Special Unipotent Representations with Half Integral Infinitesimal Characters

TL;DR

This paper extends the understanding of special unipotent representations with half-integer infinitesimal characters, proving properties previously known only for integral cases, thus broadening the theoretical framework in representation theory.

Contribution

It provides a complete proof of properties of special unipotent representations with half-integer infinitimal characters, previously established only for integral cases.

Findings

Properties of special unipotent representations are confirmed for non-integral half-integer cases.

The work generalizes previous results from integral to half-integer infinitesimal characters.

The paper solidifies the theoretical foundation for representations associated with special nilpotent orbits.

Abstract

For any special nilpotent orbit, let $\frac{1}{2}h^{\vee}$ be one half of the semisimple element of a Jacobson-Morozov triple associated to the orbit. In 1985, Barbasch and Vogan defined the notion of special unipotent representations with infinitesimal character $(\frac{1}{2}h^{\vee},\frac{1}{2}h^{\vee})$. Some properties of such representations were discovered when $\frac{1}{2}h^{\vee}$ is integral. In this manuscript, we give a complete proof of these properties when $\frac{1}{2}h^{\vee}$ is not integral.