Springer Theory for Complex Reflection Groups

Many complex reflection groups behave as though they were the Weyl groups of “nonexistent 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 Iwahori–Hecke 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 [13], root systems [17, 33] and root lattices [33], length functions [8, 9], generic degrees [28, 30], and Green functions [36, 37]. A theme in these developments is that statements that are regarded as theorems in the setting of Weyl groups are often adopted as definitions in the setting of complex reflection groups. For instance, families of representations for Weyl groups are defined in terms of the Kazhdan–Lusztig basis for the Hecke algebra, but a theorem of Rouquier [34] gives an alternate description of families in terms of blocks over a suitable coefficient ring. For cyclotomic Hecke algebras, Kazhdan–Lusztig bases are unavailable, but “Rouquier blocks” still make sense, and have been adopted as a definition [10, 23, 32]. The present paper is an exposition of how this philosophy may be applied to the theory of unipotent classes and the Springer correspondence. Many features of the geometry of the unipotent variety of an algebraic group—including the number of conjugacy classes, their dimensions and closure relations, and their local intersection cohomology—can be computed from elementary knowledge of the Weyl group. Remarkably, analogous calculations for complex reflection groups often yield sensible results, with surprisingly “geometric” integrality and positivity properties, even though there is not (yet?) an actual “unipotent variety” attached to a general complex reflection group. The ideas and results described here come from a series of joint papers by the author and A.M. Aubert [1, 2, 3]. There are no new theorems in this paper. However, the last two sections give the results of various calculations in the exceptional groups that have not previously been published. Acknowledgements. The author is grateful to Syu Kato and Susumu Ariki for having made it possible for him to visit Kyoto in October 2008, and to Hyohe Miyachi and Tatsuhiro Nakajima for the invitation to participate in the RIMS workshop “Expansion of Combinatorial Representation Theory.” The author also received support from NSF grant DMS-0500873. 1. Overview of Complex Reflection Groups 1.1. Examples and classification. The easiest example of a complex reflection group is the cyclic group Cd of order d, acting on C by multiplication by d-th roots of unity. Simiarly, the n-fold product (Cd) acts on C as a (reducible) complex reflection group. The symmetric group Sn acts on C by permuting coordinate axes, and this action is generated by reflections and normalizes the action of (Cd). The semidirect product G(d, 1, n) = (Cd) o Sn