The Dunkl Weight Function for Rational Cherednik Algebras

TL;DR

This paper proves the existence of the Dunkl weight function for all finite Coxeter groups, linking Cherednik algebra representations with braid group invariants and Hermitian forms, advancing understanding of unitarizability.

Contribution

It generalizes the Dunkl weight function to all finite Coxeter groups and connects Cherednik algebra representations with braid group invariants and Hermitian forms.

Findings

Dunkl weight function exists for any finite Coxeter group

Provides integral formula for Cherednik's Gaussian inner product

Shows KZ functor preserves signatures and unitarizability

Abstract

In this paper we prove the existence of the Dunkl weight function $K_{c, \lambda}$ for any irreducible representation $\lambda$ of any finite Coxeter group $W$, generalizing previous results of Dunkl. In particular, $K_{c, \lambda}$ is a family of tempered distributions on the real reflection representation of $W$ taking values in $\text{End}_\mathbb{C}(\lambda)$, with holomorphic dependence on the complex multi-parameter $c$. When the parameter $c$ is real, the distribution $K_{c, \lambda}$ provides an integral formula for Cherednik's Gaussian inner product $\gamma_{c, \lambda}$ on the Verma module $\Delta_c(\lambda)$ for the rational Cherednik algebra $H_c(W, \mathfrak{h})$. In this case, the restriction of $K_{c, \lambda}$ to the hyperplane arrangement complement $\mathfrak{h}_{\mathbb{R}, reg}$ is given by integration against an analytic function whose values can be interpreted as braid group invariant Hermitian forms on $KZ(\Delta_c(\lambda))$, where $KZ$ denotes the Knizhnik-Zamolodchikov functor introduced by Ginzburg-Guay-Opdam-Rouquier. This provides a concrete connection between invariant Hermitian forms on representations of rational Cherednik algebras and invariant Hermitian forms on representations of Iwahori-Hecke algebras, and we exploit this connection to show that the $KZ$ functor preserves signatures, and in particular unitarizability, in an appropriate sense.