Workshop on Integral Geometry and Group Representations

August 5 (Wed), 2009 - August 10 (Mon), 2009

Tambara Institute of mathematical Sciences, the University of Tokyo
(Tambara, Numata, Gumma)

Participants from abroad:
Mike Eastwood, Sigurdur Helgason, Angela Pasquale, Henrik Schlichtkrull

Fulton B. Gonzalez, Tomoyuki Kakehi, Toshiyuki Kobayashi, Toshio Oshima

      9:30-10:30   11:00-12:00  14:00-15:00  15:20-16:20  16:40-17:40  20:00-20:30
8/ 5 --------------------------------------------------------------   Discussion
8/ 6 Helgason1      Eastwood1    Pasquale1    Kaneyuki    Yoshino
8/ 7 Schlichtkrull1 Pasquale     Kakehi       Nakata      Takeuchi    Kumahara
8/ 8 Kobayashi      Oshima              -- Free Discussion --
8/ 9 Gonzalez       Eastwood2    Helgason2    Hiroe       Uuganbayar
8/10 Abe            Schlihitkrull2

Sigurdur Helgason
 "Support theorems and injectivity for noncompact and compact symmetric spaces"
    In this lecture we prove support theorems relative to horocycles and their interior 
    for symmetric spaces of rank one.  We consider both the X-ray tranform and the 
    horocycle Radon transform.  We also discuss injectivity and and support questions 
    for the X-ray transform for the compact symmetric spaces.

Mike Eastwood
 "The X-ray transform on complex projective space"
    The X-ray transform on complex projective space is defined by integrating
    a function or tensor field over the closed geodesics.  I shall show how 
    this transform may be related to the ordinary Radon transform and hence 
    determine the kernel of the X-ray transform acting on symmetric tensor fields. 
    The main ingredients are from representation theory and Lie algebra cohomology. 
    This is joint work with Hubert Goldschmidt.

Angela Pasquale
 "Analytic continuation of the resolvent of the Laplace-Beltrami operator on 
  symmetric spaces of the noncompact type"
    Let $\Delta$ be the Laplace-Beltrami operator on a symmetric space of the noncompact
    type $G/K$, and let $spec(\Delta)$ denote its spectrum. The resolvent 
    $R(z)=(\Delta-z)^{-1}$ is a holomorphic function on $\mathbb C \setminus spec(\Delta)$
    with values in the bounded operators on $L^2(G/K)$. We consider the meromorphic 
    continuation of $R$ as distribution valued map on a Riemann surface above 
    $\mathbb C \setminus spec(\Delta)$. In some examples, the singularities of the 
    continued resolvent possess a precise description in terms of representation theory 
    of $G$.

Soji Kaneyuki
  "On the linearity of causal automorphisms of symmetric cones"
    Let A be a simple Euclidean Jordan algebra of rank r. Then there are r+1 open 
    orbits V_0,...,V_r under the action of the identity component of the structure 
    group of A. V_0 is a Riemannian symmetric convex cone, and the others are affine 
    symmetric non-convex cones.  The closure C of V_0 is a causal cone.  Each V_i has 
    a natural causal structure obtained from the parallel transport of C.  We will 
    show that if dim A >2, then a diffeomorphism of V_i leaving the causal structure 
    invariant is necessarily a linear map.

Taro Yoshino
 "On a method to describe the topology of non-Hausdorff space"
    In this talk, we introduce a method to describe the topology of non-Hausdorff space.
    For a given non-Hausdorff space, we define a diagram consisting of Hausdorff spaces, 
    and see that the diagram determines the topology of the non-Hausdorf space completely.

Henrik Schlichtkrull
 "Holomorphic extension of eigenfunctions on Riemannian symmetric spaces"
    Let X=G/K be a Riemannian symmetric space of the non-compact type, and let
    X_C be its complexification. The COMPLEX CROWN Xi around X in X_C was 
    introduced by Akhiezer and Gindikin.  An important result in the complex 
    analysis for X asserts that every joint eigenfunction on X admits a holomorphic
    extension to Xi.  A new proof of this theorem will be presented, together with
    some generalizations (jt work with B. Kroez).

Tomoyuki Kakehi
 "Generalized Matrix Radon transform"

Fuminori Nakata
 "Wave equation, Funk transform, and the LeBrun-Mason twistor theory"
    We introduce integral transforms from functions on S^2 to functions on the de Sitter 
    3-space, and show that all the `tame' solutions of the wave equation on the de Sitter 
    3-space are obtained by using these integral transforms. This result is naturally 
    arisen in the context of the LeBrun-Mason twistor theory, which is also sketched in 
    this talk. 

Kiyoshi Takeuchi
 "Geometric Radon transforms and A-hypergeometric functions " (joint work with Y. Matsui)
    Geometric (topological) Radon transforms for constructible functions are closely 
    related to the projective duality in algebraic geometry. In particular we deduce a 
    degree (dimension) formula for A-discriminant varieties, i.e. the singular loci of 
    A-hypergeometric functions.  As a byproduct, we prove also a formula for the monodromy 
    at infinity of these functions. 

Keisaku Kumahara
 "On solutions of some type of ordinary linear differential equations”
   We state a new approach of representing fundamental solutions of homogeneous linear 
    differential equation $y^{(n)}=f(x)y$.  Using these solutions, we get (local) 
    expressions of soutions of linear differential equations of second order.

Toshiyuki Kobayashi
 "Geometric Analysis on Minimal Representations. Representation"
    Minimal representations are the smallest infinite dimensional unitary representations.
    The Weil representation for the metaplectic group, which plays a prominent role in 
    number theory, is a classic example.
    We may consider that minimal representations (from the viewpoint of groups) as 
    ''maximal symmetries (from the viewpoint of representation spaces)'', and thus 
    propose to use minimal reprn as a guiding principle to find new interactions with other
    fields of mathematics.

    Highlighting geometric analysis on minimal representations of O(p,q), I plan to discuss
    conservative quantities of ultrahyperbolic equations, the generalization of the Fourier
    -Hankel transform on the L2-model, and its deformation.

Toshio Oshima
 "Fractional calculus of Weyl algebra and its applications"
    Restrictions of zonal spherical functions and Heckman-Opdam's hypergeometric functions
    on one-dimensional singular lines through the origin satisfy interesting ordinary 
    differential equations.  By a unifying study of ordinary differential equations on 
    the Riemann sphere we have a global structure of their solutions and for example we 
    have a new proof of Gindikin-Karpelevic formula of c-functions and the Gauss 
    summation formula of Heckman-Opdam's hypergeometric functions.

Fulton B. Gonzalez
 "Conical Distributions on the Space of Flat Horocycles"

Kazuki Hiroe
 "Generalized Whittaker functions of GL(4,R) and Horn's hypergeometric functions"

Uuganbayar Zunderiya
 "Generalized Gelfand hypergeometric systems" (joint work with H. Ochiai)
    We give a combinatorial formula of the dimension of global solutions to a 
    generalization of Gelfand hypergeometric system, where the quadratic differential 
    operators are replaced by higher order operators. We also derive a polynomial 
    estimate of the dimension of global solutions for the case in 3x3 variables. 
    We show that the space of solutions of this generalized system near a generic point 
    is infinite dimensional for some cases. 

Noriyuki Abe
  "Jacquet modules of parabolic induction"
    The notion of Jacquet modules was introduced by Casselman. In this talk we 
    introduce some filtration of this generalized Jacquet modules of parabolic 
    induction and investigate this filtration.  Roughly speaking, the successive 
    quotient of this filtration is given by ``twisted induction''.

Hisaichi Midorikawa
Takaaki Nomura
Hiroyuki Ochiai
Toshihiko Matsuki
Nobukazu Shimeno
Kenji Taniguchi
Shigeru Aoki
Takeyoshi Kogiso
Chifune Kai
Atsushi Yamamori
Yoshiki Oshima


Mini-Conference "Integral Geometry and Representation Theory"
場所:東京大学大学院数理科学研究科 002号室

10:00-11:00  Sigurdur Helgason
  "Radon Transforms and some Applications"

10:20-12:20  Fulton Gonzalez
  "Mutlitemporal Wave Equations : Mean Value Solutions"

14:00-15:00  Angela Pasquale
  "Analytic continuation of the resolvent of the Laplacian in the Euclidean setting"
    We discuss the analytic continuation of the resolvent of the Laplace operator on
    symmetric spaces of the Euclidean type and some generalizations to the rational Dunkl 

15:30-16:30  Henrik Schlihitkrull
  "Decay of smooth vectors for regular representation"
    Let G/H be a homogeneous space of a Lie group, and consider the regular representation 
    L of G on E=L^p(G/H).  A smooth vector for L is a function f in E such that g mapsto L(g)
    f is smooth, G to E. We investigate circumstances under which all such functions decay 
    at infinity (jt with B. Krötz)
      東京    上野    大宮   高崎  上毛高原   セミナーハウス着
 A.  11:52   11:58   12:18  12:52   13:09       14時過ぎ
 B.  15:32  15:38   15:58  16:27   16:44       17時40分頃
高崎線       10:44   11:09  12:38 (高崎線)

      渋谷    新宿    池袋   赤羽   大宮  高崎
     10:07   10:12   10:18  10:28  10:42  11:56 (湘南新宿ライン)
     11:17   11:25   11:30  11:38  11:57        (A案:埼京線快速)

上毛高原から玉原へは,上の A. B. の新幹線の到着時間に合わせてタクシーを手配予定.



  宿泊室:宿泊棟に,3名定員の部屋が9室ある(その他,スタッフ室あり. 40名程度まで宿泊可)
  浴室  :2室(大・小),シャンプー・リンス・ドライヤー,シャワールームあり
  設備  :黒板,ホワイトボード,OHP, 液晶プロジェクタ,プリンタ,コピー機,パソコン,無線LAN
  食事  : 朝食 7:30 - 8:30(バイキング形式)    昼食 12:30 - 13:30     夕食 18:00 - 20:00

世話人: 大島利雄