Lie Groups, Lie Algebras, and Their Representations

V. S. Varadarajan (auth.)

Chapter 4 Complex Semisimple Lie Algebras And Lie Groups: Structure and Representation - all with Video Answers

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Chapter Questions

Problem 1

Determine all CSA's of $\mathrm{g}=s \mathfrak{l}(n, \mathbf{R})$

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Let $G=S O(1, n)(n \geq 2)$ and let $r(G)$ denote the maximum number of mutually nonconjugate CSA's of g. Prove that $r(G)=1$ or 2 according as $n$ is odd or even. Determine also the number of connected components of $\mathrm{g}^{\prime}$.

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Problem 3

Let $\mathrm{g}=A_l$ and $X=E_{12}+E_{23}+\cdots+E_{l, l+1}$ (notation as in §4.4). Determine the centralizer of $X$ in $g$ and verify that it is a maximal abelian subalgebra of g of dimension $l$ consisting entirely of nilpotent elements.

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Problem 4

Let $\pi,(j=0,1,2, \ldots)$ be the representations of $S L(2, C)$ constructed in §4.2. Let $j, j^{\prime}$ be two integers, $0 \leq j \leq j^{\prime}$. Prove that $\pi / \otimes \pi_j$ is the direct sum of $\pi_r, r=j^{\prime}-j+2 k, k=0,1, \ldots, j$ (Clebsch-Gordon series).

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Problem 5

Let notation and assumptions be as in Lemma 4.6.7. Prove that in this case, for any weight $\mu$ of $\pi, \operatorname{dim} V_{s \mu}=\operatorname{dim} V_\mu$ for all $s \in \mathfrak{w}$.

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Problem 6

(a) Let $n^{ \pm}$be as in Lemma 4.6.1 and let $q^{+}$(resp. $q^{-}$) be the ideal in $n^{+}$generated by all the $\theta_{i j}^{+}$(resp. ideal in $\pi^{-}$generated by all the $\theta_{i j}^{-}$). Prove that $q^{+}$and $q^{-}$are ideals in $q$. Deduce that $q=q^{+}+q^{-}$is the ideal in $g$ generated by all the $\theta_{i j}^{+}$.
(b) Let 11 be an ideal of $\mathfrak{g}, \hat{\mathfrak{g}}=\mathfrak{g} / 11, X \mapsto \hat{X}$ the natural map of $g$ onto $\hat{g}$. Assume that $u \cap \mathfrak{G}=0$ and that ad $\hat{X}_i$ and ad $\hat{Y}_i$ are locally nilpotent endomorphisms of $\hat{\mathrm{g}}$ for $1 \leq i \leq l$. For $\lambda \in \mathfrak{h}^*$, let $\hat{\mathrm{g}}_\lambda=\{X: X \in \hat{\mathrm{~g}}$, $[\hat{H}, X]=\lambda(H) X$ for all $H \in \mathfrak{h}]$. Let $\hat{\mathfrak{h}}$ be the image of $\mathfrak{h}$ in $\hat{\mathrm{g}}$, and let $\hat{\mathrm{w}}$ be the subgroup of $G L(\hat{h})$ that corresponds to w under the mapping $H \mapsto \hat{H}(H \in \hat{h})$. Prove that given any $s \in \hat{t}$, there is an "inner" automorphism $x(s)$ of $\hat{\mathrm{g}}$ such that $x(s)$ leaves $\hat{\hat{h}}$ invariant and $x(s) \mid \hat{\hat{h}}=s$. Deduce that $\operatorname{dim} \hat{\mathrm{g}}_{s \lambda}=\operatorname{dim} \hat{\mathrm{g}}_\lambda$ for $\lambda \in \mathfrak{h}^*, s \in \mathfrak{w}$.
(c) Let $\Delta=\bigcup_{1 \leq i \leq l}$ tv $\cdot \alpha_i$. For $\lambda \in \Delta$, prove that $\operatorname{dim} \hat{g}_{e \lambda}=1$ or 0 according as $c= \pm 1$ or $c \neq \pm 1$.
(d) Let $P=\Delta \cap \Gamma$. Prove that $\Delta=P \cup(-P)$ and that if $1 \leq i \leq l$, $s_i\left[P \backslash\left[\alpha_i\right]\right]=P \backslash\left\{\alpha_i\right\}$.
(e) Let $\mathfrak{h}_0$ be the linear span of the $H_i$ over $\mathbf{R}, \mathfrak{h}_0^{\prime}$ the set where no element of $\Delta$ vanishes, and $\mathfrak{h}_0^{+}$the set where all the $\alpha_i$ take positive values. Prove that $\mathfrak{h}_0^{+}$is a connected component of $\mathfrak{h}_0^{\prime}$ and that to acts transitively on the connected components of $\mathfrak{h}_0^{\prime}$.
(f) Suppose $\lambda \neq 0$ lies in $\hat{h}^* \backslash \Delta$. Prove that $\hat{\hat{G}}_\lambda=0$. Deduce that $\operatorname{dim}(\hat{\hat{G}})<\infty$.
(g) Prove that $\hat{\mathrm{g}}$ is semisimple.
(h) Prove that $q$ is the unique ideal of $g$ such that $q \cap \mathfrak{h}=0$ and $\operatorname{dim}(g / q) <\infty$, and hence that it is the unique ideal described in Lemma 4.8.2.

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Problem 7

Let $\mathrm{g}=\mathrm{s}(2, \mathrm{C}), \mathcal{G}$ the universal enveloping algebra of g , and $H, X, Y$ as in (4.2.2). For $\lambda \in \mathbf{C}$ let $\bar{\pi}_\lambda$ be the natural representation of $\mathscr{B}$ in $\mathscr{G}^{\circ} / \mathscr{G}_\lambda$, where $\zeta_\lambda=\mathcal{G} X+\mathcal{G}(H-\lambda \cdot 1)$. Prove that $\dot{\pi}_\lambda$ is irreducible if and only if $\lambda$ is not a nonnegative integer.

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Problem 8

Let $V$ be a real Hilbert space of finite dimension, $V_c$ the complexification of $V$. We regard $V_c$ as a complex Hilbert space in the natural fashion. Let $\Delta \subseteq V$ be an integral root system and to the associated finite reflection group. Let $D$ be the additive subgroup of $V$ generated by $\Delta$. For $\lambda \in V_c$, let $\mathrm{m}(\lambda)= [s: s \in \mathfrak{v}, \lambda-s \lambda \in D]$ and $\Delta(\lambda)=\{\alpha: \alpha \in \Delta, 2(\lambda, \alpha) /(\alpha, \alpha) \in \mathbf{Z}\}$. Let $P$ be a positive system $\subseteq \Delta$, and let $\Gamma$ be the set of all finite sums of elements of $P$.
(a) Prove that $\Delta(\lambda)$ is an integral root system.
(b) Prove that $m(\lambda)$ is the finite reflection group generated by the $s_\alpha(\alpha \in \Delta(\lambda))$.
(c) Let $\mathbf{Z}^{+}$be the set of positive integers, and suppose that $2(\lambda, \alpha) /(\alpha, \alpha) \notin \mathbf{Z}^{+}$ for each $\alpha \in P$. If $\lambda-s \lambda \in D \backslash\{0\}$ for some $s \in \mathfrak{w}$, prove that $\lambda-s \lambda \in-\Gamma$. (

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Problem 9

Let g be a semisimple Lie algebra over $\mathbf{C}$, and let other notation be as in §4.7. Fix $\lambda \in \mathfrak{h}^*$, and let $\mathfrak{G}_\lambda=\sum_{\alpha \in P} \mathfrak{b}_{\mathrm{g}_\alpha}+\sum_{H \in \mathfrak{G}}\left(\mathcal{B}(H-\lambda(H) 1)\right.$. Denote by $\tilde{\pi}_\lambda$ the natural representation of $\mathbf{6}$ in $V=\left(\ddot{3} / \tilde{6}_\lambda \text {. Let } \Omega(\lambda) \text { be the set of all } \mu \in \mathfrak{h}\right)^*$ such that $\mu \leqslant \lambda$ and $\mu+\delta \in \mathfrak{m}+(\lambda+\delta)$.
(a) Let $U_1$ and $U_2$ be two subspaces of $V$ invariant under $\bar{\pi}_2$ with $U_1 V_2$. Let $W=U_2 / U_1$, and let $\pi_W$ be the representation induced in $W$. Prove that there exists $\mu \in \mathfrak{h}^*$ and a nonzero $u \in W_\mu$ such that $\pi_W\left[\mathfrak{g}_\alpha\right] u=0$ for all $\alpha \in P$, and that any such $\mu$ belongs to $\Omega(\lambda)$.
(b) Take $U_1=0, U_2=V$ in (a), let $\mu$ and $u$ be as above, and put $V^{\prime}= \bar{\pi}_\lambda[$ Э $] u$. Prove that the subrepresentation of $\bar{\pi}_\lambda$ defined by $V^{\prime}$ is equivalent to $\tilde{\pi}_\mu$.
(c) Prove that any strictly monotonic sequence of invariant subspaces of $V$ is finite. Deduce that $\bar{\pi}_\lambda$ has a finite Jordan series and that the irreducible constituents of the series are of the form $\pi_\mu$, where $\mu \in \Omega(\lambda)$.
(d) Suppose $(\lambda+\delta)\left(\bar{H}_\alpha\right) \in \mathbf{C} \backslash \mathbf{Z}^{+}$for all $\alpha \in P$. Prove that $\bar{\pi}_\lambda$ is irreducible and hence equivalent to $\pi_\lambda$.
(e) Suppose $\lambda$ is such that for some simple root $\alpha, \lambda\left(\bar{H}_\alpha\right)$ is an integer $\geq 0$. Let $\mu=\lambda-\left(\lambda\left(\bar{H}_\alpha\right)+1\right) \alpha$. Prove that $\bar{\pi}_\alpha$ occurs as a subrepresentation of $\bar{\pi}_\lambda$.
(f) Suppose $\lambda$ is dominant integral. Prove that for any $s \in \mathfrak{m}, \bar{\pi}_{a(\lambda+s)-s}$ occurs as a subrepresentation of $\ddot{\pi}_\lambda$.

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Problem 10

We continue with the above setup. The partition function $\mathbf{P}$ is defined on 5$)^*$ in the following way: $\mathbf{P}(\mu)=0$ if $\mu \notin \overline{\boldsymbol{\Gamma}}=\bar{\Gamma} \cup\{0\}$; if $\mu \in \overline{\boldsymbol{\Gamma}}, \mathbf{P}(\mu)$ is the number of distinct functions $\alpha \mapsto k(\alpha)$ defined on $P$ such that $k(\alpha)$ is a nonnegative integer for all $\alpha \in P$ and $\mu=\sum_{\alpha \in P} k(\alpha) \alpha$.
(a) If $\lambda \in \mathfrak{h}^*$, the multiplicity of the weight $\mu$ in $\bar{\pi}_\lambda$ is $\mathbf{P}(\lambda-\mu)$.
(b) Prove the existence of a polynomial function $p$ on $\mathfrak{h}^*$ such that $|\mathbf{P}(\mu)| \leq|p(\mu)|$ for all $\mu$.
(c) Let $c_1, \ldots, c_p$ be constants $\geq 0$ and $v_1, \ldots, v_p$ fixed elements of $\Gamma$. Suppose $\mathbf{P}(v) \geq \sum_i c_i \mathbf{P}\left(v-v_i\right)$ for all $v \in \Gamma$. Then show that $\sum_i c_i \leq 1$.
(d) Prove that each irreducible constituent of a Jordan series of the representation $\hat{\pi}_\lambda$ occurs with multiplicity 1 .

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Problem 11

Let $\mathfrak{F}$ be the complex vector space of all formal sums of exponentials $f= \sum_{v \in v^*} c_v e^v\left(c_v \in \mathbf{C}\right)$; we write $c_v=c_v(f)$ and denote by $\operatorname{supp}(f)$ the set of all $v$ such that $c_v(f) \neq 0$. We denote by $\mathcal{E}$ the subset of all $f \in \mathcal{F}$ for which $\operatorname{supp}(f)$ has the following property: there exists a finite subset $\Phi=\Phi_f \subseteq \mathfrak{h}^*$ such that $\operatorname{supp}(f) \subseteq \bigcup_{\mu \in \Phi}(\mu-\bar{\Gamma})$.
(a) Prove that $\mathscr{E}$ is a linear subspace of $\mathscr{F}$.
(b) Given $f, g \in \mathcal{S}$, prove the existence of a unique element $h \in \mathcal{E}$ such that $c_v(h)=\sum_{v^{\prime}+v^{\prime}=v} c_v(f) c_v(g)$ for all $v \in h^*$. Writing $h=f \cdot g$, prove that $\varepsilon$ becomes an associative and commutative algebra with unit under this definition of multiplication.
(c) Let $\Delta=e^s \Pi_{\alpha \in P}\left(1-e^{-\alpha}\right)$. Prove that $\Delta$ is an invertible element of $\varepsilon$ and that $\Delta^{-1}=\sum_{\mu \in \bar{\Gamma}} \mathbf{P}(\mu) e^{-(\mu+\delta)}$.
(d) For any representation $\pi$ of $B$ with weights such that all its weights are of finite multiplicity, we define the formal character $\theta(\pi)$ to be the element $\sum_\mu m(\pi: \mu) e^\mu$ of $\mathscr{F}$ where $m(\pi: \mu)$ is the multiplicity of $\mu$ in $\pi$. If there is a finite subset $\Phi \subseteq \mathfrak{h}^*$ such that all the weights of $\pi$ are of the form $\mu-v$ with $\mu \in \Phi, v \in \bar{\Gamma}$, prove that $\theta(\pi) \in \mathcal{E}$.
(e) For any $\lambda \in \mathfrak{h}^*$, prove that $\theta\left(\bar{\pi}_\lambda\right)=e^{\lambda+s} \cdot \Delta^{-1}$.
(f) Fix $\lambda \in \mathfrak{h}^*$ and let $\mathcal{E}(\lambda)$ be the linear span of all $\theta(\pi)$ where $\pi$ is an irreducible representation which has a highest weight and whose infinitesimal character is the same as $\pi_\lambda$. Prove that $\left\{\theta\left(\tilde{\pi}_\mu\right): \mu+\delta \in \mathfrak{w} \cdot(\lambda+\delta)\right\}$ is a basis for $\varepsilon(\lambda)$.
(g) Suppose $\lambda$ is dominant integral. Prove that $\theta\left(\pi_\lambda\right)=\sum_{s \in n} \epsilon(s)^{s(\lambda+\delta)} \cdot \Delta^{-1}$.

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Problem 12

Obtain the following expressions for the Cartan-Killing forms of the classical Lie algebras:

$$
\begin{array}{ll}
\mathrm{g}=\mathrm{s}(l+1, \mathrm{C}):\langle X, Y\rangle & =2(l+1) \operatorname{tr}(X Y) \\
\mathrm{g}=\mathrm{v}(2 l, \mathrm{C}):\langle X, Y\rangle & =2(l-1) \operatorname{tr}(X Y) \\
\mathrm{g}=\mathrm{v}(2 l+1, \mathrm{C}):\langle X, Y\rangle & =(2 l-1) \operatorname{tr}(X Y) \\
\mathrm{g}=\mathrm{sp}(l, \mathrm{C}):\langle X, Y\rangle & =2(l+1) \operatorname{tr}(X Y) .
\end{array}
$$

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Problem 13

We use the notation of §4.7.
(a) Let $\lambda \in \mathscr{D}_p$, and for any integral linear function $v$ on $\mathfrak{h}$ let $m_\lambda(v)$ be the multiplicity of the weight $y$ in $\pi_\lambda$. Prove that

$$
m_\lambda(v)=\sum_{s \in m} \epsilon(s) \mathbf{P}(s(\lambda+\delta)-(v+\delta)) .
$$
(b) Let $\lambda_1 \lambda_2 \in \mathscr{D}_P$, and for $\Lambda \in \mathscr{D}_P$, let $M(\Lambda)$ be the multiplicity of $\pi_A$ in $\pi_{\lambda_1} \otimes \pi_{\lambda_1}$. Prove that

$$
M(\Lambda)=\sum_{s, t c m} \epsilon(s) \epsilon(t) \mathbf{P}\left(s\left(\lambda_1+\delta\right)+t\left(\lambda_2+\delta\right)-(\Lambda+2 \delta)\right)
$$

The formula in (a) is due to Kostant [2]; that in (b) to Steinberg (cf. Jacobson [1]).

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Problem 14

Let $\mathfrak{B}=A_3 ; S=\left\{\alpha_1, \alpha_2\right\}$ a simple system of roots. For integers $m_1, m_2 \geq 0$ let $\pi_{m_1, m_2}$ denote the irreducible representation $\pi_\lambda$ where $\lambda\left(\bar{H}_{\alpha_1}\right)=m_i(1=1,2)$. Prove the decomposition formula

$$
\pi_{m, 0} \otimes \pi_{0, m} \simeq \pi_{0,0} \oplus \pi_{1,1} \oplus \cdots \oplus \pi_{m, m}
$$

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Problem 15

(a) Let $g$ be arbitrary semisimple, $\mathfrak{y}$ a CSA, and $S$ a simple system of roots. Suppose $\lambda \in \mathfrak{b}^*$ is such that $\langle\lambda, \alpha\rangle \geq 0$ (resp. $>0$ ) for all $\alpha \in S$. Then prove that $\lambda=\sum_{\varepsilon \in S} m(\alpha) \alpha$ where the $m(\alpha)$ are all $\geq 0$ (resp. $>0$ ). If $\lambda$ is integral, prove that the $m(\alpha)$ are rational.
(b) Let $A$ be the Cartan matrix $\left(\alpha_j\left(\bar{H}_\alpha\right)\right)$. Show that all the entries of the matrix $A^{-1}$ are $\geq 0$.
(c) Let $\mathfrak{h}_{\mathbf{R}}=\sum_{1 \leq i \leq \ell} \mathbf{R} \cdot H_{\alpha_1}$ and let $\mathfrak{h}_{\mathbf{R}}$ be the set of all $H \in \mathfrak{h}_{\mathbf{R}}$ such that $\alpha_l(H)>0$ for $1 \leq i \leq l$. Prove that $\left\langle H, H^{\prime}\right\rangle \geq 0$ for all $H, H^{\prime} \in C l\left(h_{\dot{k}}\right)$.

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Problem 16

Let $g, \mathfrak{h}$ and $S=\left\{\alpha_1, \ldots, \alpha_i\right]$ be as above. Let $\Gamma$ denote, as usual, the set of all $\mu \in \mathfrak{h}^* \backslash\{0\}$ of the form $m_1 \alpha_1+\cdots+m_i \alpha_i$ where the $m_i$ are all integers $\geq 0$. Prove that for any $\lambda \in \mathfrak{D}_P$, the irreducible representation $\pi_\lambda$ has 0 as a weight if and only if $\lambda \in \Gamma \cup\{0\}$.

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Problem 17

Let notation be as above, to is the Weyl group and $\Delta$ the set of roots.
(a) Suppose $\mathfrak{h}$ is the direct sum of subspaces $\mathfrak{h}_1, \ldots, h$, which are 11 -invariant. Let $\Delta_j=\left\{\alpha: \alpha \in \Delta, H_z \in \mathfrak{h}_j\right\}$. Prove that $\Delta=\bigcup_{1 \leq j \leq r} \Delta_j$ and that $\left[\mathrm{g}_\alpha, \mathrm{g}_\beta\right]=0$ whenever $\alpha \in \Delta_j, \beta \in \Delta_k$, and $j \neq k$. Deduce that $\mathrm{g}_j=$ h) $+\sum_{\alpha \in \Delta,} g_\alpha$ are ideals of $g$ and $g$ is their direct sum.
(b) Prove that g is simple if and only if 10 acts irreducibly on $\mathfrak{h}$, and that this is equivalent to the condition that to acts irreducibly on $\mathfrak{b}_R$.
(c) Suppose g is simple. If $\alpha$ and $\beta$ are two roots, then, in order that there should exist an $s \in \mathrm{w}$ such that $s \alpha=\beta$ it is necessary and sufficient that $\langle\alpha, \alpha\rangle=\langle\beta, \beta\rangle$. (
(d) Let $n$ be the number of w -orbits in $\Delta$. Prove that $n=1$ if $\mathrm{g}=A_i(l \geq 1)$, $D_i(l \geq 3), E_i(l=6,7,8)$ while $n-2$ if $\mathrm{g}=B_i(l \geq 2), C_i(l \geq 3), G_2, F_4$.

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Problem 18

Let $\mathfrak{g}, \mathfrak{b}, P$ be as in §4.7. Let $\mathcal{G}$ be the universal enveloping algebra of $g$. Let $S=\left\{\alpha_1, \ldots, \alpha_i\right\}$ be the simple system of roots in $P, H_i=\bar{H}_{\alpha_1}(1 \leq i \leq l)$. Let $\Re$ be the subalgebra of $\mathcal{G}$ generated by 1 and the $g_{-\alpha_1}, 1 \leq i \leq l$. Prove that for any $\lambda \in D_F$

$$
\Re \cap M_\lambda=\sum_{1 \leq r \leq f} \Re Y_f^{\lambda_1+1} ;
$$

here $\lambda_l=\lambda\left(H_i\right), Y_i$ is a nonzero element of $\mathrm{g}_{-\alpha_i}$, and $\Re_2$ is the maximal left ideal in 56 corresponding to $\pi_\lambda$.

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Problem 19

We continue in the above context. Let $s_0 \in \mathfrak{v}$ be such that $s_0 \cdot P=-P$. For $\lambda \in \mathfrak{h}^*$ let us write $\lambda^*=-s_0 \lambda$. Let $X_i, Y_i(1 \leq i \leq l)$ be as in the previous exercise. Given $\mu, \nu \in \mathfrak{D}_P$, we define, for any integral linear function $\gamma$ on $\mathfrak{h}$, $V^{+}(\mu: \gamma: v)=\left[v: v\right.$ in the space of $\pi_\mu, v$ of weight $\gamma$, and $\pi_\mu\left(X_i\right)^{v_1+1} v=0$ for $1 \leq i \leq l\} ; V^{-}(\mu: \gamma ; v)$ is defined analogously, with $Y_i$ replacing $X_i$ ( $\nu_i=\nu\left(H_i\right)$ as usual).
(a) Prove that for $\mu, v \in \mathfrak{D}_P$ and $\gamma$ integral, $\operatorname{dim} V^{+}(\mu: \gamma: v)= \operatorname{dim} V^{-}\left(\mu:-\gamma^*: v^*\right)$.
(b) Let $\lambda_1, \lambda_2 \in \mathscr{D}_P, W_1, W_2$ the spaces on which $\pi_\lambda$ and $\pi_{\lambda,}$ act, $V$ the vector space of linear maps of $W_1$ into $W_2$. For $X \in g$ let $\pi(X)$ be the endomorphism of $V$ defined by $\pi(X) L=\pi_{\lambda_2}(X) L-L \pi_{\lambda i}(X), L \in V$. Prove that $\pi$ is a representation of $\beta$ equivalent to $\pi_{\lambda_1} \otimes \pi_{\lambda_2}$.
(c) With notation as in (b), let $U=\left[L: L \in V, \pi\left(Y_i\right) L=0,1 \leq i \leq l\right]$. Let $w$ be a nonzero vector of weight $\lambda_1^*$ in $W_1$. Prove that the map $\xi: L \mapsto L w$ is a linear isomorphism of $U$ onto the subspace $U^{\prime}$ of $W_2$ given by $U^{\prime}= \left\{v: v \in W_2, \pi_{\lambda a}\left(Y_i\right)^{\lambda_i+1} v=0,1 \leq i \leq l\right\}$, where $\lambda_{i i}^*=\lambda_1^*\left(H_i\right)$.
(d) Deduce from (c) that for $\gamma \in \mathscr{D}_P$, the multiplicity of $\pi_Y$ in $\pi_{\lambda_1} \otimes \pi_{\lambda z}$ is precisely $\operatorname{dim} V^{-}\left(\lambda_2:-\gamma^*+\lambda_1^*: \lambda_1^*\right)$.
(e) Show that the multiplicity of $\pi_\gamma$ in $\pi_{\lambda_1} \otimes \pi_{\lambda_2}$ is also equal to the dimension of $V^{-}\left(\gamma: \lambda_1-\lambda_2^*: \lambda_1\right)$ as well as to the dimension of $V^{+}\left(\gamma: \lambda_2-\lambda_1^*: \lambda_1^*\right)$.
(f) Let $\nu$ be the unique element of $\mathscr{D}_p$ in $w \cdot\left(\lambda_1-\lambda_2^*\right)$. Prove that $\pi_{\text {, occurs in }} \pi_{\lambda_1} \otimes \pi_{\lambda_2}$ with multiplicity 1 . Prove also that $v$ is a weight of every $\pi_7 \left(\gamma \in \mathfrak{D}_p\right)$ that enters the decomposition of $\pi_{\lambda_1} \otimes \pi_\lambda$.
(g) For fixed $\gamma \in \mathscr{D}_p$, prove that the multiplicity of $\pi_\gamma$ in $\pi_{\lambda_1} \otimes \pi_{\lambda_2}$ is the multiplicity of the weight $\lambda_1-\lambda_2^*$ in $\pi_y$ whenever $\lambda_1\left(H_i\right) \geq \operatorname{dim}\left(\pi_z\right)-1$ for $1 \leq i \leq l$, in particular if all the $\lambda_1\left(H_i\right)$ are sufficiently large.
(h) Prove that the number of irreducible constituents of $\pi_{\lambda_1} \otimes \pi_{\lambda_2}$ cannot exceed $\min \left(\operatorname{dim}\left(\pi_{j_1}\right), \operatorname{dim}\left(\pi_{\lambda t}\right)\right)$.
In this connection, see Kostant [2], Parthasarathy et al [1].

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Problem 20

Let notation be as in Exercises 18 and 19.
(a) Let $\tau$ be the adjoint representation of $g$ in (\%). Let $\mu \in \mathscr{D}_p$; let $\pi_\alpha$ be the corresponding irreducible representation of 8 acting on a vector space $V^\mu$; and let $E^\mu$ be the algebra of endomorphisms of $V^\mu$. For $X \in \mathbb{B}$, let $\tau^\mu(X) v=\left[\pi_\mu(X), v\right]\left(v \in E^\mu\right)$. Show that the map $a \mapsto \pi_\mu(a)(a \in(b)$ intertwines the representations $\tau$ and $\tau^\mu$ of g.
(b) Let $\lambda \in D_P$ be such that $\pi_\lambda$ has 0 as a weight; write $d_\lambda$ for the multiplicity of 0 in $\pi_\lambda$. Prove that for suitable $\mu, \pi_\lambda$ occurs as an irreducible constituent of $\tau^\mu$ with multiplicity $d_2$.
(c) Let $S(\mathrm{q})$ be the symmetric algebra over g . For each $X \in \mathrm{~g}$ let $\sigma(X)$ denote the derivation of $S(\mathrm{q})$ that extends ad $X$. Prove, using (a) and (b) that $\pi_\lambda\left(\lambda \in \mathfrak{D}_P\right)$ occurs as an irreducible constituent of $\sigma$ if and only if 0 is weight of $\pi_\lambda$, i.e., if and only if $\lambda \in \Gamma \cup\{0\}$.

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Problem 21

Let B be one of $A_l(l \geq 1), B_l(l \geq 1), C_l(l \geq 2), D_l(l \geq 2)$. Let notation be as in 4.4. Define the $\mu_i \in \mathscr{D}_p$ by $\mu_i\left(\bar{H}_a\right)=\delta_{i j}, 1 \leq i, j \leq 1$. Write $d_i=\operatorname{dim}\left(\pi_n\right)$, $1 \leq i \leq l$.
(a) Let $n=A_i$; then $d_i=\binom{l+1}{i}$.
(b) Let $\Omega=B_j$; then $d_i=\binom{2 l+1}{i}$ for $1 \leq i \leq l-1, d_j=2^{\prime}$. Verify also that $\operatorname{dim}\left(\pi_{2, a}\right)=\binom{2 l+1}{l}$.
(c) Let $\mathbb{n}=C_f$; then $d_i=\binom{2 /}{i}-\binom{2 /}{i-2}$ for $1 \leq i \leq l$. (For $r<0,\binom{n}{r}$ is defined to be 0 ).
(d) Let $\beta=D_j$; then $d_i=\binom{2 l}{i}$ for $1 \leq i \leq l-2, d_{l-1}=d_l=2^{l-1}$. Verify also that the linear forms $\lambda_1+\cdots+\lambda_{l-1}, \lambda_1+\cdots+\lambda_l, \lambda_1+\cdots +\lambda_{i-1}-\lambda_i$ are in $\mathfrak{D}_P$ and that the dimensions of the corresponding representations are respectively $\binom{2 l}{l}, \frac{1}{2}\binom{2 l}{l}$ and $\frac{1}{2}\binom{2 l}{l}$.

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Problem 22

Let notation be as in Exercise 21. Let $s_0$ be the element of the Weyl group that sends positive roots to negative ones.
(a) If $\mathfrak{B}=A_l, s_0 \lambda_i=\lambda_{l+2-i}(1 \leq i \leq l+1)$
(b) If $g=B_f$ or $C_l, s_0=$-identity.
(c) If $\mathrm{g}=D_j, /$ even, then $s_0=$-identity; if $l$ is odd, $s_0 \lambda_j=-\lambda_j$ for $1 \leq j \leq l-1$ and $s_0 \lambda_l=\lambda_l$.
(d) Deduce that all representations of $B_i$ and $C_i$ are self-contragredient.

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Problem 23

Let $g=B_f(I \geq 1)$ or $D_j(I \geq 2)$; other notation as in 4.4. Let $\pi_1$ be the representation of g in $V$; $\pi_k$, the representation induced by $\pi_1$ in $E_k$, the subspace of elements of degree $k$ of the exterior algebra $E=E(V)$ over $V$.
(a) Let $g=B_i$. Then the $\bar{\pi}_k(0 \leq k \leq 2 l+1)$ are all irreducible, $\bar{\pi}_k \simeq \bar{\pi}_{2 l+1-k}, \bar{\pi}_0$ is the trivial representation, $\bar{\pi}_i \simeq \pi_{p_i}$ for $1 \leq i \leq l-1$, and $\bar{\pi}_l \simeq \pi_{2 \mu_l}$.
(b) Let $\mathrm{B}=D_l$. Then $\bar{\pi}_k \simeq \pi_{2 l-k}(0 \leq k \leq 2 l)$, and $\bar{\pi}_0$ is the trivial representation; for $1 \leq k \leq l-1, \bar{\pi}_k$ is irreducible, while $\bar{\pi}_l$ splits as a direct sum of two inequivalent irreducible representations; $\pi_k \simeq \pi_{\mu_4}(1 \leq k \leq l-2), \bar{\pi}_{l-1} \simeq \pi_{\mu_{i-1}+\mu}$, and $\bar{\pi}_l$ is the direct sum of $\pi_{2 \mu_{i-1}}$ and $\pi_{2 \mu_i}$. (For the last equivalence, observe that $\lambda_1+\cdots+\lambda_{l-1} \pm \lambda_l$ are weights of $\tilde{\pi}_i$, but $\left(\lambda_1+\cdots+\lambda_{l-1} \pm \lambda_i\right)+\alpha_i$ is not a weight of $\tilde{\pi}_i$ for $1 \leq i \leq l$; for the rest, use Exercises 21 and 22.)

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Problem 24

. Let $g=C_i(l \geq 2), \bar{\pi}_i$ the representation of g in $V, \bar{\pi}_k$ the corresponding representation in the subspace $E_k$ of the exterior algebra $E$ over $V$; these and other notation as in §4.4.
(a) Prove that $\bar{\pi}_k \simeq \bar{\pi}_{2 i-k}(0 \leq k \leq 2 l)$ and that $\bar{\pi}_{2 l}$ is the trivial representation.
(b) Let $E_{k, 0}(1 \leq k \leq l)$ be the smallest $\bar{\pi}_k$-invariant subspace of $E_k$ containing $u_t \wedge \cdots \wedge u_k$, and let $\bar{\pi}_{k, 0}$ be the representation defined by $E_{k, 0}$. Prove that $\pi_{1,0}=\pi_1 \simeq \pi_{\mu 1}$ and $\pi_{k, 0} \simeq \pi_{\mu,}(1 \leq k \leq l)$.
(c) Let $\varphi=u_1 \wedge u_{l+1}+u_2 \wedge u_{l+2}+\cdots+u_l \wedge u_{2 l}$. Prove that $\mathbf{C} \cdot \varphi$ is invariant under $\pi_2$ and defines the trivial representation of $\mathfrak{B}$, and that $E_2$ is the direct sum of $E_{2,0}$ and $\mathbf{C} \cdot \varphi$.
(d) Let $2 \leq k \leq l$, and let $E_{k, s}=\underbrace{\varphi \wedge \varphi \wedge \cdots \varphi}_{s \text { factors }} \wedge E_{k-2 s, 0}, 1 \leq s \leq \frac{1}{2} k$. Prove that the $E_{k, x}\left(0 \leq s \leq \frac{1}{2} k\right)$ are all nonzero and linearly independent and span $E_k$. Deduce that $\pi_k$ is equivalent to the direct sum of $\pi_\mu(i=k$, $k-2, \ldots$, the sequence continuing as long as $i \geq 0 ; \mu_0=0$ ). (Use Exercise 21.)

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Problem 25

Suppose $G$ is a semisimple real analytic group whose Lie algebra is simple. Prove that if $G$ is not compact, $G$ has no nontrivial (finite-dimensional) unitary representation.

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Problem 26

(a) Let $G$ be a real analytic group with Lie algebra g, $H \subseteq G$ a closed subgroup, and $\hat{\eta} \subseteq \beta$ the corresponding subalgebra. Let $\mathrm{B}_c$ be the complexification of $\mathrm{g}, X_{\mapsto} \mapsto X^{\text {conj }}$ the associated conjugation of $\mathrm{g}_c$, and $\mathfrak{h}_c=\mathbf{C} \cdot \mathfrak{h}$. Prove that $G / H$ has a $G$-invariant complex structure if and only if there are subalgebras $\psi^{+}$and $\psi^{-}$of $\beta_c$ with the following properties: (i) ${p^{+}}^n{p^{-}} =\mathfrak{h}_c, \mathfrak{p}^{+}+\mathfrak{p}^{-}=\mathfrak{B}_c$, (ii) $\left(\mathfrak{p}^{+}\right)^{\text {conj }}=\mathfrak{p}^{-}$, and (iii) $\mathfrak{p}^{+}$and $\mathfrak{p}^{-}$invariant under $\operatorname{Ad}(H)$.
(b) Let $G$ be a compact semisimple analytic group, $B$ a maximal torus. Prove that $G / B$ has a unique $G$-invariant complex structure.
(c) Let $G=S L(2, \mathbf{R}), H=S O(2, \mathbf{R})$. Determine all the $G$-invariant complex structures on $G / H$.
Exercises 27-30 lead (among other things) to the proofs of existence of the exceptional simple Lie algebras. In these, $A=\left(a_{i j}\right)_{1 \leq i, j \leq i}$ is one of the matrices corresponding to the Dynkin diagrams $G_2, F_4, E_p(p=6,7,8)$ (cf. $(4.5 .7)-(4.5 .9)$ ); $V$ is a vector space over $\mathbf{C}$ with basis $\alpha_1, \ldots, \alpha_i ; s_i(1 \leq i \leq i)$ are the reflexions given by $s_i \alpha_j=\alpha_j-a_{i j} \alpha_i(1 \leq i, j \leq I) ; \mathrm{t}(A)$ is the subgroup of $G L(V)$ generated by the $s_i ; \Delta=\bigcup_{1 \leq i \leq l} b(A) \cdot \alpha_i ; \mu_i$ are the basic dominant integral linear forms.

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Problem 27

Let $l=2, A=A\left(G_2\right)$.
(a) Write $\lambda_1=\alpha_1, \lambda_2=\alpha_1+\alpha_2, \lambda_0=-\left(\lambda_1+\lambda_2\right)$. Verify that $s_1 \lambda_1= -\lambda_1, s_1 \lambda_2=-\lambda_0, s_2 \lambda_1=\lambda_2, s_2 \lambda_2=\lambda_1$, and deduce that $\mathrm{tv}(A)$ leaves the set $\left\{ \pm \lambda_i(i=0,1,2), \pm\left(\lambda_i-\lambda_j\right)(0 \leq i<j \leq 2)\right\}$ invariant.
(b) Use (a) to show that $\mathrm{tv}(A)$ is finite, and deduce the existence of a simple Lie algebra (also denoted by $G_2$ ) whose Cartan matrices are equivalent to $A\left(G_2\right)$.
(c) Prove that $\Delta$ is the set $\left\{ \pm \lambda_i(i=0,1,2), \pm\left(\lambda_i-\lambda_j\right)(0 \leq i<j \leq 2)\right\}$, and hence verify that $\operatorname{dim}\left(G_2\right)=14$.
(d) Let $\Gamma_1$ be the group of permutations of $[0,1,2]$ acting naturally on $\mathbf{C}^3= \left[\left(x_0, x_1, x_2\right)\right]$, and let $\Gamma$ be the group generated by $\Gamma_1$ and -identity. Prove that $\Gamma$ leaves the plane $x_0+x_1+x_2=0$ invariant and that there is an isomorphism of this plane with $V$ that transforms the action of $\Gamma$ into that of $1 \nu(A)$. Deduce that $\left[\nu\left(G_2\right)\right]=12$.
(e) Prove that $\mu_1=2 \alpha_1+\alpha_2$ and $\mu_2=3 \alpha_1+2 \alpha_2$, that $\delta=5 \alpha_1+3 \alpha_2$, and that $\mu_2$ is the highest root.
(f) Verify that $\operatorname{dim}\left(\pi_{\mu_1}\right)=7, \operatorname{dim}\left(\pi_{\mu 2}\right)=14$, and that $\pi_{\mu 2}$ is the adjoint representation.
(g) Prove that every representation of $G_2$ is self-contragredient and contains 0 as a weight.

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Problem 28

Let $/=4, A=A\left(F_4\right)$.
(a) Write $\lambda_1=2 \alpha_1+3 \alpha_2+2 \alpha_3+\alpha_4, \lambda_2=\alpha_2, \lambda_3=\alpha_2+\alpha_3, \lambda_4=\alpha_2+\alpha_3+\alpha_4$. Verify that $s_2, s_3, s_4$ fix $\lambda_1$ and that $s_1 \lambda_l-\frac{1}{2}\left(\lambda_1+\epsilon_{i 2} \lambda_2+\epsilon_{i 3} \lambda_3+\epsilon_{i 4} \lambda_4\right)$, where $\epsilon_{i j}=+1$ or -1 according as $j=i$ or $j \neq i(i=2,3,4)$, and $\epsilon_{i j}=+1$ for all $j$, if $i=1$. Determine the action of $s_2, s_3$, and $s_4$ on $\lambda_2$, $\lambda_3$, and $\lambda_4$. Prove that $\mathrm{v}(A)$ leaves invariant the set $\left\{ \pm \lambda_i, \pm \lambda_i \pm \lambda_j\right.$, $\left.\frac{1}{2}\left( \pm \lambda_1 \pm \lambda_2 \pm \lambda_3 \pm \lambda_4\right)\right\}(i<j, i, j=1,2,3,4)$.
(b) Use (a) to prove that there is a simple Lie algebra (denoted again by $F_4$ ) for which $A\left(F_4\right)$ is a Cartan matrix.
(c) Prove that $\Delta$ is the set described in (a) and $\operatorname{dim}\left(F_4\right)=52$.
(d) Verify that

$$
\begin{aligned}
& \mu_1=2 \alpha_1+3 \alpha_2+2 \alpha_3+\alpha_4 \\
& \mu_2=3 \alpha_1+6 \alpha_2+4 \alpha_3+2 \alpha_4 \\
& \mu_3=4 \alpha_1+8 \alpha_2+6 \alpha_3+3 \alpha_4
\end{aligned}
$$
$$
\begin{aligned}
\mu_4 & =2 \alpha_1+4 \alpha_2+3 \alpha_3+2 \alpha_4 \\
\delta & =11 \alpha_1+21 \alpha_2+15 \alpha_3+8 \alpha_4
\end{aligned}
$$

and that $\mu_4$ is the highest root of $F_4$.
(c) Verify that the dimensions of the representations $\pi_{\mu_1}(i \leq i \leq 4)$ are $26,273,1274,52$ respectively and that $\pi_{\mu_i}$ is the adjoint representation.
(f) Show that every representation of $F_4$ is self-contragredient and contains 0 as a weight.
(g) Show that $\left[p\left(F_4\right)\right]=2^7 \cdot 3^2=1152$.

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Problem 29

Let $l=8, A=A\left(E_8\right)$.
(a) Define the elements $\lambda_j(1 \leq i \leq 8)$ of $V$ as follows:

$$
\begin{aligned}
& 3 \lambda_i=3\left(\alpha_i+\alpha_{i+1}+\cdots+\alpha_5\right)+2 \alpha_6+\alpha_7+\alpha_8 \quad(1 \leq i \leq 5), \\
& 3 \lambda_6=2 \alpha_6+\alpha_7+\alpha_8, \\
& 3 \lambda_7=-\alpha_6+\alpha_7+\alpha_8, \\
& 3 \lambda_8=-\alpha_6-2 \alpha_7+\alpha_8 .
\end{aligned}
$$

Verify that the $\lambda_i$ form a basis for $V$, and that $s_j$ permutes $\lambda_i$ and $\lambda_{i \neq 1}$ leaving the others fixed $(1 \leq i \leq 7)$. Also verify that

$$
\begin{aligned}
& s_8 \lambda_i=\lambda_i+\frac{1}{3}\left(\lambda_6+\lambda_7+\lambda_8\right) \quad(1 \leq i \leq 5), \\
& s_8 \lambda_6=\frac{1}{3}\left(\lambda_6-2 \lambda_7-2 \lambda_8\right) \\
& s_8 \lambda_7=\frac{1}{3}\left(-2 \lambda_6+\lambda_7-2 \lambda_8\right), \\
& s_8 \lambda_8=\frac{1}{3}\left(-2 \lambda_6-2 \lambda_7+\lambda_8\right) .
\end{aligned}
$$

(b) Use the results of (a) to show that $\mathrm{IV}(A)$ leaves the set

$$
\begin{aligned}
\left\{ \pm\left(\lambda_i-\lambda_j\right)\right. & \pm\left(\lambda_i+\lambda_j+\lambda_k\right) \\
& \left. \pm\left(\lambda_i+\lambda_j+\lambda_k+\lambda_p+\lambda_q+\lambda_r\right), \pm\left(\lambda_j+\lambda_1+\cdots+\lambda_8\right)\right\}
\end{aligned}
$$

invariant ( $i<j<k<p<q<r$ are from ( $1, \ldots, 8$ )). Deduce the existence of a simple Lie algebra (denoted by $E_8$ ) having $A=A\left(E_8\right)$ as a Cartan matrix.
(c) Prove that $\Delta$ is the set described in (b). Deduce that $\operatorname{dim}\left(E_{\mathrm{B}}\right)=248$.
(d) Let $s_0 \in \mathrm{w}$ be such that $s_0 \cdot P=-P$. Prove that $s_0=-i d$. Deduce that all representations of $E_8$ are self-contragredient.
(e) Show that $\operatorname{det}\left(A\left(E_8\right)\right)=1$. Deduce that the additive group of integral linear forms is already generated by the roots. Hence conclude that all representations of $E_8$ contain 0 as a weight.
(f) Verify that $\mu_1=2 \alpha_1+3 \alpha_2+4 \alpha_3+5 \alpha_4+6 \alpha_5+4 \alpha_6+2 \alpha_7+3 \alpha_8$ is the highest root.

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Problem 30

(a) Let $g$ be a semisimple Lie algebra over $\mathbf{C}$, $\mathfrak{h}$ a CSA, $S=\left\{\tilde{\alpha}_1, \ldots, \tilde{\alpha}_l\right\}$ a simple system of roots, and $F$ a subset of $\{1, \ldots, l\}$. Let $\Delta_F$ be the set of all those roots of $(\beta, \mathfrak{h})$ which are linear combinations of the $\alpha_i$ with $i \in F$. Prove that $g_F=\sum_{t \in F} \mathbf{C} \cdot \bar{H}_{\alpha_1}+\sum_{\alpha \in \Delta_F} g_\alpha$ is a semisimple Lie algebra, with CSA $\mathfrak{h}_F=\sum_{i \in F} \mathrm{C} \cdot \bar{H}_{a_i}$, and Cartan matrix $\left(\bar{a}_{i j}\right)_{i, j \in F},\left(\bar{a}_{i j}\right)_{1 \leq i, j \leq l}$ being the Cartan matrix of $(g, \hat{g})$.
(b) Use (a) to prove the existence of simple Lie algebras $E_6$ and $E_7$ with respective Cartan matrices $A\left(E_6\right)$ and $A\left(E_7\right)$.
(c) Show that $\operatorname{dim}\left(E_6\right)=78$ and $\operatorname{dim}\left(E_7\right)=133$.
(d) For both $E_6$ and $E_7$ show that -id is the element of the Weyl group that sends positive roots to negative ones. Deduce that all representations of $E_6$ and $E_7$ are self-contragredient.
(e) Show that $\operatorname{det}\left(A\left(E_6\right)\right)=3$ and $\operatorname{det}\left(A\left(E_7\right)\right)-2$.
(f) Let $V_6$ be spanned by $\alpha_i(1 \leq i \leq 6)$. Define $\lambda_i(1 \leq i \leq 6)$ by $3 \lambda_i= 3\left(\alpha_i+\cdots+\alpha_3\right)+2 \alpha_4+\alpha_5+\alpha_6(1 \leq i \leq 3), 3 \lambda_4=2 \alpha_4+\alpha_5+\alpha_6$, $3 \lambda_5=-\alpha_4+\alpha_5, 3 \lambda_6=-\alpha_4-2 \alpha_5+\alpha_6$. Show that the roots of $E_6$ are

$$
\begin{aligned}
\pm\left(\lambda_I-\lambda_j\right) \pm\left(\lambda_i+\lambda_j+\lambda_k\right), \quad \pm\left(\lambda_1+\cdots+\lambda_6\right) & \\
& (1 \leq i<j<k \leq 6) .
\end{aligned}
$$

Deduce that $\mu_6=\alpha_1+2 \alpha_2+3 \alpha_3+2 \alpha_4+\alpha_5+2 \alpha_6$ is the highest root.
(g) Let $V_7$ be spanned by $\alpha_i(1 \leq i \leq 7)$. Define $\lambda_i(1 \leq i \leq 7)$ by $3 \lambda_i= 3\left(\alpha_i+\cdots+\alpha_4\right)+2 \alpha_5+\alpha_6+\alpha_7(1 \leq i \leq 4), 3 \lambda_5=2 \alpha_5+\alpha_6+\alpha_7$, $3 \lambda_6=-\alpha_5+\alpha_6+\alpha_7, 3 \lambda_7=-\alpha_5-2 \alpha_6+\alpha_7$. Show that the roots of $E_\gamma$ are

$$
\pm\left(\lambda_i-\lambda_j\right), \quad \pm\left(\lambda_i+\lambda_j+\lambda_k\right), \quad \pm\left(\lambda_i+\lambda_j+\lambda_k+\lambda_p+\lambda_q+\lambda_r\right),
$$

where $1 \leq i<j<k<p<q<r \leq 7$. Deduce that $\mu_6=\alpha_1+2 \alpha_2+ 3 \alpha_3+4 \alpha_4+3 \alpha_5+2 \alpha_6+2 \alpha_7$ is the highest root.
(h) Show that $10\left(E_6\right)=2^7 \cdot 3^4 \cdot 5,\left[10\left(E_7\right)\right]=2^{10} \cdot 3^4 \cdot 5 \cdot 7$, and $\left[10\left(E_8\right)\right]= 2^{14} \cdot 3^5 \cdot 5^2 \cdot 7$.
For explicit realizations of the exceptional Lie algebras, see Jacobson [1].

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Problem 31

Let $G=U(n, \mathbf{C}), B$ the subgroup of all diagonal matrices of $G$. We identify $B$ with $\mathrm{T}^{\varepsilon}$ via the map $\operatorname{diag}\left(a_1, \ldots, a_n\right) \mapsto\left(a_1, \ldots, a_n\right)$. If $h_1 \geq \ldots \geq h_n \geq 0$ are integers, $\left[h_1, \ldots, h_n\right]$ denotes the function $\left(a_1, \ldots, a_n\right) \mapsto \operatorname{det}\left(\left(a_i^{h_i}\right)_{1 \leq x_i, j \leq n}\right)$ on $B$.
(a) Let $h_1 \geq \cdots \geq h_n \geq 0$ be integers. Prove that $[n-1, \ldots, 0]= \prod_{1 \leq i<j \leq n}\left(a_i-a_j\right)$ and that $\left[h_1, \ldots, h_n\right]$ is divisible by $[n-1, \ldots, 0]$ in the ring of polynomials in $a_1, \ldots, a_n$. Given integers $f_1 \geq \cdots \geq f_n \geq 0$, let $h_i=f_i+n-i(1 \leq i \leq n)$, and let $\left\langle f_1, \ldots, f_n\right\rangle$ be the quotient $\left[h_1, \ldots, h_n\right] /[n-1, \ldots, 0]$. Prove that $\left\langle f_1, \ldots, f_n\right\rangle$ is a homogeneous polynomial in $a_1, \ldots, a_n$ of degree $f_1+\cdots+f_n$ with the following properties: (i) $a_1^f \cdots a_n^{f n}$ occurs in it with coefficient 1 , and (ii) if $a f^1 \cdots a_n^{f n}$ occurs with a nonzero coefficient, then $\left(g_1, \ldots, g_n\right) \leqq\left(f_1, \ldots, f_n\right)$ in the usual lexicographic ordering on $\mathbf{R}^n\left(\left(x_1, \ldots, x_n\right)<\left(y_1, \ldots, y_n\right)\right.$ if for some $i, 1 \leq i \leq n, x_i<y_i$ and $x_j=y_j$ for $j<i$ ).
(b) Prove that there is a unique irreducible character $\chi\left(f_1, \ldots, f_n\right)$ of $G$ such that $\chi\left(f_1, \ldots, f_n\right) \mid B=\left\langle f_1, \ldots, f_n\right\rangle$. Prove that $\chi^0\left(f_1, \ldots, f_n\right)= \chi\left(f_1, \ldots, f_n\right) \mid S U(n, \mathrm{C})$ is an irreducible character of $S U(n, \mathrm{C})$, that every irreducible character of $S U(n, \mathbf{C})$ is of this form, and that $\chi^0\left(f_1, \ldots, f_n\right)= \chi^0\left(f_1^{\prime}, \ldots, f_n^{\prime}\right)$ if and only if $f_i-f_{i+1}=f_i^{\prime}-f_{i+1}^{\prime}(1 \leq i \leq n-1)$.
(c) Let $\varphi(x)=\operatorname{det}(x)(x \in G)$. Prove that for any integer $s, \varphi^{\prime} \chi\left(f_1, \ldots, f_n\right)$ is an irreducible character of $G$. Prove, further, that all irreducible characters of $G$ are of this form. Prove, finally, that $\varphi^s \chi\left(f_1, \ldots, f_n\right)=\varphi^{s^{\prime}} \chi\left(f_1, \ldots, f_n^{\prime}\right)$ if and only if $f_i+s-f_i^{\prime}+s^{\prime}(1 \leq i \leq n)$.
(e) (Branching law). Identify $U(n-1, \mathrm{C})$ with the subgroup of $G$ of all elements of the form $\left(\begin{array}{ll}A & 0 \\ 0 & 1\end{array}\right)(n \geq 2)$. Denote by $\pi\left(f_1, \ldots, f_n\right)$ the representation of $G$ with character $\chi\left(f_1, \ldots, f_n\right)$. Prove that $\pi\left(f_1, \ldots, f_n\right)$ maps $a \cdot 1$ into the scalar $a^{f,+\cdots+f} \cdot 1$. Prove, further, that the irreducible constituents of the restriction of $\pi\left(f_1, \ldots, f_n\right)$ to $U(n-1, \mathbf{C})$ are precisely all the representations $\pi\left(f_1^{\prime}, \ldots, f_{n-1}^{\prime}\right)$, where $f_1^{\prime}, \ldots, f_{n-1}^{\prime}$ are integers such that $f_1 \geq f_1^{\prime} \geq f_2 \geq f_2^{\prime} \geq \cdots \geq f_{n-1} \geq f_{n-1}^{\prime} \geq f_n \geq 0$; and that each of these occurs with multiplicity 1 .
(f) Let $D\left(z_1, \ldots, z_n\right)=\prod_{1 \leq i<j \leq n}\left(z_i-z_j\right)\left(z_1, \ldots, z_n \in \mathrm{C}\right)$. Prove that

$$
\operatorname{dim}\left(\pi\left(f_1, \ldots, f_n\right)\right)=\frac{D\left(h_1, \ldots, h_n\right)}{D(n-1, \ldots, 0)} \quad\left(h_f=f_i+n-i\right) .
$$

(g) Let $V=\mathbf{C}^n$, and let $\left\{e_1, \ldots, e_n\right\}$ be the canonical basis of $V$. Let $J$ be the tensor algebra over $V, J_f(f \geq 0)$ the homogeneous subspace of 3 of degree $f$. Denote by $\lambda_f$ the natural representation of $G$ in $\mathfrak{J}_f$. If $f, g \geq 0, t \in \mathfrak{J}_f$, $t^{\prime} \in \widetilde{J}_k$, we write $t \wedge t^{\prime}$ for $(1 /(f+g)!) \sum_{\varepsilon \in \Pi_{r+g}} \epsilon(s) t \otimes t^{\prime}$, where $\prod_{f+g}$ is the group of permutations of $\{1,2, \ldots, f+g\}$. Prove that if $g_i=f_i-f_{i+1} (1 \leq i \leq n-1)$ and $g_n=f_n$, the vector

$$
\begin{aligned}
e_1 \otimes \cdots \otimes e_1 \otimes\left(e_1\right. & \left.\wedge e_2\right) \otimes \cdots \otimes\left(e_1 \wedge e_2\right) \otimes \cdots \\
& \otimes\left(e_1 \wedge \cdots \wedge e_n\right) \otimes \cdots \otimes\left(e_1 \wedge \cdots \wedge e_n\right)
\end{aligned}
$$
in which $e_1 \wedge \cdots \wedge e_k$ occurs $g_k$ times, belongs to a subspace of $J_f$ that is irreducibly invariant under $\lambda_f$ and defines the representation $\pi\left(f_1, \ldots, f_n\right)$.
(h) Prove that the $\pi\left(f_1, \ldots, f_n\right)\left(f_1+\cdots+f_n=f\right)$ are precisely all the irreducible constituents of $\lambda_f$.

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Problem 32

Let $G$ be a complex analytic group with Lie algebra $\mathrm{g}, G_0$ the $\mathbf{R}$-analytic group underlying $G$, and $\mathfrak{g}_0$ the real Lie algebra underlying $g^{\circ}$. Let $g_c$ be the complexification of $g_0, \varphi$ the canonical imbedding of $g_0$ in $g_c$ -
(a) Let $J$ be the endomorphism of $\mathrm{g}_c$ such that $\varphi(i X)=J \varphi(X)$ for all $X \in \mathrm{~g}_c$. Prove that $\pm i$ are the only eigenvalues of $J$ and that the corresponding eigenspaces $\mathrm{g}_c^{ \pm}$are ideals of $\mathrm{g}_c$ having $\mathrm{g}_c$ as their direct sum.
(b) Let $\beta^{ \pm}(X)=\frac{1}{2}(\varphi(X) \mp i J \varphi(X))\left(X \in g_0\right)$. Prove that $\beta^{+}$(resp. $\beta^{-}$) is a Lie algebra isomorphism of g (resp. the complex conjugate of g ) with $\mathrm{g}_c^{+}$ (resp. $\beta_c^{-}$).
(c) Let $A$ and $B$ be associative algebras over an algebraically closed field of characteristic 0 , and let $C=A \otimes B$. Prove that the irreducible representations of $C$ are exactly those of the form $\rho_A \otimes \rho_B$, where $\rho_A$ (resp. $\rho_B$ ) is an irreducible representation of $A$ (resp. $B$ ).
(d) Assume that $G$ is simply connected. Prove that the irreducible representations of $G_0$ in complex vector spaces are precisely all representations of the form $\pi_1 \otimes \pi_2^{\text {con } j}$, where $\pi_1$ and $\pi_2$ are irreducible complex analytic representations of $G$ and $\pi_2{ }^{\text {onj }}$ is the complex conjugate of the representation $\pi_2$.
For details, see Cartier, Exposé $\mathrm{n}^{\circ} 22$, Séminaire Sophus Lie [1].

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Problem 33

Let $G$ be a complex semisimple analytic group with Lie algebra g. Let $G_0$ (resp. $\mathrm{g}_0$ ) denote the real analytic group (resp. Lie algebra over $\mathbf{R}$ ) underlying $G$ (resp. $\beta$ ). Let $U \subseteq G_0$ be a compact real form of $G$ and $\mu \subseteq \mathfrak{g}_0$ the corresponding subalgebra. Write $p=(-1)^{1 / 2} \mathrm{LL}$.
(a) Prove that $[11, p] \subseteq p$ and $[p, p] \subseteq 11$.
(b) Prove that for any $X \in \mathfrak{p}$, ad $X$ is semisimple and has only real eigenvalues, and that $(\operatorname{ad} X)^2$ leaves $\varphi$ invariant.
(c) Let $\sigma\left(Z \mapsto Z^\sigma\right)$ be the conjugation of $\mathfrak{g}$ with respect to $u$. For $X, Y \in \mathfrak{g}$, let $(X, Y)=-\left\langle X, Y^\sigma\right\rangle$. Prove that $(\cdot, \cdot)$ is a positive definite scalar product for $g$ and that for $X \in u$ (resp. $X \in \mathfrak{p}$ ) ad $X$ is a skew-Hermitian (resp. Hermitian) endomorphism of g with respect to this scalar product.
(d) Prove that $\exp [\mathrm{p}]$ is closed in $G$.
(e) Let $\psi$ be the map $(k, X) \mapsto k \exp X$ of $U \times \psi$ into $G_0$. Let $f, g$ be the entire functions on $\mathbf{C}$ defined by $f(z)=(\cosh z-1) / z=\sum_{n \geq 0} z^{2 n+1} /(2 n+2)$ !, $g(z)=(\sinh z) / z=\sum_{0,00} z^{2 n} /(2 n+1)!(z \in \mathrm{C})$. Prove that, with appropriate identifications of Lie algebras, $(d \psi)_{i k, x}(k \in U, X \in \mathfrak{v})$ is the linear map $\operatorname{Ad}(\exp X) \circ L$ of $u \times p$ into $\mathfrak{g}_0$, where $L: u \times p \rightarrow \mathfrak{g}_0$ is the map given by

$$
L(Z, Y)=(Z+f(\operatorname{ad} X)(Y))+g(\operatorname{ad} X)(Y) \quad(Z \in \mathfrak{u}, Y \in \mathfrak{p}) .
$$

(f) Prove that $\psi$ is an analytic diffeomorphism of $U \times p$ onto $G_0$.
(g) Deduce from (f) that $U$ and $G$ have the same fundamental group.
(h) Deduce from (g) that center ( $G$ ) is finite and coincides with center ( $U$ ).
(i) Deduce from (h) that $U$ is a maximal compact subgroup of $G$. (Otherwise there would exist nonzero $X \in p$ such that all eigenvalues of $\operatorname{Ad}(\exp X) =e^{\text {ad } X}$ are of absolute value 1.)

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Problem 34

(a) Let $G_i(i=1,2)$ be a complex semisimple analytic groups, and let $U_i$ be a compact real form of $G_i$. Prove that any real analytic homomorphism of $U_1$ onto $U_2$ can be extended uniquely to a complex analytic homomorphism of $G_1$ onto $G_2$. (Use (h) of Exercise 33.)
(b) Deduce from (a) that the complex analytic group containing a given compact connected semisimple group as a real form is determined up to a complex analytic isomorphism.
(c) Let $G$ be a complex analytic semisimple group, $U$ a compact real form of $G$. Prove that the restriction map $\pi \mapsto \pi \mid U$ induces a bijection of the set of all equivalence classes of irreducible complex analytic representations of $G$ onto the set of all equivalence classes of irreducible representations of $U$.

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Problem 35

Let $G$ be a compact analytic group, $g$ its Lie algebra.
(a) Prove that $G$ is isomorphic to $\left(C \times H_1 \times \cdots \times H_s\right) / F$, where $C$ is a torus, the $H_i$ are compact, semisimple, and simply connected and have simple Lie algebras, and $F$ is a finite normal subgroup.
(b) Deduce from (a) that $\pi_1(G)$ is finite if and only if $G$ is semisimple.
(c) Prove that $G=\exp [\mathrm{g}]$.
(d) Use (a) to generalize the results involving maximal tori of compact semisimple groups to the case of arbitrary compact connected Lie groups.

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Problem 36

(a) Let $G$ be a compact topological group satisfying the second axiom of countability. Prove that given any open neighborhood $N$ of the identity, there is a closed normal subgroup $H$ of $G$ and an integer $n \geq 1$ such that $H \subseteq N$ and $G / H$ is isomorphic to a compact subgroup of $U(n, \mathrm{C})$.
(b) Suppose $G$ is a compact Lie group. Prove that $G$ has a faithful representation.
(c) Suppose $G_c$ is a complex analytic semisimple group and that $G$ is a compact real form of $G_c$. If $\pi$ is a complex analytic representation of $G_c$ whose kernel $F$ is a discrete subgroup of $G_c$, prove that $F \subseteq$ center $(G)$. Deduce that $\pi$ is faithful if and only if $\pi \mid G$ is a faithful representation of $G$.
(d) Prove that any complex analytic semisimple group has a faithful complex analytic representation.

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Problem 37

(a) Let $G$ be a compact analytic group, $\left(G_c, \gamma\right)$ its universal complexification. Prove that $\gamma$ is an isomorphism of $G$ onto a compact real form of $G_c$.
(b) Let $G_c$ be a complex analytic semisimple group, $G$ a compact real form of $G_c$, and $\gamma$ the inclusion map of $G$ into $G_c$. Prove that ( $G_c, \gamma$ ) is a universal complexification of $G$.

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Problem 38

(a) Let $G=\mathbf{T}^n, G_c=\mathbf{C}^{x n}$, and $\gamma$ the inclusion map of $G$ in $G_c$. Prove that $G_c$ is reductive, that $G$ is the largest compact subgroup of $G_c$, and that ( $G_c, \gamma$ ) is a universal complexification of $G$. Prove also that if $\pi$ is a complex analytic representation of $G_c$ such that kernel $(\pi) \cap G$ is discrete, then $\operatorname{kernel}(\pi) \subseteq G$.
(b) Let $A_c$ be a complex analytic abelian group of dimension $n$. Prove the equivalence of the following statements:
(i) $A_c$ is isomorphic to $\mathbf{C}^{x x}$ as a complex analytic group.
(ii) $A_c$ has a faithful complex analytic representation, and there exists a finite subgroup $F$ of $A_c$ such that all complex analytic representations of $A_e / F$ are semisimple.
(iii) $A_c$ is reductive.
(iv) There exists a real form $A$ of $A_c$ which is a maximal compact subgroup of $A_c$.
(v) Let $\gamma$ be the identity map of $A_c$. Then there exists a compact real form $A$ of $A_c$ such that ( $A_c, \gamma$ ) is a universal complexification of $A$.
(vi) There exists a compact real form $A$ of $A_c$ with the following property: every character $A$ extends to a complex analytic homomorphism of $A_c$ into $\mathbf{C}^x$.
$=\{1\}$. Now use (a). To prove (ii) ⟹ (iii), let $\pi$ be a complex analytic representation of $A_n$ such that $\pi \mid F$ is a character of $F$, and let $m$ be the order of $F$. Then $\underbrace{\pi \otimes C^{\circ} \mathrm{m}}_{m \text { factors }}$ is semisimple, so $\pi(x)^{2 m}$ is semisimple for all $x \in A_c$. This implies that $\pi(x)$ is semisimple for all $x \in A_c$.)

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Problem 39

Let $G_c$ be a complex analytic group with Lie algebra $\mathfrak{G}_c$. Assume that $\mathfrak{G}_c$ is reductive, and let $\mathfrak{h}_c=\left[\Omega_c, \Omega_c\right],{\mathrm{n}_c}=\operatorname{center}\left(g_c\right)$. Denote by $H_c$ and $A_c$ the complex analytic subgroups of $G_c$ defined respectively by $\mathfrak{h}_c$ and $\mathfrak{n}_c$.
(a) Prove that $H_c$ and $A_c$ are closed and $F=H_c \cap A_c$ is finite.
(b) Prove that $G_c$ is reductive if and only if $A_c$ is reductive.
(c) Suppose $G_c$ is reductive. Prove that $G_c$ has compact real forms, that these are precisely the maximal compact subgroups of $G_c$, and that any two such are conjugate in $G_{c \cdot}$

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Problem 40

Let $G_c$ be a complex analytic group, $G$ a compact real form of it. Denote by $\gamma$ the inclusion map of $G$ into $G_q$. Prove the equivalence of the following statements:
(i) $G_c$ is reductive.
(ii) $G$ is a maximal compact subgroup of $G_c$,
(iii) $\left(G_e, \gamma\right)$ is a universal complexification of $G$.

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Problem 41

Let $G$ be a compact analytic group.
(a) Let $x \in G$, and let $G_x$ be the centralizer of $x$ in $G$. Prove that $x \in G_x^0$, the component of identity of $G_x$. (Note that $x$ belongs to a torus.)
(b) Let $A \subseteq G$ be a torus, and let $x \in G$ centralize $A$. Prove that there is a maximal torus of $G$ containing $A$ and $x$. (Let $T$ be a maximal torus of $G_x^0$ containing $A$. Use (a) to prove that $x \in T$ and that $T$ is a maximal torus of $G$.)
(c) Deduce from (b) that centralizers of tori in $G$ are connected.
(d) Let $G=S O(3, \mathbf{R})$ and $x=\operatorname{diag}(1,-1,-1)$. Verify that the centralizer of $x$ in $G$ is not connected.

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Problem 42

Let $G$ be a compact semisimple analytic group, $g$ its Lie algebra, $B$ a maximal torus of $G$, and $b$ the corresponding subalgebra of $\beta . G^{\prime}$ is the set of regular points of $G ; B^{\prime}=G \cap B^{\prime} . \bar{B}$ is the normalizer of $B$ in $G$. Identify $\bar{B} / B$ with the Weyl group 1 b. $B^{+}$is a connected component of $B^{\prime} . G^*=G / B, x \mapsto x^*$ is the natural map of $G$ onto $G^*$, and $\psi^*\left(x^*, b\right)=b^x(b \in B, x \in G)$.
(a) $G^{\prime}$ is connected and $\pi_1(G)=\pi_1\left(G^{\prime}\right)$. (The main point here is that $\operatorname{dim}\left(G \backslash G^{\prime}\right)=\operatorname{dim}(G)-3$. This remarkable fact was noticed by Weyl and was exploited by both Weyl and Cartan. Sec Helgason [1], Chapter 7, for a very careful treatment of this result.)
(b) Prove that $\psi^*$ is a covering map of $G^* \times B^{+}$onto $G^{\prime}$ and that $\psi^*\left(x_1^*, b_1\right) -\psi^*\left(x_2^*, b_2\right)\left(b_i \in B^{+}, x_i \in G\right)$ if and only if there is an element $y \in \tilde{B}$ such that $y$ leaves $B^{+}$invariant and $b_2=b_1^y, x_2=x_1 y^{-1}$. (First prove the corresponding result with $B^{\prime}$ instead of $B^{+}$. Then note that $\psi^*\left[G^* \times B^{+}\right]$ is both open and closed in $G^{\prime}$.)
(c) Deduce from (b) that 10 acts transitively on the set of all connected components of $B^{\prime}$.
(d) Suppose $G$ is simply connected. Prove that both $B^{+}$and $G^*$ are simply connected and that $\psi^*$ is an analytic diffeomorphism of $G^* \times B^{+}$onto $G^{\prime}$. Deduce that in this case 11 acts simply transitively on the set of all connected components of $B^{\prime}$.
(e) Let $G$ be simply connected. Prove that the centralizers in $G$ of regular elements are maximal tori. (Let $x \in B^*$, and let $y x y^{-1}=x$. Then $y \in \tilde{B}$ and $\left(B^{+}\right)^y=B^{+}$, so $y$ centralizes $b$ by (d). Compare this with (d) of exercise 41.)

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Problem 43

Let $G$ be a compact analytic group which is semisimple, $\beta$ its Lie algebra, $B$ a maximal torus, and $\mathfrak{b}$ the corresponding subalgebra of $g$. Let $L(R), L(G), L$ and $\xi_\lambda$ be as in §4.13.
(a) Prove that the map $\lambda \mapsto \xi_\lambda$ induces an isomorphism of $L(R)$ with the group of all characters of $B$ which are trivial on center $(G)$. (Use Exercise 46 below to prove that the weights of the representations of $A D[G]$ belong to $L(R)$.)
(b) Deduce from (a) that $L(G) / L(R)$ is canonically isomorphic to the character group of center( $G$ ).
(c) Prove by similar reasoning that $L / L(G)$ is canonically isomorphic to the character group of $\pi_1(G)$.
(d) Deduce the isomorphisms (non canonical) center $(G) \approx L(G) / L(R), \pi_1(G) \approx L / L(G)$.
(e) Suppose that $G$ is simply connected. Let $A=\left(a_{i j}\right)$ be the Cartan matrix of ( $\mathrm{G}_c, \mathrm{~b}_c$ ) with respect to some simple system of roots. Prove that center( $G$ ) is isomorphic to the abelian group which has $l$ generators $(l=\operatorname{rk}(G)) \xi_1, \ldots, \xi_j$ subject to the relations $\sum_{1 \quad i, i} a_{i j} \xi_i=0(1 \leq j \leq l)$. Hence show that the order of $\operatorname{center}(G)$ is $|\operatorname{det}(A)|$.
(f) For any simple Lie algebra g over $\mathbf{C}$, let $C(1)$ denote the center of the corresponding simply connected complex (or compact) group. Obtain from (d) the following isomorphisms (here, for any integer $p \geq 2, \mathbf{Z}_p$ is the group $\mathbf{Z} / p \mathbf{Z}$ and 0 is the group having only the identity):

$$
C\left(A_i\right) \approx \mathbf{Z}_{i+1}, \quad C\left(B_i\right) \approx \mathbf{Z}_2, \quad C\left(C_i\right) \approx \mathbf{Z}_2, \quad C\left(D_i\right) \approx \mathbf{Z}_4
$$

(lodd) and $C\left(D_l\right) \approx \mathbf{Z}_2 \oplus \mathbf{Z}_2$ (/ cven), $C\left(E_6\right) \approx \mathbf{Z}_3, C\left(E_7\right) \approx \mathbf{Z}_2$, while $C\left(E_8\right), C\left(G_2\right)$, and $C\left(F_4\right)$ are all $\approx 0$.
(g) Deduce the simple connectedness of $S U(n, \mathrm{C})(n \geq 2)$ and $S p(n)(n \geq 1)$ and the relations $\pi_1(S O(n, \mathbf{R}))=\mathbf{Z}_2(n \geq 3)$.

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Problem 44

Let $G$ be a compact analytic group, $B$ a maximal torus, $E$ and $F$ two subsets of $B$. Suppose there is a $y \in G$ such that $E^y=F$. Prove that there is $z \in G$ such that $z$ normalizes $B$ and $E^z=F$.

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Problem 45

$g$ is a semisimple Lie algebra over $\mathrm{C}, \mathfrak{h}$ a CSA, $G$ a complex analytic group with Lie algebra $g$.
(a) Prove that $\exp [b]$ is the centralizer of $b$ in $G$. (Use Theorems 4.9.1 and 4.12.5 and Exercise 33(h).)
(b) Let $[$ be a subspace of $\mathfrak{b}, z$ the centralizer of $[$ in $g$. Determine $z$ in terms of the root space decomposition of $(g, h)$, and prove that $z$ is reductive. Prove also that the center of 3 consists of all $H \in \mathfrak{h}$ with the property that $\alpha(H)=0$ for every root $\alpha$ of $(g, g)$ which vanishes identically on l . Deduce that the adjoint representation of $g$ remains semisimple on restriction to 3 .
(c) Let $Z$ be the complex analytic subgroup of $G$ defined by 3 . Prove that $Z$ is the centralizer of $\dot{l}$ in $G$. (Let $y \in G$ centralize $\dot{l}$. By Theorem 4.1.3, for some $z \in Z, x=z y$ centralizes $\complement$ and $y^x=4$ ). By (4.9.5) and Theorem 4.15 .17 of the appendix, for some $z^{\prime} \in Z, z^{\prime} x$ centralizes $\emptyset$. Now use (a).)

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Problem 46

Let $G$ be a compact Lie group. Suppose $\mathbb{R}$ is a set of irreducible representations of $G$ such that (i) the trivial representation belongs to $\mathscr{R}$, (ii) if $\pi \in \mathscr{R}$, then $\mathscr{R}$ contains a representation equivalent to the contragredient of $\pi$, and (iii) if $x \in G$ and $x \neq 1$, there is a $\pi \in \mathscr{A}$ such that $\pi(x) \neq 1$. Prove that every irreducible representation of $G$ occurs as a constituent of some tensor product $\pi_1 \otimes \cdots \otimes \pi_k$ for suitable $\pi_1, \ldots, \pi_k \in \mathcal{R}$.

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Problem 47

(a) Let $G$ be a semisimple real analytic group with Lic algebra $\mathrm{G}_{\mathrm{B}}, G_c$ a complex analytic group with Lie algebra $\mathfrak{g}_c, \gamma$ an $\mathbf{R}$-analytic homomorphism of $G$ into $G_c$ such that $\mathbf{C} \cdot d \gamma[\mathfrak{g}]=g_c$. Prove that the following statements are equivalent:
(i) ( $G_c, \gamma$ ) is a universal complexification of $G$.
(ii) If $\varphi$ is any representation of $G$ in a complex vector space $V$, there exists a complex analytic representation $\pi$ of $G_c$ in $V$ such that $\varphi= \pi \circ \gamma$.
(iii) If $H_c$ is a complex semisimple group and $\varphi$ is an $\mathbf{R}$-analytic homomorphism of $G$ into $H_c$, there exists a complex analytic homomorphism $\pi$ of $G_c$ into $H_c$ such that $\varphi=\pi \circ \gamma$.
(b) Let $G$ be as in (a), and let ( $G_c, \gamma$ ) be a universal complexification of $G$. Suppose $D$ is the kernel of $\gamma$. Prove that $D$ is a discrete central subgroup of $G$.
(c) Show that $G / D$ has a faithful representation and that $D$ is the intersection of the kernels of all representations of $G$.
(d) Give an example of a complex analytic semisimple group $G_\epsilon$ and a real form $G$ of it such that $G_e$, together with the inclusion map of $G$ into it, is not a universal complexification of $G$.
(Observe the contrast with compact real forms. For another treatment of complexifications of real semisimple groups, see Harish-Chandra [3].)

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Problem 48

Let $g$ be a semisimple Lie algebra over $\mathbf{C}$, (\% its universal enveloping algebra, $\omega \in(5)$ the Casimir element,
(a) Let $\pi$ be any representation of (5). Show that $\operatorname{tr} \pi(\omega)$ is a rational number $\geq 0$.
(b) Let $\mu \in \mathbf{C}$ and let $\mathfrak{F}_\mu=\left(b(\omega-\mu \cdot 1)\right.$ (b. Prove that $\mathfrak{F}_\mu$ is a proper twosided ideal in $(\%)$.
(c) Let $\mathfrak{N}_\mu=\left(\mathfrak{N} / \tilde{d}_\mu\right.$. Prove that $\mathfrak{N}_\mu$ is a finitely generated algebra and that if $\mu$ is not a nonnegative rational number, $\mathfrak{N}_\mu$ has no finite-dimensional representation. (See Harish-Chandra [1].)
Exercises 49-51 deal with the famous reciprocity between representations of the permutation and unitary groups. For full details, see Weyl [1,5].

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Problem 49

Let $\Pi$ be a finite group, $\mathbf{A}$ its group algebra of all formal sums $\mathbf{a}=\sum_{s \in \Pi} a(s) s (a(s) \in \mathbf{C})$. For $\mathbf{a}=\sum, a(s) s \in \mathbf{A}, \mathbf{a}=\sum, a\left(s^{-1}\right)^{\text {con }} s$. We denote by $r$ the right regular representation of $A . U$ is a finite-dimensional vector space, and $\pi$ is a representation of $\Pi$ in $U$.
(a) Prove that $\pi: \mathbf{a}=\sum_s a(s) s \mapsto \sum_s a(s) \pi(s)$ is a representation of $\mathbf{A}$ in $U$. Prove further that there is a unique two-sided ideal $\mathbf{A}_0 \subseteq A$ such that $\pi$ is a bijection of $\mathbf{A}_0$ onto $\pi[\mathbf{A}]$.
(b) Let $\mathbf{B}=\pi[\mathbf{A}]^{\prime}$ ( $=$ the algebra of endomorphisms of $U$ commuting with $\pi[\mathbf{A}]$ ). Prove that the action of $\mathbf{B}$ on $U$ is semisimple and $\mathbf{B}^{\prime}=\pi[\mathbf{A}]$.
(c) For any $\mathbf{a} \in \mathbf{A}$, let $U(\mathbf{a})$ be the range of $\pi(\mathbf{a})$. Prove that $U(\mathbf{a})$ is invariant under B. Conversely, let $U^{\prime}$ be a subspace of $U$ that is invariant under $\mathbf{B}$. Prove that there is an idempotent $\mathbf{e} \in \mathbf{A}_0$ ( $\mathbf{e}^2=\mathbf{e}$ ) such that $U^{\prime}=U(\mathbf{e})$.
(d) Let $\mathbf{a}, \mathbf{b} \in \mathbf{A}_0$. Prove that $U(\mathbf{a}) \subseteq U(\mathbf{b})$ if and only if $\mathbf{a} \mathbf{A} \subseteq \mathbf{b} \mathbf{A}$. Deduce that $\rho: \mathbf{a A} \mapsto U(\mathbf{a})$ is a well-defined inclusion preserving bijection of the lattice $L\left(\mathbf{A}_0\right)$ of all right ideals of $\mathbf{A}$ that are contained in $\mathbf{A}_0$ onto the lattice $L_{\mathbf{B}}(U)$ of all $B$-invariant subspaces of $U$ and that for any $\mathbf{E} \in L\left(\mathbf{A}_0\right)$, $\rho(\mathbf{E})=\sum_{\mathbf{a} \in \mathbf{E}} U(\mathbf{a})$.
(e) For any $\mathbf{E} \in L\left(\mathbf{A}_0\right)$ (resp. $V \in L_{\mathrm{B}}(U)$ ), let $r(\mathbf{E})$ (resp. $\beta(V)$ ) be the subrepresentation of the right regular representation of $\mathbf{A}$ (resp. the representation of $\mathbf{B}$ in $U$ ) defined by $\mathbf{E}$ (resp. $V$ ). Suppose $\mathbf{E}, \mathbf{F} \in L\left(\mathbf{A}_0\right), V=\rho(\mathbf{E})$, $W=\rho(\mathbf{F})$. Prove that the following statements are equivalent:
(i) For some $\mathbf{c} \in \mathbf{A}_0, \mathbf{c E}=\mathbf{F}$.
(ii) $\beta(W)$ is equivalent to a subrepresentation of $\beta(V)$.
(iii) $r(\mathbf{F})$ is equivalent to a subrepresentation of $r(\mathbf{E})$.
(f) Let $\mathbf{e}_0$ be the idempotent such that $\mathbf{e}_0 \mathbf{A}_0=\mathbf{A}_0 \mathbf{e}_0=\mathbf{A}_0$, and let $\mathbf{e}_0= \sum \mathbf{e}_i$. where the $\mathbf{e}_i$ are primitive idempotents. Prove that the $U\left(\mathbf{e}_i\right)$ are irreducible, $U=\sum_i U\left(\mathbf{e}_i\right)$ is a direct sum, and that the subrepresentations defined by $U\left(\mathbf{e}_i\right)$ and $U\left(\mathbf{e}_i\right)$ are equivalent if and only if the irreducible representations of $\Pi$ defined on $\mathbf{e}_i \mathbf{A}$ and $\mathbf{e}_c \mathbf{A}$ by right translations are equivalent. Let $\mathcal{E}(\mathbf{B})$ be the set of equivalence classes of irreducible representations of $\mathbf{B}$ occuring in $U$, and for each $v \in \mathcal{E}(\mathbf{B})$ let $U_v$ be the linear span of all invariant subspaces of $U$ that are irreducible and transform according to $v$. Further, let $\mathbf{e}_\alpha(\alpha \in I)$ be the irreducible characters of $\Pi$ such that $\mathbf{A}_0=\sum_{x \in /} \mathbf{e}_x \mathbf{A}$ is the decomposition of $\mathbf{A}_0$ into minimal two-sided ideals. Prove that there is a bijection $\alpha \mapsto v_2$ of $l$ onto $\mathcal{E}(\mathbf{B})$ such that $U\left(\mathbf{e}_n\right)=U_{v_n}$ for all $\alpha \in I$.
(g) Convert $\mathbf{A}$ into a Hilbert space by defining $(\mathbf{a}, \mathbf{b})=\sum_{s \in n} a(s) b(s)^{\text {con } j}$. For any $\mathbf{E} \in L\left(\mathbf{A}_0\right)$ let $\mathbf{E}^{\prime}=\mathbf{A}_0 \cap \mathbf{E}^1$. Prove that $\mathbf{E}^{\prime} \in L\left(\mathbf{A}_0\right)$ too. Supposing $U$ to be a Hilbert space and $\pi$ to be a unitary representation, prove that for any $\mathbf{E} \in L\left(\mathbf{A}_0\right), \rho\left(\mathbf{E}^{\prime}\right)$ is the orthogonal complement of $\rho(\mathbf{E})$ in $U$, and further that

$$
\rho(\mathbf{E})=\left\{u ; u \in U, \pi(\hat{a}) u=0 \text { for all } a \in \mathbf{E}^{\prime}\right\},
$$

Hence deduce the result that the $\mathbf{B}$-invariant subspaces of $U$ are precisely those that are given by "symmetry conditions" of the form $\sum_{s \in \Pi} c_s \pi(s) u =0\left(c_s \in \mathbf{C}\right)$.

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Problem 50

Let $V=\mathbf{C}^n$, let $\left\{e_1, \ldots, e_n\right\}$ be the canonical basis of $V$, and let $G=U(n, \mathbf{C})$. $B$ is the subgroup of diagonal matrices of $G$ which we identify with $\mathrm{T}^{\mathrm{n}}$ via the map diag $\left(a_1, \ldots, a_n\right) \mapsto\left(a_1, \ldots, a_n\right)$. Let $f \geq 1$ be an integer, $U=V \otimes \cdots$ ( $) V$ ( $f$ factors). For any endomorphism $L$ of $V, L^{(f)}=L \otimes \cdots \otimes L (f$ factors $) . \Pi$ is the permutation group of $\{1, \ldots, f\}$, and $\pi: s \mapsto \pi(s)$ the natural representation of $\Pi$ in $U$. Let notation be as in Exercise 49.
(a) Prove that $\mathbf{A}_0=\mathbf{A}$ (i.e., $\pi$ is a faithful representation of $\mathbf{A}$ ) if and only if $n \geq f$.
(b) Prove that $\mathbf{B}$ is the linear span of the endomorphisms $x^{-\mid f^1}(x \in G)$.
(c) Let $n \geq f$. For integers $f_1 \geq \cdots \geq f_n \geq 0$, let $\pi\left(f_1, \ldots, f_n\right)$ be the irreducible representation of $G$ described in Exercise 31. Use the results of the preceding exercise to set up the one-to-one correspondence between the irreducible characters of $\Pi$ and the irreducible representations $\pi\left(f_1, \ldots, f_n\right)$ with $f_1+\cdots+f_n=f$.
(d) Prove that the spaces of symmetric and skew-symmetric tensors in $U$ are invariant and irreducible under the action of $G^{(f)}$, the corresponding representations of $G$ being respectively $\pi(f, 0, \ldots, 0)$ and $\pi(\underbrace{1, \ldots, 1}, 0, \ldots, 0)$.
(e) For any sequence ( $g_1, \ldots, g_n$ ) of integers $g_i \geq 0$ with $g_1+\cdots+g_n=f$, let $\Pi\left(g_1, \ldots, g_n\right)$ denote the subgroup of $\Pi$ of all permutations which permute the first $g_1$ numerals of $[1, \ldots, f]$ among themselves, the next $g_2$ numerals among themselves, and so on. For any conjugacy class $t$ of $\Pi$ let $c_{f_1, \ldots, f_0}(\mathfrak{t})$ be the number of elements on $\mathfrak{t} \cap \Pi\left(g_1, \ldots, g_n\right)$. Suppose $X$ is the character of an irreducible representation of $G$ occuring in $U$ and $\xi$ is the corresponding character of $\Pi$; then prove that for any diagonal matrix $a=\left(a_1, \ldots, a_n\right)$,

$$
X\left(a_1, \ldots, a_n\right)=\sum_1 \tilde{\zeta}(\mathrm{t}) \sum_{k_1 \ldots, k_n} \frac{c_{\xi}, \ldots k_n(\mathrm{t})}{g_{1}!\cdots g_{n}!} a_1^{k_1} \cdots a_n^{k_n} .
$$

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Problem 51

Let notation be as above.
(a) Prove that any $s \in \Pi$ can be written as a product of cyclic permutations no two of which involve a common numeral and that this representation is unique.
(b) Let $i_1, i_2, \ldots$ be integers $\geq 0$ such that $i_1+2 i_2+\cdots=f$. Prove that the set $\left[i_1 i_2 \ldots\right]$ of all $s \in \Pi$ in whose decomposition there are $i_1$, cycles with 1 numeral, $i_2$ cycles with 2 numerals, etc., is a conjugacy class of $\Pi$ and that every conjugacy class of $\Pi$ may be obtained this way. Deduce that if \& is any conjugacy class, \& $=$ t $^{-1}$. Prove also that the number of elements in $\left[i_1 i_2 \ldots\right]$ is $f!/ 1^{i_1} i_{1}!2^{i_2} i_{2}!3^{i i_3} i_{3}!\ldots$.
(c) Let $g_1, \ldots, g_n \geq 0$ be integers with $g_1+\cdots+g_n=f$. Prove that the number of elements in $\Pi\left(g_1, \ldots, g_n\right)$ belonging to the conjugacy class $\left[i_1 i_2 \cdots\right]$ is

$$
\frac{1}{1^{\left(i 2^{\prime a}\right.} \cdots} \sum_{(i)}\left\{\frac{\prod_\alpha g_{a}!}{i_{\alpha 1}!i_{\alpha 2}!\cdots}\right\} ;
$$

here $\alpha$ runs from 1 to $n$, and the sum is extended over all possible sequences $i_{11}, \ldots, i_{21}, \ldots, \ldots, i_{n 1}, \ldots$ of integers $\geq 0$ satisfying the equations

$$
\begin{aligned}
\sum_\alpha i_{\alpha 1}=i_1, \quad \sum_\alpha i_{\alpha 2}=i_2, \ldots \\
\sum_v v i_1=g_1, \quad \sum_v v i_{2 v}=g_2, \ldots, \quad \sum_v v i_{N v}=g_a
\end{aligned}
$$

(d) Let $\sigma_r=a_1^r+a_2^r+\cdots+a_n^r(r=1,2, \ldots)$. Prove that the characters $X$ of $G$ and the character $\xi$ of $\Pi$ of Exercise 50(e) are related by the formula

$$
X\left(a_1, \ldots, a_n\right)=\sum_{\left[i_1 i_2-1\right.}\left\{\left(\frac{\xi\left(\left[i_1 i_2 \cdots\right]\right)}{i^1 2^2 \cdots i_{1}!i_{2}!\cdots}\right) \sigma_1^i \sigma_2^i \cdots\right\}
$$
(e) Let $n(\ddagger)$ be the number of elements in the conjugacy class $\ddagger$ of $\Pi$. Prove that

$$
\sum_z \xi\left(t^{\prime}\right) \xi(t)=\delta_{t e} \frac{f!}{n(t)}
$$

the sum being extended over all irreducible characters of $\Pi$.
(f) Deduce from (e) the following formula:

$$
\sigma_1^i \sigma_2^i \cdots=\sum_\zeta \xi\left(\left[i_1, i_2 \cdots\right]\right) X\left(a_1, \ldots, a_2\right) .
$$

(g) Let $f_1 \geq \cdots \geq f_n \geq 0$ be integers with $f_1+\cdots+f_n=f, X$ the character of the representation $\pi\left(f_1, \ldots, f_n\right)$ of $G$ (cf. exercise 31 ), and $\xi$ the corresponding character of $\Pi$. Let $D\left(a_1, \ldots, a_n\right)=\Pi_{i<j}\left(a_i-a_j\right), h_i= f_i+n-i(1 \leq i \leq n)$. Show that $\xi\left(\left[i_1 i_2 \cdots\right]\right)$ is the coefficient of $a_1^{h_1} \cdots a_n^{h_n}$ in $D\left(a_1, \ldots, a_n\right) \sigma_{\vdots}^{\prime} \sigma_2^i \ldots$. Deduce that the characters of $\Pi$ are integervalued functions on $\Pi$.
In Exercises 52-55, for any $C^{\infty}$ manifold $N$ of dimension $n, H(N)$ denotes its De Rham cohomology algebra (cf. Exercises 29-34 of Chapter 2); $H^k(N)$ is the subspace of $H(N)$ spanned by elements of degree $k(0 \leq k \leq n) ; P_N(t)= \sum_{0 \leq k \leq n} \operatorname{dim} H^k(N) t^k$ is the Poincaré polynomial of $N$.

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Problem 52

Let $G$ be a compact connected Lie group of dimension $g$ acting smoothly and transitively on a $C^{-}$manifold $M$ of dimension $m$. We assume that ( $G, M$ ) is a symmetric pair (cf. Exercise 33, Chapter 2). Let $y_0 \in M$, let $H$ be the stabilizer of $y_0$ in $G$, and let $\rho(h \mapsto \rho(h))$ be the representation of $H$ induced on the tangent space $T_{y_0}(M)$ to $M$ at $y_0$. For $1 \leq k \leq m$, let $\rho_k$ be the representation of $H$ in $\Lambda_k\left(T_{y_2}(M)\right)$ obtained from $\rho . d h$ is the Haar measure on $H$ such that $\int_H d h=1$.
(a) Prove that $\operatorname{dim} H^k(M)=\int_H \operatorname{tr} \rho_k(h) d h(1 \leq k \leq m)$. (The right side is the dimension of the subspace of $\rho_k$-invariant elements in $\Lambda_k\left(T_{y_0}(M)\right)$; now use Exercise 33 of Chapter 2.)
(b) Deduce from (a) that $P_M(t)=\int_H \operatorname{det}\left(1+t_\rho(h)\right) d h$

$$
\left(\operatorname{det}\left(1+t_\rho(h)\right) \equiv 1+\sum_{1 \leq k \leq m} \operatorname{tr}\left(\rho_k(h)\right) \cdot t^k\right)
$$

(c) Prove that $M$ is orientable if and only if $\operatorname{det} \rho(h) \equiv 1(h \in H)$.
(d) Use (a) and (c) to prove that if $M$ is orientable, $\operatorname{dim} H^k(M)=\operatorname{dim} H^{m-k}(M) (0 \leq k \leq m)$.

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Problem 53

Let $G$ be a compact, semisimple, connected Lie group; $B$, a maximal torus of $G ; l=r k(G)$
(a) Prove that $P_\sigma(t) \equiv[\mathrm{m}]^{-1}(1+t)^{\mathrm{r}} \int_B D(b) \Pi_{\alpha \in \Delta}\left(1+t \xi_a(b)\right) d b$, where $D$ is defined by (4.13.9) and [ v ] is the order of the Weyl group tv .
(b) Deduce from (a) that $\sum_{0 \leq k \leq \operatorname{dim}(G)} \operatorname{dim} H^k(G)=2^{\prime}$. (Taking $t=1$ in (a), this reduces to proving that $\int_B \Pi_\alpha\left(1-\xi_\alpha^2(b)\right) d b=[1 \cup]$. Observe now that the Fourier expansions of $\prod_\alpha\left(1-\xi_{2 \alpha}\right)$ and $\prod_\alpha\left(1-\xi_\alpha\right)$ have the same constant term.)

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Problem 54

(a) Let $H=U(n, \mathbf{C}) \times U(1, \mathbf{C})$, and let $\rho$ be the representation of $H$ in $\mathbf{C}^n$ given by $\rho(x, \zeta)=(1 / \zeta) x(x \in U(n, C),|\zeta|=1)$. Prove that the representations $\rho_k$ induced by $H$ in $\Lambda_k\left(\mathbf{C}^n\right)\left(0 \leq k \leq n ; \rho_0\right.$ is the trivial representation) are all irreducible and mutually inequivalent. (Use the fact that the representations of the Lie algebra $A_{n=1}$ in $\Lambda_k\left(\mathbf{C}^n\right)(1 \leq k \leq n)$ are irreducible and mutually inequivalent.)
(b) Deduce from (a) that $\int_H\left|\operatorname{det}\left(1+t_\rho(h)\right)\right|^2 d h \equiv \sum_{0 \leq k \leq N}|t|^{2 k}$ for $t \in \mathbf{C}$, $d h$ being the Haar measure on $H$ such that $\int_H d h=1$.
(c) Let $M=\mathbf{P}_n(\mathbf{C})$ be the complex $n$-dimensional projective space ( $=$ space of one-dimensional linear subspaces of $\mathbf{C}^{n+1}$; cf. Exercise 38 of Chapter 2). Prove that $P_M(t) \equiv 1+t^2+\cdots+t^{2 n}$. (Let $y_0 \in M$ be the one-dimensional subspace of $\mathbf{C}^{n+1}$ generated by $(0, \ldots, 0,1), H_0$, the stabilizer of $y_0$ in $U(n+1, \mathrm{C})$ which acts transitively on $M$. Identify $H_0$ with $H$ in such a way that the representation of $H_0$ in $T_{y_0}(M)$ becomes equivalent to $\rho$ defined in (a), but considered as a representation of $H$ in the real vector space underlying $\mathrm{C}^{\mathrm{n}}$.)
(d) Let $S$ be the unit sphere in $\mathbf{C}^{n+1}$ given by $z_1 \bar{z}_1+\cdots+z_{n+1} \bar{z}_{n+1}=1$, and let $\pi: S \rightarrow M$ be the map which sends each point of $S$ into the one-dimensional subspace generated by it. Prove that there is a 2 -form $\boldsymbol{\Omega}$ on $M$ such that $\boldsymbol{\pi}^* \boldsymbol{\Omega}=d z_1 \wedge d \bar{z}_1+\cdots+d z_{n+1} \wedge d \bar{z}_{n+1}$ on $S$. Verify that $\Omega$ is invariant under $U(n+1, \mathrm{C})$.
(e) Prove that $1,[\Omega],[\Omega \wedge \Omega], \ldots,[\Omega \wedge \cdots \wedge \Omega]$ ( $n$ factors) are linearly independent and span $H(M)$ (here, for any exterior differential form $\omega$ on $M$ with $d \omega=0,[\omega]$ is the De Rham class in $H(M)$ that contains $\omega)$.
(f) Let $1 \leq k \leq n$, and let $M_k$ be the subset of $M$ consisting of all onedimensional subspaces lying in the subspace of $\mathbf{C}^{n+1}$ defined by $z_{k+2}= \cdots=z_{n+1}=0$. Verify that $M_k$ is a compact submanifold of $M$, and evaluate $\int_{M_k} \Omega \wedge \cdots \wedge \Omega$ ( $k$ factors) (after orienting $M_k$ ).
(g) Let $M_R$ be the subset of $M$ consisting of all one-dimensional subspaces of $\mathbf{C}^{\mathrm{n}+1}$ spanned by elements of $\mathbf{R}^{\mathrm{n}+1}$. Prove that $M_R$ is an analytic compact submanifold of $M$ of (real) dimension $n$. Deduce from (c)-(e) that $\int_{M_n} \omega=0$ for any closed $n$-form on $M$. (If $n$ is odd, this is clear because $\operatorname{dim} H^n(M)=0$ by (c). If $n=2 k$, note that $\Omega \wedge \cdots \wedge \Omega$ ( $k$ factors) is identically 0 on $M_R$.)

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Problem 55

Let $M$ be the space of projective lines in $\mathbf{P}_3(\mathbf{C})$ (= space of 2 -dimensional linear subspaces of $\mathbf{C}^4$; cf. Exercise 38 of Chapter 2). Prove that $P_M(t) \equiv 1+t^2+2 t^4+t^5+t^8$. (Let $U=U(4, \mathbf{C})$. Then $U$ acts transitively on $M$, and ( $U, M$ ) is a symmetric pair.)
For these results cf. Cartan [3]. See Weyl [1] for calculations leading to the deter-mination of the $P_G$ when $G$ is compact classical, See also Samelson [1], Borel [1].

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Problem 56

Let $g$ be a semisimple Lie algebra over $\mathbf{C}, I$ the algebra of polynomial functions on $g$ invariant under the adjoint group of $g$.
(a) Let $p_1, \ldots, p_i$ be homogeneous elements of $I$ of degrees $d_1, \ldots, d_i$ respectively $\left(d_j>0\right)$ which are algebraically independent and which generate $I$ (together with 1 ). Prove that the coefficient of $t^k$ in the formal expansion of $\Pi_{1 \leq t<t}\left(1-t^d\right)^{-1}$ in powers of the indeterminate $t$ is precisely the dimension of the subspace of homogeneous elements of degree $k$ in $I$.
(b) Deduce from (a) that the integers $d_1, \ldots, d_l$ are uniquely determined (up to a permutation) by $I$ (the $d_i$ are called the primitive degrees and the $p_i$ are said to generate I freely).
(c) Let $q_1, \ldots, q_2$ be homogeneous algebraically independent elements of $I$ such that $\operatorname{deg}\left(q_i\right)=\operatorname{deg}\left(p_i\right), 1 \leq i \leq l$. Prove that the $q_i$ freely generate $l$.
(d) Let $g_0$ be a real form of $g$. Prove the existence of homogeneous elements $p_1, \ldots, p_i$ of positive degree freely generating $I$ such that each $p_i$ is realvalued on $g_0$.

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Problem 57

Let $k$ be a field of characteristic $0, l$ an integer $\geq 1$, and $\Pi_i$ the group of permutations of $\{1, \ldots, l\}$ acting naturally on $k^l . \varepsilon=\left\{\varepsilon=\left(\varepsilon_1, \ldots, \varepsilon_i\right), \varepsilon_i= \pm 1\right.$ for all $i\} ; \delta^{+}=\left\{\varepsilon: \varepsilon_1 \cdots \varepsilon_l=+1\right\}$. $\delta$ is considered an abelian group under componentwise multiplication and is allowed to act on $k^f$ as follows: $\varepsilon:\left(a_1, \ldots, a_l\right) \mapsto\left(\varepsilon_1 a_1, \ldots, \varepsilon_l a_l\right)$. D and $\mathcal{D}^{+}$are the subgroups of $G L(l, k)$ defined by $\mathscr{D}=\mathcal{E} \Pi_{f,} \mathscr{D}^{+}=\mathcal{E}^{+} \Pi_{j-} x_f$ are the linear forms $\left(a_1, \ldots, a_f\right) \mapsto a_f$ on $k^i, J, I, I^{+}$are the respective algebras of polynomials in the $x_i$ invariant under $\Pi_l, \mathbb{D}, \mathbb{D}^{+} . t$ is an indeterminate.
(a) Prove that coefficient of the polynomial $\Pi_{1 \leq 2 d l}\left(t+x_i\right)$ freely generate $J$.
(b) Prove that the coefficients of $\Pi_{1 \leq l \leq l}\left(t+x_i^2\right)$ freely generate $I$.
(c) Let $p_j$ be the coefficient of $t^j$ in $\Pi_{1 \leq 1 \leq t}\left(t+x_i^2\right)(1 \leq j \leq l-1)$. Let $p_l=x_1 \cdots x_l$. Prove that $p_1, \ldots, p_i$ are algebraically independent and, together with 1 , generate $I^{+}$.

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Problem 58

Let $g$ be a classical simple Lie algebra over $\mathbf{C}$ and $\pi$, the basic representation of g , namely, the one which defines it. Let $F(t: X) \equiv \operatorname{det}(t+\pi(X))(X \in \mathfrak{g}$, $t$ an indeterminate). $I$ is as in Exercise 56.
(a) Let $\mathrm{g}=A_l$. Then $F(t: X) \equiv t^{l+1}+\sum_{t \leq v \leq l} p_r(X)^{v-1}$, and the $p_v (1 \leq v \leq l)$ freely generate $I$.
(b) Let $g=B_i$. Then $F(t: X) \equiv t^{2 i+1}+\sum_{1 \leq v \leq I} p_v(X) t^{2 v-1}$, and the $p_v (1 \leq v \leq l)$ freely generate $I$.
(c) Let $\mathrm{g}=C_f$. Then $F(t: X) \equiv t^{2 l}+\sum_{1 \leq v: t} p_v(X) t^{2(v-1)}$, and the $p_v (1 \leq v \leq l)$ freely generate $I$.
(d) Let $\mathrm{B}=D_i$. Prove the existence of an element $p_l \in I$ such that $p_l(X)^2= \operatorname{det} \pi(X)$ for all $X \in \mathrm{~g}$. Verify that $p_l$ is homogeneous of degree $l$.
(In the notation of §4.4, det $\pi(X)=(-1)^{\prime} \lambda_1(X)^2 \cdots \lambda_l(X)^2$ if $X=$ (diag $\left(a_1, \ldots, a_l\right), 0,0$ ). Observe now that $\lambda_1 \cdots \lambda_I$ is invariant under the Weyl group, so $\exists p_l \in I_p$ such that $p_l \mid \hat{\mathfrak{h}}=(-1)^{1 / 2} \lambda_1 \cdots \lambda_l$.)
(e) Let $g=D_l$. Then $F(t: X) \equiv t^{2 l}+p_l(X)^2+\sum_{1 \leq v \leq l-1} p_v(X) t^{2 v}$, and the $p_\nu(1 \leq \nu \leq l)$ freely generate $I$.
(f) Obtain the primitive degrees of the classical simple Lie algebras.

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Problem 59

Prove that the primitive degrees of $\beta=G_2$ are 2 and 6 , and that those of $F_4$ are $2,6,8$, and 12 .

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Problem 60

(a) Determine for which of the simple Lie algebras g over C it is true that the coefficients of the characteristic polynomial of ad $X$ generate $I$.
(b) Let $\mathrm{g}=A_l, \pi$ an irreducible representation of g with highest weight $\lambda$. Determine necessary and sufficient conditions on $\lambda$ which ensure that the coefficients of the characteristic polynomial of $\pi(X)$ generate $L$.

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Problem 61

Let $k$ be a field of characteristic $$0, g=\$(2, k)$$, and $H, X$, and $Y$ as in (4.2.1). $\rho$ is a representation g in a (finite-dimensional) vector space $V$ over $k$.
(a) Prove that the formulae (4.2.5) define an irreducible representation $\pi_j$ of $g$, that these are all the irreducible representations of $g$, and that they stay irreducible in any extension field of $k$.
(b) Let $d_k$ be the multiplicity with which $\pi_k$ occurs in $\rho$. Let $V_t=\{v: v \in V$, $\rho(H) v=t v\}(t \in \mathbf{Z})$. If $W$ is the null space of $\rho(X)$, prove that $W$ is invariant under $\rho(H)$, that $\rho(H) \mid W$ has only nonnegative integral eigenvalues, and that $d_k=\operatorname{dim}\left(V_k \cap W\right)(k=0,1,2, \ldots)$.
(c) Prove that $V$ is the direct sum of the range of $p(X)$ and the null space of $\rho(Y)$.
(d) Prove that $\rho(X)$ maps $V_k$ onto $V_{k+2}(k=0,1,2, \ldots)$, while $\rho(Y)$ maps $V_k$ onto $V_{k-2}(k=0,-1,-2, \ldots)$.
(e) Prove that $\sum_{k \geq 0} d_{2 k}=\operatorname{dim}\left(V_0\right)$.
(f) Prove that $d_k=\operatorname{dim}\left(V_k\right)-\operatorname{dim}\left(V_{k+2}\right)(k=0,1,2, \ldots)$.

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Problem 62

(a) (Theorem of Jacobson-Morozov) Let g be a semisimple Lie algebra over a field $k$ of characteristic $0, X \neq 0$ a nilpotent element of g . Prove the existence of elements $H, Y \in g$ such that $[H, X]=2 X,[H, Y]=-2 Y,[X, Y]= H$. (See Jacobson [1] and also Kostant [1]. $[H, X, Y]$ is called an S-triple.)
(b) Let $\{H, X, Y\}$ be an $S$-triple, $\mathfrak{a}=k \cdot H+k \cdot X+k \cdot Y, \mathfrak{g}_x$ the centralizer of $X$. Prove that $X$ and $Y$ are nilpotent, $H$ is semisimple, and ad $H$ has only integral eigenvalues. If $n^0(\alpha)$ (resp. $n^e(\alpha)$ ) is the number of irreducible constituents in the decomposition of the adjoint representation of $g$ restricted to a with odd dimension (resp. even dimension), prove that $n^0(\mathrm{a}) \geq l(=r k \mathrm{~g})$ and that $n(\mathrm{a})=n^0(\mathrm{a})+n^{\prime}(\mathrm{a})=\operatorname{dim} \mathrm{g}_x \geq l$. Deduce that the following statements are equivalent: (i) $n(\mathrm{a})=l$, (ii) $n^e(\mathrm{a})=0$ and $H$ is regular, and (iii) $\operatorname{dim} g_x=l$.
(c) Let $\mathrm{g}_H$ be the centralizer of $H$ and $\mathrm{g}_2=\{Z: Z \in \mathrm{~g},[H, Z]=2 Z\}$. Prove that $\left[g_2, g_H\right] \subseteq g_2$. If $g_2^{\prime}=\left\{Z: Z \in g_2,\left[Z, g_H\right]=g_2\right\}$, prove that $X \in g_2^{\prime}$.
(d) Let $k=\mathbf{R}$ or $\mathbf{C}, G$ the adjoint group of $g$, and $G_H$ the analytic subgroup of $G$ defined by $\mathfrak{g}_H$. If $k=\mathbf{C}$ and $X^{\prime} \in \mathfrak{g}_2$, prove that there exists $Y^{\prime} \in \mathfrak{g}$ such that $\left\{H, X^{\prime}, Y^{\prime}\right\}$ is an $S$-triple if and only if $X^{\prime} \in g_2^{\prime}$. Prove also that in both the real and complex cases $G_H$ acts naturally on the set $S_H$ of all $S$-triples containing $H$ and that $S_H$ splits into finitely many $G_H$ orbits. Prove, finally, that $\delta_H=G_H \cdot\{H, X, Y\}$ if $k=\mathrm{C}$. (For the first assertion, $X^{\prime} \in \beta_2^{\prime}$ is necessary by (c). Prove, by a differential calculation, that $G_H \cdot X^{\prime}$ is an open subset of $g_2^{\prime}$ for each $X^{\prime} \in g_2^{\prime}$. Deduce from Whitney's theorem that $g_2^{\prime}$ splits into finitely many orbits, since $g_2 \backslash g_2^{\prime}$ is an algebraic set. If $k=\mathbf{C}, \mathrm{g}_2^{\prime}$ is connected, so there is only one orbit.)
Exercises 63-68 study the orbit structure of the adjoint representation of a complex semisimple Lie algebra. As an outcome we also obtain the values of the primitive degrees as defined in Exercise 56. (See Kostant [1]; also, Varadarajan [1].) $\mathfrak{g}$ is a fixed semisimple Lie algebra over $\mathbf{C}, \mathfrak{O}$ is the set of nilpotent elements of $\mathfrak{g}$, and $G$ is the adjoint group of $\mathfrak{g}$. $\mathfrak{h}$ is a CSA, $P$ is a positive system of roots of $(\mathfrak{g}, \mathfrak{h})$, $\alpha_1, \ldots, \alpha_i$ are the simple roots in $P$, and $n=\sum_{x \in P} \mathfrak{G}_\alpha$.

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Problem 63

Let $0 \neq X \in g$. Denote by ʒ the centralizer of $X$ in g.
(a) Prove that $(Y, Z) \mapsto\langle Y,[X, Z]\rangle(Y, Z \in \mathfrak{g})$ induces a nonsingular skewsymmetric bilinear form on $\beta / 3 \times \mathfrak{g} / 3$.
(b) Deduce from (a) that $\operatorname{dim}([X, g])=\operatorname{dim}(g / 3)$ is even.
(c) Let $G_X$ be the stabilizer of $X$ in $G$. Prove that the map $x G_X \mapsto X^x$ is an imbedding of $G / G_X$ into $g$. Deduce that the orbit $X^{\circ}$ of $X$ in $g$ is an analytic submanifold of g of even dimension.

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Problem 64

(a) Suppose that $\{H, X, Y\}$ is an $S$-triple with $H \in \mathfrak{h}$ and that $\alpha_i(H) \geq 0$ for $1 \leq i \leq l$. Prove that $\alpha_i(H)=0,1$, or 2 for each $i=1, \ldots, l$. (Let $Y_i \in \beta-\alpha_i$ be nonzero, $Z=\left[X, Y_i\right]$. Then $Z \in \mathfrak{h}+\mathfrak{n}$. But $[H, Z]= \left(2-\alpha_i(H)\right) Z$, so $2-\alpha_i(H) \geq 0$.)
(b) Let $X$ be nilpotent. Prove that $X^x \in n$ for some $x \in G$. (Take an $S$ triple $\{H, X, Y\}$ and find $x \in G$ such that $H^x \in \mathfrak{h}$ and $\alpha\left(H^x\right) \geq 0$ for all $\alpha \in P$.)
(c) Prove that the set of nilpotent elements of a semisimple Lie algebra over $\mathbf{R}$ or $\mathbf{C}$ splits into finitely many orbits under its adjoint group. (By (a), $\exists$ only finitely many $H$ in a given CSA which form the first member of an $S$-triple. Now use (d) of Exercise 62, and note that any semisimple element can be moved by the adjoint group into one of a fixed finite set of CSA's.)

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Problem 65

A nilpotent $X \in g$ is called principal if the dimension of its centralizer is $l(=r k(\mathrm{~g}))$.
(a) Let $X \in \Re,[H, X, Y]$ an $S$-triple. Prove that $X$ is principal if and only if $H$ is regular and all eigenvalues of ad $H$ are even integers.
(b) Let $H_i \in \mathfrak{h}$ be such that $\alpha_j\left(H_i\right)=2 \delta_{i j}, H=H_1+\cdots+H_i$. Let $X_i \in \mathfrak{g}_{\alpha_i}$ be nonzero, and let $m_i$ be such that $H=m_1 H_{\alpha_1}+\cdots+m_i H_{\alpha_1}$. Prove that $m_i>0$ for all $i$. If $Y_i \in \mathfrak{g}_\alpha$, is such that $\left\langle X_i, Y_i\right\rangle=1, X=X_1+\cdots +X_l$, and $Y=m_1 Y_1+\cdots+m_l Y_l$, prove that $\{H, X, Y\}$ is an $S$-triple and $X$ is principal. ( $m_i>0$ by Exercise 15(a); $X$ is principal by (a) above.)
(c) The principal nilpotents in \# form a Zariski-open nonempty subset of tt,
(d) Prove that the set of principal nilpotents is an open, dense, connected subset of $\mathscr{H}$ and forms a single orbit under $G$. Deduce that it is a regular complex submanifold of $g$ of dimension $n-I$ ( $n=\operatorname{dimg}$ ). (If $X^{\prime}$ is a principal nilpotent, for suitable $H, Y, x \in G,\left\{H, X=X^{\prime_x}, Y\right\}$ is an $S$-triple, with $H \in \mathfrak{h}$, and $\alpha_i(H) \geq 0$ for all $i$. Then $H=H_1+\cdots+H_i$ by Exercise 64(a) and (a) above. Now use Exercise 62(d). For denseness, use (c) above and Exercise 64(b). $X^{\nu G}$ is thus locally compact; by Exercise 63(c), the imbedding of $X^\sigma$ in $\beta$ is regular.)
(e) Let $X$ be a principal nilpotent of $\mathrm{g},[H, X, Y]$ an $S$-triple. Prove that $Y$ is also a principal nilpotent. Prove further that ad $H$ leaves $g_x$ and $g_y$ invariant and that we can select bases $\left\{X_1, \ldots, X_l\right\}$ for $\mathfrak{g}_x,\left\{Y_1, \ldots, Y_i\right\}$ for $\mathfrak{g}_y$, and even integers $\lambda_1, \ldots, \lambda_i>0$ such that $\left[H, X_i\right]=\lambda_i X_i,\left[H, Y_i\right]= -\lambda_i Y_i, 1 \leq i \leq l$. Deduce that $\beta_x$ and $\beta_\gamma$ consist entirely of nilpotent elements. (By Exercise 62(d) and (d) above, we may assume that $H, X, Y$ are as in (b). If $Z \in \mathfrak{g}_x$ and $[H, Z]=0, Z \in \mathfrak{g}_x \cap \mathfrak{h}=0$. So all eigenvalues of ad $H$ in $\mathfrak{g}_x$ are $>0$. If $\mathfrak{g}_x=\{Z: Z \in \mathfrak{g},[H, Z]=s Z\}$ and $X^{\prime} \in \beta_x$, then ( $\left.\operatorname{ad} X^{\prime}\right)^q(Z) \in \sum_{\geq 2 q+1} \mathfrak{q}_4$ for $Z \in \mathfrak{g}_3, q \geq 1$; thus ad $X^{\prime}$ is nilpotent.)

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Problem 66

Let notation be as above. Let $p_i(1 \leq i \leq I)$ be homogeneous elements of $I$ (cf. Exercise 56) generating $I$. Let $\mathbf{p}$ be the map $X \mapsto\left(p_1(X), \ldots, p_i(X)\right)$ of $g$ into $\mathbf{C}^l$. For $\xi \in \mathbf{C}^l, V_\zeta$ is the set $\mathbf{p}^{-1}(\{\xi\})$. If $X \in \mathfrak{B}$ is semisimple, nilpotent, etc., the orbit $X^G$ is said to be semisimple, nilpotent, etc.
(a) Let $\overline{\mathbf{p}}=\mathbf{p} \mid \bar{h}$. Prove that $\overline{\mathbf{p}}$ is proper and $d \overline{\mathbf{p}}$ is nonsingular at all regular points of $\mathfrak{h}$. (If $\left\{\omega_i\right\}$ is a basis of $\mathfrak{y}^*, z_i$ the coordinates on $\mathbf{C}^{\prime}$, and $\pi$ the product of all $\alpha \in P$, then $\exists$ a constant $c \neq 0$ such that $\overline{\mathbf{p}}^*\left(d z_1 \wedge \cdots\right. \left.\wedge d z_l\right)=c \pi \cdot \omega_1 \wedge \cdots \wedge \omega_l \cdot \overline{\mathbf{p}}$ is proper because if $\mathbf{p}(X)$ is bounded, the coefficients of the characteristic polynomial of ad $X$ are bounded, so $\alpha(X)$ is bounded for each root $\alpha(X \in \mathfrak{h})$.)
(b) Deduce from (a) that $\overline{\mathbf{p}}$ maps $\hat{\boldsymbol{h}}$ onto $\mathbf{C}^i$, and hence prove that $\mathbf{p}$ induces a bijection of the set of all semisimple orbits onto $\mathbf{C}^f$. Deduce that for each $\xi \in \mathbf{C}^{\mathrm{l}}, V_{\xi}$ is nonempty and contains a unique semisimple orbit.
( $\overline{\mathbf{p}}[b]$ contains a Zariski-open subset of $\mathbf{C}^i$ by a standard result-cf. Jacobson [1] Chapter 9; $\overline{\mathbf{p}}[\hat{h}]$ is closed since $\overline{\mathbf{p}}$ is proper. So $\overline{\mathbf{p}}[\hat{h}]=\mathbf{C}^i$. For the second result, note that if $X, X^{\prime} \in \mathfrak{h}, \mathbf{p}(X)=\mathbf{p}\left(X^{\prime}\right)$ if and only if $X$ and $X^{\prime}$ lie in the same to-orbit, by Chevalley's theorem. Now use Theorem 4.9.1.)
(c) Let $S \in \beta$ be a semisimple element; $z$ the centralizer of $S ; z_1=D_3$ (cf. Exercise 45). Prove that if $X \in \beta_1, X$ is nilpotent in $z_1$ if and only if it is nilpotent in $g$ and that $z \cap \mathfrak{H} \subseteq z_1$. Prove also that if $\{H, X, Y\}$ is an $S$-triple in $3_1,(s+X)^{\exp t H} \longrightarrow S$ as $t \longrightarrow 0$. Deduce that the closure of any orbit contains a semisimple orbit. (For the last result, use Jordan decomposition.)
(d) Let $\xi \in \mathbf{C}^i$, and let $S \in V_{\xi}$ be semisimple. Let $z$ and $z_1$ be as in (c). Prove that $V_{\xi}=\left\{(S+N)^x: x \in G, N \in z_1 \cap \mathfrak{M}\right\}$. Prove, further, that if $N, N^{\prime} \in z_1 \cap \mathfrak{N}, S+N$ and $S+N^{\prime}$ are conjugate under $G$ if and only if $N$ and $N^{\prime}$ are conjugate under the adjoint group of $3_1$. (If $x \in G$ and $(S+N)^x=S+N^{\prime}$, then $x$ centralizes $S$ and so belongs to the analytic subgroup of $G$ defined by 3 , by Exercise 45. But then $\left.x\right|_{31}$ lies in the adjoint group of $3_1$.)
(e) Prove that each $V_Z$ splits into finitely many orbits.
(f) Prove that the semisimple orbits are precisely the closed ones. Prove also that if $X \in \beta$ is regular and $\boldsymbol{\xi}=\mathbf{p}(X)$, then $X^0=V_\zeta$. (Each orbit is $\sigma$-compact, so by (e) and the Baire category theorem, $V_{\xi}$ contains an orbit open in it. Using (e) repeatedly, we can write $V_{\xi}=\boldsymbol{\Omega}_1 \cup \cdots \cup \boldsymbol{\Omega}_k$, where each $\Omega_i$ is an orbit, and is open in $\Omega_i \cup \Omega_{i+1} \cup \cdots \cup \Omega_k$, which is closed. So $V_{\xi}$ has a closed orbit, namely $\boldsymbol{\Omega}_k$. By (c) this must be semisimple.)
(g) For any $X \in \beta$, prove that $X^\sigma$ is open in its closure, hence that it is locally compact. Deduce that $X^G$ is a regular complex submanifold of $g$ of dimension $\operatorname{dim}(\mathrm{g})-\operatorname{dim}(\mathrm{g} x), \mathrm{g} x$ being the centralizer of $X$.

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Problem 67

An element $X \in \delta$ is called principal if $\operatorname{dim}_{\beta_X}=I$ where $g_X$ is the centralizer of $X$.
(a) Prove that $X \in \mathfrak{g}$ is principal if and only if the orbit $X^\sigma$ has maximal dimension (among all orbits) and that this dimension is $n-l$.
(b) Let $X \in \mathfrak{g}$, and let $X=S+N$ be its Jordan decomposition. Prove that $X$ is principal if and only if $N$ is a principal nilpotent in the derived algebra of the centralizer of $S$. (Let $z$ be as in Exercise 66(c). Then $\mathrm{rk}(3)=\mathrm{rk}(\mathrm{g})$ and $\beta_x=3 \cap g_N$.)
(c) Let $\xi \in \mathbf{C}^i$. Prove that $V_{\varsigma}$ contains a unique principal orbit which is open and dense in $V_{\xi}$.
(d) Deduce from (c) that $\mathbf{p}$ induces a bijection of the set of all principal orbits onto $\mathbf{C}^l$.
(e) Let $X \in \beta$ be principal. Prove that $q_x$ is abelian. (Take $X_n$ regular, $n= 1,2, \ldots, X_n \rightarrow X . \beta_x$ are CSA's and → $\beta_x$ in the Grassman manifold of $l$-planes of g . Use a compactness argument to show that $\mathrm{g}_x$ is abelian; see Kostant's paper [3].)

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Problem 68

Let $p_1, \ldots, p_i$ be homogeneous generators of $I, d_i=\operatorname{deg}\left(p_i\right)$. We assume that $d_1 \leq \cdots \leq d_l, X \in \mathrm{~g}$ is a principal nilpotent; $[H, X, Y]$ an $S$-triple; $g_r$, $Y_1, \ldots, Y_l, \lambda_1, \ldots, \lambda_l$ have the same meaning as in Exercise 67(f). We assume $\lambda_1 \leq \cdots \leq \lambda_l$. Put $v_j=1+\frac{1}{2} \lambda_j . E$ is the Euler vector field on $g$ which assigns to $Z \in \beta$ the tangent vector $Z$ at $Z$. As usual, we identify the tangent spaces to $G \times \mathbf{C}^{\prime}$ (resp. g) at each point with $g \times \mathbf{C}^{\prime}$ (resp. g).
(a) Prove that $[X, \beta]+\beta y=\beta$ is a direct sum.
(b) Let $\psi$ be the map $\left(x,\left(u_1, \ldots, u_i\right)\right) \mapsto\left(X+u_1 Y_1+\cdots+u_i Y_i\right)^x$ of $G \times \mathbf{C}^i$ into g. Deduce from (a) the existence of a connected open neighborhood $N$ of $(0, \ldots, 0)$ in $\mathrm{C}^i$ such that $(d \psi)_{\left\langle x,\left\{x_1, \ldots, x, k\right\rangle\right.}$ is surjective for all $x \in G,\left(u_1, \ldots, u_l\right) \in N$.
(c) Let $Z=\frac{1}{2} H, v_j=v_j u_j\left(1 \leq j \leq l, u_j \in \mathbf{C}\right)$. Verify that

$$
(d \psi)_{\left(1,\left\{u_1, \ldots, u_i\right)\right\}}\left(Z,\left(v_1, \ldots, v_f\right)\right)=X+u_1 Y_1+\cdots+u_f Y_f,
$$

(d) Let $\Omega_N=\left[\left(X+u_1 Y_1+\cdots+u_n Y_n\right)^x:\left(u_1, \ldots, u_i\right) \in N, x \in G\right]$. Let $\mathbb{Q}$ be the algebra of all $G$-invariant holomorphic functions on $\Omega_N$ and for $f \in \mathbb{Q}$, let $\bar{f}$ be the holomorphic function on $N$ defined by $\bar{j} \mid u, \ldots, \ldots)= f\left(X+u_1 Y_1+\cdots+u_i Y_i\right), \quad\left(u_1, \ldots, u_i\right) \in N$. Prove that $t \rightarrow j$ is an injection of $\mathbb{Q}$ into the algebra of holomorphic functions on $N$ and that if $\dot{E}$ is the differential operator $\sum_{1 \leq \sum^2 l} v_j u_j\left(\partial / \partial u_j\right)$ on $N, E \hat{E} \quad E ; 1 \equiv \mathbb{Q}$ ).
(e) Let $1 \leq j \leq l$. Prove that $\tilde{p}_j$ is a linear combination of those monomials $u_1^{m_1} \ldots u_i^{m_1}$ for which $\sum_{1 \leq k \leq 1} v_k m_k=d_j$.

$$
\left(\tilde{E} \tilde{p}_j=d_j \tilde{p}_j \text { and } \tilde{E}\left(u_1^{m_1} \cdots u_l^{m_i}\right)=\left(\sum v_k m_k\right) \cdot\left(u_1^{m_1} \cdots u_l^{m_i}\right)\right) .
$$

(f) Prove that $v_j=d_j$ for $1 \leq j \leq l$. (If $v_j>d_j$ for some $j$, then $d<v_i$ for $s \leq j \leq t$, so by (e), for $s \leq j, \bar{p}_x$ is a polynomial of only $u_{\ldots} \ldots u_{j-1}$. $\tilde{p}_1, \ldots, \tilde{p}_j$ are thus algebraically dependent, contradicting 1 d 1 . (Further, $\sum_{1 \leq j \leq 2} v_j=\sum_{1 \leq j=1} d_j=\frac{1}{2}(/+\operatorname{dim}(g))$, by Chevalley's theorem and the results of the appendix.)
(g) For any $\alpha=m_1 \alpha_1+\cdots+m_l \alpha_l \in P$, let $O(\alpha)=m_1+\cdots-m_l$. For any integer $k \geq 1$, let $b_k$ be the number of $\alpha \in P$ with $O(x)=k$. Prove that $b_1 \geq b_2 \geq \cdots$ and that $b_k$ is the number of the $d$ is ishich are equal to $k+1$. (Let $H, X, Y$ be as in Exercise 65(b). Then $\left[H . X_2\right]-20(\alpha) X_\alpha$ if $\alpha \in P, X_\alpha \in \mathfrak{g}_\alpha$. By representation theory of $\mathbf{C} \cdot H \cdot C \cdot X-\mathbf{C} \cdot Y, b_k$ is the number of the $\lambda_j$ 's which are $\geq 2 k$.)

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Problem 69

$V, V_c, \Delta$, to are as in 2-5 of the appendix to Chapter 4, $\mathcal{Q}$ (resp. $S$ ) is the graded polynomial algebra (resp. symmetric algebra) over $V_c: p \cdots p$ is the algebra isomorphism of $\mathcal{P}$ onto $S$ that extends the linear isomorphism of $V^*$ with $V$ induced by the scalar product of $V . p \mapsto p^{\text {coni }}$ (resp. $u \mapsto u^{\text {con }}$ ) is the conjugation of $\mathcal{Q}$ (resp. $S$ ) corresponding to the conjugation of $V_c^*$ (resp. $V_c$ ) induced by $V^*$ (resp. $V$ ).
(a) Prove that $p, q \mapsto\langle p, q\rangle=\left(\dot{\partial}\left(\tilde{q}^{\text {conj }}\right) p\right)(0)$ is a positive definite Hermitian bilinear form on $\mathcal{Q} \times \mathcal{Q}$.
(b) Let $I$ (resp. $J$ ) be the algebra of 10 -invariant elements of $\mathcal{\theta}$ (resp. $\delta$ ), $I^{+}$ (resp. $J^{+}$) the linear span in $I$ (resp. $J$ ) of the homogeneous elements of positive degree. Let $H=\left\{p: p \in \mathscr{P}, \partial(u) p=0\right.$ for all $\left.u \in J^{+}\right\}$. Prove that $H$ is a graded subspace of $\varphi$ and that $\varphi^{\circ}=\varphi^{-} \quad H$ is an orthogonal direct sum.
(c) Deduce from (b) that $\operatorname{dim} H=[\mathrm{m}]$.
(d) Let $P$ be a positive system. Let $\pi \in(\mathcal{P}$ be such that $\pi$ is the element $\mathrm{I}_{x \in \rho} \alpha$ of S . For $s \in \mathrm{iv}$ let $\epsilon(s)=\operatorname{det}(s)$. Prove that $\pi$ is skew-symmetric, i.e., $\pi^s=\epsilon(s) \pi \forall s \in 10$, and that $I \pi$ is the space of all skew-symmetric elements of $\mathcal{P}$.
(e) Prove that $\pi \in H$. (If $u \in J^{+}, \partial(u) \pi$ is skew-symmetric and of lower degree.)
(f) Prove that $\operatorname{deg}(P) \leq \operatorname{deg}(\pi) \forall p \in H$ and that $\operatorname{deg}(p)=\operatorname{deg}(\pi)$ if and only if $p=c \pi$ for some nonzero constant $c$. (Use the Poincaré Series for $H$.)

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Problem 70

Let notation be as above, with $w=[10] ; u_1=1, u_2, \ldots, u_v$, a basis of homogeneous elements of $H . V_c^{\prime}=\left\{\lambda: \lambda \in V_c, \pi(\lambda) \neq 0\right\}$.
(a) Prove that $\sum_{1 \leq x \leq w} \operatorname{deg}\left(u_i\right)=\frac{1}{2} w[P]$. (If $P(t)$ is the Poincaré Series for $H$,

$$
\left.\sum_{1-t \in w} \operatorname{deg}\left(u_i\right)=\left(\frac{d}{d t} P(t)\right)_{t=1} .\right)
$$

(b) For $\xi \in V_c$, let $\mathcal{U}_{\xi}$ (resp. $I_{\xi}$ ) be the set of all $p$ in $\mathcal{U}^{\circ}$ (resp. $I$ ) vanishing at $\xi$. Prove that if $\xi \in V_c^{\prime}, \mathcal{P}_{\xi}=\mathcal{P} I_{\xi}$ and that $\mathcal{P}=\mathcal{P}_{\xi}+H$ is a direct sum. (Since $\varphi \simeq I \otimes H, \varphi=\varphi I_{\varepsilon}+H$ is a direct sum; observe now that $[1 v \cdot \xi]=w$.)
(c) Prove that if $\xi \in V_c^{\prime}$, the restriction map $u \mapsto u \mid 10 \cdot \xi$ is a linear isomorphism of $H$ onto the complex vector space of all functions on iv ⋅ $\xi$. Deduce that the representation of 10 in $H$ is equivalent to its regular representation.
(d) Let $U$ be the $w \times w$ matrix $\left(u_{j s}\right)_{1 \leq U \leq w, s \in w}$, where $u_{j s}=u_j^x$. Prove that $\operatorname{det}(U)=c \pi^{(1 / 2) *}$, where $c \neq 0$ is a constant. (Let $\alpha \in P$. Subtracting column $s_\alpha s$ from column $s$, prove that $\alpha^{[1 ; 2) *} \operatorname{divides} \operatorname{det}(U)$. Hence $\operatorname{det}(U)=c \pi^{(1 / 2) w}$, where $c \in \mathcal{P} . c$ is a nonzero constant by (a) and (c).)
(e) Prove that there are unique rational functions $v_1, \ldots, v_w$ such that $\sum_{s \in r} u_j^z v_k^z=\delta_{j k}(1 \leq j, k \leq w)$. (Let $Q=U^{-1}=\left(q_{r k}\right)_{t \in m, 1 \leq k \leq w} ;$ write $q_{1 k}=v_k$. Then $Q U=1 \Rightarrow \sum_{1 \leq k \leq *} v_k u_k^s=\delta_{1,}(s \in \mathfrak{t}) \Longrightarrow \sum_{1 \leq k \leq w} v_k^t u_k^t =\delta_{t t^{\prime}}\left(t, t^{\prime} \in \mathfrak{1 0}\right)$. So $q_{t k}=v_k^t$. Now use $U Q=1$.)
(f) Prove that $v_k$ is well defined on $V_c^{\prime}$ and that $\pi v_k \in \mathscr{P}(1 \leq k \leq w)$. (Argue as in (d) to prove that $\pi^{(1 / 2)_{*-1}}$ divides the cofactors in $\operatorname{det}(U)$.) (The results are due to Harish-Chandra [7, 8].)

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Problem 71

Let notation be as above. For $p \in \varphi^{\circ}$, Let $z_{i j ; p}$ be the unique elements of $I$ such that $p u_j=\sum_{t \leq i \leq *} z_{i f ; p} u_i(1 \leq j \leq w)$. Let $\Gamma(p)$ be the $w \times w$ matrix $\left(z_{j ; p}\right)$. For $\xi \in V_c$ let $\Gamma(p ; \xi)$ be the $w \times w$ matrix $\left(z_{j ; p}(\xi)\right)$.
(a) Prove that $p \mapsto \Gamma(p: \xi)$ is a representation of $\mathcal{P}$ by $w \times w$ matrices.
(b) For $\xi \in V_c^{\prime}$ and $s \in 10$, let $\psi_x(\xi)$ be the vector in $\mathbf{C}^\mu$ whose components are $v_1^x(\xi), \ldots, v_w^s(\xi)$. Prove that the $\psi_s(\xi)(s \in \mathcal{W})$ form a basis for $\mathbf{C}^w$ and that $\Gamma(p: \xi) \psi_s(\xi)=p^s(\xi) \psi_s(\xi)(s \in 10)$. Deduce that the characteristic polynomial of $\Gamma(p: \xi)$ is $\prod_{s \in r}\left(T-p^2(\xi)\right)\left(\xi \in V_c\right)$ and that $\Gamma(p: \xi)$ is semisimple if $\xi \in V_c^{\prime}, p \in \mathcal{P}^{\circ}$.
(c) Let $M$ be the matrix $\left(u_j v_k\right)_{1 \leq j, k \leq w}$ be order $w$. Prove that if $\xi \in V_c^{\prime}$, then $M^{\prime}(\xi)(s \in 10)$ are the spectral projections of $\Gamma(p: \xi)$, i.e., the projections $\mathbf{C}^* \rightarrow \mathbf{C} \cdot \psi_s(\xi)$ corresponding to the direct sum $\mathbf{C}^*=\sum_{i \in \mid 0} \mathbf{C} \cdot \psi_i(\xi)$. (See Harish-Chandra $[7,8]$.)

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