Weyl quantization for semidirect products

Let G be the semidirect product V?K where K is a connected semisimple non-compact Lie group acting linearily on a finite-dimensional real vector space V. Let O be a coadjoint orbit of G associated by the Kirillov-Kostant method of orbits with a unitary irreducible representation π of G. We consider the case when the corresponding little group K₀ is a maximal compact subgroup of K. We realize the representation π on a Hilbert space of functions on Rn where n=dim(K)−dim(K₀). By dequantizing π we then construct a symplectomorphism between the orbit O and the product R2n×O^′ where O^′ is a little group coadjoint orbit. This allows us to obtain a Weyl correspondence on O which is adapted to the representation π in the sense of [B. Cahen, Quantification d'une orbite massive d'un groupe de Poincaré généralisé, C. R. Acad. Sci. Paris Série I 325 (1997) 803-806]. In particular we recover well-known results for the Poincaré group.     

Explicit Hilbert spaces for certain unipotent representations

Summary LetG be the universal cover of the group of automorphisms of a symmetric tube domain and letP=LN be its Shilov boundary parabolic subgroup. This paper attaches an irreducible unitary representation ofG to each of the (finitely many)L-orbits onn ^*.The Hilbert space of the representation consists of functions on the orbit which are square-integrable with respect to a certainL-equivariant measure. The representation remains irreducible when restricted toP, and descends to a quotient ofG which is, at worst, thedouble cover of a linear group.If theL-orbit isnot open (inn ^*), the construction gives a unipotent representation ofG.Oblatum 28-II-1992This work was supported by an NSF grant at Princeton University, and was carried out in part during a visit to the Mehta Research Institute, Allahabad, India.     

Systems of Roots and Topology of Complex Flag Manifolds

In this paper we study the topology of a complex homogeneous space M = G/H of complex dimension n, with non vanishing Euler characteristic and G of type A, D, E by means of a topological invariant ₂, which is related to the Poincaré polynomial of M. We introduce the function Q = ₂/n and we examine how it varies as one passes from a principal orbit of the adjoint representation of a compact Lie group G to a more singular one. Moreover, it is proved that if M is a principal orbit G/T then Q depends only on the Weyl group of G.     

Symbolic dynamics and nonlinear semiflows

Summary For a transverse homoclinic orbit of a mapping (not necessarily invertible) on a Banach space, it is shown that the mapping restricted to orbits near is equivalent to the shift automorphism on doubly infinite sequences on finitely many symbols. Implications of this result for the Poincaré map of semiflows are given.This work was supported by the Air Force Office of Scientific Research under Grant #81-0198, by the National Science Foundation under Grant #MCS-8205355 and by the Army Research Office under Grant ù DAAG-29-83-K-0029.     

Gated Polling Systems with Lévy Inflow and Inter-Dependent Switchover Times: A Dynamical-Systems Approach

We study asymmetric polling systems where: (i) the incoming workflow processes follow general Lévy-subordinator statistics; and, (ii) the server attends the channels according to the gated service regime, and incurs random inter-dependentswitchover times when moving from one channel to the other. The analysis follows a dynamical-systems approach: a stochastic Poincaré map, governing the one-cycle dynamics of the polling system is introduced, and its statistical characteristics are studied. Explicit formulae regarding the evolution of the mean, covariance, and Laplace transform of the Poincaré map are derived. The forward orbit of the maps transform – a nonlinear deterministic dynamical system in Laplace space – fully characterizes the stochastic dynamics of the polling system. This enables us to explore the long-term behavior of the system: we prove convergence to a (unique) steady-state equilibrium, prove the equilibrium is stationary, and compute its statistical characteristics.     

On certain multiplicity one theorems

We prove several multiplicity one theorems in this paper. Fork a local field not of characteristic two, andV a symplectic space overk, any irreducible admissible representation of the symplectic similitude group GSp(V) decomposes with multiplicity one when restricted to the symplectic group Sp(V). We prove the analogous result for GO(V) and O(V), whereV is an orthogonal space overk. Whenk is non-archimedean, we prove the uniqueness of Fourier-Jacobi models for representations of GSp(4), and the existence of such models for supercuspidal representations of GSp(4). The first-named author was partially supported by the National Security Agency (#MDA904-02-1-0020).     

Poincare Series and Zeta Function for an Irreducible Plane Curve Singularity

The Poincaré series of an irreducible plane curve singularityequals the -function of its monodromy, by a result of Campillo,Delgado and Gusein-Zade. This fact is derived in this paperfrom a formula of Ebeling and Gusein-Zade, relating the Poincaréseries of a quasi-homogeneous complete intersection singularityto the Saito dual of a product of -functions. 2000 MathematicsSubject Classification 32S40 (primary), 14B05 (secondary).     

Plancherel Inversion as Unified Approach to Wavelet Transforms and Wigner Functions

We demonstrate that the Plancherel transform for Type-I groups provides one with a natural, unified perspective for the generalized continuous wavelet transform, on the one hand, and for a class of Wigner functions, on the other. We first prove that a Plancherel inversion formula, well known for Bruhat functions on the group, holds for a much larger class of functions. This result allows us to view the wavelet transform as essentially the inverse Plancherel transform. The wavelet transform of a signal is an L²-function on an appropriately chosen group while the Wigner function is defined on a coadjoint orbit of the group and serves as an alternative characterization of the signal, which is often used in practical applications. The Plancherel transform maps L²-functions on a group unitarily to fields of Hilbert-Schmidt operators, indexed by unitary irreducible representations of the group. The wavelet transform can essentially be looked upon as a restricted inverse Plancherel transform, while Wigner functions are modified Fourier transforms of inverse Plancherel transforms, usually restricted to a subset of the unitary dual of the group. Some known results on both Wigner functions and wavelet transforms, appearing in the literature from very different perspectives, are naturally unified within our approach. Explicit computations on a number of groups illustrate the theory. Communicated by Gian Michele Graf submitted 05/06/01, accepted: 19/09/02     

On the Complement of the Dense Orbit for a Quiver of Type 𝔸

Let 𝔸 _t be the directed quiver of type 𝔸 with t vertices. For each dimension vector d, there is a dense orbit in the corresponding representation space. The principal aim of this note is to use just rank conditions to define the irreducible components in the complement of the dense orbit. Then we compare this result with already existing ones by Knight and Zelevinsky, and by Ringel. Moreover, we compare with the fan associated to the quiver 𝔸 _t and derive a new formula for the number of orbits using nilpotent classes. In the complement of the dense orbit, we determine the irreducible components and their codimension. Finally, we consider several particular examples.     

The Virtual Poincaré Polynomials of Homogeneous Spaces

We factor the virtual Poincaré polynomial of every homogeneous space G/H, where G is a complex connected linear algebraic group and H is an algebraic subgroup, as t^2u (t²–1)^r Q_G/H(t²) for a polynomial Q_G/H with nonnegative integer coefficients. Moreover, we show that Q_G/H(t²) divides the virtual Poincaré polynomial of every regular embedding of G/H, if H is connected.     

Finite automorphisms of negatively curved Poincaré duality groups

In this paper, we show that if G is a finite p-group (p prime) acting by automorphisms on a -hyperbolic Poincaré Duality group over , then the fixed subgroup is a Poincaré Duality group over . We also provide a family of examples to show that the fixed subgroup might not be a Poincaré Duality group over . In fact, the fixed subgroups in our examples even fail to be duality groups over .     

Analysis on the minimal representation of O(p,q): III. Ultrahyperbolic equations on

For the group O(p,q) we give a new construction of its minimal unitary representation via Euclidean Fourier analysis. This is an extension of the q=2 case, where the representation is the mass zero, spin zero representation realized in a Hilbert space of solutions to the wave equation. The group O(p,q) acts as the Möbius group of conformal transformations on , and preserves a space of solutions of the ultrahyperbolic Laplace equation on . We construct in an intrinsic and natural way a Hilbert space of solutions so that O(p,q) becomes a continuous irreducible unitary representation in this Hilbert space. We also prove that this representation is unitarily equivalent to the representation on L²(C), where C is the conical subvariety of the nilradical of a maximal parabolic subalgebra obtained by intersecting with the minimal nilpotent orbit in the Lie algebra of O(p,q).     

Projective bundle ideals and Poincaré duality algebras

The study of maximal-primary irreducible ideals in a commutative graded connected Noetherian algebra over a field is in principle equivalent to the study of the corresponding quotient algebras. Such algebras are Poincaré duality algebras. A prototype for such an algebra is the cohomology with field coefficients of a closed oriented manifold. Topological constructions on closed manifolds often lead to algebraic constructions on Poincaré duality algebras and therefore also on maximal-primary irreducible ideals. It is the purpose of this note to examine several of these and develop some of their basic properties.     

Geometric quantization, reduction and decomposition of group representations

We consider a Hamiltonian action of a connected group G on a symplectic manifold (P, ω) with an equivariant momentum map and its quantization in terms of a K?hler polarization which gives rise to a unitary representation of G on a Hilbert space . If O is a co-adjoint orbit of G quantizable with respect to a K?hler polarization, we describe geometric quantization of algebraic reduction of J ⁻¹(O). We show that the space of normalizable states of quantization of algebraic reduction of J ⁻¹(O) gives rise to a projection operator onto a closed subspace of on which is unitarily equivalent to a multiple of the irreducible unitary representation of G corresponding to O. This is a generalization of the results of Guillemin and Sternberg obtained under the assumptions that G and P are compact and that the action of G on P is free. None of these assumptions are needed here. Dedicated to Vladimir Igorevich Arnold     

Low Cohomogeneity Representations and Orbit Maximal Actions

An orthogonal representation of a compact Lie group is called polar if thereexists a linear subspace which meets all orbits orthogonally.It has been shown by Conlon that one can associate a Coxeter groupto such a representation.From this, an upper bound for the cohomogeneity of an irreduciblepolar representation can be derived.Another property of irreducible polar representations isthat the action restricted to the unit spherehas maximal orbits in the sense that any action having largerorbits is transitive.We give a classification of orbit maximal actions on spheresand use it to show that irreducible polar representations arecharacterized by these two properties.     

Dirac equation and spin 1 representations,a connection with symmetries of the Maxwell equations

A unitary operator on the space of spinors that makes it possible to associate each transformation in this space with a transformation in the space of electromagnetic field strengths is found. A connection is established by means of this operator between representations in the space of spinors and the space of field strengths for the Lorentsz, Poincaré, and conformal groups. Unusual symmetries of the Dirac equation are found on this basis. It is noted that the Pauli—Gürsey symmetry operators (without the ⁵ operator) of the Dirac equation withm=0 form the same representation D(1/2, 0)D(0, 1/2) of the O(1, 3) algebra of the Lorentz group as the spin matrices of the standard spinor representation. It is shown that besides the standard (spinor) representation of the Poincaré group, the massless Dirac equation is invariant with respect to two other representations of this group, namely, the vector and tensor representations specified by the generators of the representations D(1/2, 1/2) and D(1, 0) D(0, 0) of the Lorentz group, respectively. Unusual families of representations of the conformal algebra associated with these representations of the group O(1, 3) are investigated. Analogous O(1, 2) and P(1, 2) invariance algebras are established for the Dirac equation withm>0.Institute of Nuclear Research, Ukrainian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 3, pp. 388–406, March, 1992.     

Geometric Modular Action and Spontaneous Symmetry Breaking

We study spontaneous symmetry breaking for field algebras on Minkowski space in the presence of a condition of geometric modular action (CGMA) proposed earlier as a selection criterion for vacuum states on general space-times. We show that any internal symmetry group must commute with the representation of the Poincaré group (whose existence is assured by the CGMA) and each translation-invariant vector is also Poincaré invariant. The subspace of these vectors can be centrally decomposed into pure invariant states and the CGMA holds in the resulting sectors. As positivity of the energy is not assumed, similar results may be expected to hold for other space-times.Communicated by Klaus FredenhagenDedicated to the memory of Siegfried Schliedersubmitted 25/05/04, accepted 29/10/04     

THE ACTION OF THE LINEAR CHARACTER GROUP ON THE SET OF NON-LINEAR IRREDUCIBLE CHARACTERS

Let G be a nonabelian finite group. Then Irr(G/G′) is an abelian group under the multiplication of characters and acts on the set of non-linear irreducible characters of G via the multiplication of characters. The purpose of this paper is to establish some facts about the action of linear character group on non-linear irreducible characters and determine the structures of groups G for which either all the orbit kernels are trivial or the number of orbits is at most two. Using the established results on this action, it is very easy to classify groups G having at most three nomlinear irreducible characters.     

Harmonic analysis and systems of covariance for phase space representation of the poincaré group

Continuing some earlier work on the Galilei group, the spectral resolution of phase space representations of the Poincaré group is achieved by deriving all possible decompositions into irreducible representations corresponding to reproducing, kernel Hilbert spaces. Systems of covariance related to quantum measurements performed with extended test particles are analyzed, and questions of global unitarity discussed.Supported in part by NSERC Research Grants.     

The 2-characters of a group and the group determinant

A new look at Frobenius' original papers on character theory has produced the following: (a) the group determinant determines the group (Formanek-Sibley); (b) the group is determined by the 1-, 2- and 3-characters of the irreducible representations (Hoehnke-Johnson); and (c) pairs of non-isomorphic groups exist with the same irreducible 1- and 2-characters (Johnson-Sehgal). The examples produced in Johnson and Sehgal have large orders but, recently, McKay and Sibley have proved that the ten Brauer pairs of order 256 have the same irreducible 2-characters. It is shown here that the pairs of non-isomorphic groups of order p³, p and odd prime, have the same irreducible 2-characters. Further results are given on the k-characters of the regular representation/rod a shorter proof of the result mentioned in (c) is indicated. A criterion is given which is sufficient for the 3-character of an arbitrary representation to determine the group.