Equivariant Vector Bundles on Varieties with Codimension-one Orbits

TL;DR

This paper classifies $G$-equivariant vector bundles on certain smooth algebraic varieties with two orbits, providing new insights into their structure and implications for the study of representations of semisimple Lie groups.

Contribution

It offers an algebraic description of equivariant vector bundles on varieties with two orbits, including special cases and applications to representation theory.

Findings

Classification of equivariant vector bundles on varieties with two orbits

Simplified descriptions for line bundles and local systems

New constraints on associated cycles of unipotent representations

Abstract

Let $G$ be an algebraic group and let $X$ be a smooth $G$-variety with two orbits: an open orbit and a a closed orbit of codimension $1$. We give an algebraic description of the category of $G$-equivariant vector bundles on $X$ under a mild technical hypothesis. We deduce simpler classifications in the special cases of line bundles and vector bundles which are generically local systems. We apply our results to the study of admissible representations of semisimple Lie groups. Our main result gives a new set of constraints on the associated cycles of unipotent representations.