On the Local Theory of Rankin-Selberg Convolutions for $\mathrm{SO_{2l}\times GL_{n}}$

TL;DR

This paper develops the local theory for Rankin-Selberg integrals involving special orthogonal groups and general linear groups, establishing their properties, functional equations, and connections to Shahidi's $L$-functions, with potential applications to descent methods.

Contribution

It introduces the local integrals for $SO_{2l} imes GL_{n}$, proves their functional equations, and relates the associated $ ext{g.c.d.}$ to Shahidi's $L$-function, providing new insights into their analytic properties.

Findings

The $ ext{g.c.d.}$ bounds are established and related to Shahidi's $L$-function.

The $ ext{g.c.d.}$ equals the $L$-function in the tempered case under certain assumptions.

Explicit computation of integrals with unramified data is provided.

Abstract

The Rankin-Selberg method for studying Langlands' automorphic $L$-functions is to find integral representations, involving certain Fourier coefficients of cusp forms and Eisenstein series, for these functions. In this thesis we develop the local theory for generic representations of special orthogonal groups. We study the local integrals for $SO_{2l}\times GL_{n}$, where $SO_{2l}$ is the special even orthogonal group, either split or quasi-split, over a local non-Archimedean field. These integrals admit a functional equation, which is used to define a $\gamma$-factor. We show that, as expected, the $\gamma$-factor is identical with Shahidi's $\gamma$-factor. The analytic properties of the integrals are condensed into a notion of a greatest common divisor (g.c.d.). We establish certain bounds on the g.c.d. and relate it to the $L$-function defined by Shahidi in several cases, thereby providing another point of view on the $L$-function, linking it to the poles of the integrals. In particular, in the tempered case under a reasonable assumption the g.c.d. is equal to the $L$-function. Finally, this study includes the computation of the integrals with unramified data. This work may lead to new applications of the descent method, as well as aid in analyzing the poles of the global $L$-function.