## 8/4 Sigurdur Helgason (MIT)

[pdf]  Date: August 4 (Fri), 2006, 10:30--11:30 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: 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. いつもと時刻が違いますのでご注意ください。

## 7/25 Gestur Ólafsson

[pdf]  Date: July 25 (Tue), 2006, 16:30--17:30 Room: RIMS, Kyoto University : Room 402 Speaker: Gestur Ólafsson (Louisiana State University) Title: The Image of the Segal-Bargmann transform Symmetric Spaces and generalizations 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.

## 6/20 Pavle Pandzic

[pdf]  Date: June 20 (Tue), 2006, 16:30--17:30 Room: RIMS, Kyoto University : Room 402 Speaker: Pavle Pandzic (University of Zagreb) Title: A simple proof of Bernstein-Lunts equivalence 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.

## 5/15 Anthony Dooley

[pdf]  Date: May 15 (Mon), 2006, 16:30--17:30, 17:45--18:45 Room: RIMS, Kyoto University : Room 202 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.

## 4/18 Toshihiko Matsuki

 Date: April 18 (Tue), 2006, 16:30--17:30 Room: RIMS, Kyoto University : Room 402 Speaker: 松木敏彦氏 (Toshihiko Matsuki, 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 group-theoretical methods.

## 3/17 Peter E. Trapa

 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.

## 2/24 （４回目） Hidenori Fujiwara

 Date: February 24 (Fri), 2006, 10:00-12:00 Room: RIMS, Kyoto University : Room 402 Speaker: 藤原英徳氏 (Hidenori Fujiwara) (Kinki University)

## 2/23 （３回目） Hidenori Fujiwara

 Date: February 23 (Thu), 2006, 15:30-17:30 Room: RIMS, Kyoto University : Room 005 Speaker: 藤原英徳氏 (Hidenori Fujiwara) (Kinki University)

## 2/22 （２回目） Hidenori Fujiwara

 Date: February 22 (Wed), 2006, 15:30-17:30 Room: RIMS, Kyoto University : Room 005 Speaker: 藤原英徳氏 (Hidenori Fujiwara) (Kinki University)

## 2/21 Hidenori Fujiwara

 Date: February 21 (Tue), 2006, 13:30-15:30 Room: RIMS, Kyoto University : Room 402 Speaker: 藤原英徳氏 (Hidenori Fujiwara) (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Ⅰ 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． （藤原先生には、２月２４日まで連続講演していただく予定です。２月２２日以降の 日時は追ってご連絡します）

## 2/21 Hubert Rubenthaler

 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.