_{\alpha } \), as in Sections 2.4.2 and 2.5.2, and apply Theorem 2.3.6 to the restriction of \( \pi \) to \( \mathfrak{s}\left( \alpha \right) \) to prove the first assertion. To obtain (3.14), observe that the cor... | Yes |
Lemma 3.1.17. \( P\left( \mathfrak{g}\right) \) and \( Q\left( \mathfrak{g}\right) \) are invariant under the Weyl group \( W \) . | Proof. Since \( W \) permutes the elements of \( \Phi \) and acts linearly on \( {\mathfrak{h}}^{ * } \), it leaves the root lattice invariant. Let \( \alpha \in \Phi \) and \( \mu \in P\left( \mathfrak{g}\right) \) . Then\n\n\[ \n{s}_{\alpha }\mu = \mu - \left\langle {\mu ,{h}_{\alpha }}\right\rangle \alpha \in \mu + ... | Yes |
Proposition 3.1.19. When \( G = \mathbf{SO}\left( 2l + 1, \mathbb{C} \right) \), then \( P_{+ + }\left( G \right) \) consists of all weights \( n_1 \varpi_1 + \cdots + n_{l-1} \varpi_{l-1} + n_l (2 \varpi_l) \) with \( n_i \in \mathbb{N} \) for \( i = 1, \ldots, l \). When \( G = \mathbf{SO}\left( 2l, \mathbb{C} \right... | Proof. In both cases we have \( P\left( G \right) = \sum_{i=1}^l \mathbb{Z} \varepsilon_i \). Thus the first assertion is obvious from the formulas for \( \bar{\omega}_i \). Now assume \( G = \mathbf{SO}\left( 2l, \mathbb{C} \right) \). Then every weight of the form (3.16) is in \( P_{+ + }\left( G \right) \). Converse... | Yes |
Proposition 3.1.20. For every \( \lambda \in P\left( \mathfrak{g}\right) \) there is \( \mu \in {P}_{+ + }\left( \mathfrak{g}\right) \) and \( s \in W \) such that \( \lambda = s \cdot \mu \) . The weight \( \mu \) is uniquely determined by \( \lambda \) . If \( \mu \) is regular, then \( s \) is uniquely determined by... | Proof. For a general reductive Lie algebra, the first part of the proposition follows from Proposition 3.1.12. We will prove (1)-(4) for the classical Lie algebras by explicit calculation.\n\n(1): The Weyl group is \( W = {\mathfrak{S}}_{n} \), acting on \( {\mathfrak{h}}^{ * } \) by permuting \( {\varepsilon }_{1},\ld... | Yes |
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