Exercise 13, § 1

Assume that $k=\mathbf{C}$. Put $C =H^2+ 2H-4X_-X_+$, cf. Exerc. 1. A representation $\rho$ of $\mathfrak{s}=\mathfrak{s}\mathfrak{l}(2,\mathbf{C})$ is said to be $H$-diagonalizable if the underlying space of $\rho$ has a basis consisting of eigenvectors of $\rho (H)$. Let $q\in \mathbf{C}$ and $v\in \mathbf{Q}/\mathbf{Z}$.

a) Let S be a vector space with a basis $(e_w)_{w\in\mathbf{C}}$ indexed by the elements of $\mathbf{C}$. Let $S_v=\sum_{w\in v}\mathbf{C}e_w$. There exists a unique representation $\rho_{v,q}$ of $\mathfrak{s}$ on $S_v$ such that

$^7$ The conclusion remains true even if $k$ is countable, cf. D. QUILLEN, On the

endomorphism ring of a simple module over an enveloping algebra, Proc. Amer.

Math. Soc., Vol. XXI (1969), pp. 171-172.

(We agree that, for all $z\in \mathbf{C},z^{1/2}$ is the square root of $z$ whose amplitude belongs to $0, \pi .$) Denote by $S_{v,q}$ the $\mathfrak{s}$-module defined by $\rho_{v,q}$. We have $\rho_{v,q}(C) = 4q$.

b) If $q\not=u^2+u$ for all $u\in v, S_{v,q}$ is simple.

c) Assume that $2v\not= 0$, and that $q$ is of the form $u^2+u$, where $u\in v, u\geq 0$ (which defines $u$ uniquely). Let $S^-_{v,q}$ (resp. $S^+_{v,q}$) be the vector subspace of $S_v$ generated by the $e_w$ for $w\leq u$ (resp. $w > u$). Then $S^-_{v,q}$ and $S^+_{v,q}$ are simple $\mathfrak{s}$-submodules of $S_{v,q}$.

d) Assume that $2v= 0$, and that $q$ is of the form $u^2+u$, where $u\in v, u\geq 0$ (which defines $u$ uniquely). Let $S^-_{v,q}$ (resp. $S^0_{v,q},S^+_{v,q}$) be the vector subspace of $S_v$ generated by the $e_w$ for $w <-u$ (resp. $-u\leq w\leq u, w > u$). Then $S^-_{v,q},S^0_{v,q}$ and $S^+_{v,q}$ are simple $\mathfrak{s}$-submodules of $S_{v,q}$.

e) Assume that $v=-\frac{1}{2}+\mathbf{Z}$ and $q=-\frac{1}{4}$. Let $S^-_{-1/2,-1/4}$ (resp. $S^+_{-1/2,-1/4}$) be the vector subspace of $S_{-1/2}$ generated by the $e_w$ for $w\leq -\frac{1}{2}$ (resp. $w >-\frac{1}{2}$). Then $S^-_{-1/2,-1/4}$ and $S^+_{-1/2,-1/4}$ are simple $\mathfrak{s}$-submodules of $S_{-1/2,-1/4}$.

f) Denote by $\rho^{\pm}_{v,q}, \rho^0_{v,q}$ the representations corresponding to $S^{\pm}_{v,q},S^0_{v,q}$. In case b), the elements $x\in \mathfrak{s}$ such that $\rho^-_{v,q}(x)$ admits an eigenvalue are those in $\mathbf{C}H$. In cases $c),d),e)$, the elements $x\in \mathfrak{s}$ such that $\rho^-_{v,q}(x)$ admits an eigenvalue are those in $\mathbf{C}H+\mathbf{C}X_+$; if, in addition, $x$ is nilpotent (and hence proportional to $X_+$) and non-zero, $\rho^-_{v,q}(x)$ admits 0 as its only eigenvalue, and this of multiplicity 1; on the other hand, if $x$ is semi-simple, the underlying space of $\rho^-_{v,q}$ has a basis consisting of eigenvectors of $\rho^-_{v,q}(x)$. There are analogous results for $\rho^+_{v,q}$, replacing $X_+$ by $X_-$.

g) Let V be a simple $\mathfrak{s}$-module and $\rho$ the corresponding representation. Then $\rho (C)$ is a homothety (use Exerc. 11). Assume that $\rho (C) = 4q$. Show that, if $\rho$ is $H$-diagonalizable, V is isomorphic to one of the modules $S_{v,q},S^{\pm}_{v,q},S^0_{v,q}$ considered in $b),c),d),e)$. Moreover, $\rho$ is $H$-diagonalizable if and only if $\rho (H)$ admits an eigenvalue; it suffices that $\rho (X_+)$ admits the eigenvalue 0.