Parabolic induction and perverse sheaves on h/W

TL;DR

This paper characterizes the category of perverse sheaves on the quotient of a Cartan subalgebra by the Weyl group for a complex reductive Lie group, using mixed Bruhat sheaves and parabolic decomposition techniques.

Contribution

It introduces mixed Bruhat sheaves as a new framework to describe perverse sheaves on h/W and relates them to parabolic induction and restriction processes.

Findings

Describes the category of perverse sheaves on h/W in terms of mixed Bruhat sheaves.

Establishes a connection between mixed Bruhat sheaves and parabolic induction/restriction.

Provides a geometric interpretation of these sheaves via the Bruhat decomposition.

Abstract

For a complex reductive Lie group G with Lie algebra g, Cartan subalgebra h and Weyl group W, we describe the category of perverse sheaves on h/W smooth w.r.t the natural stratification. The answer is given in terms of mixed Bruhat sheaves, which are certain mixed sheaf-cosheaf data on cells of a natural cell decomposition of h/W. Using the parabolic Bruhat decomposition, we relate mixed Bruhat sheaves with the properties of various procedures of parabolic induction and restriction that connect different Levi subgroups in G.