Harish-Chandra bimodules over rational Cherednik algebras

Abstract

Abstract. We study Harish-Chandra bimodules over the rational Cherednik algebra Hc(W ) associated to a complex reflection group W with parameter c. Our results allow us to partially reduce the study of these bimodules to smaller algebras. We classify those pairs of parameters (c, c ′ ) for which there exist fully supported Harish-Chandra bimodules, and give a description of the category of all Harish-Chandra bimodules modulo those without full support. When W is a symmetric group we are able to classify all irreducible Harish-Chandra bimodules. Our proofs are based on localization techniques, the action of the Namikawa-Weyl group on the set of parameters, and the study of partial KZ functors.

The affine Springer fiber - sheaf correspondence

“…The last bullet point is related to the Harish-Chandra property for the bimodules B ij , see [78,55]. Namely, a Harish-Chandra bimodule for a filtered algebra A is a bimodule B with an exhaustive filtration s.t.…”

“…Remark 6.22. For G = GL n , Simental [78] classified Harish-Chandra bimodules for the rational Cherednik algebra and proved that the shift bimodule is the unique Harish-Chandra bimodule which sends polynomial representation to the polynomial representation. In particular, this implies an analogue of Theorem 6.21 for the rational Cherednik algebra.…”

“…In particular, this implies an analogue of Theorem 6.21 for the rational Cherednik algebra. It would be interesting to know if the methods of [78] can be generalized to the trigonometric case to give an alternate proof of Theorem 6.21.…”