Cho-Ho Chu (Queen Mary) : Abstract. We introduce a notion of harmonic functions on infinite dimensional symmetric spaces, generalizing the usual one in finite dimensions, and discuss the consequences such as Liouville thoerem and Poisson representation. Jacques Faraut (Paris 6) :Abstract. The multivariate Meixner-Pollaczek polynomials are introduced as spherical Fourier transforms of multivariate Laguerre functions, which are orthogonal polynomials on a symmetric cone. These polynomials are orthogonal with respect to the Plancherel measure, are solutions of a difference equation, and posess a remarkable generating function. These properties are obtained from the harmonic analysis on a Hermitian symmetric space of tube type. These polynomials have been studied by Davidson, Olafsson and Zhang. We will present a different method. Simon Gindikin (Rutgers) :Abstract. There is a very systematic parallel between theories of spherical functions on compact and noncompact Riemannian symmetric manifolds. In noncompact case the essential part of the theory is connected with integral formulas: Harish-Chandra's representation of zonal functions, Poisson's integral, the horospherical transform, c-function etc. Does something corresponded to these constructions in compact case? Clerc remarked that it is possible to find the analogue of Harish-Chandra representations, using the complex extension. It turns out that it is a general phenomenon: analogues of all these integral formulas exist as parts of analysis on complex symmetric Stein manifolds (complexifications of compact ones). Sigurdur Helgason (MIT):Abstract. On a Euclidean space the inversion formula for the Radon transform is a decomposition of a function into plane waves. Then a constant coefficient differential equation Du = f has a quick solution in terms of the Radon transform of f. On a symmetric space X = G/K the inversion formula for the Radon transform should give a decomposition of a function into horocycle plane waves. Unfortunately, the formula does not amount to this. Nevertheless, with a suitable manipulation the inversion formula can be used to write down a formula for a solution of Du = f. The method works also for the Cauchy problem and for the multitemporal wave equation on X. Hideyuki Ishi (Nagoya) :Abstract. T-algebra is a non-associative formal matrix algebra with vector entries, introduced by Vinberg in order to study a general homogeneous cone. The Euclidean Jordan algebra associated to a symmetric cone is naturally obtained from the T-algebra corresponding to the cone. In this talk, we see that every T-algebra is realized as a vector space of ordinaly real matrices, where the product is a composition of matrix muliplication with a projection. Then the triangular action of a solvable group on the homogeneous cone is easily described. Soji Kaneyuki (Sophia) :Abstract. We associate a certain G-structure with a causal structure on a manifold, and show the coincidence of the automorphism groups of both structures. Applying this to the Shilov boundaries of tube type irreducible bounded symmetric domains or Cayley type symmetric spaces, we determine the maximal extensions of local causal automorphisms and the (global) causal automorphism groups. Similar results are valid also for the generalized conformal structures on symmetric R-spaces and on parahermitian symmetric spaces. Michael K. Kinyon (Denver) :Abstract. Leibniz algebras are non-anticommutative generalizations of Lie algebras introduced by Loday. The coquecigrue problem for Leibniz algebras is to find the (possibly mythical) Lie grouplike object whose tangent algebra structure is a given Leibniz algebra. In this talk, I will first describe the coquecigrue for split Leibniz algebra (direct products of a Lie subalgebra and a subspace on which the Lie subalgebra acts). This is the notion of a Lie digroup, a special type of bisemigroup. For arbitrary Leibniz algebras, Didry's dissertation constructs another object analogous to a formal group. This is also a reasonable candidate for a coquecigrue, but its properties are still not well understood. I will discuss the relationship between this two notions of coquecigrue. For the remainder of the talk, I will discuss the Jordan analogs of Leibniz algebras, which are quasi-Jordan algebras, as introduced by Velasquez and Felipe. I will describe some symmetric space-like structures associated to quasi-Jordan algebras, such as on the set of invertible elements. Toshiyuki Kobayashi (Harvard) :Abstract.Visible actions are holomorphic actions on complex manifolds having certain nice properties. Likewise, polar actions are defined for isometric actions on Riemannian manifolds, and coistropic action are for symplectic manifolds. In this talk, I plan to discuss some basic properties of these actions on Kahler manifolds in relation with multiplicity-free actions. Adam Korányi (CUNY) :Abstract. A bounded operator T on a Hilbert space is called homogeneous if its spectrum is contained in the closed unit disc D and if g(T) is unitarily equivalent to T for every holomorphic automorphism g of D. This notion is of interest in operator theory; it is also related to the "inductive algebras" of Steger-Vemuri. In joint work with G. Misra a large new class of homogeneous operators has been constructed with the aid of holomorphic Hermitian vector bundles that are homogeneous under the universal covering group of SL(2,R). This class can be described fairly explicitly and can be shown to contain all the homogeneous operators that belong to the Cowen-Douglas class. Some of the results extend to n-tuples of commutative operators associated to bounded symmetric domains. Bernhard Krötz (Max-Planck) :Abstract. In this talk I will report on a new short proof of the Casselman-Wallach smooth globalization theorem (joint with Joseph Bernstein) and about novel approaches to the analytic globaliaztion theorem of Kashiwara-Schmid (work in progress with Henrik Schlichtkrull). Stéphane Mérigon (Nancy 1) :Abstract. We construct a homotopy invariant index for paths in the set of invertible tripotents of a JB*-triple that satisfy a Fredholm type condition with respect to a fixed invertible tripotent. First we review the definitions of the Maslov index for paths in the Shilov boundary of a (finite dimensional) bounded symmetric domain given in Clerc and Koufany, then, we give an interpretation of the Maslov index as a spectral flow. This interpretation enable us to construct the Maslov index for paths in the infinite dimensional setting. Karl-Hermann Neeb (Darmstadt) :Abstract. In this talk we describe a systematic approach to unitary representations
of infinite-dimensional Lie groups in terms of boundedness conditions
on spectra in the derived representations. For any unitary
representation π : G → U(H)
which is smooth in the sense that it has a dense
set of smooth vectors (which is automatic for finite-dimensional groups),
one can associate its momentum set l Abstract. We will survey recent work on universal central extensions of Lie algebras related to generalizations of affine Lie algebras (root-graded Lie algebras, multi-loop algebras). Takaaki Nomura (Kyushu):Abstract. Let Ω be a regular homogeneous open convex cone in a finite dimensional real vector space V. Let H ⊂ GL(Ω) (linear automorphism group of Ω) be a split solvable Lie group acting simply transitively on Ω. In 2001 Ishi gave an algorithm to extract basic H-relative invariant polynomials on V. We first show that these basic relative invariants are just the irreducible factors of the determinant of the right multiplication operators in the complexified clan associated to Ω. Moreover, some examples of non-selfdual homogeneous open convex cones with interesting (properties of) basic relative invariants are presented. Joint work with Hideyuki Ishi. Gestur Ólafsson (Baton Rouge) : Abstract. The Fourier transform F of a function f on a compact symmetric space M=U/K are given by integration of f against matrix coefficients of irreducible representations of U. The coefficients depend on a spectral parameter µ, which determines the representation, and they can be represented by elements F(µ) in a common Hilbert space H. We discuss a resent work with H. Schlichtkrull which describes the size of the support of f by means of the exponential type of a holomorphic H-valued extension of F, provided f is K-finite and of sufficiently small support. Hubert Rubenthaler (Strasbourg 1) : Abstract. If Q is a non degenerate quadratic form on C Abstract. The classical method of constructing group representations on spaces of global sections of holomorphic line bundles over homogeneous manifolds G/P can be generalized to root graded Banach Lie groups. Important examples of such Lie groups are those whose Lie algebras are of the form Abstract. I will summarize the main results obtained with M. Slupinski on this topic over the past several years. The relationship with certain Jordan algebras as appeared in Clerc's work will be elucidated. Many open problems will be identified, as well as areas for future investigation. André Unterberger (Reims) :Abstract. The analysis of one dimensional pseudodiferential operators by means of their diagonal matrix elements against families of coherent states of an arithmetic nature leads to new interpretations of the zeta function, if not to a new approach towards one of your favorite conjectures. L-functions (of interest to number theorists) show themselves when the Weyl calculus is replaced by the appropriate (horocyclic) symbolic calculus associated to the discrete series of SL(2;R). Harald Upmeier (Marburg):Abstract. For any bounded symmetric domain we construct a holomorphic Hilbert space bundle associated with a discrete Wallach parameter, which generalizes the classical Fock spaces, and find a projectively flat connection on this bundle which gives rise to an invariant construction of the generalized metaplectic representation of the
holomorphic automorphism group and a representation theoretic interpretation of the Maslov index. Compared to other approaches towards these problems, we work in the bounded Harish-Chandra realization instead of the unbounded model, and do not use an embedding of the automorphism group into a higher-dimensional symplectic group. Joseph A. Wolf (Berkeley) : Abstract. Riemannian symmetric spaces show up in many branches of mathematics. Commutative spaces are a generalization of some interest in geometry, analysis and number theory. The finite dimensional ones
are now classified, and the simplest infinite dimensional ones are direct limits G/K = lim G Abstract. Let X=H/L be a real bounded symmetric domain realied as a real form in an Hermitian symmetric domain D=G/K. The intersection S of the Shilov boundary of D with X defines a distinguished subset of the topological boundary of X and is invariant under H and can also be realized as S=H/P for certain parabolic subgroup P of H. We study the induced shperical representation Ind |
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