Calculus of archimedean Rankin--Selberg integrals with recurrence relations

TL;DR

This paper explicitly describes archimedean Rankin--Selberg integrals for certain pairs of principal series representations of GL(n,F) and GL(n',F), utilizing recurrence relations, with applications to automorphic L-functions.

Contribution

It provides explicit formulas for Rankin--Selberg integrals at minimal K-types using recurrence relations, advancing understanding of automorphic L-functions over complex fields.

Findings

Explicit descriptions of integrals at minimal K-types

Recurrence relations for principal series representations

Applications to critical values of automorphic L-functions

Abstract

Let $n$ and $n'$ be positive integers such that $n-n'\in \{0,1\}$. Let $F$ be either $\mathbb{R}$ or $\mathbb{C}$. Let $K_n$ and $K_{n'}$ be maximal compact subgroups of $\mathrm{GL}(n,F)$ and $\mathrm{GL}(n',F)$, respectively. We give the explicit descriptions of archimedean Rankin--Selberg integrals at the minimal $K_n$- and $K_{n'}$-types for pairs of principal series representations of $\mathrm{GL}(n,F)$ and $\mathrm{GL}(n',F)$, using their recurrence relations. Our results for $F=\mathbb{C}$ can be applied to the arithmetic study of critical values of automorphic $L$-functions.