Transfer of characters in the theta correspondence with one compact member

Abstract

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

arXiv comment29 pages