Exercise 18, § 7

Choose a positive non-degenerate symmetric bilinear form $(\cdot | \cdot )$ on $\mathfrak{h}^*_{\mathbf{R}}$ invariant under W. Let $\lambda \in P_{++}$.

a) Let $\mu$ be a weight of $E(\lambda )$. Write $\lambda -\mu$ in the form $\sum_{\alpha\in B}k_{\alpha}\alpha$. Let $\alpha \in B$ be

such that $k_{\alpha}\not= 0$. Show that there exists $\alpha_1, . . . , \alpha_n\in B$ such that $(\lambda |\alpha_1)\not= 0$, $(\alpha_1|\alpha_2)\not= 0, . . . ,(\alpha_{n-1}|\alpha_n)\not= 0,(\alpha_n|\alpha )\not= 0$.

b) Let $v$ be a primitive element of $E(\lambda )$, and let $\alpha_1, . . . , \alpha_n\in B$ satisfy the following conditions:

(i) $(\alpha_i|\alpha_{i+1})\not= 0$ for $i= 1,2, . . . , n-1$;

(ii) $(\alpha_i|\alpha_j) = 0$ for $j > i+ 1$;

(iii) $\lambda (H_{\alpha_1})\not= 0$ and $\lambda (H_{\alpha_2}) =\cdots =\lambda (H_{\alpha_n}) = 0$.

Show that $X_{-\alpha_n}X_{-\alpha_{n-1}}. . . X_{-\alpha_1}v\not= 0$. (Observe that, for $1\leq s\leq n$,

is not a weight of $E(\lambda )$, and deduce that $X_{\alpha_s}X_{-\alpha_s}X_{-\alpha_{s-1}}. . . X_{-\alpha_1}v\not= 0$ by induction on $s.)$ If $\sigma \in \mathfrak{S}_n$ and $\sigma \not= 1$, then $X_{-\alpha_{\sigma(n)}}X_{-\alpha_{\sigma(n-1)}}. . . X_{-\alpha_{\sigma(1)}}v= 0$. (Let $r$ be the smallest integer such that $\sigma (r)\not=r$. Use $a$) to show that $\lambda -\alpha_{\sigma(1)}-. . .-\alpha_{\sigma(r)}$ is not a weight of $E(\lambda ).)$

c) Let $\lambda '\in P_{++}$. A chain joining $\lambda$ to $\lambda '$ is a sequence $(\alpha_1, . . . , \alpha_n)$ of elements of B such that $n\geq 1, (\lambda |\alpha_1)\not= 0,(\alpha_1|\alpha_2)\not= 0, . . . ,(\alpha_{n-1}|\alpha_n)\not= 0$, $(\alpha_n|\lambda ')\not= 0$. Such a chain is called minimal if no strict subsequence of $(\alpha_1, . . . , \alpha_n)$ joins $\lambda$ to $\lambda '$. In that case, we have $(\alpha_i|\alpha_j) = 0$ if $|i-j| \geq 2$, $(\lambda |\alpha_i) = 0$ if $i\geq 2$ and $(\lambda '|\alpha_i) = 0$ if $i\leq n-1$.

d) Let $(\alpha_1, . . . , \alpha_n)$ be a minimal chain joining $\lambda$ to $\lambda '$. If $v'$ is a primitive vector in $E(\lambda ')$, put:

$a_0= (\lambda |\alpha_1), a_n= (-1)^n(\lambda '|\alpha_n), a_s= (-1)^{s+1}(\alpha_s|\alpha_{s+1}), 1\leq s\leq n-1$. Show that

is a primitive element of $E(\lambda )\otimes E(\lambda ')$ of weight $\lambda +\lambda '-\alpha_1- \cdots -\alpha_n$, and that it is unique up to homothety.

(Use $b$) and c) to show that every element of $E(\lambda )\otimes E(\lambda ')$ of weight

is a linear combination of the $v_s\otimes v'_s$. Then write down the condition for such a linear combination to be a primitive vector.)

Deduce that $E(\lambda )\otimes E(\lambda ')$ contains a unique $\mathfrak{g}$-submodule isomorphic to

(When $n= 1$ we recover Exerc. 17.)

e) Let $w$ be a primitive element of $E(\lambda )\otimes E(\lambda ')$. Assume that the weight $\nu$ of $w$ is distinct from $\lambda +\lambda '$. Show that there exists a chain $(\alpha_1, . . . , \alpha_n)$ joining $\lambda$ to $\lambda '$ such that

(Let C be the set of $\alpha \in B$ such that the coordinate of index $\alpha$ of $\lambda +\lambda '-\nu$ is $\not= 0$. Let D (resp. $D'$) be the set of $\alpha \in C$ such that there exists $\gamma_1, . . . , \gamma_t\in C$ satisfying $(\lambda |\gamma_1)\not= 0$ (resp. $(\lambda '|\gamma_1)\not= 0$)$, (\gamma_1|\gamma_2)\not= 0, . . . ,(\gamma_{t-1}|\gamma_t)\not= 0$, $(\gamma_t|\alpha )\not= 0$. Let Y (resp. $Y'$) be the set of weights of $E(\lambda )$ (resp. $E(\lambda ')$) of the form $\lambda -\sum_{\alpha\in C}k_{\alpha}\alpha$ (resp. $\lambda '-\sum_{\alpha\in C}k_{\alpha}\alpha$ ), with $k_{\alpha}\in \mathbf{N}$. Show that $w$ belongs to $((\sum_{\mu\in Y}E(\lambda )^\mu)\otimes ((\sum_{\mu'\in Y'}E(\lambda ')^{\mu'})$. Using a) and the fact that $\nu \not=\lambda +\lambda '$,

show that $D\cap D'\not=\emptyset .)$

f) Show the equivalence of the following properties:

(i) $E(\lambda )\otimes E(\lambda ')$ is isomorphic to $E(\lambda +\lambda ')$.

(ii) $E(\lambda )\otimes E(\lambda ')$ is a simple module.

(iii) There is no chain joining $\lambda$ to $\lambda '$.

(iv) R is the direct sum of two root systems $R_1$ and $R'_1$ such that $\lambda \in P(R_1)$ and $\lambda '\in P(R'_1)$.

(v) $\mathfrak{g}$ is the product of two ideals $\mathfrak{s}$ and $\mathfrak{s}'$ such that $\mathfrak{s}'.E(\lambda ) = 0$ and $\mathfrak{s}.E(\lambda ') = 0$.

(Use $d$) to prove the equivalence of (ii) and (iii).)$^{15}$