Proposition 7

For any class $\gamma$ of finite dimensional simple $\mathfrak{g}$-modules, let $U_{\gamma}$ be the isotypical component of type $\gamma$ of the $\mathfrak{g}$-module U (for the adjoint representation of $\mathfrak{g}$ on U). Let $\gamma_0$ be the class of the trivial $\mathfrak{g}$-module of dimension 1. Let $[U,U]$ be the vector subspace of U generated by the brackets of pairs of elements of U.

(i) U is the direct sum of the $U_{\gamma}$.

(ii) $U_{\gamma_0}= Z$, and $\sum_{\gamma\not=\gamma_0}U_{\gamma}= [U,U]$.

(iii) Let $u \rightarrow u^{\natural}$ be the projection of U onto Z defined by the decomposition $U = Z\oplus [U,U]$. If $u\in U$ and $v\in U$, we have $(uv)^{\natural}= (vu)^{\natural}$. If $u\in U$ and $z\in Z$, we have $(uz)^{\natural}=u^{\natural}z$.

(iv) Let $\varphi$ be the Harish-Chandra homomorphism. Let $\lambda \in P_{++}$, and let E be a finite dimensional simple $\mathfrak{g}$-module of highest weight $\lambda$. For all $u\in U$, we have

dim E1 Tr($u_E$) $= (\varphi (u^{\natural}))(\lambda )$.

The $\mathfrak{g}$-module U is a direct sum of finite dimensional submodules. This implies (i).

It is clear that $U_{\gamma_0}= Z$. Let $U'$ be a vector subspace of U defining a subrepresentation of class $\gamma$ of the adjoint representation. Then either $[\mathfrak{g},U'] = U'$ or $[\mathfrak{g},U'] = 0$. Thus, if $\gamma \not=\gamma_0$ then $[\mathfrak{g},U'] = U'$, so $\sum_{\gamma\not=\gamma_0}U_{\gamma}\subset [U,U]$. On the other hand, if $u\in U$ and $x_1, . . . , x_n\in \mathfrak{g}$, then

Hence $[U,U]\subset [\mathfrak{g},\sum_{\gamma}U_{\gamma}]=[\mathfrak{g},\sum_{\gamma\not=\gamma_0}U_{\gamma}]\subset \sum_{\gamma\not=\gamma_0}U_{\gamma}$. This proves (ii). Under these conditions, (iii) follows from Chap. I, §6, no. 9, Lemma 5.

Finally, let $E, \lambda$ be as in (iv). Then

Tr($u_E$) $=$ Tr(($u^{\natural}$)$_E$) since $u-u^{\natural}\in [U,U]$

= Tr($\varphi (u^{\natural})(\lambda ).1$) (§$6$, no$.4$, Prop$.7$)

= (dim $E$)$.\varphi (u^{\natural})(\lambda )$.