# Characters of some unitary highest weight representations via the theta correspondence

## TL;DR

This paper explicitly computes the characters of certain unitary highest weight representations in the context of the Howe correspondence for dual pairs in symplectic groups, focusing on their restrictions to regular points.

## Contribution

It provides explicit character formulas for representations related by Howe correspondence in the setting of dual pairs with compact groups.

## Key findings

- Explicit character restrictions on regular points of compact Cartan subgroups.
- Detailed computations for representations in the Howe correspondence.
- Enhanced understanding of unitary highest weight representations via theta correspondence.

## Abstract

In this article, we consider a dual pair $(G, G')$ in the symplectic group $Sp(W)$ with $G$ compact and let $(\tilde{G}, \tilde{G}')$ be the preimages of $G$ and $G'$ in the metaplectic group $\widetilde{Sp(W)}$. For every irreducible representation $\Pi$ of $\tilde{G}$ appearing in Howe correspondence, we compute explicitly the restriction of the character $\Theta_{\Pi'}$ of the associated representation $\Pi'$ of $\tilde{G}'$ on the set of regular points on the compact Cartan subgroup $\tilde{H}'$ of $\tilde{G}'$.