Toshihiko Matsuki, Anthony Dooley (1), Anthony Dooley (2), Pavle Pandzic, Gestur Ólafsson, Sigurdur Helgason, Katsuhiko Kikuchi, Pierre Pansu, Sai Kee Yeung, Oksana Yakimova (4 lectures), Kyo Nishiyama, Michael Pevzner (3 lectures), Hadi Salmasian, Yiannis Sakellaridis, Ali Baklouti, Taro Yoshino, Gen Mano, Joseph Bernstein (1), Joseph Bernstein (2) (4 lectures), Tilmann Wurzbacher (6 lectures), Peter Trapa, Herve Sabourin, R. Stanton
Date:  April 18 (Tue), 2006, 16:3017:30 
Room:  RIMS, Kyoto University : Room 402 
Speaker:  Toshihiko Matsuki (ΌΨqF) (Kyoto University) 
Title:  An introduction to Gindikin's horospherical Cauchy transform 
Abstract: [pdf] 
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 grouptheoretical methods. 
Date:  May 15 (Mon), 2006, 16:3017:30 
Room:  RIMS, Kyoto University : Room 202 
Speaker:  Anthony Dooley (University of New South Wales) 
Title:  Intertwining operators, the Cayley transform, and the contraction of K to NM 
Abstract: [pdf] 
If G=KAN is the Iwasawa decomposition of a rank one semisimple 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 representationtheoretic version of the CowlingHaagerup theorem on the approach to the identity by uniformly bounded representations. In order to establish the BaumConnes 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 
Date:  May 15 (Mon), 2006, 17:4518:45 
Room:  RIMS, Kyoto University : Room 202 
Speaker:  Anthony Dooley (University of New South Wales) 
Title:  Orbital convolution theory for semidirect productsy 
Abstract: [pdf] 
Dooley and Wildberger in the setting of compact groups, introduced
the wrapping map Φ. This map associates, to each Adinvariant
distribution μ of compact support on the Lie algebra g, a
central distribution Φμ on the Lie group G, via the
formula, for f ∈ C_{c}^{∞}(G),
Φ provides a convolution homomorphism between the Euclidean convolution structure on g and the group convolution on G, that is 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 g. We have extended these ideas to semidirect 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 Adinvariant distributions of compact support — so the convolutions in formula (2) need careful interpretation. 
Date:  June 20 (Tue), 2006, 16:3017:30 
Room:  RIMS, Kyoto University : Room 402 
Speaker:  Pavle Pandzic (University of Zagreb) 
Title:  A simple proof of BernsteinLunts equivalence 
Abstract: [pdf] 
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 Kinjective
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 Kprojectives 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:  July 25 (Tue), 2006, 16:3017:30 
Room:  RIMS, Kyoto University : Room 402 
Speaker:  Gestur Ólafsson (Louisiana State University) 
Title:  The Image of the SegalBargmann transform Symmetric Spaces and generalizations 
Abstract: [pdf] 
Let Δ = ∑ ∂^{2}/∂x_{i}^{2} be the Laplace operator on R^{n}.
The heat equation is The Heat equation has a natural generalization to all Riemannian manifolds. The solution is again given by the Heat transform 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 Ginvariant complexification of G/K, called the crown, and describe the image of the SegalBargmann 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 HeckmannOpdam setting for positive multiplicity functions. The main tools here are the spherical Fourier transform and the Abel transform. 
Date:  August 4 (Fri), 2006, 10:3011:30 
Room:  RIMS, Kyoto University : Room 402 
Speaker:  Sigurdur Helgason (MIT) 
Title:  Recent results and problems on the Fourier and Radon Transform on Symmetric Spaces 
Abstract: [pdf] 
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. 
Date:  September 1 (Fri), 2006, 11:0012:00 
Room:  RIMS, Kyoto University : Room 402 
Speaker:  Katsuhiko Kikuchi (enF) (Kyoto University) 
Title:  Invariant polynomials and invariant differential operators for multiplicityfree actions of rank 3 
Abstract: [pdf] 
Let V be a finitedimensional vector space over C, and
K a compact Lie group acting on V linearly. We call (K,V) a multiplicityfree action if each irreducible component appears at most
one in the (holomorphic) polynomial ring \mathcal{P}(V) on V. If (K,V) is multiplicityfree, then there exists a number r such that the
ring \mathcal{P}(V_{R})^{K} =\mathcal{P}(V) \otimes \overline{\mathcal{P}(V)} of Kinvariant polynomials on the underlying real vector space
V_{R} of V is isomorphic to the polynomial ring of r
variables. The number r is called the rank of (K,V).
For each highest weight λ which appears in the irreducible decomposition of \mathcal{P}(V) there exist, up to a scalar, a unique Kinvariant polynomial p_{λ}(z,\overline{z}) and a unique Kinvariant differential operator p_{λ}}(z,∂). In this talk, we describe all Kinvariant polynomials {p_{λ}(z,\overline{z})} and Kinvariant differential operators {p_{λ}(z,∂)} for a rank 3 multiplicityfree 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 Kinvariant polynomials, and that of the other indicates the symmetry of the eigenvalues of the Kinvariant differential operators. 
Date:  September 29 (Fri), 2006, 16:3018:00 
Room:  RIMS, Kyoto University : Room 402 
Speaker:  Pierre Pansu (ParisSud) 
Title:  L^{p}cohomology and negative curvature 
Abstract: [pdf] 
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.

Date:  October 17 (Tue), 2006, 16:3017:30 
Room:  RIMS, Kyoto University : Room 402 
Speaker:  Sai Kee Yeung (Purdue Univ) 
Title:  Geometric superrigidity, integrality of lattice and classification of fake projective planes 
Abstract: [pdf] 
I would explain geometric approach to rigidity problems for lattices in semisimple 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. 
Date: 
October 23 (Mon), 2006, 10:0012:00 October 24 (Tue), 2006, 16:3018:00 October 25 (Wed), 2006, 9:0010:30 October 27 (Fri), 2006, 13:0015:00 
Room:  RIMS, Kyoto University : Room 402 
Speaker:  Oksana Yakimova (Humboldt fellow, Cologne) 
Title:  Gelfand pair, definitions, main properties, and generalisations 
Abstract: [pdf] 
The concept of a Gelfand pair is a natural generalisation of a
symmetric Riemannian homogeneous space. It plays an important role 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 Gelfand pair and the homogeneous space X is said to be commutative if the algebra D(X)^{G} of Ginvariant 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 Ginvariant 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 σ of G such that each double coset of K is σstable. If condition (GS) is satisfied, then X is said to be 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(C) of X is a spherical G(C)variety. Spherical variety is a wellstudied object of algebraic geometry. If X(C) is homogeneous and affine, then the complete classification was obtained by Krämer, 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 twostep nilpotent, and K ⊂ L. We will present an effective commutativity criterion in terms of representations of L and K on 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). 
Date:  October 31 (Tue), 2006, 14:0015:00 
Room:  RIMS, Kyoto University : Room 402 
Speaker:  Kyo Nishiyama (ΌR ) (Kyoto University) 
Title:  Resolution of null fiber and its quotient as a conormal bundle over Lagrangean Grassmannian 
Abstract: [pdf] 
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 × V* by the action of GL(V).
Then, the categorical quotient by O(U) × 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. 
Date: 
November 21 (Tue), 2006, 16:3018:00 November 22 (Wed), 2006, 9:0010:30 November 24 (Fri), 2006, 10:3012:00 
Room:  RIMS, Kyoto University : Room 402 
Speaker:  Michael Pevzner (Reims University, France) 
Title:  Quantization of symmetric spaces: spectral approach 
Abstract: [pdf] 
In this series of talks we shall introduce a family of covariant
symbolic calculi on a particular class of semisimple symmetric spaces, to wit
the paraHermitian symmetric spaces (PHSS).
Beyond the formal definition and basic properties of such "pseudodifferential 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:

Date:  December 15 (Fri), 2006, 10:3011:30 
Room:  RIMS, Kyoto University : Room 402 
Speaker:  Hadi Salmasian (Queen's University, Canada) 
Title:  Rank, Kirillov's Orbit method, and Small Representations 
Abstract: [pdf] 
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. 
Date:  February 6 (Tue), 2007, 16:3017:30 
Room:  RIMS, Kyoto University : Room 402 
Speaker:  Yiannis Sakellaridis (Tel Aviv) 
Title:  On the spectrum of spherical varieties over padic fields 
Abstract: [pdf] 
Spherical varieties are a very important class of spaces which includes all symmetric varieties. Over a padic 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 CasselmanShalika formula for eigenfunctions of the Hecke algebra on an arbitrary spherical variety. 
Date:  February 8 (Thu), 2007, 10:0011:00 
Room:  RIMS, Kyoto University : Room 402 
Speaker:  Ali Baklouti (Sfax) 
Title:  On Hardy's Theorem on nilpotent Lie groups 
Abstract: [pdf] 
It is well known that Hardy's uncertainty principle for 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 noncompact centre. We show however that Hardy's theorem fails for a connected nilpotent Lie group which is not simply connected. 
Date:  February 8 (Thu), 2007, 11:1512:15 
Room:  RIMS, Kyoto University : Room 402 
Speaker:  Taro Yoshino (gμΎY) (RIMS) 
Title:  Deformation spaces of compact CliffordKlein forms of homogeneous spaces of Heisenberg groups 
Abstract: [pdf] 
T. Kobayashi introduced the deformation space of CliffordKlein forms, which is a natural generalization of deformation spaces of geometric structures. SelbergWeil's local rigidity theorem claims that the deformation space is discrete for Riemannian irreducible symmetric spaces M if the dimension d(M) ≥ 3. In contrast to this theorem, Kobayashi proved that local rigidity does not hold (even in higher dimensional case) in the nonRiemannian 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 CliffordKlein forms of homogeneous spaces of Heisenberg groups. 
Date:  February 8 (Thu), 2007, 12:1513:00 
Room:  RIMS, Kyoto University : Room 402 
Speaker:  Gen Mano (^μ³) (RIMS) 
Title:  The unitary inversion operator for the minimal representation of the indefinite orthogonal group O(p,q) 
Abstract: [pdf] 
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 KobayashiØrsted (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. 
Date:  February 14 (Wed), 2007, 10:0011:30 
Room:  RIMS, Kyoto University : Room 005 
Speaker:  Joseph Bernstein (Tel Aviv) 
Title:  On the support of the Plancherel measure 
Abstract: [pdf] 
In 1970s 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 × G
(Plancherel decomposition).
The remarkable feature of this formula was the fact that only some of the unitary representations of the group Gcontributed to this formula (so called 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 selfadjoint 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 Gspaces 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/Γ, where Γ is a discrete lattice in G. (iv) G a reductive padic 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. 
Date: 
February 9 (Fri), 2007, 10:0011:30 February 13 (Tue), 2007, 13:0014:30 February 16 (Fri), 2007, 10:0011:30 February 20 (Tue), 2007, 13:0014:30 
Room:  RIMS, Kyoto University : Room 402 
Speaker:  Joseph Bernstein (Tel Aviv) 
Title:  Applications of representation theory to problems in analytic number theory (mini course) 
Abstract: [pdf] 
In this minicourse I will describe a general approach which allows
to use methods of analytic representation theory in order to prove some highly
nontrivial 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 = Γ\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. Abstract. In this lecture I will discuss automorphic forms and Maass forms on upper halfplane. 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. Abstract. I will discuss the problem of estimates for triple product of automorphic functions and its connection to estimates of automorphic Lfunctions. 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. I will continue the investigation of triple products and show how one can prove a nontrivial subconvexity bound for triple products using a combination of geometric and spectral estimates. 
Date: 
March 7 (Wed), 2007, 10:0011:30 (Room 005) March 8 (Tur), 2007, 10:0011:30 (Room 005) March 9 (Fri), 2007, 10:0011:30 (Room 402) March 13 (Tue), 2007, 10:0011:30 (Room 402) March 14 (Wed), 2007, 10:0011:30 (Room 005) March 16 (Fri), 2007, 10:0011:30 (Room 402) 
Room:  RIMS, Kyoto University : Room 005 and 402 
Speaker:  Tilmann Wurzbacher (Metz) 
Title:  Coisotropic actions (mini course) 
Abstract: [pdf] 
The first aim of this course is to explain the context and the
basic properties of coisotropic actions of Lie groups on symplectic manifolds
(i.e. actions having generically coisotropic orbits), as well as of spherical
varietes (i.e. complexalgebraic 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 complexalgebraic setup.
Finally, we give applications of this theorem to, e.g., geometric quantization
of Kähler manifolds and remark on connections to related subjects.
I. Symplectic reminders II. Geometric quantization in 90 minutes III. Lagrangian actions IV. Coisotropic actions V. Applications, remarks and outlook For details, see the pdf file. 
Date:  March 13 (Tue), 2007, 16:3017:30 
Room:  RIMS, Kyoto University : Room 402 
Speaker:  Peter TrapaiUtah, USA) 
Title:  On the Matsuki correspondence for sheaves 
Abstract: [pdf] 
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 orderreversing bijection between the sets of K_{C} and G orbits on X. Later MirkovicUzawaVilonen extended this to an equivalence of K_{C}equivariant and Gequivariant 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 MirkovicUzawaVilonen equivalence, one obtains an isomorphism between the topdimensional homology of the conormal variety for K_{C} orbits on X and the topdimensional homology of the conormal variety for G orbits on X. Schmid and Vilonen proved that this isomorphism is compatible with the KostantSekiguchi 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. 
Date:  March 20 (Tue), 2007, 16:3017:30 
Room:  RIMS, Kyoto University : Room 402 
Speaker:  Herve Sabourin (Université de Poitiers) 
Title:  Unipotent representations of a real simple Lie group attached to small nilpotent orbits 
Abstract: [pdf] 
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 g* of its Lie algebra. When G is nilpotent, we know that there is a bijection between the set of Gcoadjoint 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 semisimple, there is a natural way to associate to O an unitary representation Π(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 Π(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. 
Date:  March 23 (Fri), 2007, 10:3011:30 
Room:  RIMS, Kyoto University : Room 402 
Speaker:  R. Stanton (Ohio) 
Title:  Symplectic constructions for extraspecial parabolics 
Abstract: [pdf] 
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. Röhrle ['93] referred to the corresponding JacobsonMorozov 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 reexamine 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}. 
Organizer:  Toshiyuki Kobayashi 
© Toshiyuki Kobayashi