Translation of Dolbeault representations on reductive homogeneous spaces

TL;DR

This paper adapts techniques from the study of the cubic Dirac operator to analyze the Dolbeault operator on elliptic coadjoint orbits, establishing key properties of cohomologically induced representations and their relation to translation functors.

Contribution

It introduces a geometric interpretation of the Zuckerman translation functor and proves that cohomological induction commutes with this functor for reductive homogeneous spaces.

Findings

Cohomologically induced representations have an infinitesimal character.

Cohomological induction and Zuckerman translation functor commute.

Provides a geometric interpretation of the Zuckerman translation functor.

Abstract

We adapt techniques used in the study of the cubic Dirac operator on homogeneous reductive spaces to the Dolbeault operator on elliptic coadjoint orbits to prove that cohomologically induced representations have an infinitesimal character, that cohomological induction and Zuckerman translation functor commute and give a geometric interpretation of the Zuchkerman translation functor in this context.