A Derived Decomposition for Equivariant $D$-modules

Abstract

We show that the adjoint equivariant derived category of $D$-modules on a reductive Lie algebra $\mathfrak{g}$ carries an orthogonal decomposition in to blocks indexed by cuspidal data (in the sense of Lusztig). Each block admits a monadic description in terms a certain monad related to the homology of Steinberg varieties; this monad carries a filtration (the Mackey filtration) whose associated graded functor is given by the action of the relative Weyl group. Furthermore, we show that the Mackey filtration is generally non-split and thus the Springer-theoretic description of the entire equivariant derived category of $D$-modules appears to be substantially more subtle than either the case of the abelian category in earlier work of the author, or the derived category of nilpotent orbital sheaves in work of Rider and Russell. One notable feature of this setting is that the parabolic induction and restriction functors depend on the choice of parabolic subgroup containing a given Levi factor.