Takeshi Kawazoe, Alexander Alldridge, Troels Johansen, Andreas Nilsson, Misha Pevzner (1), Misha Pevzner (2), Jun O'Hara, Ivan Cherednik, Simon Gindikin (1), Simon Gindikin (2), Chifune Kai, Joachim Hilgert, Hubert Rubenthaler, Hidenori Fujiwara (4 lectures), Peter Trapa
Date: | September 2 (Fri), 2005, 15:00-16:00 |
Room: | RIMS Room 402 |
Speaker: | Takeshi Kawazoe (河添健) (Keio University) |
Title: | On Hardy's theorem on SU(1,1) |
Abstract: [pdf] |
The classical Hardy theorem asserts that f and its Fourier transform \hat f can not be very rapidly decreasing. This theorem was generalized on Lie groups by various people, and also for the Fourier-Jacobi transform. Especially, the heat kernel plays an essential role, which is a "good" function in the sense that f and a generalised Fourier transform both have good decay. However, on SU(1,1) there are infinitely many "good" functions. In this talk, we shall consider a characterization of "good" functions on SU(1,1). |
Date: | September 2 (Fri), 2005, 13:30-14:30 |
Room: | RIMS Room 402 |
Speaker: | Alexander Alldridge (University of Paderborn) |
Title: | The Embedding of Discrete Series Representations of Facial Subgroups |
Abstract: [pdf] |
Consider a Hermitian symmetric domain B with connected automorphism group G. The boundary of the convex domain B decomposes into lower-rank Hermitian symmetric spaces \bar B with connected automorphism groups \bar G ⊂ G. It is natural to ask for embeddings of discrete series representations (or more general irreducible unitary representations) of \bar G into corresponding representations of G. If B is an irreducible classical domain, we exhibit an explicit unitary embedding of all discrete series representations of \bar G (holomorphic or non-holomorphic), such that the highest weight vectors of the lowest K-types correspond. The construction uses Knapp-Wallach's Szegö operators, and can be extended to all representations in the support of the Plancherel measure of \bar G. |
Date: | October 4 (Tue), 2005, 16:30-17:30 |
Room: | RIMS, Kyoto University : Room 402 |
Speaker: | Troels Johansen (University of Paderborn) |
Title: | Geometry of Orbits and the Plancherel Decomposition |
Abstract: [pdf] |
For a class of affine symmetric spaces G/H, we construct a tube domain whose Silov boundary S allows for a G-equivariant identification of the associated L^{2}-spaces. The adjoint orbits of H in the tangent space of G/H are realized as open orbits in the abelian group S, and we associate natural unitary representations to these orbits. In the rank one case we thus obtain the Plancherel decomposition. Furthermore the orthogonal projection onto 'the most continuous part' may be described by the (un-normalized) orbital integral associated to the Cartan subspace with noncompact centralizer group in H. |
Date: | October 11 (Tue), 2005, 16:30-17:30 |
Room: | RIMS, Kyoto University : Room 402 |
Speaker: | Andreas Nilsson (RIMS) |
Title: | Characterization of discrete Riesz transforms |
Abstract: [pdf] |
The Riesz transforms in R^{n} have been characterized to be the only translation invariant linear operators satisfying some condtion of relative invariance under the conformal group CO(n,R). The purpose of this talk is to investigate to what extent the corresponding statement is true for their discrete analogues. It turns out that it does not always hold. |
Date: | October 21 (Fri), 2005, 10:30-11:30 |
Room: | RIMS, Kyoto University : Room 402 |
Speaker: | Misha Pevzner (Reims) |
Title: | Kontsevich Quantization and Duflo Isomorphism |
Abstract: [pdf] |
Since the fundamental results by Chevalley, Harish-Chandra and
Dixmier one knows that
the set of ad-invariant polynomials on the dual of a Lie algebra of a
particular type (solvable, simple or nilpotent)
is isomorphic, as an algebra, to the center of the enveloping algebra. This
fact was generalized to an arbitrary finite-dimensional
real Lie algebra by M. Duflo in late 70's. His proof is based on the
Kirillov's orbits method that parametrizes infinitesimal characters
of unitary irreducible representations of the corresponding Lie group in
terms of co-adjoint orbits.
The Kontsevich' Formality theorem implies not only the existence of the Duflo map but shows that it is canonical. We shall describe this construction and indicate how does this construction extend to the whole Poisson cohomology of an arbitrary finite-dimensional real Lie algebra. |
Date: | October 25 (Tue), 2005, 13:10-13:55 |
Room: | RIMS, Kyoto University : Room 402 |
Speaker: | Misha Pevzner (Reims) |
Title: | Rankin-Cohen brackets and quantaziation of causal symmetric spaces |
Abstract: [pdf] |
We consider semi-simple real Lie groups G such that the associated
Riemannian symmetric space G/K is a Hermitian symmetric space of
tube type and the non-Riemannian one G/H is a para-hermitian
symmetric space. Such symmetric spaces are usually called
causal symmetric spaces of Cayley type.
The first requirement implies that the Lie group G has
holomorphic discrete series representations acting on the space of
square integrable holomorphic functions on G/K. The fact that
G/H is para-hermitian, i.e. has a G-invariant splitting of the
tangent bundle into two isomorphic sub-bundles, allows us to build up
on G/H a symbolic calculus. It turns out that one can define a
G-covariant symbolic calculus on G/H generalizing the so-called
convolution first symbolic calculus on R^{2}.
In the present talk we discuss two different ring structures on the set of holomorphic discrete series. First one comes from the convolution of functions on the symmetric cone underlying the Hermitian symmetric space of tube type G/K. The second one is non-commutative and is induced by the composition of operators whose symbols belong to the discrete series representations of the causal symmetric space of Cayley type G/H. We also discuss the relationship that exists between these ring structures, intertwining operators for tensor products of holomorphic discrete series representations and the so-called Rankin-Cohen brackets. |
Date: | November 1 (Tue), 2005, 16:30-17:30 |
Room: | RIMS, Kyoto University : Room 402 |
Speaker: | Jun O'Hara (Tokyo Metropolitan University) |
Title: | Conformal geometry of curves |
Abstract: [pdf] |
We study the space S(q,n) of (q+2)-dimensional vector subspaces of the
(n+2)-dimensional Minkowski space which intersect the light cone transversely.
It is a subset of an indefinite Grassmann manifold.
This space can be identified with the space of q-spheres in S^{n}.
I will explain the notion of a pencil, which is one-parameter family of
codimension 1 spheres in S^{k}.
Using pencils, I will give a pseudoorthonormal basis S(q,n).
The pseudo-Riemannian structure of S(q,n) allows us to give an interpretation of the "infinitesimal cross ratio", which is a complex valued 2-form on the two point configuraion space of a knot K, K × K-(diagonal): The real part of it can be interpreted as an area element of a surface in S(q,n). |
Date: | November 8 (Tue), 2005, 18:00-19:30 |
Room: | RIMS, Kyoto University : Room 402 |
Speaker: | Ivan Cherednik (University of North Carolina) |
Title: | On unitary representations of rational dahas |
Abstract: [pdf] |
The talk will be an attempt to show the importance of the analytic aspects of the rational dahas via the the simplest case. I will begin with discussing the eigenvalue problem for the Dunkl operator (one x), and then consider in full detail a simple example, that may be viewed as a starting point of the generaly theory of unitary representations of the rational dahas. This example is directly connected with the harmonic analysis on SL(2,R) and OSP(2|1); Dunkl operator is a square root of the radial part of the Laplace operator in this case, a kind of the Dirac operator. |
Date: | November 11 (Fri), 2005, 10:30-11:30 |
Room: | RIMS, Kyoto University : Room 402 |
Speaker: | Simon Gindikin (Rutgers) |
Title: | Integral geometry and representations |
Abstract: [pdf] |
The old project of Gelfand was to obtain the harmonic analysis on
symmetric and other homogeneous spaces using the horospherical
transform. The realization of
this program had an enormous obstruction since the initial version
of the horospherical transform annulates discrete series of representations.
Several years ago I suggested a modified version - horospherical
Cauchy transform - which is nontrivial at least for some discrete series.
I want to consider several examples, starting of SL(2,R), and to discuss hoe realistic looks today this Gelfand's project. |
Date: | November 22 (Tue), 2005, 16:30-17:30 |
Room: | RIMS, Kyoto University : Room 402 |
Speaker: | Simon Gindikin (Rutgers) |
Title: | Holomorphic language for Cauchy-Riemann cohomology and representations |
Abstract: [pdf] |
It turns out that in many explicit computations with \bar ∂-cohomlogy it is unconvenient to use by the traditional languages of Cech or Dolbeault. It was developped a conception of smoothly parameterized Cech cohomolgy, different versions of which were developpped in the collaboration with Eastwood and Wang. One of possibilities is to build a purely holomorphic language. I will give a review of constructions and several illustrating examples basically from theory of reprentations. |
Date: | November 29 (Tue), 2005, 16:30-17:30 |
Room: | RIMS, Kyoto University : Room 402 |
Speaker: | Chifune Kai (Kyoto University) |
Title: | A Characterization of symmetric Siegel domains by convexity of Cayley transform images |
Abstract: [pdf] |
A homogeneous Siegel domain is a higher dimensional analogue of the
right (or upper) half plane, and is mapped to a bounded domain by the
Cayley transform. Among homogeneous Siegel domains, we have an important
subclass consisting of symmetric ones, which we characterize in this
talk using the parametrized family of Cayley transforms defined by
Nomura. This family includes the Cayley transforms associated with the
Bergman kernel and Szegö kernel, and if the domain is symmetric,
(the inverse of) the Cayley transform introduced by Korányi and Wolf.
In this talk, we show that the Cayley transform image is convex if and only if the domain is symmetric and the parameter is a specific one so that the Cayley transform coincides with the Korányi-Wolf Cayley transform. |
Date: | December 20 (Tue), 2005, 16:30-17:30 |
Room: | RIMS, Kyoto University : Room 402 |
Speaker: | Joachim Hilgert (Paderborn) |
Title: | Hecke Operators on Spaces of Period Functions |
Abstract: [pdf] |
Period functions are the analog of period polynomials, wellknown in the theory of modular forms, for Maass cusp forms. Before they were considered in the context of Maass forms they came up in physicists work on the transfer operator of geodesic flows on hyperbolic surfaces. The structure of these operators suggested the existence of Hecke type operators on spaces of period functions. It turns out that the constructions given in the context of transfer operators can be related to the standard Hecke operators on Maass forms in a precise way. |
Date: | February 21 (Tue), 2006, 16:30-17:30 |
Room: | RIMS, Kyoto University : Room 402 |
Speaker: | Hubert Rubenthaler (IRMA, Strasbourg) |
Title: | Local Zeta functions for a class of real symmetric spaces |
Abstract: [pdf] |
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. |
Date: | February 21 (Tue), 2006, 13:30-15:30 |
Room: | RIMS, Kyoto University : Room 402 |
Speaker: | Hidenori Fujiwara (藤原英徳) (Kinki University) |
Title: | 指数型可解リー群のユニタリ表現 I |
Abstract: [pdf] |
1970年代初頭 Auslander-Kostant は軌道の方法を用いて，連結かつ単
連結な型可解リー群のユニタリ双対を構成することに成功し，この結果は
Pukanszky により非 I 型の可解リー群に拡張された．これらの仕事は可解
リー群の表現論における画期的な成果である．ただ，正則誘導表現やその
応用を詳しく研究することは今でも困難である．
例えば，誘導表現や部分群へ制限された表現について，既約分解し，繋絡 作用素を構成し，また関連する不変微分作用素環を調べたい．このような 状況に直面すると，たとえ指数型可解リー群の場合ですら，我々はごく僅 かなことしか知らない．より多くの道具を手にできるのは冪零リー群に対 してのみである．リー群の表現論は半単純リー群と可解リー群の間でかな り異なった様相を見せている. 半単純リー群の豊富な代数構造は多くの研 究材料と結果を提供し，可解リー群の貧弱な構造は帰納法を唯一の有効な 手段としている．いずれにしても可解リー群のユニタリ表現論において軌 道の方法が非常に実り多いことは疑いのないところである．既約ユニタリ 表現に余随伴軌道を対応させるという Kirillov の革新的なアイデアはその 価値ある成果の数々を誇っているように見える．それは Mackey 理論の可 解リー群への見事な応用であるが，ひとたびこの枠組みが採用されると， 解析学における多くの対象物を余随伴軌道の代数的または幾何的性質を用 いて研究することができる． 冪零リー群の場合を中心に指数型可解リー群に対する軌道の方法の現状を 紹介する． |
Date: | February 22 (Wed), 2006, 15:30-17:30 |
Room: | RIMS, Kyoto University : Room 005 |
Speaker: | Hidenori Fujiwara (藤原英徳) (Kinki University) |
Date: | February 23 (Thu), 2006, 15:30-17:30 |
Room: | RIMS, Kyoto University : Room 005 |
Speaker: | Hidenori Fujiwara (藤原英徳) (Kinki University) |
Date: | February 24 (Fri), 2006, 10:00-12:00 |
Room: | RIMS, Kyoto University : Room 402 |
Speaker: | Hidenori Fujiwara (藤原英徳) (Kinki University) |
Date: | March 17 (Fri), 2006, 17:00-18:00 |
Room: | RIMS, Kyoto University : Room 402 |
Speaker: | Peter E. Trapa (University of Utah) |
Title: | Shimura correspondences for split real groups |
Abstract: [pdf] |
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. |
Organizer: | Toshiyuki Kobayashi |
© Toshiyuki Kobayashi