## Lie Groups and Representation Theory Seminar at the University of Tokyo 2000

 Date: April 18 (Tue), 2000, 16:30-17:30 Speaker: Toshiyuki Kobayashi (яrs) (University of Tokyo) Title: a generalization of the Kostant-Schmid formula Abstract: LEΏ̗̈̕fl̂Ɋւ􉽓IȐ, Đj̐ f邱Ƃɂ, ̒藝ؖł(): uXJ[^̍ōEFCg\̑Ώ̑΂Ɋւ镪򑥂͏dx1łv , SL(2) ɑ΂Clebsh-Gordan ̌≪cɂL\ ̕򑥂ɂdx1̐͂, Riemann Ώ̋Ԃ Hermite Ώ̋ԏ̒ɑ΂ Plancherel ^藝 (Helgason, Harish-Chandra, , Heckman) ɂ閳\̏dx1̐, GL_m x GL_n o΂ɂL\̏dx1̐Ȃǂ ɐ钊ۓIȒ藝łB ̂bł, Unɑ΂, ̕򑥂̓IȈʌƂ ؖB ꂽ, QRpNg̏ꍇ, Hua, Kostant, Schmid, Johson ɂēꂽ K-type ʉƂȂĂB ܂, QRpNgȏꍇ, e񐬕\ɂȂ, SU(2,2) Ȃǂʂ̏ꍇɂ Jakobsen, Vergne, Xie Xɂ 򑥂mĂB Date: May 16 (Tue), 2000, 16:30-17:30 Speaker: Yoshihisa Saito (֓v) (Hiroshima University) Title: Double loop 㐔̕\_̌ Abstract: P Lie 㐔 2 ϐ Laurant  tensor ς𒆐Sg債ē Lie 㐔 Double loop 㐔itoroidal Lie 㐔jƌĂԁB Double loop 㐔͒AP Lie 㐔 1 ϐ Laurant  tensor ς𒆐Sg債ē affine Lie 㐔̊gƌ鎖ł邪A̍\y \_͑SقȂl悷B uł Double loop 㐔̕\_ǂ affine Lie 㐔̕\_ƈقȂ ̂ɂȂĂ܂̂HƂn߂āA Double loop 㐔̕\_ ĉǂ܂ł킩Ă̂ЉB Date: May 23 (Tue), 2000, 16:30-18:00 Speaker: Tibor Odor (University of Tokyo) Title: On Kervaire's conjecture on equations over groups Abstract: We present some old and new results related to the longstanding Kerver conjecture over (that is in some extension of) groups. This conjecture is a group theoretic analogy of polynomial equations over algebraically closed fields. Date: June 6 (Tue), 2000, 16:30-18:00 Speaker: Hideyuki Ishi (ɎtpV) (Yokohama City University) Title: Siegel ̈̔ԂƔpf Abstract: Siegel ̈ D ƁȀ affine ϊƂĒPړIɍp  Lie Q G lB̈ D ̐ȂĐj Hilbert Ԃ Q G unitary \̂́Aoΐ̕Ɋ܂܂lXȋO Εsϑx Fourier - Laplace ϊɂč\B ̍uł́Aoΐ܂ރxNgԏɏs񎮂̈ʉɑ 񑽍AeȎ㐔Iȓt ̏̑Εsώ̍\ ĐBO҂̌ʂ璼ɔԂ̖A ҂ unitary \̊Ԃ intertwining pfB Date: June 13 (Tue), 2000, 16:30-18:00 Speaker: Toshio Oshima (哇Y) (University of Tokyo) Title: s̒Pq̗ʎq Abstract: $C^n$̐^ϊ n-s A ŕ\邪A̎Ɉ˂Ȃs ʂȂ킿 $J_A = \bigcup_{g\in G} Ad(g)A$ (G=GL(n,C)) JordanW^ Pq̗ʎqlBȂ킿A$J_A$̒A邢 G-sς CfAɑ΂Asς G ̔pf̃CfAΉBp [^ "ÓTɌ" ƓɍlđΉ𖾂炩ɂBCapelli identity, Verma module Ƃ̊֘AAŏ̗ʎqA\_Ȃǂւ̉p qׂB Date: June 20 (Tue), 2000, 16:30-18:00 Speaker: Hiroshi Oda (Dc ) (University of Tokyo) Title: [ɂ Generalized Capelli Operators ̃AiW[ Abstract: A ^ɂ哇 Generalized Capelli Operators ƓlȖ pfC̏ꍇɕāC$U({\mathfrak so}(n))$ ɍ\D ̓Iɂ́CPfaffian ^̍pfɂC o--o-- --x--x--x=>x o--o-- --x--x--x--x                  |                 x 邢́C x--x-- --x--x--x--o                  |                 x ̌̑މnɑΉ̂\łix މĂ郋[gjD Pfaffian ^ȊOɁCA^̏ꍇƂ悭 determinant ^̍pf邪C炩 炻̑̑މnɑΉ̂\łƍlD݌́C determinant ^̍pfɂĂC܂œʂ񍐂D Date: June 27 (Tue), 2000, 16:30-18:00 Speaker: Kenji Taniguchi (J) (Aoyama Gakuin University) Title: Weyl Qsςȉpf̈Ӑɂ Abstract: 哇--֌ɂĒׂꂽ Weyl QsςȔpf̉́A K̍pfiH ƂjƉȔp̂ȂAƂtقڂ B ł͂̓tłȂꍇlB ̓Iɂ @ H Ɖ W-sςłȂpf݂邽߂ H ̏ Â悤Ȃ̂̍\AɃ|eVȉ~֐łƂ́A "shift type pf" i[H, D] = a(x) D 𖞂pf Dj̋̓Iȍ\ ɂďqׂB Date: July 4 (Tue), 2000, 16:30-18:00 Speaker: Jae-Hyun Yang (Inha University and PIMS) Title: On the group $SL(2,{\Bbb R}) \ltimes {\Bbb R}^{(m,2)}$ Abstract: In this talk, we present some basic properties of the group $SL_{2,m}({\Bbb R}):=SL(2,{\Bbb R}) \ltimes {\Bbb R}^{(m,2)}$ and study automorphic forms related to the group $SL_{2,m}({\Bbb R}).$ Date: July 11 (Tue), 2000, 17:00-18:00 Speaker: Toshiyuki Tanisaki (JrV) (Hiroshima University) Title: Radon transforms on flag manifolds Abstract: I will talk about my joint work with C. Marastoni [Math. RT/9911095] concerning Radon transforms for quasi-equivariant $D$-modules on generalized flag manifolds. Date: September 13 (Wed), 2000, 16:30-18:00 Speaker: Gerhard Roehrle (Bielefeld University) Title: Spherical Orbits and Abelian Ideals Abstract: This is a report on recent joint work with Dmitri Panyushev \cite{PR}. Let $G$ be a reductive complex Lie group with Lie algebra $\Lie G = \frakg$. Let $B$ be a Borel subgroup with Lie algebra $\bb$. Let $P$ be a parabolic subgroup of $G$ containing $B$ with unipotent radical $P_u$. We denote the Lie algebra of $P$ and $P_u$ by $\mathfrak p$ and $\mathfrak p_u$, respectively. The group $P$ acts on any ideal of $\pp$ in $\pp_u$ by means of the adjoint representation. Let $\mathfrak a$ be an abelian ideal of $\pp$ in $\pp_u$. It was shown in \cite[Thm.\ 1.1]{Ro3} that $P$ operates on $\aaa$ with a finite number of orbits. The proof of this result in \cite{Ro3} involved long and tedious case by case considerations. The original proof of this theorem went as follows: it readily reduces to the case of a Borel subalgebra $\bb$ of $\frakg$. It then suffices to only consider the maximal abelian ideals of $\bb$. These were classified in \cite{Ro3} and then it was shown that the number of $B$--orbits is finite in each of these instances. Using the structure theory for spherical nilpotent orbits \cite{Pa4} we found a short conceptual proof of this fact. The finiteness result for the number of $P$--orbits on such an abelian ideal $\aaa$ is a consequence of one of the main results in \cite{PR}: namely that for $\aaa$ an abelian ideal in $\bb$ and $\OO$ any nilpotent orbit in $\frakg$ meeting $\aaa$ the orbit $\OO$ is a spherical $G$--variety. A $G$--variety is called {\em spherical}, whenever $B$ acts on it with an open dense orbit. By a fundamental theorem, due to M.\ Brion \cite{Br1} and E.B.\ Vinberg \cite{Vi2} independently, $B$ acts on a spherical $G$--variety with a finite number of orbits. Besides presenting this conceptual proof of the finiteness result, I shall exhibit a natural connection between abelian ideals of $\bb$ and $\ZZ$--gradings of $\frakg$. In particular, all the maximal abelian ideals of $\bb$ stem from certain $\ZZ$--gradings of $\frakg$. Further, I shall also address the set of maximal abelian ideals $\AAA_{max}$ of $\bb$. I shall present the classification of $\AAA_{max}$ from \cite{Ro3} and will explain the existence of a canonical bijection between $\AAA_{max}$ and the set of long simple roots \begin{thebibliography}{99} \bibitem{Br1} {\sc M.~Brion}, {\em Quelques propri\'et\'es des espaces homog\'enes sph\'eriques}, Man.\ Math.\ {\bf 99} (1986), 191--198. \bibitem{Pa4} D.~Panyushev, {\em On spherical nilpotent orbits and beyond}, Annales de l'Institut Fourier, {\bf 49}(5) (1999), 1453--1476. \bibitem{PR} {\sc D.~Panyushev, G.R\"ohrle}, {\em Spherical orbits and abelian Ideals}, Preprint 00--052 SFB 343, Diskrete Strukturen in der Mathematik, Universit\"at Bielefeld (2000). \bibitem{Ro3} {\sc G.~R\"ohrle}, {\em On Normal Abelian Subgroups of Parabolic groups}, Annales de l'Institut Fourier, {\bf 48}(5), (1998), 1455--1482. \bibitem{Vi2} {\sc E.\,B.~Vinberg}, {\em Complexity of actions of reductive groups}, Funkt. Analiz i Prilozhen. {\bf 20}(1986), {\rus N0}\,1, 1--13 (Russian). English translation: Funct.\ Anal.\ Appl.\ {\bf 20}, (1986), 1--11. \end{thebibliography} Date: December 5 (Tue), 2000, 16:30-18:00 Speaker: Kouichi Takemura (|) (Yokohama City University) Title: ȉ~^ Calogero-Moser ͌^̌ŗLlAŗL֐ɂ Abstract: ϕȗʎq͊w̖͌^łȉ~^ Calogero-Moser ͌^̌ŗLlA ŗL֐ Bethe ݖ@pĒׂƁABethe ꍇɁA LlAŗL֐OpɌœOpI Calogero-Sutherland ͌^̂ ƐɂȂĂ邱Ƃ킩B ܂ȀXƂ̋ɂAKato-Rellich ̐ۓ̗_Ȃǂ Kpł邱Ƃʂ̏ꍇŏL̐ۓ\B ɂ Jack ̑ȉ~ό̂P̌₪݂֐Ƃčl@ ƂłB Date: March 9 (Fri), 2001, 15:30-17:00 Speaker: Wolfgang Bertram (Université Nancy I) Title: From linear algebra via affine algebra to projective algebra Abstract: http://math1.uibk.ac.at/jordan/ vvg܂B ȉ́AɌfڂẴRs[łB We introduce an algebraic formalism, called "affine algebra", which corresponds to affine geometry over a field or ring K in a similar way as linear algebra corresponds to affine geometry with respect to a fixed base point. In a second step, we describe projective geometry over K by a similar formalism, called "projective algebra". We observe that this formalism not only applies to ordinary projective geometry, but also to several other geometries such as, e.g., Grassmannian geometry, Lagrangian geometry and conformal geometry. These are examples of `generalized projective geometries". The axiomatic definition and general theory of such geometries is given in Part II of this work.