µþÅÔÂç³Ø¿ôÍý²òÀϸ¦µæ½ê

Title

Symplectic constructions for extraspecial parabolics.

Date

March 23 (Fri), 2007, 10:30--11:30

Room

Room 402 RIMS, Kyoto University

Speaker

R. Stanton (Ohio)

Abstract

The minimal nilpotent orbit in a simple, say, complex Lie algebra has interaction with several topics. In work joint with M. Slupinski, we are investigating the Heisenberg grading associated to any element of the orbit. R\"ohrle ['93] referred to the corresponding Jacobson-Morozov parabolic as an extraspecial parabolic, and parametrized the orbits of the Levi subgroup acting on the nilradical modulo the center. Using exclusively methods from symplectic geometry, we shall re-examine this representation of the Levi subgroup. We shall classify orbits using the moment map; examine the symplectic nature of each of the orbits; give symplectic constructions of distinguished subgroups that occur in Rubenthaler's list of reductive dual pairs. In particular, we give a symplectic construction of the exceptional simple group $G_2$.

Title

Unipotent representations of a real simple Lie group attached to small nilpotent orbits

Date

March 20 (Tue), 2007, 16:30--17:30

Room

Room 402 RIMS, Kyoto University

Speaker

Herve Sabourin (Universite de Poitiers)

Abstract

It is a classical idea of Kirillov and Kostant that irreducible representations of a real simply connected Lie group $G$ are related to the orbits of $G$ in the dual ${\mathfrak g}^*$ of its Lie algebra. When $G$ is nilpotent, we know that there is a bijection between the set of $G$-coadjoint orbits and the unitary dual $\widehat G$ of $G$. When $G$ is solvable, a similar correspondence is due to Auslander and Kostant. For other groups, there are complications even with regard to what is true. Let us suppose now that $G$ is simple and let $O$ be a coadjoint orbit. If $O$ is semi-simple, there is a natural way to associate to $O$ an unitary representation $\Pi(O)$, but the problem is much more difficult if $O$ is nilpotent. Nevertheless, when $O$ is a minimal nilpotent orbit, one can define a notion of representation ``associated'' to $O$ and develop a strategy to construct explicitly $\Pi(O)$.Our goal is to show how this strategy can be extended to the non minimal case and what kind of new results it yields.

Title

On the Matsuki correspondence for sheaves.

Date

March 13 (Tue), 2007, 16:30--17:30

Room

Room 402 RIMS, Kyoto University

Speaker

Peter Trapa ¡ÊUtah, USA)

Abstract

Suppose G is a real reductive group with maximal compact subgroup K. Let X denote the flag variety for the complexified Lie algebra of G, and let K_C denote the complexification of K. Nearly thirty years ago, Matsuki established an order-reversing bijection between the sets of K_C and G orbits on X. Later Mirkovic-Uzawa-Vilonen extended this to an equivalence of K_C-equivariant and G-equivariant sheaves on X (a result originally conjectured by Kashiwara). Meanwhile, to each such kind of sheaf, Kashiwara showed how to attach a Lagrangian cycle in the cotangent bundle of X. Composing this characteristic cycle construction with the Mirkovic-Uzawa-Vilonen equivalence, one obtains an isomorphism between the top-dimensional homology of the conormal variety for K_C orbits on X and the top-dimensional homology of the conormal variety for G orbits on X. Schmid and Vilonen proved that this isomorphism is compatible with the Kostant-Sekiguchi correspondence of nilpotent orbits. The purpose of this talk is to give a finer explicit computation of a suitable "leading part" of the isomorphism in homology.

Title

Co-isotropic actions

Date

March 7 (Wed) 10:00-11:30 (Room 005)
March 8 (Tur) 10:00-11:30 (Room 005)
March 9 (Fri) 10:00-11:30 (Room 402)
Mach 13 (Tue) 10:00-11:30 (Room 402)
March 14 (Wed) 10:00-11:30 (Room 005)
March 16 (Fri) 10:00-11:30 (Room 402)

Room

Room 005 and 402 RIMS, Kyoto University

Speaker

Tilmann Wurzbacher (Metz)

Abstract

The first aim of this course is to explain the context and the basic properties of co-isotropic actions of Lie groups on symplectic manifolds (i.e. actions having generically co-isotropic orbits), as well as of spherical varietes (i.e. complex-algebraic varieties with an action of a complex reductive Lie group such that all Borel subgroups thereof have an open orbit). After interludes on geometric quantization resp. on lagrangian actions, we prove the equivalence of the two above conditions in the complex-algebraic set-up. Finally, we give applications of this theorem to, e.g., geometric quantization of K\"ahler manifolds and remark on connections to related subjects.

I. Symplectic reminders
II. Geometric quantization in 90 minutes
III. Lagrangian actions
IV. Co-isotropic actions
V. Applications, remarks and outlook

For details, see the pdf file

Title

Applications of representation theory to problems in analytic number theory

Date

February 9 (Fri) 10:00--11:30
February 13 (Tue) 13:00--14:30
February 16 (Fri) 10:00--11:30
February 20 (Tue) 13:00--14:30

Room

Room 402, RIMS, Kyoto University

Speaker

Joseph Bernstein (Tel Aviv)

Abstract

In this minicourse I will describe a general approach which allows to use methods of analytic representation theory in order to prove some highly non-trivial estimates in analytic number theory.
This minicourse is based on my works with Andre Reznikov.
I will study representations of the group $G = SL(2, \R)$ (and closely related groups) in the space of functions on the automorphic space $X = \Gam \backslash G$.
My aim is to describe relations of this problem to analytic number theory and to show how using methods of representation theory one can get very powerful estimates of different quantities important in number theory.

The plan of the minicourse.

    Lecture 1. Automorphic forms on the upper half-pane.
Abstract. In this lecture I will discuss automorphic forms and Maass forms on upper half-plane.
I will show that many problems about such forms are better expressed in the language of automorphic representations.
I will illustrate this on the model example which gives bounds for Sobolev norms of the automorphic functional.
    Lecture 2. Triple product problem. Convexity bound.
Abstract. I will discuss the problem of estimates for triple product of automorphic functions and its connection to estimates of automorphic $L$-functions.
Using the language of automorphic representations described in first lecture I will show how to explain the exponentially decaying factor in the triple product and then I will describe how to prove the convexity bound for these products.
    Lecture 3. Subconvexity bound for triple products.
I will continue the investigation of triple products and show how one can prove a non-trivial subconvexity bound for triple products using a combination of geometric and spectral estimates.

Title

On the support of the Plancherel measure

Date

February 14 (Wed), 2007, 10:00--11:30

Room

005 RIMS, Kyoto University

Speaker

Joseph Bernstein (Tel Aviv)

Abstract

In 1970-s Harish Chandra finished his work on harmonic analysis on real reductive groups $G$. In particular, he proved the Plancherel formula for $G$ which describes the decomposition of the regular representation of $G$ as an integral of irreducible unitary representations of the group $G \times G$ (Plancherel decomposition).
The remarkable feature of this formula was the fact that only some of the unitary representations of the group $G$ contributed to this formula (so called {\bf tempered}¡¡representations). ¡¡ In fact this phenomenon was already known in PDE. Namely in this case it was known that one can describe the spectral decomposition of an elliptic self-adjoint differential operator $D$ in terms of eigenfunctions which have moderate growth (i.e. they almost lie in $L^2$). The general result of this sort was proven by Gelfand and Kostyuchenko in 1955.
In my paper "On the support of Plancherel measure" (1988) I have applied the ideas of Gelfand and Kostyuchenko and gave an a priori proof of the fact that only tempered representations contribute to the Plancherel decomposition.
Moreover, I have shown that a similar statement holds for decompositions of $L^2(X)$ for a large class of homogeneous $G$-spaces $X$.
Examples are:
(i) $X = G/K$ , where $K$ is the maximal compact subgroup
(ii) more generally, $X = G/H$, where $H$ is a symmetric subgroup (subgroup of fixed points of some involution of $G$);
(iii) $X = G / \Gamma$ , where $\Gamma$ is a discrete lattice in¡¡$G$.
(iv) $G$ a reductive $p$-adic group, $X = G/H$, where $H$ is either an open compact subgroup or a symmetric subgroup.
I have discovered that the corresponding statement depends on some geometric structure on the space $X$ (I called it "the structure of large scale space") and that this structure has the same properties in all the cases listed above.
In my lecture I will discuss all these questions.

Title

The unitary inversion operator for the minimal representation of the indefinite orthogonal group $O(p,q)$.

Date

February 8 (Thu), 2007, 12:15--13:00

Room

Room 402, RIMS, Kyoto University

Speaker

Gen Mano (RIMS)

Abstract

The $L^2$-model of the minimal representation of the indefinite orthogonal group $O(p,q)$ ($p+q$ even, greater than four) was established by Kobayashi-Orsted (2003). In this talk, we present an explicit formula for the unitary inversion operator, which plays a key role for the analysis on this $L^2$-model. Our proof uses the Radon transform of distributions supported on the light cone.

Title

Deformation spaces of compact Clifford-Klein forms of homogeneous spaces of Heisenberg groups

Date

February 8 (Thu), 2007, 11:15--12:15

Room

Room 402, RIMS, Kyoto University

Speaker

Taro Yoshino (RIMS)

Abstract

T. Kobayashi introduced the deformation space of Clifford-Klein forms, which is a natural generalization of deformation spaces of geometric structures. Selberg-Weil's local rigidity theorem claims that the deformation space is discrete for Riemannian irreducible symmetric spaces $M$ if the dimension $d(M)\geq 3$. In contrast to this theorem, Kobayashi proved that local rigidity does not hold (even in higher dimensional case) in the non-Riemannian case. Then, this opens a new problem to find explicity such deformation spaces in high dimensions. However, such explicit forms have not been obtained except for a few cases now. In this talk, I will give an explicit description of the deformation spaces of compact Clifford-Klein forms of homogeneous spaces of Heisenberg groups.

Title

Deformation spaces of compact Clifford-Klein forms of homogeneous spaces of Heisenberg groups

Date

February 8 (Thu), 2007, 11:15--12:15

Room

Room 402, RIMS, Kyoto University

Speaker

Taro Yoshino (RIMS)

Abstract

T. Kobayashi introduced the deformation space of Clifford-Klein forms, which is a natural generalization of deformation spaces of geometric structures. Selberg-Weil's local rigidity theorem claims that the deformation space is discrete for Riemannian irreducible symmetric spaces $M$ if the dimension $d(M)\geq 3$. In contrast to this theorem, Kobayashi proved that local rigidity does not hold (even in higher dimensional case) in the non-Riemannian case. Then, this opens a new problem to find explicity such deformation spaces in high dimensions. However, such explicit forms have not been obtained except for a few cases now. In this talk, I will give an explicit description of the deformation spaces of compact Clifford-Klein forms of homogeneous spaces of Heisenberg groups.

Title

On Hardy's Theorem on nilpotent Lie groups

Date

February 8 (Thu), 2007, 10:00--11:00

Room

Room 402, RIMS, Kyoto University

Speaker

Ali Baklouti (Sfax)

Abstract

It is well known that Hardy's uncertainty principle for $\mathbb{R}^n$ was generalized to connected and simply connected nilpotent Lie groups. In this work, we extend it further to connected nilpotent Lie groups with non-compact centre. We show however that Hardy's theorem fails for a connected nilpotent Lie group which is not simply connected.

Title

On the spectrum of spherical varieties over p-adic fields

Date

February 6 (Tue), 2007, 16:30--17:30

Room

Room 402, RIMS, Kyoto University

Speaker

Yiannis Sakellaridis (Tel Aviv)

Abstract

Spherical varieties are a very important class of spaces which includes all symmetric varieties. Over a p-adic field, their representation theory seems to admit some description through a "Langlands dual" group. I will discuss results on the unramified component of their spectrum which point to this direction. If time permits, I will show how this theory can be used to obtain a completely general Casselman-Shalika formula for eigenfunctions of the Hecke algebra on an arbitrary spherical variety.

Title

Rank, Kirillov's Orbit method, and Small Representations

Date

December 15 (Fri), 2006, 10:30-11:30

Room

Room 402, RIMS, Kyoto University

Speaker

Hadi Salmasian (Queen's University, Canada)

Abstract

I will survey results and applications of the theory of rank for unitary representations of reductive groups. I will explain how this concept is related to Kirillov's method of orbits. Finally, I will describe the impllications of these ideas in the study of representations of exeptional groups.

Title

Quantization of symmetric spaces: spectral approach

Date

November 21 (Tue) , 2006, 16:30-18:00
November 22 (Wed), 2006, 9:00-10:30
November 24 (Fri), 2006, 10:30-12:00

Room

Room 402, RIMS, Kyoto University

Speaker

Michael Pevzner (Reims University, France)

Abstract

In this series of talks we shall introduce a family of covariant symbolic calculi on a particular class of semi-simple symmetric spaces, to wit the para-Hermitian symmetric spaces (PHSS).
Beyond the formal definition and basic properties of such "pseudo-differential analyses" we shall focus on some fruitful applications of these techniques to the harmonic analysis on PHSS and more broadly to the representation theory of their transformation groups.
Main topics that will be discussed are:

• Composition formul\ae\, and branching laws for tensor products of highest weight modules - an alternative definition of Rankin-Cohen brackets.
• $H^*$-algebra structure of $L^2$ spaces on PHSS.
• Square root method and the Plancherel formula for the rank-one PHSS.
• Star-products versus sharp-products.

Title

Resolution of null fiber and its quotient as a conormal bundle over Lagrangean Grassmannian

Date

October 31 (Tue), 2006 14:00-15:00

Room

Room 402, RIMS, Kyoto University

Speaker

À¾»³µý Kyo Nishiyama (Kyoto University)

Abstract

We give an explicit realization of the resolution of singularities of each irreducible component of the null fiber of standard contraction map of $ U \otimes V + U \times V^{\ast} $ by the action of $ GL(V) $.
Then, the categorical quotient by $ O(U) \times O(U) $ of the resolution turns out to be a conormal bundle of a certain closed $ GL(V) $-orbit in the Lagrangean Grassmannian. The moment map image of the conormal bundle is the closure of a spherical nilpotent orbit, which is the theta lift of the trivial nilpotent orbit for a certain indefinite orthogonal group in the stable range.
We will explain the construction in detail, and relationships with representation theory and prehomogeneous spaces.

Title

Gelfand pair, definitions, main properties, and generalisations

Date

October 23 (Mon), 2006, 10:00-12:00
October 24 (Tue) , 2006, 16:30-18:00
October 25 (Wed), 2006, 9:00-10:30
October 27 (Fri), 2006, 13:00-15:00

Room

Room 402, RIMS, Kyoto University

Speaker

Oksana Yakimova (Humboldt fellow, Cologne)

Abstract

The concept of a Gelfand pair is a natural generalisation of a symmetric Riemannian homogeneous space. It plays an important r\^{o}le in representation theory, differential geometry, symplectic geometry, and functional analysis.
Let $X=G/K$ be a connected Riemannian homogenous space of a real Lie group $G$. Then $(G,K)$ is called a {\it Gelfand pair\/} and the homogeneous space $X$ is said to be {\it commutative\/} if the algebra $D(X)^G$ of $G$-invariant differential operators on $X$ is commutative; or, equivalently, if the representation of $G$ on $L^2(X)$ is multiplicity free. In this lectures we will consider other characterisations of Gelfand pairs. For example, $X$ is commutative if and only if the action of G on the cotangent bundle $T^*X$ is coisotropic with respect to the standard G-invariant symplectic structure.
In 1956, Gelfand and Selberg independently introduced two sufficient conditions for commutativity. These conditions turned out to be equivalent and can be formulated as follows:
(GS) there is a group antiautomorphism $\sigma$ of $G$ such that each double coset of $K$ is $\sigma$-stable.

If condition (GS) is satisfied, then $X$ is said to be {\it weakly symmmetric}. This condition is not necessary for commutativity. In 1998, Lauret constructed the first example of a commutative but not weakly symmetric homogeneous space.
Suppose that $G$ is reductive. Then, by a result of Akhiezer and Vinberg, $(G,K)$ is a Gelfand pair if and only if $X$ is weakly symmetric. Moreover, $(G,K)$ is a Gelfand pair if and only if the complexification $X(\cp)$ of $X$ is a {\it spherical} $G(\cp)$-variety. Spherical variety is a well-studied object of algebraic geometry. If $X(\cp)$ is homogeneous and affine, then the complete classification was obtained by Kr\"amer, Brion, and Mikityuk. Classification of Gelfand pairs with reductive $G$ easily follows from their results.
In general, if $(G,K)$ is commutative, then, up to a local isomorphism, $G$ has a factorisation $G=N\rtimes L$, where $L$ is reductive, $N$ is commutative or two-step nilpotent, and $K\subset L$. We will present an effective commutativity criterion in terms of representations of $L$ and $K$ on $\gt n=\Lie N$; and discuss main ideas of the classification of Gelfand pairs.
The notion of a Gelfand pair can be generalised in different ways. If $K$ is not compact, then one can give various reasonable definitions that are not equivalent. In this way, we will obtain different objects, which belong to several areas of mathematics, like spherical varieties in case $G$ is reductive (invariant theory, algebraic geometry); generalised Gelfand pairs (harmonic analysis); coisotropic actions (symplectic geometry, integrable Hamiltonian systems).

Title

Geometric superrigidity, integrality of lattice and classification of fake projective planes

Date

October 17 (Tue), 2006 16:30-17:30

Room

Room 402, RIMS, Kyoto University

Speaker

Sai Kee Yeung (Purdue Univ)

Abstract

I would explain geometric approach to rigidity problems for lattices in semi-simple Lie group, the difficulties encountered and results known for the complex rank one cases, and the relation to the recent classification of fake projective planes given by Gopal Prasad and myself. I would also explain the new examples of fake projective planes and fake projective fourfolds we constructed. The techniques used in the first part are mainly geometrical, while the second part are mainly algebraic group and number theoretical.

Title

L^p -cohomology and negative curvature

Date

September 29 (Fri), 2006, 16:30--18:00

Room

Room 402, RIMS, Kyoto University

Speaker

Pierre Pansu (Paris-Sud)

Abstract

L^p -cohomology of a Riemannian manifold is the cohomology of the (de Rham) complex of differential forms which are L^p -integrable. We explain the role played by L^p -cohomology in three problems related to negatively curved manifolds and groups.
- Hopf's conjecture on the sign of Euler characteristic of compact negatively curved manifolds (specificly, the Kahler case).
- Cannon's conjecture on hyperbolic groups whose ideal boundary is a 2-sphere.
- Optimal sectional curvature pinching for rank one symmetric spaces.

Title

Invariant polynomials and invariant differential operators for multiplicity-free actions of rank 3

Date

September 1 (Fri), 2006, 11:00--12:00

Room

Room 402, RIMS, Kyoto University

Speaker

µÆÃÏ ¹îɧ»á (Katsuhiko Kikuchi)(Kyoto University)

Abstract

Let $V$ be a finite-dimensional vector space over $\mathbb{C}$, and $K$ a compact Lie group acting on $V$ linearly. We call $(K, V)$ a {\it multiplicity-free action} if each irreducible component appears at most one in the (holomorphic) polynomial ring ${\cal P}(V)$ on $V$. If $(K, V)$ is multiplicity-free, then there exists a number $r$ such that the ring ${\cal P}(V_{\mathbb{R}})^K ={\cal P}(V)\otimes \overline{{\cal P}(V)}$ of $K$-invariant polynomials on the underlying real vector space $V_{\mathbb{R}}$ of $V$ is isomorphic to the polynomial ring of $r$ variables. The number $r$ is called the {\it rank} of $(K, V)$.
For each highest weight $\lambda$ which appears in the irreducible decomposition of ${\cal P}(V)$ there exist, up to a scalar, a unique $K$-invariant polynomial $p_{\lambda}(z, \overline{z})$ and a unique $K$-invariant differential operator $p_{\lambda}(z, \partial )$. In this talk, we describe all $K$-invariant polynomials $\{p_{\lambda}(z, \overline{z})\}$ and $K$-invariant differential operators $\{p_{\lambda}(z, \partial )\}$ for a rank 3 multiplicity-free action $(K, V)$ which is not derived from a Hermitian symmetric space. Moreover, we give two `symmetric' slices for visibility of the action $(K, V)$. We show that the action of the stabilizer of one indicates the symmetry of the $K$-invariant polynomials, and that of the other indicates the symmetry of the eigenvalues of the $K$-invariant differential operators.

Title

Recent results and problems on the Fourier and Radon Transform on Symmetric Spaces

Date

August 4 (Fri), 2006, 10:30--11:30

Room

Room 402, RIMS, Kyoto University

Speaker

Sigurdur Helgason (MIT)

Abstract

In this lecture we discuss problems concerning the Fourier transform for integrable functions on symmetric spaces and some problems concerning its topological properties.
We shall also discuss some geometric problems about the Radon transform on symmetric spaces, including refinements of its support properties.

Title

The Image of the Segal-Bargmann transform Symmetric Spaces and generalizations

Date

July 25 (Tue), 2006, 16:30--17:30

Room

Room 402, RIMS, Kyoto University

Speaker

Gestur Ólafsson (Louisiana State University)

Abstract

Let \Delta =\sum \partial^2/\partial x_i^2 be the Laplace operator on R^n.
The heat equation¡¡is
\Delta u(x,t)=\frac{\partial\, }{\partial t}¡¡u(x,t)¡¡\lim_{t\to 0^+}u(x,t)¡¡ =¡¡f(x)
where f is a L^2-function or a distribution. The solution
u(x,t)=e^{t\Delta }f(x)=H_tf(x)
is given by
H_tf(x)=¡¡\int_{R^n}f(y)h_t(x-y) ¡¡¡¡¡¡
=¡¡\frac{1}{(4\pi t)^{n/2}}¡¡¡¡ \int_{R^n}¡¡f(y) e^{-(x-y)¡¦ (x-y)/4t} dy
where h_t(x)=1/(4\pi t)^{n/2} e^{-x\cdot x/4t} is the heat kernel, i.e. the solution corresponding to¡¡f=\delta_0. It can be read of from this explicit formula,
that \R^n\ni x\mapsto H_tf(x) has a holomorphic extension to all of C^n. The transform
f\mapsto H_tf\in \mathcal{O}(\mathbb{C}^n)
is the Segal-Bargmann transform. Its image is the space of holomorphic functions
F:C^n\to C, such that
|F|_t^2:=¡¡(2\pi t)^{-n/2}¡¡\int¡¡ |F(x+iy)|^2¡¡e^{-|y|^2/2t}¡¡ dxdy¡¡<¡¡\infty
and
|f|=|H_tf|.
The Heat equation has a natural generalization to all Riemannian manifolds. The solution is again given by the Heat transform
u(x,t)=H_tf(x)=\int f(y)h_t(y)\, dy
where h_t is the heat kernel, but as there is no natural complexification in general it is not clear how to realize the image in a space of holomorphic functions.
An exception is the class of symmetric spaces on noncompact type. In this talk, we start by a short discussion of the Heat transform on R^n to motivate the main part of the talk and introduce the concepts and ideas that are needed for the Riemannian symmetric spaces of the form G/K where G is a connected noncompact semisimple Lie group and K a maximal compact subgroup.
We introduce the natural G-invariant complexification of G/K, called the crown, and describe the image of the Segal-Bargmann transform as a Hilbert space of holomorphic functions on the crown. If time allows, then we will give a different realization of the image space of L^2(G/K)^K. That results has a natural formulation for the Heckmann-Opdam setting for positive multiplicity functions.
The main tools here are the spherical Fourier transform and the Abel transform.

Title

A simple proof of Bernstein-Lunts equivalence

Date

June 20 (Tue), 2006, 16:30--17:30

Room

Room 402, RIMS, Kyoto University

Speaker

Pavle Pandzic (University of Zagreb)

Abstract

In a paper in J.Amer.Math.Soc., Bernstein and Lunts proved that the equivariant derived category of (g,K) modules is equivalent to the ordinary derived category of (g,K) modules. Their proof is quite complicated; it uses K-injective resolutions and a few dualizing arguments. In addition, it works only for bounded derived categories. I will present a simple proof of this result using K-projectives and some basic properties of triangulated categories. This requires the group K to be reductive, but this assumption is easily eliminated. No boundedness assumptions are necessary in my approach.

Date

May 15 (Mon), 2006, 16:30--17:30, 17:45--18:45

Room

Room 202, RIMS, Kyoto University

Speaker

Anthony Dooley (University of New South Wales)

Time

16:30--17:30

Title

Intertwining operators, the Cayley transform, and the contraction of K to NM

Abstract

If $G=KAN$ is the Iwasawa decomposition of a rank one semi-simple Lie group, it is interesting to use harmonic analysis on $N$ together with the $N$ picture of the principal and exceptional series to analyse the representation theory. In particular, the author recently proved a representation-theoretic version of the Cowling-Haagerup theorem on the approach to the identity by uniformly bounded representations. In order to establish the Baum-Connes conjecture ``with coefficients", one needs information about the $K$ picture, and it turns out that this can be obtained from this result together with the study of the contraction of $K$ to $NM$

Time

17:45--18:45

Title

Orbital convolution theory for semi-direct productsy

Abstract

Dooley and Wildberger in the setting of compact groups, introduced the wrapping map $\Phi$. This map associates, to each Ad-invariant distribution $\mu$ of compact support on the Lie algebra $\gfrak$, a central distribution $\Phi \mu$ on the Lie group $G$, via the formula, for $f \in C_c^{\infty}(G),$
\la \Phi\mu,f\ra=\la \mu,j \cdot f\circ\exp\ra, (1)
where $j$ is the square root of the Jacobian of $\exp : \gfrak\to G$.
$\Phi$ provides a convolution homomorphism between the Euclidean convolution structure on $\gfrak$ and the group convolution on $G$, that is
\Phi(\mu\ast_{\gfrak}\nu)=\Phi\mu\ast_G \Phi\nu. (2)
This mapping is a global version of the Duflo isomorphism --- there are no conditions on the supports of $\mu$ and $\nu$ (they need not, for example, lie in a fundamental domain).We may interpret the dual of $\Phi$, a map from the Gelfand space of $M_G(G)$ to that of $M_G(\gfrak)$, in such a way as to obtain the Kirillov character formula for $G$.
In a recent paper Andler, Sahi and Torrosian have extended the Duflo isomorphism to arbitrary Lie groups. Their results give a version of equation which holds for germs of hyperfunctions with support at the identity.
Our result can be viewed as a statement that, for compact Lie groups, the results of hold for invariant distributions of compact support, and hold globally in the sense that the restriction that the supports are compact is needed only in order to ensure that the convolutions exist. This observation allows one to develop calculational tools for invariant harmonic analysis based on convolutions of orbits and distributions in the Euclidean space $\gfrak$.
We have extended these ideas to semi-direct product groups $G=V\rtimes K$, where $V$ is a vector space and $K$ a compact group. There are several significant differences between this case and the compact case --- firstly, there is no identification between the adjoint and coadjoint pictures as the Killing form is indefinite, and secondly, perhaps more significantly, the fact that the orbits are no longer compact means that there are few Ad-invariant distributions of compact support --- so the convolutions in formula (2) need careful interpretation.

Title

An introduction to Gindikin's horospherical Cauchy transform

Date

April 18 (Tue), 2006, 16:30--17:30

Room

Room 402, RIMS, Kyoto University

Speaker

¾¾ÌÚÉÒɧ»á (Toshihiko Matsuki, Kyoto University)

Abstract

Recently S. Gindikin introduced a notion of the horospherical Cauchy transform from $\mathcal{O}(G_C/K_C)$ to $\mathcal{O}(G_C/N_C)$ and its inverse. He also showed the isomorphism between the space of hyperfunctions on the compact symmetric space $U/K$ and the space of holomorphic functions on some domain in $G_C/N_C$ by this transform. In this talk I would like to explain his idea by using elementary examples and group-theoretical methods.

Title

Shimura correspondences for split real groups

Date

March 17 (Fri), 2006, 17:00--18:00

Room

Room 402, RIMS, Kyoto University

Speaker

Peter E. Trapa (University of Utah)

Abstract

Suppose G is a split real reductive Lie group (like SL(2,R), for instance). Then G admits an essentially unique nonlinear two-fold double cover. In the past few years, Barbasch and Barbasch-Ciubotaru have given a striking description of the spherical unitary dual of G. The point of this talk is to recall their results and then show how they can be used to give very precise information about the unitary dual of the double cover of G. The relationship between unitary representations of G and its double cover is the Shimura correspondence of the title. This talk represents joint work with Jeffrey Adams, Dan Barbasch, Annegret Paul, and David Vogan.

Title

»Ø¿ô·¿²Ä²ò¥ê¡¼·²¤Î¥æ¥Ë¥¿¥êɽ¸½ I

Date

February 21 (Tue), 2006, 13:30-15:30
February 22 (Wed), 2006, 15:30-17:30
February 23 (Thu), 2006, 15:30-17:30
February 24 (Fri), 2006, 10:00-12:00

Room

Room 402, 005, RIMS, Kyoto University

Speaker

Æ£¸¶±ÑÆÁ (Hidenori Fujiwara) (Kinki University)

Abstract

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Title

Local Zeta functions for a class of real symmetric spaces

Date

February 21 (Tue), 2006, 16:30-17:30

Room

Room 402, RIMS, Kyoto University

Speaker

Hubert Rubenthaler (IRMA, Strasbourg)

Abstract

Let G/H be a symmetric space which is embedded as an open set in R^n, let P be a polynomial invariant of the action of G on G/H and let \pi be a representation of G admitting a generalized H-invariant vector u. Then for f \in S(R^n) one can form the Zeta function:
Z(f,\pi,s)=\int_{G/H} f( . g) |P( . g)|^s \pi( .g) u d ( .g).
For a class of symmetric spaces we will make this definition precise in the case where \pi belongs to the spherical minimal series, and we will prove a functional equation.