Parameters for Generalized Hecke Algebras in Type B

TL;DR

This paper computes the parameters for generalized Hecke algebras associated with certain parabolic subgroups in type B Coxeter groups, extending understanding of their representation theory.

Contribution

It provides explicit parameter calculations for generalized Hecke algebras in previously unresolved cases related to type B Coxeter groups.

Findings

Parameters for generalized Hecke algebras in new cases are explicitly determined.

Extends the classification of irreducible representations in rational Cherednik categories.

Bridges gaps in the understanding of algebraic structures associated with type B Coxeter groups.

Abstract

The irreducible representations of full support in the rational Cherednik category $\mathcal{O}_c(W)$ attached to a Coxeter group $W$ are in bijection with the irreducible representations of an associated Iwahori-Hecke algebra. Recent work has shown that the irreducible representations in $\mathcal{O}_c(W)$ of arbitrary given support are similarly governed by certain generalized Hecke algebras. In this paper we compute the parameters for these generalized Hecke algebras in the remaining previously unknown cases, corresponding to the parabolic subgroup $B_n \times S_k$ in $B_{n+k}$ for $k \geq 2$ and $n \geq 0$.