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Lie Groups and Representation Theory Seminar

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Upcoming talks

Place: Usually Room 126 (this year)
Graduate School of Mathematical Sciences, the University of Tokyo [ Access ]
Date: May 19 (Tue), 2015, 17:00-18:30
Place: Room 122, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Anton Evseev (University of Birmingham)
Title: RoCK blocks, wreath products and KLR algebras
Abstract:
[ pdf ]
The so-called RoCK (or Rouquier) blocks play an important role in representation theory of symmetric groups over a finite field of characteristic $p$, as well as of Hecke algebras at roots of unity. Turner has conjectured that a certain idempotent truncation of a RoCK block is Morita equivalent to the principal block $B_0$ of the wreath product $S_p\wr S_d$ of symmetric groups, where $d$ is the "weight" of the block. The talk will outline a proof of this conjecture, which generalizes a result of Chuang-Kessar proved for $d < p$. The proof uses an isomorphism between a Hecke algebra at a root of unity and a cyclotomic Khovanov-Lauda-Rouquier algebra, the resulting grading on the Hecke algebra and the ideas behind a construction of R-matrices for modules over KLR algebras due to Kang-Kashiwara-Kim.
Date: May 26 (Tue), 2015, 17:00-18:30
Place: Room 122, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Takeyoshi Kogiso (小木曽岳義) (Josai University)
Title: Local functional equations of Clifford quartic forms and homaloidal
Abstract:
[ pdf ]
It is known that one can associate local functional equation to the irreducible relative invariant of an irreducible regular prehomogeneous vector spaces. We construct Clifford quartic forms that cannot obtained from prehomogeneous vector spaces, but, for which one can associate local functional equations. The characterization of polynomials which satisfy local functional equations is an interesting problem. In relation to this characterization problem (in a more general form), Etingof, Kazhdan and Polishchuk raised a conjecture. We make a counter example of this conjecture from Clifford quartic forms. (This is based on the joint work with F.Sato)
局所関数等式が正則概均質ベクトル空間の基本相対不変式とその双対空間の多項式のペアから与えられることは知られている。我々は Clifford quartic form と呼ばれるある4次斉次多項式を構成し、それが概均質ベクトル空間の相対不変式ではないも関わらず 局所関数等式を満たすことを示した。局所関数等式を満たす多項式を特徴付ける問題は興味深い未解決問題であるが、この問題に関連し、 Etingof, Kazhdan, Polishchuk が(もっと一般的な設定で)ある予想を提示した。我々は、 Clifford quartic form を用いて、この予想に反例があることを示した。(この講演は佐藤文広氏との共同研究に基づいている。)
Analytic Representation Theory of Lie Groups

Date: July 1 (Wed)-July 4 (Sat), 2015
Place: Kavli IPMU, the University of Tokyo

Speaker: Masatoshi Kitagawa (Univ of Tokyo)
Title: On the irreducibility of U(g)^H-modules
Abstract: I will report on the irreducibility of U(g)^H-modules arising from branching problems. It is well-known that a U(g)^K-module Hom_K(W,V) is irreducible for any irreducible (g, K)-module V and K-type W. For a non-compact subgroup H, the same statement is not true in general. In this talk, I will introduce a positive example and negative example for the irreducibility of Hom_H(W,V).

Speaker: Bent Ørsted (Aarhus University)
Title: Generalized Fourier transforms
Abstract: In these lectures, based on joint work with Salem Ben Said and Toshiyuki Kobayashi, we shall define a natural family of deformations of the usual Fourier transform in Euclidian space. The main idea is to replace the standard Laplace operator by a two-parameter family of deformations in such a way, that it still is a member of a triple generating the three-dimensional simple Lie algebra. In particular we shall describe
  1. Coxeter groups and Dunkl operators
  2. Holomorphic semigroups generalizing the Hermite semigroup
  3. Generalized Fourier transforms and some applications
  4. Hecke algebras and interpolations between minimal representations.
Speaker: Anatoly Vershik (St. Petersburg State University)
Title: Cohomology with the coefficients in the unitary representations of the Lie groups and representations of current groups
Abstract: Accordingly to the old papers by H.Araki, (and related continuaion by Vershik-Gelfand-Graev in 70-th) the non-local irreducible representations of the group of functions with values in Lie groups $G$ depends on the existence of the cocycles of this group with values in the unitary representation of it. Such cocycle do exist for $O(n,1)$ and $U(n,1)$ but not for the rest semsimple groups.We put the same question for Iwasawa subgroup $P$ of the groups of type $U(p,q)$ and $O(p,q)$ is it true that $H^1(P;\pi)\ne 0$, where $\pi$ is exact irreducible representation of $P$. This positive answer on this question allows to find out a nontrivial cohomology for nonunitary representations for semi-simple groups and then to construct nonunitary representations of the corresponding current groups.

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