Fake degrees for reflection actions on roots

TL;DR

This paper investigates the fake degrees associated with reflection actions on roots of finite irreducible real reflection groups, revealing divisibility properties and explicit formulas in simply-laced types.

Contribution

It establishes divisibility of fake degrees by a q-analog of the Coxeter number and provides explicit formulas in simply-laced cases, linking to exponents and codegrees.

Findings

Fake degrees are divisible by [h]_q.

In simply-laced types, fake degrees equal [h]_q times sum of q^{e_i - 1}.

Connections between fake degrees, exponents, and codegrees are established.

Abstract

A finite irreducible real reflection group of rank l and Coxeter number h has root system of cardinality h*l. It is shown that the fake degree for the permutation action on its roots is divisible by [h]_q = 1+q+q^2+...+q^{h-1}, and that in simply-laced types, it equals [h]_q times the summation of q^{e_i - 1} where e_i runs through the exponents, so that e_i - 1 are the codegrees.