Green functions via hyperbolic localization

TL;DR

This paper introduces a new functor connecting perverse sheaves on the nilpotent cone to locally constant sheaves on the flag variety, providing fresh insights into Ext-groups and stalks of perverse sheaves.

Contribution

It constructs an exact functor linking perverse sheaves on the nilpotent cone to sheaves on the flag variety, offering new proofs and perspectives in geometric representation theory.

Findings

Established a functor from perverse sheaves on N to sheaves on G/B

Revealed relationships between Ext-groups and cohomology of G/B

Provided new proofs of known results on stalks of perverse sheaves

Abstract

Let G be a reductive algebraic group, with nilpotent cone N and flag variety G/B. We construct an exact functor from perverse sheaves on N to locally constant sheaves on G/B, and we use it to study Ext-groups of simple perverse sheaves on N in terms of the cohomology of G/B. As an application, we give new proofs of some known results on stalks of perverse sheaves on N.