The Dunkl weight function for rational Cherednik algebras

Abstract

Abstract In this paper we prove the existence of the Dunkl weight function $$K_{c, \lambda }$$Kc,λ for any irreducible representation $$\lambda $$λ of any finite Coxeter group W, generalizing previous results of Dunkl. In particular, $$K_{c, \lambda }$$Kc,λ is a family of tempered distributions on the real reflection representation of W taking values in $$\text {End}_\mathbb {C}(\lambda )$$EndC(λ), with holomorphic dependence on the complex multi-parameter c. When the parameter c is real, the distribution $$K_{c, \lambda }$$Kc,λ provides an integral formula for Cherednik’s Gaussian inner product $$\gamma _{c, \lambda }$$γc,λ on the Verma module $$\Delta _c(\lambda )$$Δc(λ) for the rational Cherednik algebra $$H_c(W, \mathfrak {h})$$Hc(W,h).queryPlease check and confirm the inserted city name ‘Stanford’ for the affiliation is correct. In this case, the restriction of $$K_{c, \lambda }$$Kc,λ to the hyperplane arrangement complement $$\mathfrak {h}_{\mathbb {R}, reg}$$hR,reg is given by integration against an analytic function whose values can be interpreted as braid group invariant Hermitian forms on $$KZ(\Delta _c(\lambda ))$$KZ(Δc(λ)), 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.