A Product of Tensor Product $L$-functions of Quasi-split Classical Groups of Hermitian Type

TL;DR

This paper constructs and analyzes global integrals representing products of tensor product L-functions for automorphic representations of quasi-split classical groups of Hermitian type, extending previous work and providing explicit unramified local factors.

Contribution

It establishes the Eulerian property of these integrals and computes unramified local L-factors in general, advancing the theory of automorphic L-functions for classical groups.

Findings

Global integrals are Eulerian.

Explicit unramified local L-factors are calculated.

Extends previous results to more general classical groups.

Abstract

A family of global integrals representing a product of tensor product (partial) $L$-functions: $ L^S(s,\pi\times\tau_1)L^S(s,\pi\times\tau_2)... L^S(s,\pi\times\tau_r) $ are established in this paper, where $\pi$ is an irreducible cuspidal automorphic representation of a quasi-split classical group of Hermitian type and $\tau_1,...,\tau_r$ are irreducible unitary cuspidal automorphic representations of $\GL_{a_1},...,\GL_{a_r}$, respectively. When $r=1$ and the classical group is an orthogonal group, this was studied by Ginzburg, Piatetski-Shapiro and Rallis in 1997 and when $\pi$ is generic and $\tau_1,...,\tau_r$ are not isomorphic to each other, this is considered by Ginzburg, Rallis and Soudry in 2011. In this paper, we prove that the global integrals are eulerian and finish the explicit calculation of unramified local $L$-factors in general. The remaining local and global theory for this family of global integrals will be considered in our future work.