Hecke group algebras as quotients of affine Hecke algebras at level 0

TL;DR

This paper demonstrates that the Hecke group algebra of a classical Weyl group can be realized as a quotient of the affine Hecke algebra at level 0, revealing connections in their representation theories.

Contribution

It provides an alternative construction of the Hecke group algebra as a quotient of the affine Hecke algebra at level 0 for classical Weyl groups.

Findings

Hecke group algebra is a quotient of affine Hecke algebra at level 0

Level 0 representation is a calibrated principal series representation

The quotient factors through the principal central specialization

Abstract

The Hecke group algebra $HW_0$ of a finite Coxeter group $W_0$, as introduced by the first and last author, is obtained from $W_0$ by gluing appropriately its 0-Hecke algebra and its group algebra. In this paper, we give an equivalent alternative construction in the case when $W_0$ is the classical Weyl group associated to an affine Weyl group $W$. Namely, we prove that, for $q$ not a root of unity, $HW_0$ is the natural quotient of the affine Hecke algebra through its level 0 representation. We further show that the level 0 representation is a calibrated principal series representation for a suitable choice of character, so that the quotient factors (non trivially) through the principal central specialization. This explains in particular the similarities between the representation theory of the classical 0-Hecke algebra and that of the affine Hecke algebra at this specialization.