Calculus of archimedean Rankin–Selberg integrals with recurrence relations

Abstract

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