Calculus of archimedean Rankin–Selberg integrals with recurrence relations

Abstract

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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