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 $G'$.
Categories: math.RT