Associated cycles of local theta lifts of unitary characters and unitary lowest weight modules

TL;DR

This paper constructs filtrations on theta lifts for real reductive dual pairs to compute associated cycles and varieties of certain Harish-Chandra modules, revealing new properties of these representations.

Contribution

It introduces natural filtrations on theta lifts and calculates associated cycles and varieties for specific modules, confirming conjectures about their unipotent nature.

Findings

Some representations are shown to be special unipotent

Confirmed a K-type formula conjecture of Vogan

Computed associated cycles and varieties for theta lifts

Abstract

In this paper we first construct natural filtrations on the full theta lifts for any real reductive dual pairs. We will use these filtrations to calculate the associated cycles and therefore the associated varieties of Harish-Chandra modules of the indefinite orthogonal groups which are theta lifts of unitary lowest weight modules of the metaplectic double covers of the real symplectic groups. We will show that some of these representations are special unipotent and satisfy a K-type formula in a conjecture of Vogan.