Garth Warner

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 ).

Multiplicities of the integrable discrete series: The case of a nonuniform lattice in an R-rank one semisimple group

Let G be a noncompact connected simple Lie group of split-rank 1. Assume that G possesses a compact Cartan subgroup so that the discrete series for G is not empty. Let Γ be a nonuniform lattice in G. In this paper, we give an explicit formula for the multiplicity with which an integrable discrete series representation of G occurs in the space of cusp forms in L2(GΓ).

The Selberg trace formula. V. Questions of trace class

The purpose of this paper is to develop criteria which will ensure that the K-finite elements of C^°(G) are represented on L¿ia(G/T) by trace class operators.

The Selberg trace formula. VIII. Contribution from the continuous spectrum

The purpose of this paper is to isolate the contribution from the continuous spectrum to the Selberg trace formula.

Topology on the Dual Plancherel Measure

The purpose of the present chapter is to set down in a precise fashion the main facts as regards the structure of the unitary dual Ĝ of a locally compact group G (which satisfies the second axiom of countability, say). Since these topics are dealt with more than adequately by Dixmier [15], we shall give very few proofs; instead we shall be content to look at a number of examples which serve to illustrate the abstract theory, full details being provided in the case of a semi-simple G.

The Universal Enveloping Algebra of a Semi-Simple Lie Algebra

Let g be a reductive Lie algebra over R, G a connected Lie group with Lie algebra g. In the hands of Harish-Chandra (and others) the universal enveloping algebra G of gc (the complexification of g) plays a prominent role in the representation theory of G. This will become apparent in Chapter 4 where we shall see, for instance, how a representation of G on a Banach space E, say, gives rise in a natural manner to various linear representations of G on subspaces of E. Moreover the study of these representations of G yields important information about the given representation of G. This chapter, then, which is algebraic in character, deals on the one hand with the structure of G and, on the other, with various aspects of its representation theory. [Actually it will be just as easy to carry out most of the discussion in the context of a reductive Lie algebra g over an algebraically closed field k of characteristic zero — unless specifically stated to the contrary, this will be the underlying assumption in what follows.]

The Discrete Series for a Semi-Simple Lie Group — Existence and Exhaustion

Let G be an acceptable connected semi-simple Lie group with finite center, Ĝ its unitary dual. Let G d be the discrete series for G. Given U ∊ G d , let To denote its character, d U its formal dimension — then, as is known, the distribution

Spherical Functions on a Semi-Simple Lie Group

{T_{d}} = \sum\limits_{{\hat{U} \in {{\hat{G}}_{d}}}} {{d_{{\hat{U}}}}{T_{{\hat{U}}}}}