Adeles and the spectrum of compact nilmanifolds

Jeffrey Stephen Fox

Vol. 140 (1989), No. 2, 233–250

DOI: 10.2140/pjm.1989.140.233

Abstract

Let G be a nilpotent Lie group and Γ a discrete cocompact subgroup of G. A basic problem in harmonic analysis is to determine the structure of L²(G∕Γ). We apply adelic techniques to determine the decomposition of L²(G∕Γ). To do so, we first develop a “rational” Kirillov theory for the adele group G_A. Once this is done, the decomposition and multiplicity formulas follow from elementary considerations.

Mathematical Subject Classification 2000

Primary: 22E25

Secondary: 22E40, 22E55

Milestones

Received: 11 January 1988

Published: 1 December 1989