Frederick P. Greenleaf

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 ƒ0 ϵ N∗, the multiplicity of π(ƒ0) in U is zero unless the Ad′(N) orbit of ƒ0 meets Λ⊥ = {h ϵ N∗: ⊆ Z}. If ƒ0 ϵ Λ⊥, then the multiplicity is a sum over representatives of certain Ad′(Γ)-orbits in, m(π(ƒ0),U) = ∑Ad′(N)ƒ0∩Λ⊥Ad′(Γ)k(ƒ) . The constants k(ƒ) are given both algebraic and geometric interpretations that lead to simple and effective calculations. Similar formulas hold if Γ is not a lattice subgroup.

Group structure and the pointwise ergodic theorem for connected amenable groups

Abstract Let G be a connected amenable group (thus, an extension of a connected normal solvable subgroup R by a connected compact group K = G R ). We show how to explicitly construct sequences { U n } of compacta in G in terms of the structural features of G which have the following property: For any “reasonable” action G × L p ( X , μ ) ↓ L p ( X , μ ) on an L p space, 1 p f ∈ L p ( X , μ ), the averages A n f= 1 |U n | ∫ U n T g −1f dg (|E|= left Haar measure in G) converge in L p norm, and pointwise μ-a.e. on X , to G -invariant functions f∗ in L p ( X , μ ). A single sequence { U n } in G works for all L p actions of G . This result applies to many nonunimodular groups, which are not handled by previous attempts to produce noncommutative generalizations of the pointwise ergodic theorem.

Integral formulas with distribution kernels for irreducible projections in L2 of a nilmanifold

Abstract Let N be a simply connected nilpotent Lie group and Γ a discrete uniform subgroup. The authors consider irreducible representations σ in the spectrum of the quasi-regular representation N × L2(Γ/N) → L2(Γ→) which are induced from normal maximal subordinate subgroups M ⊆ N. The primary projection Pσ and all irreducible projections P ⩽ Pσ are given by convolutions involving right Γ-invariant distributions D on Γ→, Pf(Γn) = D ∗ f(Γn) = all f ϵ C ∞ (Γ/N) , where n · f(ζ) = f(ζ · n). Extending earlier work of Auslander and Brezin, and L. Richardson, the authors give explicit character formulas for the distributions, interpreting them as sums of characters on the torus Tκ = (Γ ∩ M) · [M, M]⧹M. By examining these structural formulas, they obtain fairly sharp estimates on the order of the distributions: if σ is associated with an orbit O ⊆ n ∗ and if V ⊆ n ∗ is the largest subspace which saturates θ in the sense that f ϵ O ⇒ f + V ⊆ O . As a corollary they obtain Richardsons criterion for a projection to map C0(Γ→) into itself. The authors also resolve a conjecture of Brezin, proving a Zero-One law which says, among other things, that if the primary projection Pσ maps Cr(Γ→) into C0(Γ→), so do all irreducible projections P ⩽ Pσ. This proof is based on a classical lemma on the extent to which integral points on a polynomial graph in Rn lie in the coset ring of Zn (the finitely additive Boolean algebra generated by cosets of subgroups in Zn). This lemma may be useful in other investigations of nilmanifolds.

Fourier transforms of smooth functions on certain nilpotent Lie groups

The authors consider irreducible representations π ϵ N of a nilpotent Lie group and define a Fourier transform for Schwartz class (and other) functions φ on N by forming the kernels Kφ(x, y) of the trace class operations πφ = ∝N φ(n)πn dn, regarding the π as modeled in L2(Rk) for all π in general position. For a special class of groups they show that the models, and parameters λ labeling the representations in general position, can be chosen so the joint behavior of the kernels Kφ(x, y, λ) can be interpreted in a useful way. The variables (x, y, λ) run through a Zariski open set in Rn, n = dim N. The authors show there is a polynomial map u = A(x, y, λ) that is a birational isomorphism A: Rn → Rn with the following properties. The Fourier transforms F1φ = Kφ(x, y, λ) all factor through A to give “rationalized” Fourier transforms Fφ(u) such that Fφ ∘ A = F1φ. On the rationalized parameter space a function f(u) is of the form Fφ = f ⇔ f is Schwartz class on Rn. If polynomial operators T ϵ P(N) are transferred to operators T on Rn such that F(Tφ) = T(Fφ), P(N) is transformed isomorphically to P(Rn).

Spectral decomposition of invariant differential operators on certain nilpotent homogeneous spaces

Abstract If K is a connected subgroup of a nilpotent Lie group G, the irreducible decompositionof the action on L2(KG) has either pure infinite or boundedly finite multiplicities. In the finite case the authors recently proved that the algebra D(KG) of G-invariant differential operators on KG is commutative, even if the action is not multiplicity free, and produced evidence for the conjecture that D(KG) is isomorphic to the algebra of all Ad∗(K)-invariant polynomials on the annihilator , where is the Lie algebra of K. Here the conjecture is proved for a large class of data (K, G). For such pairs an explicit construction of the isomorphism can be found; it is a type of Fourier transform with some unusual nonlinear aspects. Furthermore the operators in D(KG) have tempered fundamental solutions.

Central idempotent measures on connected locally compact groups

Abstract A measure μ of finite total variation on a locally compact group G is idempotent if μ ∗ μ = μ, and is central if invariant under all inner automorphisms of G . Recent results of D. Rider and D. Ragozin concerning compact groups are combined with results of the authors for noncompact groups to determine all central idempotent measures on a connected G in terms of the structural features of G .

Geometry of coadjoint orbits and noncommutativity of invariant differential operators on nilpotent homogeneous spaces

A monomial representation τ = Ind(H G, χ) induced from a character on a connected subgroup H of a nilpotent Lie group G has a primary decomposition whose multiplicities are either purely infinite (m(τ) = ∞) or uniformly bounded (m(τ) < ∞). The multiplicities are completely determined by the geometry of coadjoint orbits in 𝔤*, and there are strong indications that orbit geometry also determines the structure of the algebra Dτ of τ-invariant differential operators on smooth sections. One unresolved conjecture says that Dτ is commutative m(τ) < ∞; () is well known, and in this note we report significant progress toward the converse by proving that () holds when CASE I: m(τ0) < ∞, m(τ) = ∞, and CASE II: D ≠ Dτ, where τ0 = Ind(H G0, χ) and G0 ⊇ H is a codimension-1 subgroup. When m(τ) = ∞, one can always reduce to Case I; all evidence so far suggests that II is always valid when I holds (which would resolve the conjecture), but no general proof is known. Similar results have been reported recently by H. Fujiwara, G. Lion, and S. Medhi [5] using traditional methods of induction on dimension. Our methods are completely noninductive and rest entirely on analysis of coadjoint orbit geometry. The same methods may prove useful in an ultimate orbital description of Dτ, along the lines of the structure theorems known to hold when m(τ) < ∞.