Summary: SPRINGER THEORY FOR COMPLEX REFLECTION GROUPS
PRAMOD N. ACHAR
Abstract. Many complex reflection groups behave as though they were the Weyl groups of "nonexis-
tent algebraic groups": one can associate to them various representation-theoretic structures and carry
out calculations that appear to describe the geometry and representation theory of an unknown object.
This paper is a survey of a project to understand the geometry of the "unipotent variety" of a complex
reflection group (enumeration of unipotent classes, Springer correspondence, Green functions), based
on the author's joint work with A.-M. Aubert.
A complex reflection group is a finite group of automorphisms of a finite-dimensional complex vector
space V that is generated by reflections, i.e., linear transformations that fix some hyperplane pointwise.
Some complex reflection groups can actually be realized on a real vector space, and a famous theorem
of Coxeter states that these are precisely the finite Coxeter groups. Among those, the reflection groups
that can be realized on a Q-vector space are particularly important: these are the groups that occur as
Weyl groups of reductive algebraic groups.
Since the early 1990's, there has been a growing awareness that many complex reflection groups
that cannot be realized over Q nevertheless behave as though they were the Weyl groups of certain
"nonexistent" algebraic groups. The first important step was the discovery [4, 5, 13] that their group
algebras admit deformations resembling IwahoriHecke algebras of Coxeter groups. Those deformations
are now known as cylcotomic Hecke algebras. Subsequent work by a number of authors showed that
complex reflection groups admit analogues of Coxeter presentations , root systems [17, 33] and root