Abstract. We give a construction of a wide class of modular symbols attached to reductive groups. As an application we construct a p-adic distribution interpolating the special values of the twisted Rankin-Selberg L-function attached to cuspidal automorphic representations π and σ of GLn and GL n−1 over a number field k. If π and σ are ordinary at p, our distribution is bounded and yields analyticity of the associated p-adic L-function.

We study Hecke algebras for pairs (g, K) over arbitrary fields E of characteristic 0, define the Bernstein functor and give another definition of the Zuckerman functor over E. Building on this and the author's previous work on rational structures on automorphic representations, we show that hard duality remains valid over E and apply this result to the study of rationality properties of Sun's cohomologically induced functionals. Our main application are period relations for the special values of standard L-functions of automorphic representations of GL(2n) admitting Shalika models.

This paper proves the existence of global rational structures on spaces of cusp forms of general reductive groups. We identify cases where the constructed rational structures are optimal, which includes the case of GL(n). As an application, we deduce the existence of a natural set of periods attached to cuspidal automorphic representations of GL(n). This has consequences for the arithmetic of special values of L-functions that we discuss in [30,31].In the course of proving our results, we lay the foundations for a general theory of Harish-Chandra modules over arbitrary fields of characteristic 0. In this context, a rational character theory, translation functors and an equivariant theory of cohomological induction are developed. We also study descent problems for Harish-Chandra modules in quadratic extensions, where we obtain a complete theory over number fields.

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