Lawrence Corwin

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.

Tempered distributions on Heisenberg groups whose convolution with Schwartz class functions is Schwartz class

Abstract Let N be a nilpotent Lie group and Q a tempered distribution on N. We say that Q is a left J -multiplier if convolution on the left by Q takes Schwartz class functions to Schwartz class functions; there is a similar definition for right J -multipliers. We show that if ϱ is an irreducible unitary representation of N, then one can define ρ(Q): H ∞:(ρ)→ H ∞:(ρ) whenever Q is a left J -multiplier. The main results of the paper characterize left J -multipliers Q on Heisenberg groups in terms of the transform operators ϱ(Q) and show how this characterization can be used to find fundamental solutions of some left invariant differential operators. There is also an example of a left J -multiplier which is not a right J -multiplier.

A note on the exponential map of a real or p-adic Lie group

where the right-hand side is an absolutely convergent series, involving higher-order brackets of X and Y, and (1.2) is valid for all small X and Y. For an exposition of this and any other facts Soncer3ing real Lie groups see [l]. The reason that (1.1) holds is that if X and Y commute so do tX and tY for real t so that by (1.2), exp’ tX mexp t Y = expt(X + Y) for small t, since all the other terms in the formula are zero. But then, by the identity theorem for real analytic functions, this holds for ali t. Taking t = 1 gives the result. Now, since the log inverts exp locally at the origin, and X and Y are small in (1.2), it follows that an absolutely convergent series of these brackets is also small. Hence

Order estimates for irreducible projections in L2 of a nilmanifold

Let π be an irreducible representation occurring in L2(Г⧹N), where N is a nilpotent Lie group and Γ is a discrete, cocompact subgroup. The projection onto the π-equivariant subspace is given by convolution against a distribution Dπ. For certain π, we obtain an estimate on the order of Dπ. The condition on π involves an extension of the “canonical objects” associated to elements of the Kirillov orbit of π; there does not appear to be an example in the literature where it is not satisfied.

Necessary and sufficient conditions hypoellipticity of certain left invariant operators on nilpotent lie groups

Let G be a 2-step nilpotent lie Group of type (H) such that there is a subalgebra of the Lie algebra g which is maximal subordinate for all nontrivial linear functionals on the center g2 of the Lie algebra g— Suppose that L is a left invariant operator on G such tha the term or highest homogeneous degree is elliptic in the generating directions on G. Then L is hypoelliptic ⇔ the elements l ∈ (g2)*C for which пl(L*L) is not invertible move away from g*faster than any multiple of log ∣l∣. The implication ⇐ holds for all Lie groups of type (H) ⇔

Constructing the Supercuspidal Representation of GL n (F), F p—ADIC

In this paper I give a description of a construction of the supercuspidal representations of GL n (F). This construction, the result of work done in late 1988 and early 1989, was first given in [6], and the description here follows the same lines.

Kirillov Orbits and Direct Integral Decompositions on Certain Quotient Spaces

A fundamental problem in representation theory is that of describing the unitary representations of a locally compact group G as a direct integral of simpler representations. The “abstract” problem (i.e., the problem of determining when such decompositions exist) takes up a good part of Mackey’s Chicago lecture notes ↑l6←; it is shown there that if G is Type I, then the problem has a satisfactory solution. The “concrete” problem (given G and a representation ρ, decompose ρ) can be considerably harder. For instance, if G is a semisimple Lie group andΓ is a discrete, cocompact subgroup, then the quasi-regular representation of G on L2(Γ\G) is known to be a direct sum of irreducibles, each occurring with finite multiplicity, but little is known about which irreducibles appear. (See ↑l6←.)