宮﨑 直

<p>After introducing the notion of uniform integrality of critical values of the Rankin–Selberg -functions for , we study it when the base field is totally imaginary. For this purpose, we adopt specific models with highest weight representations of the general linear groups, construct the Eichler–Shimura classes for and in an explicit manner, and then evaluate the cohomological cup product of them, by making the best use of Gel′fand–Tsetlin basis.</p>

<p>Let and be positive integers such that . Let be either or . Let and be maximal compact subgroups of and , respectively. We give the explicit descriptions of archimedean Rankin–Selberg integrals at the minimal - and -types for pairs of principal series representations of and , using their recurrence relations. Our results for can be applied to the arithmetic study of critical values of automorphic -functions.</p>

<p>In this article, we give explicit formulas of archimedean Whittaker functions on and . Moreover, we apply those to the calculation of archimedean zeta integrals for , and show that the zeta integral for appropriate Whittaker functions is equal to the associated -factors.</p>

This is an announcement of a new result which is a generalization of Popa's result in [Po]. Popa gives explicit formulas of Whittaker functions on GL(2,C) at the minimal K-types, and shows that the local zeta integrals for GL(2,C) defined from some Whittaker functions are equal to the associated L-factors. In this article, we will give explicit formulas of Whittaker functions on GL(2,C) at all K-types, and show that the local zeta integrals for GL(2,C) × GL(2,C) defined from some Whittaker functions are equal to the associated L-factors. Proofs will appear elsewhere.