On representations of rational Cherednik algebras

Abstract

This thesis introduces and studies two constructions related to the representation theory of rational Cherednik algebras: the refined filtration by supports for the category O and the Dunkl weight function. The refined filtration by supports provides an analogue for rational Cherednik algebras of the Harish-Chandra series appearing in the representation theory of finite groups of Lie type. In particular, irreducible representations in the rational Cherednik category O with particular generalized support conditions correspond to irreducible representations of associated generalized Hecke algebras. An explicit presentation for these generalized Hecke algebras is given in the Coxeter case, classifying the irreducible finite-dimensional representations in many new cases. The Dunkl weight function K is a holomorphic family of tempered distributions on the real reflection representation of a finite Coxeter group W with values in linear endomorphisms of an irreducible representation of W. The distribution K gives rise to an integral formula for the Gaussian inner product on a Verma module in the rational Cherednik category O. At real parameter values, the restriction of K to the regular locus in the real reflection representation can be interpreted as an analytic function taking values in Hermitian forms, invariant under the braid group, on the image of a Verma module under the Knizhnik-Zamolodchikov (KZ) functor. This provides a bridge between the study of invariant Hermitian forms on representations of rational Cherednik algebras and of Hecke algebras, allowing for a proof that the KZ functor preserves signatures in an appropriate sense.