Contemporary Mathematics

Volume 602, 2013

http://dx.doi.org/10.1090/conm/602/12031

Kostka systems and exotic t-structures for reflection groups

Pramod N. Achar

Abstract. Let W be a complex reflection group, acting on a complex vector

space h. Kato has recently introduced the notion of a “Kostka system,” which

is a certain collection of finite-dimensional W -equivariant modules for the sym-

metric algebra on h. In this paper, we show that Kostka systems can be used

to construct “exotic” t-structures on the derived category of finite-dimensional

modules, and we prove a derived-equivalence result for these t-structures.

1. Introduction

1.1. Overview. In the early 1980’s, Shoji [S1, S2] and Lusztig [L3] showed

that Green functions—certain polynomials arising in the representation theory of

finite groups of Lie type—can be computed by a rather elementary procedure, now

often known as the Lusztig–Shoji algorithm. This algorithm can be interpreted as

a computation in the Grothendieck group of the derived category of mixed -adic

complexes on the nilpotent cone of a reductive algebraic group, with the simple

perverse sheaves playing a key role; see [A3].

In recent work [K1], Kato has proposed an alternative interpretation of Green

functions in terms of the Grothendieck group of the (derived) category of graded

modules over the ring AW = C[W ] #

C[h∗],

where W is the Weyl group, and h is

the Cartan subalgebra. In place of simple perverse sheaves, the key objects are now

projective AW -modules. Thus, Kato’s viewpoint is “Koszul dual” to the geometric

one. A prominent place is given to certain collections of finite-dimensional AW -

modules (denoted by Kχ in [K1] and by

¯

∇

χ

here), called Kostka systems.

In this paper, we study Kostka systems as generators of the derived category

Dfd(AW

b

) of finite-dimensional AW -modules. We prove that they form a dualizable

quasi-exceptional sequence, which implies that they determine a new t-structure

on Dfd(AW b ), called the exotic t-structure. The heart of this t-structure, denoted

by ExW , is a finite-length weakly quasi-hereditary category. The main result (see

Theorem 6.9) states that there is an equivalence of triangulated categories

(1.1)

DbExW

∼

→ Dfd(AW

b

).

Of course, projective AW -modules cannot belong to ExW , since they are not finite-

dimensional. Nevertheless, in some ways, they behave as though they were tilting

objects of ExW . Thus, in a loose sense, which we do not attempt to make precise

2010 Mathematics Subject Classification. Primary 20F55, 18E30.

The author received support from NSF Grant DMS-1001594.

c 2013 American Mathematical Society

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