Rankin–Selberg integrals for principal series representations of GL(n)

We prove that the local Rankin–Selberg integrals for principal series representations of the general linear groups agree with certain simple integrals over the Rankin–Selberg subgroups, up to certain constants given by the local gamma factors.

Rankin–Selberg L-functions via good sections

In this article, we revisit Rankin–Selberg integrals established by Jacquet, Piatetski-Shapiro and Shalika. We prove the equality of Rankin–Selberg local factors defined with Schwartz–Bruhat functions and the factors attached to good sections, introduced by Piatetski-Shapiro and Rallis. Moreover, we propose a notion of exceptional poles in the framework of good sections. For cases of Rankin–Selberg, Asai and exterior square L-functions, the exceptional poles are consistent with well-known exceptional poles which characterize certain distinguished representations.

Local Rankin–Selberg integrals for Speh representations

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 (extended) 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 and 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.

Integrals Derived from the Doubling Method

In this note, we use a basic identity, derived from the generalized doubling integrals of [2], in order to explain the existence of various global Rankin–Selberg integrals for certain $L$-functions. To derive these global integrals, we use the identities relating Eisenstein series in [11], together with the process of exchanging roots. We concentrate on several well-known examples, and explain how to obtain them from the basic identity. Using these ideas, we also show how to derive a new global integral.