La formule de Plancherel pour les groupes de Lie presque algébriques réels

We give a proof of the Plancherel formula for real almost algebraic groups in the philosophy of the orbit method, following the lines of the one given by M. Duflo and M. Vergne for simply connected semisimple Lie groups. Main ingredients are: (1) Harish-Chandra's descent method which, interpreting Plancherel formula as an equality of semi-invariant generalized functions, allows one to reduce it to a neighbourhood of zero in the Lie algebra of the centralizer of any elliptic element; (2) character formula for representations constructed by M. Duflo, we recently proved; (3) Poisson-Plancherel formula near elliptic elements s in good position, a generalization of the classical Poisson summation formula expressing the Fourier transform of the sum of a series of Harish-Chandra type elliptic orbital integrals in the Lie algebra centralizing s as a generalized function supported on a set of admissible regular forms in the dual of this Lie algebra.     

L~2-SPACES AND SUB-LAPLACIANS ON HOMO GENEOUS GROUPS

In this paper,we introduce a special class of nilpotent Lie groups defined by hermitianmaps,which includes all the groups of affine holomorphic automorphisims of Siegel domains oftype Ⅱ,in particular,the Heisenberg group.And we study harmonic analysis on these groupsas spectral theory of the associated Sub-Laplacian instead of the group representation theoryin usual way.     

Hua System and Pluriharmonicity for Symmetric Irreducible Siegel Domains of Type II

We consider here a generalization of the Hua system that was proved by Johnson and Korányi to characterize Poisson-Szegő integrals for Siegel domains of tube-type. We show that the situation is completely different when dealing with non-tube-type symmetric irreducible symmetric domains: then all functions that are annihilated by this second-order system and satisfy an H²-type integrability condition are pluriharmonic functions.     

Admissible convergence for the Poisson-Szegö integrals

We prove almost everywhere semirestricted admissible convergence of the Poisson-Szegö integrals ofL ^p functions (1 <p ≤ ∞) on the Bergman-Shilov boundary of a Siegel domain. In the case of symmetric domains our theorem can be deduced from the results by Peter Sjögren on admissible convergence to the boundary of Poisson integrals on symmetric spaces, although semirestricted admissible convergence means here a more general approach to the boundary then originally defined for symmetric spaces.     

3-Filiform Leibniz algebras of maximum length,whose naturally graded algebras are Lie algebras

In this article we present the classification of the 3-filiform Leibniz algebras of maximum length, whose associated naturally graded algebras are Lie algebras. Our main tools are a previous existence result by Cabezas and Pastor [J.M. Cabezas and E. Pastor, Naturally graded p-filiform Lie algebras in arbitrary finite dimension, J. Lie Theory 15 (2005), pp. 379–391] and the construction of appropriate homogeneous bases in the connected gradation considered. This is a continuation of the work done in Ref. [J.M. Cabezas, L.M. Camacho, and I.M. Rodríguez, On filiform and 2-filiform Leibniz algebras of maximum length, J. Lie Theory 18 (2008), pp. 335–350].     

Generalizations of the image conjecture and the Mathieu conjecture

We first propose a generalization of the image conjecture Zhao (submitted for publication) [31] for the commuting differential operators related with classical orthogonal polynomials. We then show that the non-trivial case of this generalized image conjecture is equivalent to a variation of the Mathieu conjecture Mathieu (1997) [21] from integrals of G-finite functions over reductive Lie groups G to integrals of polynomials over open subsets of Rⁿ with any positive measures. Via this equivalence, the generalized image conjecture can also be viewed as a natural variation of the Duistermaat and van der Kallen theorem Duistermaat and van der Kallen (1998) [14] on Laurent polynomials with no constant terms. To put all the conjectures above in a common setting, we introduce what we call the Mathieu subspaces of associative algebras. We also discuss some examples of Mathieu subspaces from other sources and derive some general results on this newly introduced notion.     

Projective Curvature and Integral Invariants

In this paper, an extension of all Lie group actions on R ²to coordinates defined by potentials is given. This provides a new solution to the equivalence problems of curves under the projective group and two of its subgroups. The potentials correspond to integrals of higher and higher-order producing an infinite number of independent integral invariants. Applications to computer vision are discussed.     

Lie Bialgebras of Generalized WITT Type,II

In an article by Michaelis, a class of infinite-dimensional Lie bialgebras containing the Virasoro algebra was presented. This type of Lie bialgebras was classified by Ng and Taft. In a recent article by Song and Su, Lie bialgebra structures on graded Lie algebras of generalized Witt type with finite dimensional homogeneous components were considered. In this article we consider Lie bialgebra structures on the graded Lie algebras of generalized Witt type with infinite dimensional homogeneous components. By proving that the first cohomology group H¹(𝒲, 𝒲 ? 𝒲) is trivial for any graded Lie algebras 𝒲 of generalized Witt type with infinite dimensional homogeneous components, we obtain that all such Lie bialgebras are triangular coboundary.     

On integrality of Eisenstein liftings

Letf be a holomorphic Siegel modular form of integral weightk for Sp_2r (Z). Forn≥r, let[f] _r ⁿ be the lift off to Sp_2n (Z) via the Klingen type Eisenstein series, which is defined under some conditions onk. We study an integrality property of the Fourier coefficients of[f] _r ⁿ . A common denominator for them is described in terms of a critical value of the standardL-function attached tof, some Bernoulli numbers, and a certain ideal depending only onf. The result specialized to the caser=0 coincides with the Siegel-Böcherer theorem on the Siegel type Eisenstein series.     

Sharp Norm Estimates for Weighted Bergman Projections in the Mixed Norm Spaces

In this paper, we show that the norm of the Bergman projection on L^p,q-spaces in the upper half-plane is comparable to csc(π/q). Then we extend this result to a more general class of domains, known as the homogeneous Siegel domains of type II.     

Hardy-Hausdorff spaces on the Heisenberg group

In this paper, by using the tent spaces on the Siegel upper half space, which are defined in terms of Choquet integrals with respect to Hausdorff capacity on the Heisenberg group, the Hardy-Hausdorff spaces on the Heisenberg group are introduced. Then, by applying the properties of the tent spaces on the Siegel upper half space and the Sobolev type spaces on the Heisenberg group, the atomic decomposition of the Hardy-Hausdorff spaces is obtained. Finally, we prove that the predual spaces of Q spaces on the Heisenberg group are the Hardy-Hausdorff spaces.     

Almost simple p-obstructions in odd characteristic lie type groups ∗

In this note we investigate the length of a strictly descending chain of maximum length in simple groups of Lie type in odd characteristic p. We desire information regarding which maximal subgroups can support (serve as the initial proper step in a longest chain) a longest chain in in G. Considered herein are those M possessing a simple Generalized Fitting subgroup. We show that if M supports a longest chain and has length exceeding that of every maximal parabolic subgroup of G, then either F*(M) is a Lie type group in the same characteristic as C, or Alt(5) = F^?(M) = M and G≌L2(p)     

Noncommutative Integrability,Moment Map and Geodesic Flows

The purpose of this paper is to discuss the relationship betweencommutative and noncommutative integrability of Hamiltonian systemsand to construct new examples of integrable geodesic flows onRiemannian manifolds. In particular, we prove that the geodesic flowof the bi-invariant metric on any bi-quotient of a compact Lie group isintegrable in the noncommutative sense by means of polynomial integrals, andtherefore, in the classical commutative sense by means ofC -smooth integrals.     

Characterization of radon integrals as linear functionals

The problem of characterization of integrals as linear functionals is considered in this paper. It has its origin in the well-known result of F. Riesz (1909) on integral representation of bounded linear functionals by Riemann?CStieltjes integrals on a segment and is directly connected with the famous theorem of J. Radon (1913) on integral representation of bounded linear functionals by Lebesgue integrals on a compact in ? ⁿ . After the works of J. Radon, M. Fréchet, and F. Hausdorff, the problem of characterization of integrals as linear functionals has been concretized as the problem of extension of Radon??s theorem from ? ⁿ to more general topological spaces with Radon measures. This problem turned out to be difficult, and its solution has a long and abundant history. Therefore, it may be naturally called the Riesz?CRadon?CFréchet problem of characterization of integrals. The important stages of its solution are connected with such eminent mathematicians as S. Banach (1937?C38), S. Saks (1937?C38), S. Kakutani (1941), P. Halmos (1950), E. Hewitt (1952), R. E. Edwards (1953), Yu. V. Prokhorov (1956), N. Bourbaki (1969), H. K¨onig (1995), V. K. Zakharov and A. V. Mikhalev (1997), et al. Essential ideas and technical tools were worked out by A. D. Alexandrov (1940?C43), M. N. Stone (1948?C49), D. H. Fremlin (1974), et al. The article is devoted to the modern stage of solving this problem connected with the works of the authors (1997?C2009). The solution of the problem is presented in the form of the parametric theorems on characterization of integrals. These theorems immediately imply characterization theorems of the above-mentioned authors.     

Sharp inequalities and geometric manifolds

Sharp constants for function-space inequalities over a manifold encode information about the geometric structure of the manifold. An important example is the Moser-Trudinger inequality where limiting Sobolev behavior for critical exponents provides significant understanding of geometric analysis for conformal deformation on a Riemannian manifold [5, 6]. From the overall perspective of the conformal group acting on the classical spaces, it is natural to consider the extension of these methods and questions in the context of SL(2, R), the Heisenberg group, and other Lie groups. Among the principal tools used in this analysis are the linear and multilinear operators mapping L^p(M) to L^q(M) defined by the Stein-Weiss integral kernels which extend the Hardy-Littlewood-Sobolev fractional integrals\(\mathcal{H}^1 (\mathbb{R}^d )\) conformal geometry, and the notion of equimeasurable geodesic radial decreasing rearrangement. To illustrate these ideas, four model problems will be examined here: (1) logarithmic Sobolev inequality and the uncertainty principle, (2) SL(2,R) and axial symmetry in fluid dynamics, (3) Stein-Weiss integrals on the Heisenberg group, and (4) Morpurgo’s work on zeta functions and trace inequalities of conformally invariant operators.     

Ramanujan's Master Theorem for Hermitian Symmetric Spaces

In this paper we generalize Ramanujan's Master Theorem to the context of a Siegel domain of Type II, which is the Hermitian symmetric space of a real non-compact semisimple Lie group of finite center. That is, the spherical transform of a spherical series can be expressed in terms of the coefficients of this series.     

On Cayley-Transform Methods for the Discretization of Lie-Group Equations

In this paper we develop in a systematic manner the theory of time-stepping methods based on the Cayley transform. Such methods can be applied to discretize differential equations that evolve in some Lie groups, in particular in the orthogonal group and the symplectic group. Unlike many other Lie-group solvers, they do not require the evaluation of matrix exponentials. Similarly to the theory of Magnus expansions in [13], we identify terms in a Cayley expansion with rooted trees, which can be constructed recursively. Each such term is an integral over a polytope but all such integrals can be evaluated to high order by using special quadrature formulas similar to the construction in [13]. Truncated Cayley expansions (with exact integrals) need not be time-symmetric, hence the method does not display the usual advantages associated with time symmetry, e.g., even order of approximation. However, time symmetry (with its attendant benefits) is attained when exact integrals are replaced by certain quadrature formulas. March 7, 2000. Final version received: August 10, 2000. Online publication: January 2, 2001.