Characters of some unitary highest weight representations via the theta correspondence

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 $\tilde{G}'$.