Spherical functions on some $p$-adic classical groups

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...

REPRESENTATIONS OF CLASSICAL p-ADIC GROUPS

Preface 1 1. Classical groups 4 2. Parabolic induction 10 3. Admissible representations 16 4. Jacquet modules and cuspidal representations 24 5. Composition series of induced representations of SL(2, F ) and GL(2, F ) 34 6. Some examples 39 7. Parabolically induced representations of SL(2, F ) and GL(2, F ) 45 8. Some general consequences 52 9. GL(n, F ) 55 10. GSp(n, F ) 62 11. On the reducibi...

Semisimple characters for p-adic classical groups

Let G be a unitary, symplectic or orthogonal group over a non-archimedean local field of residual characteristic different from 2, considered as the fixed point subgroup in a general linear group G̃ of an involution. Following [7] and [13], we generalize the notion of a semisimple character for G̃ and for G. In particular, following the formalism of [4], we show that these semisimple characters h...

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...

SPHERICAL FUNCTIONS ON A p-ADIC CHEVALLEY GROUP