Exercise 14, § 1

The notations are those of the preceding exercise. Denote by $B_q$ the quotient of $U(\mathfrak{s})$ by the two-sided ideal generated by $C-4q$, and denote by $u \rightarrow u^{\bullet}$ the canonical map $U(\mathfrak{s})\rightarrow B_q$. Consider the representations of Exerc. 12 and 13 as representations of $B_q$.

a) Every element of $B_q$ can be expressed uniquely in the form

where the $p_r$ and the $q_s$ are polynomials. If two elements of $B_q$ generate the same left ideal, they are proportional. b) Consider case b) of Exerc. 13. Let $a$ be a non-zero element of $S_{v,q}$. If $a$ is an eigenvector of $\rho_{v,q}(H)$, with eigenvalue $\lambda$, the annihilator of $a$ in $B_q$ is the left ideal generated by $H^{\bullet}-\lambda$; if $a$ is not an eigenvector of $\rho_{v,q}(H)$, its annihilator is a non-monogenic left ideal.

c) Consider the case of the representations $\rho^{\pm}_{v,q}$ of Exerc. 13. If $a$ is a non-zero element of the space of such a representation, its annihilator in $B_q$ is a non-monogenic left ideal.

d) Consider the representation $\rho_q$ of Exerc. 12. The annihilator of $e_0$ in $B_q$ is generated by $X_+^{\bullet}-1$. If $a\in W$ is not proportional to $e_0$, its annihilator is a non-monogenic left ideal.

e) Every automorphism of $\mathfrak{s}$ extends to $U(\mathfrak{s})$ and defines by passage to the quotient an automorphism of $B_q$; let $A'$ be the subgroup of A = Aut(B$_q$) thus obtained. Show that $A'\not= A$. (Let $\varphi$ be the endomorphism $x \rightarrow [X_+^{\bullet 2}, x]$ of $B_q$; this automorphism is locally nilpotent, and $e^{\varphi}\in A, e^{\varphi}\notin A'.$)

f) The group A operates by transport of structure on the set of classes of simple representations of $B_q$. Let $\pi_1$ (resp. $\pi_2, \pi_3$) be a representation of type $\rho_{v,q}$ of Exerc. $13b)$ (resp. of type $\rho^{\pm}_{v,q}$ of Exerc. 13, resp. of type $\rho_q$ of Exerc. 12). Then $A\pi_1,A\pi_2$ and $A\pi_3$ are pairwise disjoint. (Use $a$)$,b),c),d).)$ If $\psi \in A$ is such that $\psi \pi_1$ is of type $\rho_{v,q}$ of Exerc. $13b)$, then $\psi \in A'$ (use $a$) and $b))$. Deduce that, if $\psi$ is the automorphism $e^{\varphi}$ of e), the representation $\sigma =\pi_1\circ \psi$ has the following property: for all $x\in \mathfrak{s}$-- $\{0\},\sigma (x)$ has no eigenvalue.$^8$