Rankin-Selberg integrals for $\mathrm{GSpin}$ groups with application to the global Gan-Gross-Prasad conjecture

TL;DR

This paper develops Rankin-Selberg integrals for GSpin groups using Bessel models to study tensor product L-functions and applies these results to prove a key case of the global Gan-Gross-Prasad conjecture for generic representations.

Contribution

It constructs new integral representations for tensor product L-functions on GSpin groups and applies them to verify part of the global Gan-Gross-Prasad conjecture.

Findings

Constructed Rankin-Selberg integrals for GSpin groups.

Proved one direction of the global Gan-Gross-Prasad conjecture.

Established connections between Bessel models and L-functions.

Abstract

We construct Rankin-Selberg integrals using Bessel models for a product of tensor product partial $L$-functions \begin{equation*} L^S(s,\pi\times\tau_1) L^S(s,\pi\times\tau_2)\cdots L^S(s,\pi\times\tau_r) \end{equation*} where $\pi$ is an irreducible cuspidal automorphic representation of a quasi-split $\mathrm{GSpin}$ group, and $\tau_1, \cdots, \tau_r$ are irreducible unitary cuspidal automorphic representations of $\mathrm{GL}_{k_1}, \cdots, \mathrm{GL}_{k_r}$ respectively. As an application, we prove one direction of the global Gan-Gross-Prasad conjecture for generic representations of quasi-split $\mathrm{GSpin}$ groups.