Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type

TL;DR

This paper introduces new odd double affine Hecke algebras involving Weyl groups and skew-polynomial generators, establishing their Morita equivalences, PBW properties, and Verma-type representations.

Contribution

It constructs novel odd double affine Hecke algebras and demonstrates their Morita equivalences and representation theory properties.

Findings

Established PBW properties for the new algebras

Constructed Verma-type representations using Dunkl operators

Proved Morita equivalences among different algebraic realizations

Abstract

We introduce an odd double affine Hecke algebra (DaHa) generated by a classical Weyl group W and two skew-polynomial subalgebras of anticommuting generators. This algebra is shown to be Morita equivalent to another new DaHa which are generated by W and two polynomial-Clifford subalgebras. There is yet a third algebra containing a spin Weyl group algebra which is Morita (super)equivalent to the above two algebras. We establish the PBW properties and construct Verma-type representations via Dunkl operators for these algebras.