Lorentz Center

International center for scientific workshops

International center for scientific workshops

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## Analysis, Geometry and Group Representations for Homogeneous Spaces |

“Holomorphic discrete series representations on Siegel-Jacobi domains” We present holomorphic discrete series representations of the Jacobi group on Siegel-Jacobi domains. Explicit polynomial orthonormal bases of the Fock spaces based on the Siegel-Jacobi disk are obtained. The scalar holomorphic discrete series of the Jacobi group for the Siegel-Jacobi disk is constructed and polynomial orthonormal bases of the representation spaces are given. By S.Berceanu, A. Gheorghe ---
“Real structures on wonderful varieties” This is a work in progress jointly with Dmitry Akhiezer. We consider a nice class of complex algebraic varieties (endowed with an action of a reductive complex algebraic group G) generalizing that of De Concini-Procesi compactifications of symmetric spaces, the so-called wonderful varieties. We investigate the existence of real structures (invariant under a split real form of G) on these varieties and their associated real parts. For instance, consider the variety of complex quadrics X in the complex projective n-space and the complex conjugate for matrices as real structure. Then, it is well-known that the real part of the set of non-degenerate quadrics is the set of real quadrics and that its connected components are indexed by the signatures (p,q) with p+q=n+1. Further, such a parametrization can be generalized to the whole X. My talk shall deal with results generalizing this example. ---
“Constant term of Eisenstein integrals of reductive p-adic symmetric spaces” Click here for abstract ---
“A geometric proof of the Karpelevich-Mostow theorem” I will explain a short and geometric proof of Karpelevich-Mostow's theorem which assert that a semisimple subgroup of the isometry group of a symmetric space of noncompact type has a totally geodesic orbit. As an application we will use Karpelevich-Mostow's theorem to find all finite dimensional irreducible representations of connected Lie groups preserving an inner product of signature (1,n) or (2,n). ---
“On the polynomial conjectures” Click here for abstract ---
I will discuss a direct limit topology on split Kac-Moody groups over non-discrete locally compact \sigma-compact topological fields F which is induced by the Lie group topology on the fundamental SL_2(F) subgroups. This topology turns the corresponding twin building into a topological building in the sense of Hartnick. The geometry of orbits under Borel subgroups of the Kac-Moody group has properties very similar to the situation of Lie groups; in particular, there exists a unique open-dense big cell. Moreover, it is possible to prove Mostow-type rigidity results for (S-)arithmetic subgroups. The results presented in my talk have been obtained in collaboration with Andreas Mars. ---
“On Shahidi's tempered L-function conjecture” The aim of the talk is to present Shahidi's tempered L-function conjecture and its general proof obtained recently in common with E. Opdam. Special cases (covering all but one case) had been known before thanks to the work of Casselman-Shahidi, Muic-Shahidi, H. Kim, H. Kim-W. Kim and W. Kim. The proof uses my previous characterisation of the infinitesimal character of a discrete series representation of a p-adic group. ---
"Wigner and Patterson-Sullivan distributions for locally symmetric spaces" Anantharam and Zelditch observed a remarkable connection between Wigner and Patterson-Sullivan distributions on compact hyperbolic surfaces. These are distributions associated with the eigenvalues of the Laplace-Beltrami operators and satisfy invariance properties under the geodesic flow. A key tool to establish this connection is a specific pseudodifferential calculus adapted to the symmetries of the situation. Together with M. Schroeder we reformulated these results in terms of group and representation theory and generalized them to rank 1 locally symmetric spaces. In this talk I will sketch how this goes and explain the connection with Selberg zeta functions. Further I will explain the problems one has to overcome in order to also treat higher rank cases. ---
"The Berezin transforms associated to homogeneous Kaehler metrics on a homogeneous bounded domain" We consider a family of parametrized homogeneous Kaehler metrics on a homogeneous bounded domain and the associated Berezin transforms. We compute the Berezin transform of relatively invariant functions on the domain with a convergence condition, and present several applications of the relevant integral formulas. ---
"Universal
central extensions of Gauge groups" We prove that a
certain central extension of the gauge group (the group of vertical symmetries
of a smooth principal fibre bundle over a compact
base) is universal. This is joint work
with Christoph Wockel." ---
“ Almost everywhere convergence of Bochner-Riesz means in Jacobi Analysis” Abstract: We derive the mapping properties of the maximal operator for Bochner-Riesz means associated with the Jacobi transform, including sharp Lorentz space estimates at the critical index of integrability. Of special interest is the Jacobi-analogue of the Euclidean disc multiplier, since its local structure is closely connected with the Carleson--Hunt Theorem for one-dimensional Fourier series. Its global behavior is highly non-Euclidean, however, due to the Kunze-Stein phenomenon.The resulting statement on almost everywhere convergence of the Bochner-Riesz means was previously known for rank one symmetric spaces of the noncompact type. Some of the results extend at once to noncompact Chebli-Trimeche hypergroups. ---
"Modular properties for quantum groups" For locally compact quantum groups analogues of the Duflo-Moore operators have been introduced in 2003 by Pieter Desmedt. We discuss how these operators can be used in order to describe modular properties of matrix elements of corepresentations of quantum groups. Several examples will be discussed, including the the quantum group analogue of SU(1,1). (joint work with Marijn Caspers) ---
“Braid group actions on quantum symmetric pair coideal subalgebras” It was noted recently by Molev and Ragoucy, and idependently by Chekhov, that the nonstandard quantum enveloping algebra of so(N) allows an action of the Artin braid group. We interpret and generalize this action within the theory of quantum symmetric spaces. ---
“Nonnegatively curved homogeneous metrics” Abstract: We study invariant Riemannian metrics on compact homogeneous spaces G/H where an intermediate subgroup K between G and H exists. We determine the triples (G,K,H) having the property that scaling up the fibers of the Riemannian submersion defined by K retains the nonnegative sectional curvature of a normal homogeneous metric. We obtain a full list of such triples (G,K,H) in particular for the case where H is a subgroup of full rank. Joint work with Megan Kerr. ---
“Invariant measures on homogeneous spaces, with applications to function spaces and lattice counting” Let G be a real reductive group and G/H a unimodular homogeneous G space with a closed connected subgroup H. We establish estimates for the invariant measure on G/H. Using these, we prove that all smooth vectors in the Banach representation L^p(G/H) of G are functions that vanish at infinity if and only if G/H is of reductive type. An application to lattice counting on G/H is presented. (joint with Eitan Sayag and Henrik Schlichtkrull) ---
“Radon transformation on reductive symmetric spaces: support theorems” Click here for abstract ---
“Connection formulas of the solutions of Fuchsian differential equations and intersection numbers of twisted cycles” ---
"Remarks on elliptic Schur functions" In this talk I will consider a class of elliptic Schur functions in the framework of Macdonald's ninth variation of Schur functions. This class of elliptic Schur functions share various properties with ordinary Schur functions. I will discuss in particular the hook-length formula for the elliptic Schur functions and related summation formulas. (Kobe University, Japan) ---
“Positivity of an alpha determinant” The alpha determinant is one of one-parameter interpolations of the determinant and the permanent. This interpolation is different from the quantum determinant, and has some role in statistics and probability, and recently in representation theory of general linear groups. In this talk, we will discuss the problem raised by T. Shirai and Y. Takahashi on the positivity of the alpha determinant on the positive cone of positive definite hermitian matrices. The alpha determinant of some special matrix has an expression in terms of the special values of the generalized hypergeometric series $_3 F_2$ by 3F2 and this fact enables us to determine the signature of the alpha determinant by an estimate of such an special value. ---
“Spectral approach to composition formulas” We shall discuss an alternative approach to the composition formulas (*-products) of quantized operators based on the representation theory of the underlying Lie groups. Two examples leading to interactions with number theory will be presented. ---
"Synthesis properties of orbits in the unitary dual of nilpotent Lie groups" Click here for abstract ---
“Visible actions on multiplicity-free spaces” The notion of (strongly) visible actions on complex manifolds has been introduced by T. Kobayashi. Under the assumption of strongly visible actions on base spaces, Kobayashi shows propagation theorem in the representation theory, namely, multiplicity-free property propagates from fibers to the space of holomorphic sections. In this talk, we consider the linear actions on complex vector spaces. Let GC be a connected complex reductive Lie group and V a complex vector space. Given a holomorphic representation of GC on V , we naturally define a representation on the polynomial ring C[V ]. Our main theorem is that we decompose C[V ] into the multiplicity-free sum of irreducible representations of GC if and only if the action of a maximal compact subgroup Gu of GC on V is strongly visible. This gives a classification (GC; V ) such that Gu acts on V in a strongly visible fashion. Atsumu Sasaki (Waseda University, Tokyo)
"Distinction, Gelfand property and base change" Let G be a reductive group defined over a local field F and let H be an algebraic subgroup of G. We will consider (admissible) representations of G(F) that admits a linear functional that is H(F) invariant. Such are called H-distinguished representations. Their study is a central theme in the theory of periods of automorphic forms. Base change is part of Langlands functoriality, a vast paradigm that predicts connections beteen representations of various reductive groups. More specifically, if E/F is a field extension there should be a so-called Base Change lifting that sends representations of G(F) to representations of G(E). We will consider the following question: to what extent does distinction correlates with Langlands functoriality. Based on joint works with O. Offen we describe some cases where a very satisfying answer to such a question can be provided. In the talk I will sketch our results in the example of the pair (GL_{2n},Sp_{2n}) and some related Gelfand pairs, a notion that will play an important role in the lecture. We will also explain how these results fits into a conjectural Langlands correspondence for spherical varities.
---
“On the symplectic structure of hyperbolic co-adjoint orbits” In the talk I will discuss a result concerning the existence of a global Lagrangian fibration on hyperbolic co-adjoint orbits of real semisimple Lie groups. This result, obtained in joint work with H. Azad and I. Biswas, generalizes a result for complex orbits formulated by V.I. Arnold. ---
“Hermitian symmetric spaces of tube type and multivariate Meixner-Pollaczek polynomials” Harmonic analysis on Hermitian symmetric spaces of tube type is a natural framework for introducing multivariate Meixner-Pollaczek polynomials. Their main properties are established in this setting: generating and determinant formulas, difference equations, etc. Starting by the Euler operator, we will arrive these formulas by constructing appropriate intertwining operators between several function spaces. The talk is based on the joint work with Jacques Faraut. ---
“Nilpotent Gelfand pairs and spherical transform of Schwartz functions” A pair (N,K) of a nilpotent Lie group N and a compact subgroup K of the automorphisms of N is said to be a Gefand pair if the algebra D(N)^K of K-invariant differential operators on N is commutative. K-invariant common eigenvectors of D(N)^K are called spherical functions and the set of corresponding eigenvalues is called the Gelfand spectrum of (N,K). Fulvio Ricci has conjectured that the spherical transform (scalar-valued analogue of the Fourier transform) on (N,K) provides an isomorphism between the space of K-invariant Schwartz functions on N and the space of Schwartz functions restricted to the Gelfand spectrum properly embedded in a Euclidean space. The conjecture is known to be true for N being a Heisenberg Lie group. We will discuss recent progress in establishing the conjecture outside of the Heisenberg case and representation theory aspects of our approach. (Based on a joint project with Veronique Fischer and Fulvio Ricci.) ---
“Topological blow-up” We introduce a method 'topological blow-up', which heals a 'scar' in a topological space. Let $S$ be a closed subset of a topological space $X$. We say $S$ is a scar if for any distinguished points $x$ and $y$ in $X$, one can take disjoint neighbourhoods if at least one of $x, y$ is outside of $S$. Roughly speaking, a 'scar' means lack of Hausdorffness. In this talk, we define the 'healed space' $\tilde{X}$ for given $(X,S)$. In many cases, this space $\tilde{X}$ is Hausdorff. Moreover, the space $\tilde{X}$ has the complete information of the topology of $X$. More precisely, one can recover the original space $X$ from $(\tilde{X},S)$. --- [Back] |