Moment sets and unitary dual for the diamond group

The diamond group G is a solvable group, semi-direct product of R with a (2n+1)-dimensional Heisenberg group Hn. We consider this group as a first example of a semi-direct product with the form R?N where N is nilpotent, connected and simply connected.Computing the moment sets for G, we prove that they separate the coadjoint orbits and its generic unitary irreducible representations.Then we look for the separation of all irreducible representations. First, moment sets separate representations for a quotient group G⁻ of G by a discrete subgroup, then we can extend G to an overgroup G⁺, extend simultaneously each unitary irreducible representation of G to G⁺ and separate the representations of G by moment sets for G⁺.     

Kähler geometry of the universal Teichmüller space and coadjoint orbits of the Virasoro group

The Kähler geometry of the universal Teichmüller space and related infinite-dimensional Kähler manifolds is studied. The universal Teichmüller space T may be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The classical Teichmüller spaces T(G), where G is a Fuchsian group, are contained in T as complex Kähler submanifolds. The homogeneous spaces Diff₊(S ¹)/Möb(S ¹) and Diff₊(S ¹)/S ¹ of the diffeomorphism group Diff₊(S ¹) of the unit circle are closely related to T. They are Kähler Frechet manifolds that can be realized as coadjoint orbits of the Virasoro group (and exhaust all coadjoint orbits of this group that have the Kähler structure).     

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.     

On the structure of the monotone asymmetric travelling salesman polytope I: hypohamiltonian facets

The monotone asymmetric travelling salesman polytope P?ⁿ_T is defined to be the convex hull of the incidence vectors of all hamiltonian circuits and all subsets of these in a complete diagraph of order n. We prove that certain hypohamiltonian diagraphs G=(V,E), i.e. diagraphs which are not hamiltonian but such that G–υ is hamiltonian for all υ?V, induce facets x(E)?n–1 of P?ⁿ_T. This result indicates that P?ⁿ_T has very complicated facets and that it is very unlikely that an explicit complete characterization of P?ⁿ_T can ever be given.     

Long paths and cycles in subgraphs of the cube

Let Q _n denote the graph of the n-dimensional cube with vertex set {0, 1} ⁿ in which two vertices are adjacent if they differ in exactly one coordinate. Suppose G is a subgraph of Q _n with average degree at least d. How long a path can we guarantee to find in G? Our aim in this paper is to show that G must contain an exponentially long path. In fact, we show that if G has minimum degree at least d then G must contain a path of length 2 ^d ? 1. Note that this bound is tight, as shown by a d-dimensional subcube of Q _n . We also obtain the slightly stronger result that G must contain a cycle of length at least 2 ^d .     

On Flat Flag-Transitive c.c *-Geometries

We study flat flag-transitive c.c ^*-geometries. We prove that, apart from one exception related to Sym(6), all these geometries are gluings in the meaning of [6]. They are obtained by gluing two copies of an affine space over GF(2). There are several ways of gluing two copies of the n-dimensional affine space over GF(2). In one way, which deserves to be called the canonical one, we get a geometry with automorphism group G = 2²ⁿ · L _n(2) and covered by the truncated Coxeter complex of type D ₂ ⁿ . The non-canonical ways give us geometries with smaller automorphism group (G ≤ 2²ⁿ · (2 ^n?1)n) and which seldom (never ?) can be obtained as quotients of truncated Coxeter complexes.     

The symplectic normal space of a cotangent-lifted action

For the cotangent bundle T^∗Q of a smooth Riemannian manifold acted upon by the lift of a smooth and proper action by isometries of a Lie group, we characterize the symplectic normal space at any point. We show that this space splits as the direct sum of the cotangent bundle of a linear space and a symplectic linear space coming from reduction of a coadjoint orbit. This characterization of the symplectic normal space can be expressed solely in terms of the group action on the base manifold and the coadjoint representation. Some relevant particular cases are explored.     

Laguerre expansion on the Heisenberg group and Fourier-Bessel transform on ℂ n

Given a principal value convolution on the Heisenberg group H _n = ? ⁿ × ?, we study the relation between its Laguerre expansion and the Fourier-Bessel expansion of its limit on ? ⁿ . We also calculate the Dirichlet kernel for the Laguerre expansion on the group H _n .     

A class of Sasakian 5-manifolds

We obtain some general results on Sasakian Lie algebras and prove as a consequence that a (2n + 1)-dimensional nilpotent Lie group admitting left-invariant Sasakian structures is isomorphic to the real Heisenberg group H _{2n + 1}. Furthermore, we classify Sasakian Lie algebras of dimension five and determine which of them carries a Sasakian α-Einstein structure. We show that a five-dimensional solvable Lie group with a left-invariant Sasakian structure and which admits a compact quotient by a discrete subgroup is isomorphic to either H ₅ or a semidirect product ? ? (H ₃ × ?). In particular, the compact quotient is an S ¹-bundle over a four-dimensional Kähler solvmanifold.     

Most primitive groups have messy invariants

Suppose G is a finite group of complex n × n matrices, and let R^G be the ring of invariants of G: i.e., those polynomials fixed by G. Many authors, from Klein to the present day, have described R^G by writing it as a direct sum Σ^δ_j=1 η_j$\text{C}$[θ₁ ,…, θ_n]. For example, if G is a unitary group generated by reflections, δ = 1. In this note we show that in general this approach is hopeless by proving that, for any ? > 0, the smallest possible δ is greater than | G |^n-1-? for almost all primitive groups. Since for any group we can choose δ ? | G |^n-1, this means that most primitive groups are about as bad as they can be. The upper bound on δ follows from Dade's theorem that the θ_i can be chosen to have degrees dividing | G |.     

On local newforms for unramified U(2, 1)

Let G be the unramified unitary group in three variables defined over a p-adic field F with p ≠ 2. In this paper, we investigate local newforms for irreducible admissible representations of G. We introduce a family of open compact subgroups {K _n } _n≥0 of G to define the local newforms for representations of G as the K _n -fixed vectors. We prove the existence of local newforms for generic representations and the multiplicity one property of the local newforms for admissible representations.     

Cauchy integral decomposition for harmonic forms

Let ann-dimensional differential form Ω be defined at points of aC ¹-smooth boundary π of a domainG ? ? ⁿ . Under what condition can Ω be represented as Ω = Ω₊ + Ω₊ + Ω_-, where Ω_± are forms insideG and outsideG, harmonic in the sense of Hodge? A necessary condition is that both restrictions Ω{inπ and *Ω{inπ be closed in the sense of currents. This condition, with an additional smoothness assumption, turns out to be sufficient as well. This is an analogue of the Cauchy integral decomposition of functions in the plane.     

Sequences of contractions on L1-spaces

Let T_n, n = 1,2,… be a sequence of linear contractions on the space where is a finite measure space. Let M be the subspace of L¹ for which T_ng → g weakly in L¹ for g?M. If T_n1 → 1 strongly, then T_nf → f strongly for all f in the closed vector sublattice in L¹ generated by M.This result can be applied to the determination of Korovkin sets and shadows in L¹. Given a set G ? L¹, its shadow S(G) is the set of all f?L¹ with the property that T_nf → f strongly for any sequence of contractions T_n, n = 1, 2,… which converges strongly to the identity on G; and G is said to be a Korovkin set if S(G) = L¹. For instance, if 1 ?G, then, where M is the linear hull of G and $\text{B}$_M is the sub-σ-algebra of $\text{B}$ generated by {x?X: g(x) > 0} for g?M. If the measure algebra is separable, has Korovkin sets consisting of two elements.     

Isoliertheit und Stabilität von Flächen konstanter mittlerer Krümmung

In the Sobolev space H^m(B,?³), B the open unit disc in ?², we consider the set M_n of all conformally parametrized surfaces of constant mean curvature H with exactly n simple interior branch points (and no others). We denote by M*_n the set of all xεM_n with the following properties:

1. in every branch point the geometrical condition K_G¦x_Z¦≡O holds (K_G is the Gauss curvature and x_z is the complex gradient of the surface x).
2. the corresponding boundary value problem Δh+×_z{²(2H²-K_G)h=O,^hδB=O, is uniquely solvable.

We prove then, that the manifold M*=UM*_n is open and dense in the set of all surfaces of constant mean curvature H and that all x εM*_n are isolated and stable solutions of the Plateau problem corresponding to their boundary curves. In addition, the submanifold M*_n contains exactly all surfaces x for which the space of Jacobi fields is transversal (with exception of the 3-dimensional space of conformai directions) to the tangent space T_xM_n.     

Joint eigenfunctions of invariant differential operators on the quaternion Heisenberg group

Let L be the sublaplacian on the quaternion Heisenberg group N and T the Dirac type operator with respect to central variables of N.In this article,we characterize the H c-valued joint eigenfunctions of L and T having eigenvalues from the quaternionic Heisenberg fan.     

Commutators of Calderón-Zygmund operators related to admissible functions on spaces of homogeneous type and applications to Schrödinger operators

Let X be an RD-space. In this paper, the authors establish the boundedness of the commutator Tbf = bTf-T(bf) on Lp , p∈(1,∞), where T is a Calderón-Zygmund operator related to the admissible function ρ and b∈BMOθ(X)BMO(X). Moreover, they prove that Tb is bounded from the Hardy space H1ρ(X) into the weak Lebesgue space L1weak(X). This can be used to deal with the Schrdinger operators and Schrdinger type operators on the Euclidean space Rn and the sub-Laplace Schrdinger operators on the stratified Lie group G.     

On the characteristic connection of gwistor space

We give a brief presentation of gwistor spaces, which is a new concept from G ₂ geometry. Then we compute the characteristic torsion T ^c of the gwistor space of an oriented Riemannian 4-manifold with constant sectional curvature k and deduce the condition under which T ^c is ?^c-parallel; this allows for the classification of the G ₂ structure with torsion and the characteristic holonomy according to known references. The case of an Einstein base manifold is envisaged.     

Extremals in some classes of Carnot groups

Let G be a Carnot group and D={e 1,e 2 } be a bracket generating left invariant distribution on G.In this paper,we obtain two main results.We first prove that there only exist normal minimizers in G if the type of D is (2,1,...,1) or (2,1,...,1,2).This immediately leads to the fact that there are only normal minimizers in the Goursat manifolds.As one corollary,we also obtain that there are only normal minimizers when dim G 5.We construct a class of Carnot groups such as that of type (2,1,...,1,2,n 0,...,n a) with n 0 1,n i 0,i=1,...,a,in which there exist strictly abnormal extremals.This implies that,for any given manifold of dimension n 6,we can find a class of n-dimensional Carnot groups having strictly abnormal minimizers.We conclude that the dimension n=5 is the border line for the existence and nonexistence of strictly abnormal extremals.Our main technique is based on the equations for the normal and abnormal extremals.     

Clifford Algebra-Valued Wavelet Transform on Multivector Fields

This paper presents a construction of the n = 2 (mod 4) Clifford algebra Cl _n,0-valued admissible wavelet transform using the admissible similitude group SIM(n), a subgroup of the affine group of \mathbbRⁿ{\mathbb{R}^{n}} . We express the admissibility condition in terms of the Cl _n,0 Clifford Fourier transform (CFT). We show that its fundamental properties such as inner product, norm relation, and inversion formula can be established whenever the Clifford admissible wavelet satisfies a particular admissibility condition. As an application we derive a Heisenberg type uncertainty principle for the Clifford algebra Cl _n,0-valued admissible wavelet transform. Finally, we provide some basic examples of these extended wavelets such as Clifford Morlet wavelets and Clifford Hermite wavelets.