Semi-Simplicity of Invariant Holonomic Systems on a Reductive Lie Algebra

Let$\germ{g}$be a reductive, complex Lie algebra, with adjoint group G, let G act on the ring of differential operators${\cal D}(\germ{g})$via the adjoint action and write$\tau \colon \germ{g}\rightarrow {\cal D}(\germ{g})$for the differential of this action. Fix$\lambda \in \germ{h}^{\ast}$. Generalizing work of Hotta and Kashiwara, we prove that the invariant holonomic system${\cal N}_{\lambda}={\cal D}(\germ{g})/({\cal D}(\germ{g})\tau (\germ{g})+\sum_{p\in S(\germ{g})^{G}}{\cal D}(\germ{g})(p-p(\lambda)))$is is semisimple. The simple summands of${\cal N}_{\lambda}$are parametrized by the irreducible representations of$W_{\lambda}$, the stabilizer of λ in the Weyl group. Consequently, the subcategory generated by${\cal N}_{\lambda}$is equivalent to the category of finite dimensional representations of$W_{\lambda}$.