Aluthge iteration in semisimple Lie group

We extend, in the context of connected noncompact semisimple Lie group, two results of Antezana, Massey, and Stojanoff: Given 0<λ<1, (a) the limit points of the sequence are normal, and (b) , where ‖X‖ is the spectral norm and r(X) is the spectral radius of X∈C_n×n and Δ_λ(X) is the λ-Aluthge transform of X.     

Foliations invariant under Lie group transverse actions

In this paper we study (smooth and holomorphic) foliations which are invariant under transverse actions of Lie groups. Authors’ address: Alexandre Behague and Bruno Scárdua, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21945-970 Rio de Janeiro, RJ, Brazil     

Global behavior of the heat kernel associated with certain sub-Laplacians on semisimple Lie groups

We prove global sharp estimates for the heat kernel related to certain sub-Laplacians on a real semisimple Lie group, from which we deduce an estimate for the corresponding Green function.     

Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces

In this paper we give global characterisations of Gevrey–Roumieu and Gevrey–Beurling spaces of ultradifferentiable functions on compact Lie groups in terms of the representation theory of the group and the spectrum of the Laplace–Beltrami operator. Furthermore, we characterise their duals, the spaces of corresponding ultradistributions. For the latter, the proof is based on first obtaining the characterisation of their α-duals in the sense of Köthe and the theory of sequence spaces. We also give the corresponding characterisations on compact homogeneous spaces.     

Riesz transforms on a solvable Lie group of polynomial growth

We study Riesz transforms associated with a sublaplacian H on a solvable Lie group G, where G has polynomial volume growth. It is known that the standard second order Riesz transforms corresponding to H are generally unbounded in L^p(G). In this paper, we establish boundedness in L^p for modified second order Riesz transforms, which are defined using derivatives on a nilpotent group G_N associated with G. Our method utilizes a new algebraic approach which associates a distinguished choice of Cartan subalgebra with the sublaplacian H. We also obtain estimates for higher derivatives of the heat kernel of H, and give a new proof (without the use of homogenization theory) of the boundedness of first order Riesz transforms. Our results can be generalized to an arbitrary (possibly non-solvable) Lie group of polynomial growth.     

Hardy-Sobolev type inequalities on the H-type group

Motivated by the idea of Badiale and Tarantello who have found Hardy-Sobolev inequalities on Rⁿ, a class of Hardy-Sobolev type inequalities on H-type groups is proved via a new representation formula for functions. Extremal functions realizing equality in the inequalities are discussed by refined Concentration-Compactness principles. Finally, some sharp constants for Hardy type inequalities are given. The project supported by National Natural Science Foundation of China, Grant No. 10371099.     

Subelliptic operators on Lie groups: Variable coefficients

Let G be a Lie group with Lie algebra g and a _i,...,a _d and algebraic basic of g. Futher, if A _i=dL(a_i) are the corresponding generators of left translations by G on one of the usual function spaces over G, let% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamXvP5wqonvsaeHbfv3ySLgzaGqbciab-Heaijaab2dadaaeqbqa% aiaadogadaWgaaWcbaqedmvETj2BSbacgmGae4xSdegabeaakiaadg% eadaahaaWcbeqaaiab+f7aHbaaaeaacqGFXoqycaGG6aGaaiiFaiab% +f7aHjaacYhatuuDJXwAK1uy0HMmaeXbfv3ySLgzG0uy0HgiuD3BaG% Wbbiab9rMiekaaikdaaeqaniabggHiLdaaaa!5EC1!\[H{\rm{ = }}\sum\limits_{\alpha :|\alpha | \le 2} {c_\alpha A^\alpha } \] be a second-order differential operator with real bounded coefficients c . The operator is defined to be subelliptic if% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiGacMgacaGGUbGaaiOzamXvP5wqonvsaeHbfv3ySLgzaGqbaKaz% aasacqWF7bWEcqWFTaqlkmaaqafabaGaam4yamaaBaaaleaarmWu51% MyVXgaiyWacqGFXoqyaeqaaaqaaiab+f7aHjaacQdacaGG8bGae4xS% deMaaiiFaiabg2da9iaaikdaaeqaniabggHiLdGccqWFOaakiuGacq% qFNbWzcqWFPaqkcqaH+oaEdaahaaWcbeqaamaaBaaameaacqGFXoqy% aeqaaaaakiaacUdacqqFNbWzcqGHiiIZcqqFhbWrcqqFSaalcqqFGa% aicqaH+oaEcqGHiiIZrqqtubsr4rNCHbachaGaeWxhHe6aaWbaaSqa% beaacqqFKbazcqqFNaWjcqaFaC-jaaGccaGGSaGaaiiFaiabe67a4j% aacYhacqGH9aqpjqgaGeGae8xFa0NccqGH+aGpcaaIWaGaaiOlaaaa% !7884!\[\inf \{ - \sum\limits_{\alpha :|\alpha | = 2} {c_\alpha } (g)\xi ^{_\alpha } ;g \in G, \xi \in ^{d'} ,|\xi | = \} > 0.\]We prove that if the principal coefficients {c ; ||=2} of the subelliptic operator are once left differentiable in the directions a ₁,...,a _d with bounded derivatives, then the operator has a family of semigroup generator extensions on the L _p-spaces with respect to left Haar measure dg, or right Haar measure d, and the corresponding semigroups S are given by a positive integral kernel,% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamXvP5wqonvsaeHbfv3ySLgzaGqbaiab-HcaOGqbciab+nfatnaa% BaaaleaacaWG0baabeaaruqqYLwySbacgiGccaqFgpGae8xkaKIae8% hkaGIae43zaCMae8xkaKIae8xpa0Zaa8qeaeaacaqGKbaaleaacqGF% hbWraeqaniabgUIiYdGcceWGObGbaKaacaWGlbWaaSbaaSqaaiaads% haaeqaaOGae8hkaGIae43zaCMae43oaSJae4hAaGMae8xkaKIaa0NX% diab-HcaOiab+HgaOjab-LcaPiab-5caUaaa!5DFA!\[(S_t \phi )(g) = \int_G {\rm{d}} \hat hK_t (g;h)\phi (h).\]The semigroups are holomorphic and the kernel satisfies Gaussian upper bounds. If in addition the coefficients with ||=2 are three times differentiable and those with ||=1 are once differentiable, then the kernel also satisfies Gaussian lower bounds.Some original features of this article are the use of the following: a priori inequalities on L in Section 3, fractional operator expansions for resolvent estimates in Section 4, a parametrix method based on reduction to constant coefficient operators on the Lie group rather than the usual Euclidean space in Section 5, approximation theory of semigroups in Section 11 and time dependent perturbation theory to treat the lower order terms of H in Sections 11 and 12.     

Riordan group involutions

We study involutions in the Riordan group, especially those with combinatorial meaning. We give a new determinantal criterion for a matrix to be a Riordan involution and examine several classes of examples. A complete characterization of involutions in the Appell subgroup is developed. In another direction we find several examples that generalize the RNA matrix but are of independent interest.     

Mock tridiagonal systems

We introduce the notion of a mock tridiagonal system. This is a generalization of a tridiagonal system in which the irreducibility assumption is replaced by a certain nonvanishing condition. We show how mock tridiagonal systems can be used to construct tridiagonal systems that meet certain specifications. This paper is part of our ongoing project to classify the tridiagonal systems up to isomorphism.     

A classification of sharp tridiagonal pairs

Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A:V→V and A^∗:V→V that satisfy the following conditions: (i) each of A,A^∗ is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A^∗V_i⊆V_i-1+V_i+V_i+1 for 0?i?d, where V_-1=0 and V_d+1=0; (iii) there exists an ordering of the eigenspaces of A^∗ such that for 0?i?δ, where and ; (iv) there is no subspace W of V such that AW⊆W, A^∗W⊆W, W≠0, W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0?i?d the dimensions of coincide. The pair A,A^∗ is called sharp whenever . It is known that if F is algebraically closed then A,A^∗ is sharp. In this paper we classify up to isomorphism the sharp tridiagonal pairs. As a corollary, we classify up to isomorphism the tridiagonal pairs over an algebraically closed field. We obtain these classifications by proving the μ-conjecture.     

Nonlinear and multiplicative inequalities in Sobolev spaces associated with Lie algebras

This paper is devoted to multiplicative inequalities in some generalized Sobolev spaces associated with Lie algebras. These Lie algebras are generated by the differential operator of variable coefficients or by pseudo-differential operators having non-regular symbols. Under geometrical assumptions we show that the norms of two suitable classes of generalized Sobolev spaces are equivalent. This leads to the proof that the composition operator u→|u|^p$u\to {\left|u\right|}^p$ acts on such spaces.     

Lie algebras and recurrence relations IV: Representations of the Euclidean group and Bessel functions

Representations of a complex form of the Lie algebra of the Euclidean group of the plane are found. Bases for the corresponding Hilbert spaces are given in terms of Lommel polynomials and Bessel functions. An element of the Lie algebra is interpreted as a quantum random variable leading to the Lommel distributions. The tails of these distributions and limit theorems are studied. Applications to electronic music are noted, as well as connection with quantum mechanics on a one-dimensional lattice.     

Fix-mahonian calculus III; a quadruple distribution

A four-variable distribution on permutations is derived, with two dual combinatorial interpretations. The first one includes the number of fixed points “fix”, the second the so-called “pix” statistic. This shows that the duality between derangements and desarrangements can be extended to the case of multivariable statistics. Several specializations are obtained, including the joint distribution of (des, exc), where “des” and “exc” stand for the number of descents and excedances, respectively. Authors’ addresses: Dominique Foata, Institut Lothaire, 1 rue Murner, F-67000 Strasbourg, France; Guo-Niu Han, Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, P.R. China; I.R.M.A. UMR 7501, Université Louis Pasteur et CNRS, 7, rue René-Descartes, F-67084 Strasbourg, France     

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.     

Invariant Orders on Lie Groups and Coverings of Ordered Homogeneous Spaces

 In this paper we give a characterization of those reductive or solvable connected, not necessarily simply connected, Lie groups which permit a non-degenerate group order. A non-degenerate group ordering on G always defines a pointed generating invariant convex cone W in the Lie algebra of G, but not every such cone arises in this way. The cones that do are called global. To decide whether a given cone is global or not is a difficult problem which for simply connected groups and invariant cones has completely been solved by Gichev. (Received 22 October 1999; in revised form 3 March 2000)     

Hölder continuous homomorphisms between infinite-dimensional Lie groups are smooth

Let f:G→H be a homomorphism between smooth Lie groups modelled on Mackey complete, locally convex real topological vector spaces. We show that if f is Hölder continuous at 1, then f is smooth.     

Minimal faithful representations of reductive Lie algebras

Let G and H be Lie groups with Lie algebras and . Let G be connected. We prove that a Lie algebra homomorphism is exact if and only if it is completely positive. The main resource is a corresponding theorem about representations on Hilbert spaces. This article summarizes the main results of [1]. Received: 6 December 2005