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arXiv · 2006.04095

Calculus of archimedean Rankin--Selberg integrals with recurrence relations

Taku Ishii·Tadashi Miyazaki
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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.

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Taku Ishii, Tadashi Miyazaki. 2020-06-07. Calculus of archimedean Rankin--Selberg integrals with recurrence relations. https://arxiv.org/abs/2006.04095

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