Summary: ON SPECIAL PIECES, THE SPRINGER CORRESPONDENCE,
AND UNIPOTENT CHARACTERS
PRAMOD N. ACHAR AND DANIEL S. SAGE
Abstract. Let G be a connected reductive algebraic group over the algebraic
closure of a finite field Fq of good characteristic. In this paper, we demonstrate
a remarkable compatibility between the Springer correspondence for G and the
parametrization of unipotent characters of G(Fq). In particular, we show that
in a suitable sense, "large" portions of these two assignments in fact coincide.
This extends earlier work of Lusztig on Springer representations within special
pieces of the unipotent variety.
Let G be a connected reductive algebraic group over the algebraic closure of a
finite field Fq of good characteristic, and let W be its Weyl group. The Springer
correspondence, which we denote by , assigns to each irreducible representation
of W an irreducible equivariant local system on a unipotent class of G. On the
other hand, part of Lusztig's parametrization of unipotent characters of finite re-
ductive groups (or of his parametrization of unipotent character sheaves) is a map
assigning to each irreducible representation of W an element of a certain finite
set, whose members may be thought of as irreducible equivariant local systems on
certain finite groups.