Springer correspondences for dihedral groups

TL;DR

This paper extends the Springer correspondence framework to dihedral groups, providing computational methods and evidence supporting the conjecture that complex reflection groups relate to geometric structures akin to algebraic groups.

Contribution

It develops the Lusztig--Shoji algorithm for dihedral groups, demonstrating that the resulting Green functions satisfy key properties, and identifies Springer correspondences in this context.

Findings

Green functions satisfy integrality and positivity conditions

Springer correspondences for dihedral groups are determined

Unipotent varieties are rationally smooth

Abstract

Recent work by a number of people has shown that complex reflection groups give rise to many representation-theoretic structures (e.g., generic degrees and families of characters), as though they were Weyl groups of algebraic groups. Conjecturally, these structures are actually describing the representation theory of as-yet undescribed objects called ''spetses'', of which reductive algebraic groups ought to be a special case. In this paper, we carry out the Lusztig--Shoji algorithm for calculating Green functions for the dihedral groups. With a suitable set-up, the output of this algorithm turns out to satisfy all the integrality and positivity conditions that hold in the Weyl group case, so we may think of it as describing the geometry of the ''unipotent variety'' associated to a spets. From this, we determine the possible ''Springer correspondences'', and we show that, as is true for algebraic groups, each special piece is rationally smooth, as is the full unipotent variety.