The Weyl upper bound on the discrete spectrum of locally symmetric spaces

Abstract

Introduction1.1.Let G be a reductive Lie group with finitely many connected components, and T a cofinite volume discrete subgroup of G. Let K C G be a maximal compact subgroup, and X = G/K be the associated symmetric space, which is the product of a symmetric space of noncompact type and a possible Euclidean space.Then r\X is a locally symmetric space of finite volume.For simplicity, we assume, unless otherwise specified, that there exists a reductive algebraic group G defined over Q satisfying the conditions in [18, p. 1] such that G = G (R), and r C G(Q) is an arithmetic subgroup.Any finite dimensional unitary representation a of K defines a homogeneous bundle E a on X and hence a locally homogeneous bundle E a on r\X.The bundle E a admits a locally invariant connection y which is the push forward of the invariant connection on the homogeneous bundle E a .The connection y defines a quadratic form D on sections of E a : For any f G CQ°(T\X, a), D(f)= Z \vf(x)\ 2 dx. r\xThis quadratic form D defines an elliptic operator A on L 2 (T\X, a), called the Laplace operator, where L 2 (T\X, a) denotes the space of Lsections of E a .If a is irreducible, A is equal to a shift of the restriction of

Locations

• Journal of Differential Geometry - View - PDF