On branching laws of Speh representations

TL;DR

This paper investigates the branching laws of Speh representations of general linear groups over p-adic fields, constructing zeta integrals via the Shalika model to understand their restriction behavior.

Contribution

It introduces a new approach using zeta integrals and the Shalika model to analyze the branching laws of Speh representations, advancing the local theory of Miyawaki liftings.

Findings

Constructed zeta integrals for Speh representations.

Established nonzero restriction maps between Speh representations and tensor products.

Contributed to the local theory of Miyawaki liftings for unitary groups.

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 $\tau\boxtimes\tau^\vee\chi_\pi\times\mathrm{Sp}(\pi, l)$ for any irreducible representation $\tau$ of $\mathrm{GL}_n$. These results form part of the local theory of the Miyawaki lifting for unitary groups.