Character formulas and spectra of compact nilmanifolds

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.