Volume 602, 2013
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.
1.1. 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 ] #
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
here), called Kostka systems.
In this paper, we study Kostka systems as generators of the derived category
) 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
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