Steinberg's theorem for crystallographic complex reflection groups

TL;DR

This paper investigates the applicability of Steinberg's fixed point theorem to crystallographic complex reflection groups, identifying conditions under which the theorem holds for these infinite groups and those built on finite Coxeter groups.

Contribution

It determines when Steinberg's theorem applies to crystallographic complex reflection groups, extending understanding beyond finite and affine cases.

Findings

Steinberg's theorem holds for certain crystallographic complex reflection groups.

The paper characterizes conditions for the theorem's validity in these groups.

Includes analysis of groups built on finite Coxeter groups.

Abstract

Popov classified crystallographic complex reflection groups by determining lattices they stabilize. These analogs of affine Weyl groups have infinite order and are generated by reflections about affine hyperplanes; most arise as the semi-direct product of a finite complex reflection group and a full rank lattice. Steinberg's fixed point theorem asserts that the regular orbits under the action of a reflection group are exactly the orbits lying off of reflecting hyperplanes. This theorem holds for finite reflection groups (real or complex) and also affine Weyl groups but fails for some crystallographic complex reflection groups. We determine when Steinberg's theorem holds for the infinite family of crystallographic complex reflection groups. We include crystallographic groups built on finite Coxeter groups.