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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中文翻译：

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具有递推关系的阿基米德 Rankin-Selberg 积分的微积分

摘要：令$n$和$n'$为正整数，使得$n-n'\in \{0,1\}$。令 $F$ 为 $\mathbb {R}$ 或 $\mathbb {C}$。令$K_n$ 和$K_{n'}$ 分别是$\mathrm {GL}(n,F)$ 和$\mathrm {GL}(n',F)$ 的最大紧子群。对于 $\mathrm {GL}(n,F)$ 和 $\ mathrm {GL}(n',F)$，使用它们的递归关系。我们对 $F=\mathbb {C}$ 的结果可以应用于自守 $L$-函数的临界值的算术研究。

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