Spherical functions on the p-adic group PGL(2). II

SPHERICAL FUNCTIONS ON A p-ADIC CHEVALLEY GROUP

Remarks on Macdonald’s book on p-adic spherical functions

When Ian Macdonald’s book Spherical functions on a group of p-adic type first appeared, it was one of a very small number of publications concerned with representations of p-adic groups. At just about that time, however, the subject began to bewidely recognized as indispensable in understanding automorphic forms, and the literature on the subject started to grow rapidly. Since it has by now gro...

On p-adic Artin L-functions II

Let p be a prime. Iwasawa’s famous conjecture relating Kubota-Leopoldt p-adic L-functions to the structure of certain Galois groups has been proven by Mazur and Wiles in [10]. Wiles later proved a far-reaching generalization involving p-adic L-functions for Hecke characters of finite order for a totally real number field in [14]. As we discussed in [5], an analogue of Iwasawa’s conjecture for p...

GENERALIZED SPHERICAL FUNCTIONS ON REDUCTIVE p-ADIC GROUPS

Let G be the group of rational points of a connected reductive p-adic group and let K be a maximal compact subgroup satisfying conditions of Theorem 5 from Harish-Chandra (1970). Generalized spherical functions on G are eigenfunctions for the action of the Bernstein center, which satisfy a transformation property for the action of K. In this paper we show that spaces of generalized spherical fu...

CONSTANT TERM OF SMOOTH Hψ-SPHERICAL FUNCTIONS ON A REDUCTIVE p-ADIC GROUP

Let ψ be a smooth character of a closed subgroup, H, of a reductive p-adic group G. If P is parabolic subgroup of G such that PH is open in G, we define the constant term of every smmoth fubction on G which transforms by ψ under the right action of G. The example of mixed models is given: it includes symmetric spaces and Whittaker models. In this case a notion of cuspidal function is defined an...