A Casselman-Osborne theorem for rational Cherednik algebras

TL;DR

This paper extends the Casselman-Osborne theorem to rational Cherednik algebras by defining Lie algebra cohomology via half-Dirac operators and establishing a Vogan's conjecture analogue.

Contribution

It introduces a novel Lie algebra cohomology framework for rational Cherednik algebras and proves a Vogan's conjecture analogue for half-Dirac operators.

Findings

Established a version of the Casselman-Osborne theorem for rational Cherednik algebras

Proved a Vogan's conjecture analogue for half-Dirac operators

Analyzed the relationship between Lie algebra cohomology and Dirac cohomology

Abstract

We define Lie algebra cohomology associated with the half-Dirac operators for representations of rational Cherednik algebras and show that it has property described in the Casselman-Osborne Theorem by establishing a version of the Vogan's conjecture for the half-Dirac operators. Moreover, we study the relationship between Lie algebra cohomology and Dirac cohomology in analogy of the representations for semisimple Lie algebras.