Semi-simplicity of invariant holonomic systems on a reductive Lie algebra

Let [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] be a reductive, complex Lie algebra, with adjoint group G, let G act on the ring of differential operators [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] via the adjoint action and write τ: [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] → [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /] for the differential of this action. Fix λ ∈ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /]*. Generalizing work of Hotta and Kashiwara, we prove that the invariant holonomic system [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="06i" /] is semisimple. The simple summands of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="07i" /] are parametrized by the irreducible representations of W_λ, the stabilizer of λ in the Weyl group. Consequently, the subcategory generated by [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="08i" /] is equivalent to the category of finite dimensional representations of W_λ.