论文标题
Archimedean Rankin的微积分 - 与复发关系的积分
Calculus of archimedean Rankin--Selberg integrals with recurrence relations
论文作者
论文摘要
令$ n $和$ n'$为正整数，以便$ n-n \ in \ {0,1 \} $。令$ f $为$ \ mathbb {r} $或$ \ mathbb {c} $。令$ k_n $和$ k_ {n'} $分别为$ \ mathrm {gl}（n，f）$和$ \ mathrm {gl}（n'，f）$的最大紧凑型亚组。我们对Archimedean Rankin-selberg积分的明确描述为最低$ k_n $ - 和$ k_ {n'} $ - 类型 - $ \ mathrm {gl}（gl}（n，f）$和$ \ mathrm {gl}（gl}（n'，f）$，使用其recurrence的一对主要系列表示形式。我们对$ f = \ mathbb {c} $的结果可以应用于对自动形态$ l $ functions的关键值的算术研究。
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.
