The Auslander-Gorenstein property for Z-algebras

TL;DR

This paper develops a homological framework for Z-algebras, extending the Auslander-Gorenstein condition, and applies it to representations of Lie algebras, quantum symplectic resolutions, and Cherednik algebras.

Contribution

It introduces a new homological approach for Z-algebras and their generalizations, enabling analysis of their Auslander-Gorenstein properties and applications to representation theory.

Findings

Proves equidimensionality of characteristic varieties of irreducible representations.

Establishes the Auslander-Gorenstein condition for Z-algebras and related rings.

Answers a previously open question for Cherednik algebras of type A.

Abstract

We provide a framework for part of the homological theory of Z-algebras and their generalizations, directed towards analogues of the Auslander-Gorenstein condition and the associated double Ext spectral sequence that are useful for enveloping algebras of Lie algebras and related rings. As an application, we prove the equidimensionality of the characteristic variety of an irreducible representation of the Z-algebra, and for related representations over quantum symplectic resolutions. In the special case of Cherednik algebras of type A, this answers a question raised by the authors.