## Lie Groups and Representation Theory Seminar at the University of Tokyo 2003

 Date: October 14 (Tue), 2003 Speaker: Hisayosi Matumoto ({v) (University of Tokyo) Title: scalar ^̈ʉꂽ Verma Q̊Ԃ̏^ɂ Abstract: fP Lie 㐔lB̕^㐔̊L\̗U\͈ʉꂽ Verma QƌBʉꂽ Verma Q̊Ԃ̏^𕪗ނ ʉꂽl̂̏̓σxNg̊Ԃ̓ϔpf𕪗ނ邱ƂƓłABaston ɂĎnꂽ Parabolic geometry ̊ϓ_悤łB ł́A^㐔̈ꎟ\̗U\̏ꍇɖčl@B(͒̏ꍇɂB) vȌʂ͈ȉ̒ʂłB (1) ^㐔ɑ̏ꍇ̏^̕ށB (2) (1)ő݂ꂽ^Ar藝ɂĈʂ̕^㐔̏ꍇɂ^\ł邱ƁB Date: October 21 (Tue), 2003, 17:00-18:00 Speaker: Joseph A. Wolf (University of California, Berkeley) Title: The Double Fibration Transform for Flag Domains Date: November 11 (Tue), 2003, 16:30-18:00 Speaker: Hiroshi Oda (Dc ) (Takushoku University) Title: ŏƈ Verma Q̗뉻CfA Abstract: fȖ Lie g ̈ Verma Q MΘ(λ) ̗뉻CfA Ann(MΘ(λ)) Ɋւ哇搶Ƃ̋D g ̒ȗL\ π ɑ΂Ē܂ U(g) ̐s F ́CAnn(MΘ(λ)) @Ƃŏ q(x;λ) Cp[^ λ generic ȏꍇɋLqD Ann(MΘ()) ͓̉ւ̉pƂƂɁCπ P Lie ̏ꍇ adjoint \Cminuscule \Cŏ\̏ꍇɂāC q(x) ̋̌LqD Date: January 20 (Tue), 2004, 16:30-18:00 Speaker: Pavle Pandzic (University of Zagreb & RIMS) Title: Dirac cohomology of Harish-Chandra modules Abstract: Let g be a semisimple complex Lie algebra and r a reductive subalgebra to which the Killing form restricts non-degenerately. Kostant has defined a cubic Dirac operator D attached to r. If X is a g-module, then D acts on X tensored with a space of spinors. Dirac cohomology of X is the kernel of D divided by (a part of) the image. It is an invariant similar to Lie algebra cohomology. Indeed, when r is a Levi subalgebra of a parabolic subalgebra with nilradical u, then one can relate D to u-homology and \bar u-cohomology differentials. I will present some results and some counterexamples regarding this relationship in case when X is a Harish-Chandra module for a pair (g,K) and the parabolic is θ-stable. Date: January 27 (Tue), 2004, 16:30-18:00 Speaker: Jiro Sekiguchi (֌Y) (Tokyo University of Agriculture and Technology) Title: On the geometry of unimordular congruence classes of bilinear forms (with D. Z. Djokovic and Kaiming Zhao) Abstract: Let V  be an n-dimensional vector space over an algebraically closed field K of characteristic 0. We consider the natural action of the unimodular group SLn  on the space B  of bilinear forms f:V~V → K. Denote by B/SLn  the categorical quotient, which is known to be an affine space of dimension m+1. We study the canonical projection :B → KB/ SLn  and its fibers. We prove that each fiber of , and in particular the zero fiber, i.e., the null-cone N = -1((0)),  is irreducible, reduced, and has dimension n2 - m - 1. Furthermore, we show that each fiber of contains a unique SLn-orbit which is open and dense in the fiber. Date: March 8 (Mon), 2004, 16:00-17:00 Speaker: Munibur R. Chowdhury (the University of Dhaka, Bangladesh) Title: Arthur Cayley and his contribution to abstract group theory Abstract: We critically reexamine in considerable detail Cayley's first three papers on group theory (1854-59), with special reference to his formulation of the (abstract) group concept. We show convincingly (we hope) that Cayley, writing his first paper on November 2, 1853, was in full and conscious possession of the abstract group concept, and that - as far as finite groups are concerned - his definition was complete and unequivocal, refuting opinion expressed by some earlier writers. Already in the first paper Cayley classified the abstract groups of orders upto 6, and suggested that there might exist composite numbers n such that the only abstract group of order n is the cyclic group of that order. We also discuss Cayley's motivation for generalizing the then current concept of a permutation group. Cayley extended the classification to groups of order 8 in the third paper. There he also initiated the study of groups in terms of generators and relations (a procedure usually attributed to Walter Dyck), and in this way constructed the abstract dihedral group of order 2n. However, these pioneering studies were swept away by the then burgeoning surge of permutation groups, and apparently went completed unheeded by his contemporaries.