Semisimplicity of the category of admissible D-modules

Abstract

Using a representation theoretic parametrization for the orbits in the enhanced cyclic nilpotent cone, derived by the authors in a previous article, we compute the fundamental group of these orbits. This computation has several applications to the representation theory of the category of admissible D-modules on the space of representations of the framed cyclic quiver. First, and foremost, we compute precisely when this category is semi-simple. We also show that the category of admissible Dmodules has enough projectives. Finally, the support of an admissible D-module is contained in a certain Lagrangian in the cotangent bundle of the space of representations. Thus, taking characteristic cycles defines a map from the K-group of the category of admissible D-modules to the Z-span of the irreducible components of this Lagrangian. We show that this map is always injective, and a bijection if and only if the monodromicity parameter is integral. Contents 1. Introduction 1 2. Monodromic D-modules 5 3. Quantum Hamiltonian reduction 11 4. Admissible D-modules 17 5. The framed cyclic quiver 23 6. Semi-simplicity 28 References 35

Invariant holonomic systems on symmetric spaces and other polar representations

“…Since we have shown in Proposition 12.1 that eu V acts locally finitely on every object of C χ ,0 , it follows that strongly admissibility in our situation is the same as χ-admissibility as defined in [BB,Section 4.1]. Therefore, it follows from [BB,Proposition 1.4] that C χ ,0 has enough projectives. The same result holds for the corresponding category C op χ,0 of right modules.…”

“…which implies that χ • δ := χ 0 + • • • + χ ℓ−1 = 0. By [BB,Theorem 6.7], G 0 is not semisimple in this case. In fact, the same conclusion holds for any ς for which the Hecke algebra is semisimple; the condition χ • δ = 0 holds for all ς so G 0 is never semisimple.…”