In this largely expository paper we extend properties of the homological duality functor $RHom_{\mathcal H}(-,{\mathcal H})$ where ${\mathcal H}$ is Hecke algebra a reductive $p$-adic group, to case it finite central extension group. The most important being that concentrated in single degree for irreducible representations and gives rise Schneider--Stuhler Ext groups (a Serre like property). A...

This paper is an expanded version of my lecture delivered at the 97 Seattle summer research conference on finite fields. It gives a quick exposition of Dwork’s conjecture about p-adic meromorphic continuation of his unit root L-function arising from a family of algebraic varieties defined over a finite field of characteristic p. As a simple illustration, we discuss the classical example of the ...

Founding harmonic analysis on classical simple complex groups, I.M. Gelfand and M.A. Naimark in their classical book [GN] posed three basic questions: unitary duals, characters of irreducible unitary representations and Plancherel measures. In the case of reductive p-adic groups, the only series of reductive groups where unitary duals are known are general linear groups. In this paper we reduce...

Let G be the group of points of a split reductive group over a finite extension of Qp. In this paper, we compute the dimensions of certain classes of locally analytic G-representations. This includes principal series representations and certain representations coming from homogeneous line bundles on p-adic symmetric spaces. As an application, we compute the dimensions of the unitary GL2pQpq-rep...

The specializations of the motivic elliptic polylogarithm on the universal elliptic curve to the modular curve are referred to as Eisenstein classes. In this paper, we prove that the syntomic realizations of the Eisenstein classes restricted to the ordinary locus of the modular curve may be expressed using p-adic Eisenstein-Kronecker series, which are p-adic modular forms defined using the two-...