of any algebraic group. See [S3, S4, GM] for variations and conjectures on the
Lusztig–Shoji algorithm.
One of Kato’s aims in [K1] was to provide a categorical framework for inter-
preting the output of the algorithm in this more general setting, where geometric
tools like perverse sheaves are not available. In the present paper, we try to preserve
this goal. Most definitions and constructions in this paper make sense for arbitrary
complex reflection groups and arbitrary preorders on Irr(W ). We do invoke some
results of Kato whose proofs involve the geometry of the nilpotent cone, and are
thus valid only for Weyl groups. However, outside of Section 4, we treat these
results as axioms: if, in the future, non-geometric proofs of these results become
available for other complex reflection groups, then the main results of this paper
will extend to those complex reflection groups as well.
1.4. Acknowledgments. The author is grateful to Syu Kato for a number
of helpful comments. This paper has, of course, been deeply influenced by his
ideas. I would also like to thank the organizers of the ICM 2010 satellite conference
on Algebraic and Combinatorial Approaches to Representation Theory for having
given me the opportunity to participate.
2. Notation and preliminaries
2.1. Graded rings and vector spaces. If R is a noetherian graded C-
algebra, we write R-gmod (resp. R-gmodfd) for the category of finitely-generated
(resp. finite-dimensional) graded left R-modules. For any M R-gmod, we write
grk V for its k-th graded component. We define M 1 to be the new graded module
grk(M 1 ) = grk−1 M.
The operation M M 1 also makes sense for chain complexes of modules over
R. If M and N are (complexes of) graded R-modules, we define HomR(M, N) (or
simply Hom(M, N)) to be the graded vector space given by
grk HomR(M, N) = Hom(M, N−k ).
We use the term grade to refer to the integers k such that grk M = 0, reserving
the term degree for homological uses, such as indexing the terms in a chain complex.
Thus, a module M is said to have grades n if grk M = 0 for all k n. If M is
a chain complex of modules, we say that M has grades n if all its cohomology
have grades n.
If M and N are objects in a derived category of R-modules, we employ the usual
N) = Hom(M, N[i]), as well as
N) = Hom(M, N[i]).
2.2. Reflection groups and phyla. Throughout the paper, W will be a
fixed complex reflection group, acting on a finite-dimensional complex vector space
h. Let Sh be the symmetric algebra on h, regarded as a graded ring by declaring
elements of h Sh to have degree 1. Our main object of study is the ring
AW = C[W ] # Sh.
Let AW -gmod be the category of finitely-generated graded AW -modules. Hence-
forth, all AW -modules are assumed to be objects of AW -gmod.
Let Irr(W ) denote the set of irreducible complex characters of W . For χ
Irr(W ), let ¯ χ denote the complex-conjugate character. If W is a Coxeter group,
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