TY - JOUR

T1 - Character formulas and spectra of compact nilmanifolds

AU - Corwin, Lawrence

AU - Greenleaf, Frederick P.

N1 - Funding Information:
v Author is a Sloan Foundation Fellow; this research was supported in part by NSF Research Grant GP-30673. Current address: Department of Mathematics, Rutgers University, New Brunswick, N. J., 08903. + Research supported in part by NSF Research Grant GP-19258.

PY - 1976/2

Y1 - 1976/2

N2 - 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*: ⊆ 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.

AB - 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*: ⊆ 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.

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U2 - 10.1016/0022-1236(76)90074-4

DO - 10.1016/0022-1236(76)90074-4

M3 - Article

AN - SCOPUS:0347299772

SN - 0022-1236

VL - 21

SP - 123

EP - 154

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 2

ER -