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 \(H_c(W, \mathfrak {h})\).queryPlease check and confirm the inserted city name ‘Stanford’ for the affiliation is correct. 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.
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Acknowledgements
This paper arose from a project which was developed by Pavel Etingof and benefited from his conversations with Larry Guth, Richard Melrose, Leonid Polterovich, and Vivek Shende. I would like to express my deep gratitude to Pavel Etingof for suggesting this project, for many useful conversations, and for his countless ideas and insights that permeate this paper and led to the conjectures appearing in Sect. 6. I would also like to thank Semyon Dyatlov for providing essentially all of the content of Sects. 5.1 and 5.2, notably including the crucial Lemma 5.2.2 and its proof, and for explaining to me the techniques from semiclassical analysis used in those sections.
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