Mackey formulas for smooth stacks

Abstract

We study parabolic induction and restriction functors on derived categories of constructible sheaves, D-modules, and other sheaf theories on smooth stacks with good or adequate moduli spaces. We prove a commutation rule between them in the form of a Mackey filtration, indexed by the set of relative positions of two elements in the component lattice, a combinatorial structure that characterizes inductions and restrictions on a given stack. This generalizes similar results for adjoint quotient stacks studied in generalized Springer theory and the theory of character sheaves, and for stacks of quiver representations appearing in Lusztig's categorification of quantum groups. We also apply our result to the stack of semistable principal bundles on a curve, as a first step towards Mackey formulas in geometric Langlands correspondences.