Character formulas and spectra of compact nilmanifolds

Lawrence Corwin, Frederick P. Greenleaf

Research output: Contribution to journalArticle

Abstract

The authors give a new method for calculating the spectrum and multiplicities of the irreducible unitary representations appearing in the quasi-regular representation U: N × L2(ΓβN) → L2(ΓβN) on a compact nilmanifold ΓβN. They proceed by decomposing the trace of U into traces of irreducible representations. The basic calculations in the paper deal with lattice subgroups (Λ = log Γ an additive lattice in the Lie algebra N), essentially using the Poisson summation formula. Let Ad′ be the contragredient adjoint action of N on N*. If f{hook}0 ε{lunate} N*, the multiplicity of π(f{hook}0) in U is zero unless the Ad′(N) orbit of f{hook}0 meets Λ = {h ε{lunate} N*: <h, Λ> ⊆ Z}. If f{hook}0 ε{lunate} Λ, then the multiplicity is a sum over representatives of certain Ad′(Γ)-orbits in, m(π(f{hook}0),U) = ∑ Ad′(N)f{hook}0∩Λ Ad′(Γ)k(f{hook}). The constants k(f{hook}) are given both algebraic and geometric interpretations that lead to simple and effective calculations. Similar formulas hold if Γ is not a lattice subgroup.

Original languageEnglish (US)
Pages (from-to)123-154
Number of pages32
JournalJournal of Functional Analysis
Volume21
Issue number2
DOIs
StatePublished - Feb 1976

ASJC Scopus subject areas

  • Analysis

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