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> Khalid Koufany

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Speakers / Titles / Abstracts

Cho-Ho Chu (Queen Mary) : Harmonic functions on symmetric spaces

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) : Harmonic analysis on Hermitian symmetric spaces and multivariate Meixner-Pollaczek polynomials

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) : Analysis on complex symmetric manifolds

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): Solving Differential Equations by Radon Transforms

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) : Matrix T-algebras and Jordan algebras

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) : Liouville-type theorems for causal structures on symmetric spaces

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) : Leibniz algebras, quasi-Jordan algebras and their associated Lie group-like and symmetric space-like objets

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) : Visible action, polar action and coisotropic action

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) : Homogeneous vector bundles in operator theory

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) : Globalizations of Harish-Chandra moduls

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) : Finite and infinite dimentional Maslov index

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) : Semi-bounded unitary representations of infinite dimensional Lie groups

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 lπ which is a convex weak-*-closed subset of the dual space L(G)', invariant under the coadjoint action. It encodes the information on the spectral bounds of the derived representation. We call π bounded if lπ is equicontinuous and semi-bounded if lπ has a weaker property which we call semi-equicontinuity and which implies in particular that the convex cone B(lπ) of all elements in L(G) for which the spectrum of idπ(x) is bounded from below has interior points, which leads to an invariant open convex cone in L(G). For finite-dimensional groups, the semi-bounded representations are precisely the unitary highest weight representations and only groups with compact Lie algebras have bounded representations. For infinite-dimensional groups, the picture is much more colorful. There are many interesting bounded representations, in particular all those coming from representations of C*-algebras, and most of the unitary representations appearing in physics are semibounded. We present old and new results connecting convexity properties of the coadjoint representation and unitary representations.

Erhard Neher (Ottawa) : Central extention of Lie algebras

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): Homogeneous convex cones and basic relative invariants

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) : Local Paley-Wiener theorem for compact symmetric spaces

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) : Invariant differential operators and an infinite dimensional Howe type correspondence

Abstract. If Q is a non degenerate quadratic form on Cn, it is well known that the differential operators X = Q(x), Y = Q(∂), and H = E + n/2 , where E is the Euler operator, generate a Lie algebra isomorphic to sl2. Therefore the associative algebra they generate is a quotient of the universal enveloping algebra U(sl2). This fact is in some sense the foundation of the metaplectic representation. In this talk we will discuss the case where Q(x) is replaced by Δ0(x), where Δ0(x) is the ”determinant” function of a simple Jordan algebra V over C, or equivalently where Δ0 is the relative invariant of a prehomogeneous vector space of commutative parabolic type (g,V). We will show several structure results for the associative algebra generated by X =Δ0(x), Y = Δ0(∂). In particular we will show that the associative algebra generated by X, Y and D(V )G (G is the structure group) is a generalized Smith algebra over a commutative subring A. The Smith algebras (over C) were introduced by P. Smith as ”natural” generalizations of U(sl2). Let L be the Lie algebra of differential operators generated by X, Y and the image of gl(V ), let A be the Lie sub-algebra which is generated by X and Y , and let B be the image of [g, g] (where g is the structure algebra). Then A and B are commuting subalgebras of L. The restriction of the natural representation of L on polynomials on V to A×B gives rise to a correspendence between some highest weight modules of A and the "harmonic" representation of B, which generalizes the Howe correspondence between highest weight modules of sl2 and ordinary spherical harmonics. The Lie algebras L and A are infinitedimensional except if Δ0 is a quadratic form, and in this case L is the usual symplectic algebra, A = sl2 and the corresponding representation of L is the infinitesimal metaplectic representation.

Henrik Seppänen (Darmstadt) : Borel-Weil theory for root graded Lie groups

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 g⊗A, where g is a simple complex Lie algebra and A is a commutative Banach algebra. I will present some recent work on the particular case when A is a finite dimensional local algebra. It turns out that the geometry is related to the classical case in an interesting way. This geometric structure can be used to characterize those line bundles which admit global holomorphic sections.The talk is based on a joint project with Karl-Hermann Neeb.

Robert Stanton (Ohio): Heisenberg graded Lie algebras: a survey

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) : Quantization and arithmetic

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): Holomorphic Hilbert bundles and flat connections

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.
This is joint work with Karina Bischoff, University of Marburg

Joseph A. Wolf (Berkeley) : Infinite dimentional Gelfand pairs

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 Gn/Kn where the Gn/Kn are finite dimensional commutative spaces. Here "commutative space" means that the action of G on a suitable Hilbert space L2(G/K) is multiplicity free, and we study several cases where that multiplicity free property holds. The strongest results are for cases where (i) the Gn/Kn are compact riemannian symmetric spaces and (ii) the Gn are semidirect products Nn×Kn with Nn nilpotent, In case (ii) the Nn are commutative or 2-step nilpotent (not so different from a Heisenberg group); then we usually can construct Gn-equivariant isometric maps ζn : L2(Gn/Kn) → L2(Gn+1/Kn+1) and prove that the left regular representation of G on the Hilbert space L2(G/K) : = lim{L2(Gn/Kn),ζn} is a multiplicity free direct integral of irreducible unitary representations. In all cases the isometric injections ζn : L2(Gn/Kn) → L2(Gn+1/Kn+1), which define L2(G/K), make use of Frobenius-Schur orthogonality.

Genkai Zhang (Göteborg) : Realization of unitary spherical representations in singular holomorphic representations

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 IndHP(λ) of H and their realizations as discrete summand in the branching of singular holomorphic representations of G. We find formulas for the spherical functions in terms of the Macdonald's 2F1 hypergeometric function. This generalizes the earlier result of Faraut-Koranyi for Hermitian symmetric spaces D. We consider a class of H-invariant integral intertwining operators from the representations IndHP(λ) on L2(S) to the holomorphic representations of G on D restricted to H. We construct a new class of complementary series for the groups H =SO(n,m), SU(n;m) (with n-m>2) and Sp(n,m) (with n-m >1) and find their realization in the branching rule of holomorphic representations of G=SU(n,m), SU(n,m)×SU(n,m) and SU(2n,2m) respectively.