Local Rankin--Selberg integrals for Speh representations

TL;DR

This paper develops local Rankin--Selberg integrals for Speh representations of GL(n) over p-adic fields, connecting them to classical integrals and introducing new models and structures.

Contribution

It constructs new local integrals for Speh representations using the Shalika model, linking them to classical integrals and establishing a unitary structure.

Findings

Constructed local Rankin--Selberg integrals for Speh representations.

Connected local integrals to classical Jacquet--Piatetski-Shapiro--Shalika integrals.

Introduced a unitary structure for Speh representations on the Shalika model.

Abstract

We construct analogues of Rankin--Selberg integrals for Speh representations of the general linear group over a $p$-adic field. The integrals are in terms of the Shalika model and are expected to be the local counterparts of (suitably regularized) global integrals involving square-integrable automorphic forms and Eisenstein series on the general linear group over a global field. We relate the local integrals to the classical ones studied by Jacquet--Piatetski-Shapiro--Shalika. We also introduce a unitary structure for Speh representation on the Shalika model, as well as various other models including Zelevinsky's degenerate Whittaker model.