  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
$\tilde{G}'$.
