Transfer of characters in the theta correspondence with one compact member

For an irreducible dual pair $(G,G^{\mathrm{\prime }})\in Sp(W)$ with one member compact and two representations $\mathrm{\Pi }\leftrightarrow {\mathrm{\Pi }}^{\mathrm{\prime }}$ appearing in the Howe duality, we give an expression of the character ${\mathrm{\Theta }}_{{\mathrm{\Pi }}^{\mathrm{\prime }}}$ of ${\mathrm{\Pi }}^{\mathrm{\prime }}$ via the character of $\mathrm{\Pi }$. 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 $(G=U(1,\mathbb{C}),G^{\mathrm{\prime }}=U(1,1,\mathbb{C}))$, we verify directly a result of H. Hecht saying that the character has the same value on both Cartan subgroups of $G^{\mathrm{\prime }}$.