## Lie Groups and Representation Theory Seminar at the University of Tokyo 1996

 Date: April 16 (Tue), 1996, 16:30-17:30 Place: Room 122, Graduate School of Mathematical Sciences, Komaba Speaker: Yoshitaka Koyama (RF) (RIMS) Title: Bosonization of Admissible Representations of $U_q(\widehat{sl}_2)$ at Level $-\frac{1}{2}$ and $q$-Vertex Operators Abstract: $U_q(\widehat{sl}_2)$ ̃x $-\frac{1}{2}$ Admissible\ƃxNg\ɕt q-vertex operator Boson pč\B Date: May 7 (Tue), 1996, 16:30-18:00 Place: Room 122, Graduate School of Mathematical Sciences, Komaba Speaker: Kenji Taniguchi (J) (University of Tokyo) Title: pfn̍pf̈Ӑɂ Abstract: Weyl Qsςȉpfn̂QK̍pf (Laplacian) Ɖȍpf͂̌nɊ܂܂邩AƂɂĘ_B Date: June 11 (Tue), 1996, 16:30-18:00 Place: Room 122, Graduate School of Mathematical Sciences, Komaba Speaker: Manabu Yamaguchi (R w) (University of Tokyo) Title: ffQ̎ˉe\Ƃ̎wW Abstract: ffQ $G_{ m, n } = {\Bbb Z}/m{\Bbb Z}\wr{\frak S}n$ ̐ \͂悭mĂB$m$ Œ肵, $n$ 𓮂 family $\{ G_{ m, n } \}_n$ l, ɑ΂ ''wW'' ƌĂ΂̂, ̊\͑Ώ̊֐ $m$ Ke\ ςƓ^ɂȂ. , ˉe\܂ōLʂɂĘb . ̓Iɂ, wW͏̓^ Schur ֐ĩe\ ρjɑΉĂ邪, ˉe\̏ꍇ, Schur Q ֐ĩe \ρjɑΉĂ. Date: June 18 (Tue), 1996, 16:30-18:00 Place: Room 122, Graduate School of Mathematical Sciences, Komaba Speaker: Sergey Fomin (MIT) Title: Parametrizations of canonical bases and totally positive matrices This lecture is supported by a grant from the Research Institute of Aoyama Gakuin University. Abstract: Let N be the group of n × n unipotent upper-triangular matrices. G. Lusztig discovered a close connection between the following two problems: studying the piecewise-linear maps that relate different labelings of the canonical basis in the quantized enveloping algebra of the Lie algebra of N; studying the variety of totally positive elements of N. Jointly with A. Berenstein and A. Zelevinsky, we obtained: explicit closed formulas for these piecewise-linear maps; formulas for decomposing an upper-triangular matrix into a product of elementary Jacobi matrices, and related total positivity criteria. Date: June 25 (Tue), 1996, 16:30-18:00 Place: Room 122, Graduate School of Mathematical Sciences, Komaba Speaker: Kazuhiro Hikami (aO) (Dept. Physics, Faculty of Science, University of Tokyo) Title: Topics in Long-Ranged Interacting Models Abstract: Recent developments in the theory of one-dimensional integrable systems with inverse square interactions are reviewed. the infinite dimensional representation for solutions of the Yang-Baxter equation and the reflection equation is given. Based on the representation, the elliptic Dunkl operators associated with the classical root systems are constructed. The Macdonald polynomial and its generalization are discussed in connection with the eigenstates for the trigonometric case. Classical case, Gaudin magnets, ..... etc. Date: July 2 (Tue), 1996, 16:30-18:00 Place: Room 122, Graduate School of Mathematical Sciences, Komaba Speaker: Hiroyuki Ochiai ([V) (Rikkyo University) Title: Euler-Poisson-Darboux $sl_2$ Ώ̐ Abstract: Euler-Poisson-Darboux ̉ɂ͖I $sl_2$ Ώ̐ȊOɏ~ZqɂΏ̐B $sl_2$ ̑Ώ̐ŕ\AX E-P-D pf ̂ $sl_2$ ̌tŏ\B Date: July 9 (Tue), 1996, 16:30-18:00 Place: Room 122, Graduate School of Mathematical Sciences, Komaba Speaker: Nobukazu Shimeno (M) (Okayama University of Science) Title: Ώ̋ $U(p,q)/(U(p)\times U(q))$ ̎X̋Eɑ΂鋫El Abstract: Ώ̋ $U(p,q)/(U(p)\times U(q))$ ̎X̋E ɕsމn\ Poisson ϊlC ̑ÂnɂĂbD n͑哇Yɂ̓Iɍ\ꂽ $U(\frak g\frak l(p+q,\Bbb C))$ ̍CfAɂ ^D܂ Shilov E̓ʂȋEɂ ĒmĂ錋ʂƂ̊֘AɂĂqׂD Date: October 8 (Tue), 1996, 16:30-18:00 Place: Room 122, Graduate School of Mathematical Sciences, Komaba Speaker: Toshiyuki Kobayashi (яrs) (University of Tokyo) Title: j^\̐Ƃ̉pɂ Abstract: 傫ȌQ G ̊񃆃j^\ȌQ H iQjɐƁCʂɊł͂ȂCH ̊񃆃j^\̒ϕiUXyNgƘAXyNgjƂĕDł́CG H Ȗ񃊁[Q̏ꍇl邪C̏ꍇł񕪉ǂ̂悤ɂȂ邩𒲂ׂ鎖͈ʂɔɓłD UXyNgȂ悤ȏꍇ͂ƂȒPłCLxȗႪ邱Ƃ𐔔NOɏЉCꂽ\ [Invent. Math. '94] ͕Kvł邱Ƃׂ̂D ɁCj^\̐Ɋւʘ_̐VpƂāC ǏΏ̋Ԃ̃g|W[ — [}Ώ̋Ԃ̎ZpƂē modular symbols ̏Œ藝 iʂȏꍇƂāC_𓮋@ƂDcFK̗\zؖj L2-a — Ώ̂Ƃ͌ȂȖ^Ԃ̗Un\̍\ Matsuki decomposition ̉p ȂǂTD Date: October 15 (Tue), 1996, 16:30-17:30 Place: Room 122, Graduate School of Mathematical Sciences, Komaba Speaker: Nanhua Xi (RIMS, Institute of Mathematics, Academia Sinica) Title: Irreducible modules of quantized enveloping algebras at roots of 1 Abstract: We give a concrete realization of finite dimensional irreducible modules of quantized enveloping algebras at roots of 1 and of irreducible rational modules of a semisimple algebraic group over an algebraically closed field of prime characteristic. The realization should lead to a better understanding of the irreducible modules. Date: October 22 (Tue), 1996, 16:30-18:00 Place: Room 122, Graduate School of Mathematical Sciences, Komaba Speaker: Yoshihisa Saito (ꎓv) (RIMS) Title: crystal base Weyl Q̕\_ւ̉pɂāi^̏ꍇj Abstract: Beilinson-Bernstein Ήɂ $frac g$ QɑΉl ̂cQlA̓l̂lB̓l͈̂ʂɂ͊ ȂȂA^̏ꍇɂ͊ł낤Ƃ Kazhdan-Lusztig ɂ\z B̖ɑ΂ crystal base ̗_p邱ƂɂAႪ ݂邱Ƃ̂ŕ񍐂B_ł͊񐫂̔ crystal base ̌tŏƂ͂łȂB Date: November 12 (Tue), 1996, 16:30-18:00 Place: Room 122, Graduate School of Mathematical Sciences, Komaba Speaker: Zhu Chengbo (National University of Singapore) Title: Degenerate Principal Series and Local Theta Correspondence Abstract: In this talk, we study some natural $U(n,n)$ (Respectively $Sp(2n,\Bbb R)$) modules which arise from the formalism of theta correspondence. We show that they can be imbedded into some degenerate series representations and determine their module structure. Moreover, we show that certain irreducible constituents in the degenerate seies can be identified with these modules or their irreducible quotients, in a very precise way. Date: December 10 (Tue), 1996, 16:30-18:00 Place: Room 122, Graduate School of Mathematical Sciences, Komaba Speaker: Toshiyuki Kobayashi (яrs) (University of Tokyo) Title: Ȗ^l̗̂Un\ Abstract: G Ȗ񃊁[QCH ̕Qł G ɂĊȖƂDႦ΁C(G,H) = (GL(n,R), SO(p,q)), (Sp(m,R), GL(n,R)) Ȃǂ̓T^łD̂ƂCl G/H ɂ G-sϑx݂D̑xɊւāCG/H ̂Qϕ֐̂Ȃ Hilbert L2(G/H) ͎R G ̃j^\ƂȂDL2(G/H) ̕ԂɎ G/H ̊񃆃j^\ul G/H ̗Un\vƌD Ql̂┼PΏ̋Ԃ̗Un\̑ݏ́CHarish-ChandraCFlensted-JensenC哇-؂ȂǂɂăNɂ锻mĂ邪Cƈʂ̊Ȗ񓙎l̂ɂẮCsl Stiefel l̂ȂǁCꂽꍇāCUn\݂邩ǂ͒mĂȂD Z~i[ł́CѓcC؎ɂČĂQ involutions ɂ郊[Q̗̊􉽂pċOԏ̒a͂l@Dj^\́uU\ȕ򑥁v̌ʂp邱ƂɂāCUn\Ώ̂ȓl̂ɂ ̐VႪ邱ƂbD Date: January 28 (Tue), 1997, 15:00-16:30 Place: Room 122, Graduate School of Mathematical Sciences, Komaba Speaker: Yutaka Saburi ( L) (Chiba Junior College) Title: RpNg [Q̃t[G_̕f͓Il@ Abstract: RpNg [Q͓̎Iђ t[GWJ̌W̑xɂt, {-FC̒藝ɊÂf͓Iȕ@ ĂƂbłB Date: January 28 (Tue), 1997, 16:40-18:10 Place: Room 122, Graduate School of Mathematical Sciences, Komaba Speaker: Yuji Satoh (E) (University of Tokyo) Title: String theory in three dimensional black hole geometry Abstract: RʑΘ_ɂ black hole AߔN Banados ɂ ꂽB̎͋ǏIɂ͂R anti-de Sitter space Ƃ ȒPȍ\Ă邽߁Ablack hole ̐TLpȃf ĂBɁÂR black hole ͌_̌Ȕwi ł邱ƂmĂBZ~i[ł́AR black hole wǐ _ꗝ_̗ꂩc_B́AdRx black hole wiŗʎqŏ݂̎łBRȎ󒆂̌_ȂǁA _̊{IȂƂɂȒPɐG\łB