## Lie Groups and Representation Theory Seminar at the University of Tokyo 1997

 Date: May 7 (Tue), 1997, 16:30-18:00 Place: Room 122, Graduate School of Mathematical Sciences, Komaba Speaker: Kenji Taniguchi (J) (University of Tokyo) Title: Commutants of r -2-type Hamiltonian Abstract: r -2 ^̃|eVn~gjAƉȔpf̑ɂāA݂܂łɒmĂ邱ƂTB Ƀ|eV֐L֐̏ꍇɁADunkl pfp sl(2) oscillator \̗ގ邱Ƃɂ肱̑(̈ꕔ)R ɍ\B Weyl Qsςȉpfn̂QK̍pf(Laplacian)Ɖȍpf͂̌nɊ܂܂邩AƂɂĘ_B Date: May 13 (Tue), 1997, 16:30-18:00 Place: Room 122, Graduate School of Mathematical Sciences, Komaba Speaker: Takahiro Hayata (cF) (University of Tokyo) Title: Ώ̑ (SU(2,2), Sp(2,R)) ̐VJ֐ Abstract: $G$ 𔼒P[Q, $\pi$ admissible \, involution ŌŒ肳镔Q $H$ Ƃ̃j^\ $\eta$ ƂƂ, $\rom{Hom}\,(\pi, C_{\eta}^{\infty}(H\backslash G))$ 邢, $K$ LxNĝɂ鑜 ł, Ώ̑ $(G,H)$ ̐VJ֐ƌĂł. $\eta$ Pʕ\̎ɂ, Flensted-Jensen ɂ錋ʂ , ł, $(G,H)$ ^Cg̑Ώ̑΂̂Ƃ, $\eta$ Ƃ ĐUnƂꍇ̌vZЉ. (Survey seminar) Date: May 20 (Tue), 1997, 16:30-18:00 Place: Room 122, Graduate School of Mathematical Sciences, Komaba Speaker: Hisayosi Matumoto ({v) (University of Tokyo) Title: Howe duality ̉ (noncompact case) Abstract: ̏ reductive dual pair ɂĂ Howe duality łʓI 󋵂̂ĊmAR. Howe ̘_ Transcending classical invariant theory, J. of Amer. Math. Soc. 2 (1989), 535-552. ̏Љ BN̐_T}[XN[łׂ̂ȂA藝̏ؖ̕j Ďɉ\łB (Joint seminar of Geometry Colloquium-Lie Groups and Representation Theory Seminar) Date: May 22 (Thu), 1997, 16:30-18:30 Place: Room 128, Graduate School of Mathematical Sciences, Komaba Speaker: Toshiyuki Kobayashi (яrs) (University of Tokyo) Title: [[}l̂ɂsAQƕό Abstract: ̂Q̘bЉ\łF [[}l̂ɂŗLsA̔ [[} RpNg NtH[h=NC̘Aό̈ [[}l̂ɓϊƂčp闣UQ͕KŗLsAɍpȂD̎́C[cl̂ɂJr=}NXہi1962jɂĂ悤ɁCǏIɋ[[}l̂Ɠ^ł悤ȑl̂̊{Qɂ͋񂪂肤邱ƂɑΉĂD(1) ł͌ŗLsA̔𔼒P[Q̍\pďؖD C񃊁[}Ώ̋Ԃ̃RpNgȃNtH[h=NCŘAόł̂́CQCȂ킿[}ʂ̏ꍇɌiSelberg-Weil ̌ÓTIȋǏ藝jC[[}vʂꍇ͍łAόłႪ݂邱Ƃ킩D(1) ̔p, Ǐό̗ʓI]^D ɁCRȗ[cl̂ɂ Goldman ̒NƗ\zi1985j́ił̈ʉ܂߂j莮Ƃ̏ؖ^D Date: May 27 (Tue), 1997, 16:30-18:00 Place: Room 122, Graduate School of Mathematical Sciences, Komaba Speaker: Tibor Ódor (University of Tokyo & University of Szeged (Hungary); Mathematical Institute of the Hungarian Academy of Sciences) Title: On the Pompeiu problem and its equivalent forms Abstract: Let $f$ be a continuous function and let $\Omega$ be a bounded open domain with connected Lipschitzian boundary in the $d$ dimensional real vector space $R^d$ having the property that the integral of the function $f$ is zero on every congruent copy of the domain $\Omega$. If $f$ must be zero, then we say that the domain $\Omega$ has the Pompeiu property. The only known bounded open domains which fails to have the Pompeiu property are the balls. The longstanding conjecture of Raleigh, Pompeiu and Schiffer that there are no other $\Omega$'s failing the Pompeiu property. The problem has non trivial connections to Lie group representations, overdetermined Neumann problems and even to combinatorics. It has several equivalent forms. In the first part of the lecture we overview some of the most important results, and in the second part I would like to talk about my work on the problem. Date: June 3 (Tue), 1997, 16:30-18:00 Speaker: Soji Kaneyuki (ss) (Sophia University) Title: pG~[gΏ̋Ԃ̎ȓ^Q Abstract: A (simple) parahermitian symmetric space is a simple symmetric space which admits an invariant symplectic structure and a pair of invariant, completely integrable Lagrangian distributions. By the automorphism group of such a space, we mean the group of diffeomorphisms which leave the two Lagrangian distributions invariant. A fundamental problem is to determine the automorphism group for each simple parahermitian symmetric space. This was settled by Noburu Tanaka for classical parahermitian symmetric spaces. In this talk, by using graded Lie algebras, we solve the problem for arbitrary simple parahermitian symmetric spaces. Date: June 10 (Tue), 1997, 16:30-18:00 Speaker: Hiroshi Tamaru (c۔m) (Sophia University) Title: Orbit types of s-representations and graded Lie algebras Abstract: PΏ̋Ԃ̐ isotropy \ s-representation ƌĂ. S-representation ̑SĂ̋ǏIȋO^, ܂SĂ̌Œ Q Lie , Ώ̋Ԃ̐[gnpČł. ܂l̕@ɂ, e Graded Lie ̂Q̕ [gn݂̂猈ł. Date: June 17 (Tue), 1997, 17:00-18:30 Speaker: Ian Grojnowski Title: Affine Hecke algebras and their representations Abstract: I will define the affine Hecke algebra associated to a reductive group G, and describe the parametrisation and dimensions of its irreducible representations, even at roots of unity. Date: June 24 (Tue), 1997, 17:00-18:30 Speaker: Toshihiko Matsuki (ؕqF) (Kyoto University) Title: [Q̂Q involution Ɋւ闼]ޕ (noncompact case) Abstract: G reductive [QAH L G ̑Ώ̕QiG ̂ involution ւŒ蕔Q̊JQjƂƂA]ޕ H\G/L ---- (*) lBG, H, L ̂ꂩRpNĝƂ́A"maximal torus" A p G=HAL ƕ\BłȂƂA(*) ̍\͈ʂɕGł邪A [Q̋ނ̗_Ɠl Jordan ACartan "subsets" pċLq łBȒPȗpĂl@B Date: July 1 (Tue), 1997, 16:30-18:00 Speaker: Toshinori Oaku (刢vr) (Yokohama City University) Title: Computing with D-modules using Groebner bases --- a user's guide to Kan Abstract: D-Q(Δn)̗_ł͓ĺAdxA b-֐(indicial polynomial)Ȃǂ̕sϗʂAAϕAe\ρA local cohomology Ȃǂ̎X functor dvȖB ]̓IɌvZ邱Ƃ͍łA ߔNpfɂ Groebner p邱ƂɂāA ̂vZASY邱Ƃ킩ĂB Groebner ɂēĺAb-֐AȂǂǂ vZł邩A_IȃASYƁARMBɂč쐬ꂽ VXe Kan pۂ̌vZ@̗ʂB Date: October 7 (Tue), 1997, 16:30-17:30 Place: Room 202, RIMS, Kyoto University Speaker: Eric Opdam Title: On the Knizhnik Zamolodchikov equations for complex reflection groups Date: October 9 (Thu), 1997, 16:30-17:30 Place: Room 102, RIMS, Kyoto University Speaker: Grigori Olshanski Title: Shifted Schur functions and their applications Date: October 14 (Tue), 1997, 16:30-17:30 Place: Room 202, RIMS, Kyoto University Speaker: Eric Opdam Title: Symmetries for fake degrees of complex reflection groups Abstract: The second talk is an application of the first, and in the first talk I can explain some topological results of Broue, Malle, and Rouquier for orbit spaces, and results/conjectures for Dunkl theory for complex reflection groups. Date: November 11 (Tue), 1997, 16:30-17:30 Place: Room 202, RIMS, Kyoto University Speaker: Grigori Olshanski Title: Applications of shifted symmetric functions in representation theory Date: November 13 (Thu), 1997, 16:30-17:30 Place: Room 102, RIMS, Kyoto University Speaker: Andreas Nilsson Title: Multipliers on symmetric spaces Date: November 25 (Tue), 1997, 16:30-17:30 Place: Room 202, RIMS, Kyoto University Speaker: Eric Opdam Title: Residue calculus for root systems Date: December 9 (Tue), 1997, 16:00-17:00 Speaker: Andreas Nilsson (RIMS) Title: Fourier multipliers on pseudo-Riemannian symmetric spaces Abstract: In this talk we will consider multipliers for K-invariant functions on non-Riemannian symmetric spaces. They seem to have similar behavior as the multipliers in the Riemannian case, even if only partial results are known so far. We will give sufficient and also in some cases necessary conditions for Lp-boundedness of the corresponding operators. Date: December 9 (Tue), 1997, 17:15-18:15 December 11 (Thu), 1997, 16:30-18:00 Place: Room 122, Graduate School of Mathematical Sciences, Komaba Speaker: Grigori Olshanski (RIMS, Kyoto Univ., and Dobrushin Math. Lab., IPIT, Moscow) Title: Distinguished central basis for enveloping algebras of classical type and Capelli identities (2 lectures) Abstract: Let $G$ be any of the complex classical groups $GL(n)$, $O(2n+1)$, $Sp(2n)$, $O(2n)$, and let $Z$ be the center of the corresponding universal enveloping algebra. The purpose of the talks is to discuss a distinguished linear basis in the algebra $Z$. This basis arises, in a natural way, from the binomial formula for $G$ (a Taylor--type expansion of finite--dimensional irreducible characters of $G$). The eigenvalues of the basis elements in irreducible $G$-modules are certain factorial Schur polynomials --- new combinatorial functions, discovered by Biedenharn and Louck and studied by Macdonald and other authors. One of the remarkable properties of the distinguished basis is its connection with generalized Capelli identities. Date: December 19 (Fri), 1997, 16:30-18:00 Place: Room 118, Graduate School of Mathematical Sciences, Komaba Speaker: Corrado Marastoni (University of Padova) Title: Integral transforms for sheaves and D-modules: Radon-Penrose correspondence, Grassmann duality and the action of real forms of SL_n(C) Abstract: after introducing the general framework of integral transforms in the language of sheaves and D-modules, I will concentrate on Grassmannians to study the Radon- Penrose p-plane correspondence (joint work with A. D'Agnolo) and the "Grassmann duality" (my recent Ph.D. thesis at Paris VI). The final aim is to examine the action of real forms of SL_n(C) on these transforms, and to discuss the links with representation theory. Date: January 26 (Mon)-28 (Wed), 1998, 14:40-16:10 Speaker: Toshiyuki Kobayashi (яrs) (University of Tokyo) Title: Unitary Highest Weight Modules and Multiplicity Free Branching Laws (3 lectures) Abstract: Đj̗_p G-ςȐ̐Ȑؒf̋ԂɎꂽj^\dxPŕ邽߂̈ʓIȏ\ؖB z肵Ă󋵂͌Q G ̒Ԃւ̍pړIƂ͌ȂʓI ݒłBp̊􉽓IȐƂ炦邱Ƃɂ, G-pɊւ S̉ؖB̊{IȃACfBA Gelfand, Faraut ȂǂɑkB P[Q̃j^ōEFCg\ɂĊȒPȃT[xCB Weil \␳Un\, 邢 RpNgQ̕\Ȃǂ̗łB ōEFCg\̕Qւ̐ɊւĂ, Howe dual pair ̗_, , Vergne, Jakobsen, Kostant, Schmid, Adams ̑̋̓IȌvZ mĂB ܂L\̏dxRȕ򑥂ɊւĂ, ΐ-R pc̕ (ÓT^) Ȃǂ̑gݍ킹_̗ꂩ̋̓I mĂB ̍uł Ώ̑΂Ɋւ鐧 () _B ܂ʓIȏ󋵂ŏdxRȕ򑥂߂̊􉽓Iȏ\ؖ, ̎@ŏؖdxR̕򑥂^ݒ̃Xg^ (ɖ̏ꍇ邪, L\̏ꍇV܂)B ɎԂ, Kostant-Schmid ̌̈ʉ (ɑRpNg Q-->RpNgQ) ƂȂ镪򑥂^̏ؖЉB Date: February 3 (Tue), 1998, 16:30-18:00 Speaker: Jean-Pierre Labesse (Centre International de Rencontres Mathématiques) Title: Selberg trace formula and index theorem Abstract: I compare the Selberg trace formula and the Index theorem; I show that both objects coincide for invariant elliptic operators on locally symetric manifolds (this is more or less well known) using that pseudo-coefficients are closely related to heat-kernels techniques. The point is that various generalizations and problems arise naturally: in particular this gives $L^2$ index theorems if the manifold is only of finite volume; moreover one can consider equivariant index formula that correspond to twisted trace formula and also to the action of Hecke operators explicit computation of the various terms that show up is not known in general but examples show that this contains in particular twisted characters formula and also eta-invariants etc...