  For an irreducible dual pair $(G, G') \in Sp(W)$ with one member compact and
two representations $\Pi \leftrightarrow \Pi'$ appearing in the Howe duality,
we give an expression of the character $\Theta_{\Pi'}$ of $\Pi'$ via the
character of $\Pi$. We make computations for the dual pair $(G = U(n,
\mathbb{C}), G' = U(p, q, \mathbb{C}))$, which are explicit in low dimensions.
For $(G = U(1, \mathbb{C}), G' = U(1, 1, \mathbb{C}))$, we verify directly a
result of H. Hecht saying that the character has the same value on both Cartan
subgroups of $G'$.
