 
Summary: GREEN FUNCTIONS VIA HYPERBOLIC LOCALIZATION
PRAMOD N. ACHAR
Abstract. Let G be a reductive algebraic group, with nilpotent cone N and
flag variety B. We construct an exact functor from perverse sheaves on N to
locally constant sheaves on B, and we use it to study Extgroups and stalks
of simple perverse sheaves on N in terms of the cohomology of B.
1. Introduction
Let G be a connected reductive algebraic group over an algebraically closed field
k of good characteristic. Let N denote the nilpotent cone in its Lie algebra g,
and let W denote its Weyl group. An explicit description of the stalks of simple
perverse sheaves on N has been given by Lusztig [L], building on earlier ideas of
Shoji [S1, S2]. For most such perverse sheaves (those appearing in the Springer cor
respondence), this description involves the representation theory of W, and specif
ically its coinvariant algebra. The coinvariant algebra of W is also isomorphic to
the cohomology ring H·
(B) of the flag variety B. However, that cohomology ring
does not appear in the proofs in [L], which rely instead on orthogonality properties
of character sheaves coming from the geometry of semisimple classes.
The present paper is an attempt to understand Lusztig's results directly in terms
of the geometry of B. Consider the diagram
