
Journal of Lie Theory 30 (2020), No. 4, 9971026 Copyright Heldermann Verlag 2020 Transfer of Characters in the Theta Correspondence with One Compact Member Allan Merino Dept. of Mathematics, National University of Singapore, Singapore 119076 matafm@nus.edu.sg [Abstractpdf] \newcommand{\Sp}{\textrm{Sp}\,} For an irreducible dual pair $(G, G') \subseteq \Sp(W)$ 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 compute the value of $\Theta_{\Pi'}$ on the maximal compact torus $T'$ of $G'$ 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 determine the value of the character on both Cartan subgroups of $G'$. Keywords: Howe correspondence, characters, oscillator semigroup, reductive dual pairs. MSC: 22E45, 22E46, 22E30. [ Fulltextpdf (251 KB)] for subscribers only. 