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

Date: October 14 (Tue), 2003
Speaker: Hisayosi Matumoto (松本久義) (University of Tokyo)
Title: scalar 型の一般化された Verma 加群の間の準同型について
Abstract: 複素半単純 Lie 代数を考える。その放物型部分代数の既約有限次元表現からの誘導表現は一般化された Verma 加群と言われる。一般化された Verma 加群の間の準同型を分類する問題は 一般化された旗多様体の上の同変ベクトル束の間の同変微分作用素を分類することと等価であり、Baston らによって始られた Parabolic geometry の観点からも興味があるようである。 ここでは、放物型部分代数の一次元表現からの誘導表現の場合に問題を限って考察する。(これは直線束の場合にあたる。) 主要な結果は以下の通りである。
(1) 放物型部分代数が極大の場合の準同型の分類。
(2) (1)で存在が示された準同型から、比較定理によって一般の放物型部分代数の場合にも準同型が構成できること。
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 (織田 寛) (Takushoku University)
Title: 最小多項式と一般 Verma 加群の零化イデアル
Abstract: 複素簡約 Lie 環 g の一般 Verma 加群 MΘ(λ) の零化イデアル Ann(MΘ(λ)) に関する大島先生との共同研究.
g の忠実な有限次表現 π に対して定まる U(g) 成分の正方行列 F の,Ann(MΘ(λ)) を法とする最小多項式 q(x;λ) を,パラメータ λ が generic な場合に記述する.
Ann(MΘ(λ)) の解析等への応用とともに,π が単純 Lie 環の場合の adjoint 表現,minuscule 表現,最小次元表現の場合について, q(x) の具体形を記述する.
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 (関口次郎) (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×VK. Denote by B/SLn  the categorical quotient, which is known to be an affine space of dimension m+1. We study the canonical projection π:BKB/ SLn  and its fibers. We prove that each fiber of π, and in particular the zero fiber, i.e., the null-cone N = π-1(π(0)),  is irreducible, reduced, and has dimension n2 - m - 1. Furthermore, we show that each fiber of π 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.

© Toshiyuki Kobayashi