Ramesh Gangolli

Harmonic Analysis of Spherical Functions on Real Reductive Groups

Contents: The Concept of a Spherical Function Structure of Semisimple Lie Groups and Differential Operators on Them The Elementary Spherical Functions The Harish-Chandra Series for and the c-Function Asymptotic Behaviour of Elementary Spherical Functions The L2-Theory. The Harish-Chandra Transform on the Schwartz Space of G//K LP-Theory of Harish-Chandra Transform. Fourier Analysis on the Spaces CP(G//K) Bibliography Subject Index.

Zeta functions of Selberg's type for some noncompact quotients of symmetric spaces of rank one

In a previous paper [5], one of the present authors has worked out a theory of zeta functions of Selberg’s type for compact quotients of symmetric spaces of rank one. In the present paper, we consider the analogues of those results when G/K is a noncompact symmetric space of rank one and Γ is a discrete subgroup of G such that G/Γ is not compact but such that vol( G/Γ ) Γ is a non-uniform lattice. Certain mild restrictions, which are fulfilled in many arithmetic cases, will be put on Γ , and we shall consider how one can define a zeta function Z Γ of Selberg’s type attached to the data ( G, K, Γ ).

On the symmetry of L1 algebras of locally compact motion groups, and the Wiener Tauberian theorem

Abstract Let G be a locally compact motion group, i.e., it is a semidirect product of a compact subgroup with a closed abelian normal subgroup, the action of the compact subgroup on the other one being by conjugation. The main result of this paper is that the group algebra of such a group is symmetric. This result is then used to prove that a generalization of the Wiener-Tauberian theorem holds for such groups. Precisely, it is shown that every proper closed two-sided ideal in L 1 ( G ) is annihilated by an irreducible unitary representation of G , lifted to L 1 ( G ).

The Elementary Spherical Functions

The main reason that harmonic analysis on semisimple groups can be developed in great depth is of course that the irreducible representations and their matrix coefficients can be constructed quite explicitly. In the setting that is of interest to us this means the construction of the principal series representations which in turn leads to integral representations of the associated elementary spherical functions. The fundamental theorem is that all elementary spherical functions are obtained by this method. This theorem sets the stage for the definition and study of the Harish-Chandra spherical transform of functions and distributions on G//K.

Asymptotic Behaviour of Elementary Spherical Functions

This chapter, as well as the next one, will be devoted to the formulation and proofs of the main theorems of the L2 harmonic analysis of spherical functions. At the center of the theory is the Harish-Chandra transform (see §3.3)

L p -Theory of Harish-Chandra Transform. Fourier Analysis on the Spaces ℓ P ( G//K )

f \mapsto Hf

The Harish-Chandra Series for φ λ and the c-Function

where