Exercise 14, § 9

a) Let $\mu, \nu \in P_{++}$ and let $\gamma$ be a weight of $E(\mu)$. Let $\rho_\mu$ be the representation of $\mathfrak{g}$ on $E(\mu)$. Let $X_{\alpha}\in \mathfrak{g}^{\alpha}$ -- $\{0\}, Y_{\alpha}\in \mathfrak{g}^{-\alpha}$ -- $\{0\}$. If $\alpha \in B$, put $\nu_{\alpha}=\nu (H_{\alpha})$, and

$E^+(\mu, \gamma , \nu ) = E(\mu)^{\gamma}\cap \bigcap_{\alpha\in B}$ Ker $\rho_\mu(X_{\alpha})^{\nu_{\alpha}+1}$,

$E^-(\mu, \gamma , \nu ) = E(\mu)^{\gamma}\cap \bigcap_{\alpha\in B}$ Ker $\rho_\mu(Y_{\alpha})^{\nu_{\alpha}+1}$,

$d^+(\mu, \gamma , \nu ) =$ dim $E^+(\mu, \gamma , \nu ),d^-(\mu, \gamma , \nu ) =$ dim $E^-(\mu, \gamma , \nu )$.

For all $\lambda \in \mathfrak{h}^*$, put $\lambda^*=-w_0\lambda$, where $w_0$ is the element of W that takes B to $-B$. Show that

b) Let $\lambda_1, \lambda_2\in P_{++}, V$ the $\mathfrak{g}$-module Hom$_k(E(\lambda^*_1),E(\lambda_2)), U$ the set of $\varphi \in V$ such that $Y_{\alpha}.\varphi = 0$ for all $\alpha \in B$, and $w$ a primitive vector in $E(\lambda^*_1)$. Show that $\varphi \rightarrow \varphi (w)$ is an isomorphism from U to the set of $v\in E(\lambda_2)$ such that $Y_{\alpha}^{\lambda^*_1(H_{\alpha})+1}.v= 0$ for all $\alpha \in B$. (Use Exerc. 15 of §7 to prove surjectivity.) c) With the notations of Prop. 2, prove that, for $\lambda_1, \lambda_2, \lambda \in P_{++}$,

(Observe that $m(\lambda_1, \lambda_2, \lambda )$ is the dimension of the space of $\mathfrak{g}$-invariant elements of

and hence is equal to $m(\lambda^*_1, \lambda , \lambda_2).)$

d) Let $\lambda_1, \lambda_2\in P_{++}$, and let $\lambda$ be the unique element of $P_{++}\cap W.(\lambda_1-\lambda^*_2)$. We have

(Use $c$)$.)$ Deduce that $E(\lambda_1)\otimes E(\lambda_2)$ contains a unique submodule isomorphic to $E(\lambda )$.

e) We retain the notations of d). Show the equivalence of the following conditions:

(i) $\lambda =\lambda_1+\lambda_2$

(ii) $\|\lambda \|=\|\lambda_1+\lambda_2\|$

(iii) $\lambda_1$ and $\lambda^*_2$ are orthogonal

(iv) $\lambda_1$ and $\lambda^*_2$ are disjoint (Exerc. 13)

(v) $\lambda_1$ and $\lambda_2$ are disjoint.

Conclude that $E(\lambda_1)\otimes E(\lambda_2)$ is not a simple module unless $\lambda_1$ and $\lambda_2$ are disjoint (and hence another proof of Exerc. $18f$) of §7).