On branching laws of Speh representations

Abstract

In this paper, we consider the branching law of the Speh representation $\mathrm{Sp}(\pi ,n+l)$ of ${\mathrm{GL}}_{2n+2l}$ with respect to the block diagonal subgroup ${\mathrm{GL}}_n\times {\mathrm{GL}}_{n+2l}$ for any irreducible generic representation $\pi$ of ${\mathrm{GL}}_2$ over any $p$-adic field. We use the Shalika model of $\mathrm{Sp}(\pi ,n)$ to construct certain zeta integrals, which were defined by Ginzburg and Kaplan independently, and study them. Finally, using these zeta integrals, we obtain a nonzero ${\mathrm{GL}}_n\times {\mathrm{GL}}_{n+2l}$-map from $\mathrm{Sp}(\pi ,n+l)$ to $\tau ⊠{\tau }^{\vee }{\chi }_{\pi }\times \mathrm{Sp}(\pi ,l)$ for any irreducible representation $\tau$ of ${\mathrm{GL}}_n$. These results form part of the local theory of the Miyawaki lifting for unitary groups.