Equivariant analogues of Zuckerman functors

TL;DR

This paper introduces equivariant analogues of Zuckerman functors within derived categories of Harish-Chandra modules, providing explicit formulas and connecting them to classical functors via cohomology, enhancing localization techniques.

Contribution

It presents explicit formulas for equivariant Zuckerman functors and demonstrates their relation to classical functors through cohomology, extending the Beilinson-Ginzburg framework.

Findings

Explicit formula for equivariant Zuckerman functors

Recovery of classical Zuckerman functors via cohomology

Applicability to modules with specified infinitesimal characters

Abstract

We review the Beilinson-Ginzburg construction of equivariant derived categories of Harish-Chandra modules, and introduce analogues of Zuckerman functors in this setting. They are given by an explicit formula, which works equally well in the case of modules with a given infinitesimal character, which is important if one wants to apply Beilinson-Bernstein localization. We also show how to recover the usual Zuckerman functors from the equivariant ones by passing to cohomology.