@MISC{Fox_adelesand, author = {Jeffrey Stephen Fox and Jeffrey Fox}, title = {ADELES AND THE SPECTRUM OF COMPACT NILMANIFOLDS}, year = {} }

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Abstract

Let G be a nilpotent Lie group and Γ a discrete cocompact sub-group 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 "ratio-nal " Kirillov theory for the adele group GΆ Once this is done, the decomposition and multiplicity formulas follow from elementary con-siderations. Let G be a locally compact group and Γ c G a closed subgroup such that G/T is compact. For x e G, we can define Λ(JC), a unitary operator on L2{G/Γ) by (λ(x)f)(y) = f(x~ιy). The representation x--> λ{x) is called the quasi-regular representation of G. A fundamental problem in representation theory is to decompose λ into irreducible representa-tions. A theorem of Fell [F] says that λ will be discretely decomposable and each irreducible will occur with finite multiplicity. Thus we can write: