Rankin-Selberg L-Functions for GSpin x GL Groups

TL;DR

This paper develops an integral representation for Rankin-Selberg L-functions involving general spin and linear groups, extending previous work and providing insights into their pole structure and Langlands transfer.

Contribution

It constructs a new integral representation for L(s, π × τ) for general spin groups over arbitrary number fields, generalizing prior results and including all ranks and forms.

Findings

Established the location of poles of L(s, π × τ).

Implications for the Langlands functorial transfer from GSpin to GL.

Extended integral representations to all ranks and forms of GSpin groups.

Abstract

We construct an integral representation for the global Rankin-Selberg (partial) $L$-function $L(s, \pi \times \tau)$ where $\pi$ is an irreducible globally generic cuspidal automorphic representation of a general spin group (over an arbitrary number field) and $\tau$ is one of a general linear group, generalizing the works of Gelbart, Piatetski-Shapiro, Rallis, Ginzburg, Soudry and Kaplan among others. We consider all ranks and both even and odd general spin groups including the quasi-split forms. The resulting facts about the location of poles of $L(s, \pi \times \tau)$ have, in particular, important consequences in describing the image of the Langlands funtorial transfer from the general spin groups to general linear groups.