The freeness and trace conjectures for parabolic Hecke subalgebras

TL;DR

This paper proves the freeness and trace conjectures for parabolic Hecke subalgebras of complex reflection groups of rank 2, extending known results from real reflection groups to certain complex cases.

Contribution

It establishes the validity of the freeness and trace conjectures for all rank 2 complex reflection groups where the BMM symmetrising trace conjecture is confirmed.

Findings

Proved the conjectures for all rank 2 complex reflection groups with known BMM trace conjecture.

Extended the validity of fundamental Hecke algebra conjectures beyond real reflection groups.

Confirmed the structure and trace properties of parabolic Hecke subalgebras in new complex cases.

Abstract

The two most fundamental conjectures on the structure of the generic Hecke algebra $\mathcal{H}(W)$ associated with a complex reflection group $W$ state that $\mathcal{H}(W)$ is a free module of rank $|W|$ over its ring of definition, and that $\mathcal{H}(W)$ admits a canonical symmetrising trace. The first conjecture has recently become a theorem, while the second conjecture, known to hold for real reflection groups, has only been proved for some exceptional non-real complex reflection groups (all of rank $2$ but one). The two most fundamental conjectures on the structure of the parabolic Hecke subalgebra $\mathcal{H}(W')$ associated with a parabolic subgroup $W'$ of $W$ state that $\mathcal{H}(W)$ is a free left and right $\mathcal{H}(W')$-module of rank $|W|/|W'|$, and that the canonical symmetrising trace of $\mathcal{H}(W')$ is the restriction of the canonical symmetrising trace of $\mathcal{H}(W)$ to $\mathcal{H}(W')$. Until now, these two conjectures have only be known to be true for real reflection groups. We prove them for all complex reflection groups of rank $2$ for which the BMM symmetrising trace conjecture is known to hold.