Caractères des groupes de Lie

Let G be a real Lie group with Lie algebra $\text{G}$. M. Duflo has constructed irreducible unitary representations of G associated to some G-orbits Ω in the dual $\text{G}$¹ of $\text{G}$. We prove a character formula when Ω is tempered, closed, and of maximal dimension.     

Order estimates for irreducible projections in L2 of a nilmanifold

Let π be an irreducible representation occurring in $\text{L}$²(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 D_π. For certain π, we obtain an estimate on the order of D_π. 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.     

Factorial representations of path groups

The reduction of the energy representation of the group of mappings from I = [0, 1], S¹1, $\text{R}$₊ or $\text{R}$ into a compact semisimple Lie group G is given. For G = SU(2), the factoriality of the representation, which is of type III in the case I=$\text{R}$, is proved.     

Sur les semi-caracteres des groupes de Lie resolubles connexes

Let G be a connected solvable Lie group, π a normal factor representation of G and ψ a nonzero trace on the factor generated by G. We denote by $\text{D}$(G) the space of C^∞ functions on G which are compactly supported. We show that there exists an element u of the enveloping algebra U$\text{G}$_$\text{c}$ of the complexification of the Lie algebra of G for which the linear form $\text{? ψ(π(u 1 ?))}$ on $\text{D}$(G) is a nonzero semiinvariant distribution on G. The proof uses results about characters for connected solvable Lie groups and results about the space of primitive ideals of the enveloping algebra U$\text{G}$_$\text{c}$.     

Riemann-Lebesgue subsets of Rn and representations which vanish at infinity

The main theorem of this paper is that if χ is a character of a connected closed normal subgroup of a connected Lie group, then every matrix element of the induced representation U^χ vanishes at infinity modulo the kernel of that representation. As a consequence, it is shown that every faithful irreducible unitary representation of a connected motion group vanishes at infinity. In the course of the development a generalization of the classical Riemann-Lebesgue lemma is proved. Suppose $\text{M}$ is an analytic submanifold of $\text{R}$ⁿ which is not contained in any proper hyperplane. Then the Fourier transform of any measure, which is concentrated on $\text{M}$ and which is absolutely continuous with respect to the “Lebesgue” measure on $\text{M}$, vanishes at infinity.     

Classification projective des espaces d’opérateurs différentiels agissant sur les densités

One classifies the representation D_λμ of sl(m + 1, R) obtained by letting its projective embedding in the Lie algebra of vector fields of R^m, m > 1, act by Lie derivatives on the space of differential operators between densities of weight λ and μ. For each μ - λ, there is only finitely many isomorphism classes, most frequently one, in which case D_λμ is isomorphic to its graded space relative to the order of differentiation.     

A characterization of certain weak1-closed subalgebras of L∞

Let $\text{R}$ denote the real line and L^∞(R), the class of all Borel measurable L^∞-functions of $\text{R}$. Let S ≠ {0} or φ, be a linear subspace of L^∞(R) which is (i) translation invariant, (ii) weak¹-closed, (iii) self-adjoint, i.e., f?S implies f?S, and (iv) an algebra. Then either (a) S = all constant functions in L^∞; or (b) S = L^∞; or (c) there is a unique c > 0 such that S consists of all L^∞-functions which are periodic of period c.Extension of the above characterization of periodic subalgebras of L^∞ to LCA groups are presented. Also it is shown that the above characterization is in various ways best possible.     

Gauge invariance and implementability of the S-operator for spin-0 and spin-12 particles in time-dependent external fields

It is proved that the classical S-operator for relativistic spin-0 and spin-$\text{1}\text{2}$ particles in time-dependent external fields is gauge invariant, and that S_+- and S_?+ are entire functions of the coupling constant in the Hilbert-Schmidt norm. As a result the Fock space S-operator exists for any real value of the coupling constant, and is gauge invariant. The external fields and the gauge function are assumed to be real-valued resp. complex-valued functions in S(R⁴).     

On the decomposition of the tensor algebra of the classical Lie algebras

Let $\text{g}$_n be the Lie algebra gl_n(C), let S($\text{g}$_n) be the symmetric algebra of $\text{g}$_n, and let T($\text{g}$_n) be the tensor algebra of $\text{g}$_n. In a recent paper, R. K. Gupta studied certain sequences of representations R_∞ = (R_n)^∞_{n = 1}, where R_n is a representation of $\text{g}$_n. These sequences have the property that every irreducible representation occurring in S($\text{g}$_n) is in exactly one of these sequences. Fixing f, she considers s(R_∞, f) which is the limit on n of the multiplicity of R_n in S^f($\text{g}$_n), the fth-graded piece of S($\text{g}$_n). She and R. P. Stanley independently showed that the limit s(R_∞, f) exists and is given by an amazingly elegant formula. They call s(R_∞, f) the stable multiplicity of R_n in S^f($\text{g}$_n). In this paper, an entirely different approach is used to extend the above result in several directions. Appropriately defined sequences R_∞ for all of the classical Lie algebras $\text{g}$_n are studied, and a simple formula for the stable multiplicity m(R)_∞, ψ, f, $\text{g}$_∞) of R_n in the ψ-isotypic component of T^f($\text{g}$_n), where ψ is any irreducible character of the symmetric group tS_f, is obtained. As in the work of Gupta and Stanley, the expressions for m(R)_∞, ψ, f, $\text{g}$_∞) are amazingly simple. Special cases include the stable decomposition of the tensor algebra, the symmetric algebra and the exterior algebra of $\text{g}$_n. As a byproduct of our proof, a “stable” decomposition of every isotypic component of T($\text{g}$_n) is obtained. This combinatorial decomposition is in some sense a generalization of Kostant's decomposition of S($\text{g}$_n) into direct sum of the harmonics and the ideal generated by the invariants of positive degree. To be precise, for f <n the combinatorial decomposition of T^f($\text{g}$_n) projects onto Kostant's decomposition of S^f($\text{g}$_n).     

Distributions de type K-positif sur l'espace tangent

Let K be a compact subgroup of the isometry group of $\text{R}$ⁿ. A distribution T is said to be of K-positif type if it is K-invariant and if $\text{〈T, ? 1}\text{}\text{?}\text{gj〉 = ∝∝ ?(x + y)}\text{?(Y) dT(x) ? 0}$ for every K-invariant $\text{b}$^∞ function ? with compact support. We look for an integral representation of these distributions (i.e., an analog of the Bochner-Schwartz theorem). In this paper we obtain such a representation for distributions with growth of exponential type in the following case: K is the maximal compact subgroup of a semi-simple connected Lie group G with finite center, acting by the adjoint action on the tangent space of $\text{G}\text{K}$. The main step is to prove that it suffices to work with distributions of W-positif type (where W is the Weyl group associated with $\text{G}\text{K}$). This is achieved following ideas of a paper of S. Helgason [Advan. in Math.36 (1980) 297]. The end of the proof follows from the case where K is finite [N. Bopp, in “Analyse harmonique sur les groupes de Lie,” Lecture Notes in Mathematics No. 739, p. 15, Springer-Verlag, Berlin/New York 1979].     

The class A(R,S) of (0, 1)-matrices

Let R and S be two vectors whose components are m and n non-negative integers, respectively. Let $\text{A}$(R, S) be the class consisting of all (0, 1)-matrices of size m by n with row sum vector R and column sum vector S. In this paper we derive a lower bound to the cardinality of the class $\text{A}$(R, S), which can be computed readily.     

Limit of the spectra of the interior Neumann problems when a solid domain shrinks to a plane one

Let (Δ + λ) u = 0 in $\text{D}$c$\text{R}$^d, $\text{?u}\text{?N}\text{=0}$ on ?$\text{D}$. How do the eigenvalues λ_j behave when $\text{D}$ shrinks to a domain Δ ? R^{d ? 1} ? The answer depends not only on Δ but on the way $\text{D}$ shrinks to Δ. The limit of λ_j is found. Examples are given.     

Reconstruction of a factor from measures on Takesaki's unitary equivalence relation

Let $\text{Ol}$?L^∞(S, μ) be a maximal abelian subalgebra of the factor $\text{F}$ on separable Hilbert space with modular involution J. ($\text{Ol}$∪ J$\text{Ol}$J)″ is represented naturally as L^∞(S × S, λ). If Takesaki's unitary equivalence relation $\text{R}$ ? S × S is not λ-null, it is a measure groupoid. If it is conull, and ($\text{Ol}$∪ J$\text{Ol}$J)″ is maximal abelian, $\text{F}$ and $\text{Ol}$ are reconstructed by the σ-left regular representation procedure. Examples show that these hypotheses are not always satisfied. An application shows that the L^∞ spectrum of a properly infinite ergodic transformation is null with respect to the L² spectrum.     

An analogue of the thom isomorphism for crossed products of a C1 algebra by an action of R

In this paper we show the existence and uniqueness of a natural isomorphism ø^j_α of K_j(A) with K_j+1(A ?_α$\text{R}$), j ? $\text{Z}$/2 where (A, $\text{R}$, α) is a $\text{C1}$ dynamical $\text{R}$-system, K is the functor of topological K theory and A ?_α$\text{R}$ is the crossed product of A by the action of $\text{R}$. The Pimsner-Voiculescu exact sequence is obtained as a corollary. We show that given an α-invariant trace τ on A, with dual trace $\text{}\text{?}$gt, one has $\text{}\text{?}\text{gtø}{}^1{}_{\mathrm{\alpha }}\text{[u] = (}\text{1}\text{2}\text{iπ) τ(δ(u)u1)}$ for any unitary u in the domain of the derivation δ of A associated to the action α. Finally, we show that the crossed product of C(S³) (continuous functions on the 3 sphere) by a minimal diffeomorphism is a simple $\text{C1}$ algebra with no nontrivial idempotent.     

Spectra of ergodic transformations

We study the group properties of the spectrum of a strongly continuous unitary representation of a locally compact Abelian group G implementing an ergodic group of ¹-automorphisms of a von Neumann algebra $\text{R}$. It is shown that in many cases the spectrum equals the dual group of G; e.g. if G is the integers and $\text{R}$ not finite dimensional and Abelian, then the spectrum is the circle group.     

The fixed-point algebra of tensor-product actions of finite Abelian groups on UHF-algebras

Let G be a finite Abelian group acting by tensor-product automorphisms on a UHF-C¹-algebra $\text{D}$. Extending a result of A. Kishimoto it is shown that the number of extremal traces on the fixed-point algebra $\text{D}$^G equals the cardinality of the subgroup K of automorphisms in G which are weakly inner in the trace representation of $\text{D}$.     

On admissible shifts of generalized white noises

Let $\text{S}$ be the Schwartz space of rapidly decreasing real functions. The dual space $\text{S}$¹ consists of the tempered distributions and the relation $\text{S}$ ? L²($\text{R}$) ? $\text{S}$¹ holds. Let γ be the Gaussian white noise on $\text{S}$¹ with the characteristic functional $\text{γ}\text{(ξ) =}\text{exp}\text{{?∥ξ∥}{}^2\text{/2}}$, ξ ∈ $\text{S}$, where ∥·∥ is the L²($\text{R}$)-norm. Let ν be the Poisson white noise on $\text{S}$¹ with the characteristic functional $\text{ν}\text{(ξ)}$ = exp?_$\text{R}$∫_$\text{R}$ {[exp(iξ(t)u)] ? 1 ? (1 + u²)^?1(iξ(t)u)} dη(u)dt), ξ ∈ $\text{S}$, where the Lévy measure is assumed to satisfy the condition ∫_$\text{R}$u²dη(u) < ∞. It is proved that γ1ν has the same dichotomy property for shifts as the Gaussian white noise, i.e., for any ω ∈ $\text{S}$¹, the shift $\text{(γ1ν)}{}_{\mathrm{\omega }}$ of γ1ν by ω and γ1ν are either equivalent or orthogonal. They are equivalent if and only if when ω ∈ L²($\text{R}$) and the Radon-Nikodym derivative is derived. It is also proved that for the Poisson white noice ν_ω is orthogonal to ν for any non-zero ω in $\text{S}$¹.     

Function algebra invariants and the conformal structure of planar domains

Let A(S) be the sup-normed Banach algebra of analytic functions with continuous boundary values on the compact bordered Riemann surface S.For (?) in $\text{A(S)}{}^{\mathrm{?1}}\text{exp}\text{(A(S))}$, the colength of (?) is defined by $\text{∥(?)∥ =}\text{1}\text{2}\text{log inf}\text{{∥ g ∥ ∥ g}{}^{\mathrm{?1}}\text{∥; g ? (?)}}$. Colength is shown to induce a norm on the cohomology group H¹(S,R) dual to the norm induced on the homology group H₁(S,R) by harmonic length, or, equivalently, dual to the norm on Re A(S)^⊥.The existence and uniqueness of extremal functions for the colength functional is demonstrated. The aforementioned norms are shown to determine the conformal structure of S (up to reflection) and to be related to the mapping properties of S.     

Spectral theory of finite volume domains in Rn

Let Ω be an arbitrary open subset of $\text{R}$ⁿ of finite positive measure, and assume the existence of a subset Λ ? $\text{R}$ⁿ such that the exponential functions e_λ = exp i(λ₁x₁ + … + λ_nx_n), λ = (λ₁,…, λ_n) ∈ Λ, form an orthonormal basis for $\text{L}{}_2\text{(Ω)}$ with normalized measure. Assume 0 ∈ Λ and define subgroups K and A of ($\text{R}$ⁿ, +) by K = Λ⁰ = {γ ∈ $\text{R}$ⁿ:γ·λ ∈ 2π$\text{Z}$}, A = {a ∈ $\text{R}$ⁿ:U_a$\text{m}$ U1_a = $\text{m}$}, where U_t is the unitary representation of $\text{R}$ⁿ on $\text{L}{}_2\text{(Ω)}$ given by U_te = e^itλe_λ, t ∈ $\text{R}$ⁿ, λ ∈ Λ, and where $\text{m}$ is the multiplication algebra of $\text{L}{}_{\mathrm{\infty }}\text{(Ω)}$ on L₂. Assume that A is discrete. Then there is a discrete subgroup D ? A of dimension n, a fundamental domain $\text{D}$ for D, and finite sets of representers R_Λ, R_Γ, $\text{R}{}_{\mathrm{\Omega }}$, each containing 0, R_Λ for $\text{A}\text{K}$ in K⁰, and $\text{R}{}_{\mathrm{\Omega }}$ for $\text{A}\text{K}$ in A such that Ω is disjoint union of translates of $\text{D}$: Ω = ∪_a∈RΩ (a + $\text{D}$), neglecting null sets, and Λ = R_Λ ⊕ D⁰. If R_Γ is a set of representers for $\text{D}\text{A}$ in D, then Γ = R_Γ ⊕ K is a translation set for Ω, i.e., Ω ⊕ Γ = $\text{R}$ⁿ, direct sum, (neglecting null sets). The case A = $\text{R}$ⁿ corresponds to Ω = $\text{D}$, Λ = D⁰ and Γ = K. This last case corresponds in turn to a function theoretic assumption of Forelli.     

Functions and derivations of C1-algebras

It is shown that there is a closed symmetric derivation δ of a C¹-algebra with dense domain $\text{D}$(δ), an element A = A¹ ?$\text{D}$(δ), and a C¹-function $\text{f}$ such that $\text{f}$(A)?$\text{D}$(δ). Some estimates are derived for $\text{∥ δ(¦ A ¦)∥}\text{and}\text{∥ δ(A}{}_{\mathrm{+}}{}^{\mathrm{\alpha }}\text{)∥}$, where 0 < α < 1. It is shown that there exists a family of one-one self-adjoint operators S(t) in $\text{L}$(H) which depends linearly on t, while $\text{¦ S(t)¦}$ is not differentiable. It is also shown that there exists $\text{L}$(H) which is not C¹-self-adjoint even though it satisfies $\text{∥}\text{exp}\text{(itT)∥ ? C(1 + ¦ t ¦)}$ for all t ? $\text{R}$