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Lie Groups and Representation Theory Seminar

[ Seminar 2015 | Past Seminars | Related conferences etc. ]

Upcoming talks

Place: Usually Room 126 (this year)
Graduate School of Mathematical Sciences, the University of Tokyo [ Access ]
数理科学広域演習
1月19日(月): 16:00-18:00 126号室
奥田隆幸氏 「実半単純リー環の冪零軌道の分類理論 I」
1月20日(火): 14:50-16:50 126号室
奥田隆幸氏 「実半単純リー環の冪零軌道の分類理論の解説 II」
17:00-19:00 126 号室
大島芳樹氏 「自己共役な二階常微分作用素のスペクトル分解 I」
1月21日(水): 14:45-15:45 126号室
大島芳樹氏 「自己共役な二階常微分作用素のスペクトル分解 II」
1月22日(木): 16:00-18:00 118号室
大島芳樹氏 「自己共役な二階常微分作用素のスペクトル分解 III」
Speaker: 奥田隆幸氏 (広島大学)
Title: 実半単純リー環の冪零軌道の分類理論 I, II
Abstract: 実半単純リー環内の冪零(随伴)軌道の分類の手法について解説する. 今回紹介するのは Djokovic による主に例外型を想定した手法で, 関口-Kostant 対応を経由して weighted Dynkin diagram の形で冪零軌道を特 徴付けるというものである. また, Jacobson-Morozov, Kostant らの結果から, 実半単純リー環内の冪零軌道の分類は sl(2,R) の埋め込みの分類と対応するこ とにも注意しておく.
Speaker: 大島芳樹氏 (東京大学 IPMU)
Title: 自己共役な二階常微分作用素のスペクトル分解 I, II, III
Abstract: 自己共役な二階常微分作用素に対する固有関数への展開の理論 はWeyl-Stone-小平-Titchmarshによって完成し、表現論をはじめ様々な分野で使われてきた。今回はスペクトル測度と固有関数の漸近挙動とを結びつける小平- Titchmarshの定理を目標として、小平邦彦による証明に基づいてお話しする。
Winter School 2015 on Representation Theory of Real Reductive Groups
Graduate School of Mathematical Sciences, the University of Tokyo
January 24 (Sat)-26 (Mon), 2015
January 24 (Sat) 13:00-14:00, Room 126
Peter Trapa
"Unitary representations of reductive Lie groups I"
14:30-15:30, Room 126
Raul Gomez
"The Tor and Ext functors for smooth representations of real algebraic groups"
16:30-17:30, Room 126
Benjamin Harris
"The Geometry of Tempered Characters"
January 25 (Sun) 9:00-10:00, Room 126
Raul Gomez
"Generalized and degenerate Whittaker models associated to nilpotent orbits"
10:30-11:30, Room 126
Benjamin Harris
"The Geometry of Harmonic Analysis"
12:00-13:00, Room 126
Peter Trapa
"Unitary representations of reductive Lie groups II"
January 26 (Mon) 9:30-10:30, Room 122
Benjamin Harris
"The Geometry of Nontempered Characters"
11:00-12:00, Room 122
Raul Gomez
"Local Theta lifting of generalized Whittaker models"
12:30-13:30, Room 128
Peter Trapa
"Unitary representations of reductive Lie groups III"
Speaker Benjamin Harris (Oklahoma State University)
Title The Geometry of Tempered Characters
Abstract In this introductory talk, we will briefly recall parts of Harish-Chandra's theory of characters for reductive groups and the geometric formula of Rossmann and Duflo for tempered characters of reductive groups. Examples will be given in the case G=SL(2,R).
Title The Geometry of Harmonic Analysis
Abstract In this talk, we will present recent joint work with Tobias Weich. When G is a real, reductive algebraic group and X is a homogeneous space for G with an invariant measure, we will completely describe the regular, semisimple asymptotics of the support of the Plancherel measure for L^2(X). We will give concrete examples of this theorem, describing what can and cannot be deduced from this result.
Title The Geometry of Nontempered Characters
Abstract In this talk, we will survey the results of Rossmann and Schmid-Vilonen on geometric formulas for nontempered characters of reductive groups, and we will mention an old result of Barbasch-Vogan on the special case A_q(lambda). We will discuss what nontempered character formulas would be necessary to generalize the main formula of the second talk, and we will make conjectures.
Speaker Peter Trapa (University of Utah)
Title Unitary representations of reductive Lie groups I, II, III
Abstract Let G be a real reductive group. I will describe an algorithm to determine the unitary dual of G. More precisely, I will describe an algorithm to determine if an irreducible (g,K) module (specified in the Langlands classification) is unitary in the sense that it admits a positive definite invariant Hermitian form. This is joint work with Jeffrey Adams, Marc van Leeuwen, and David Vogan.
Speaker Raul Gomez (Cornel University)
Title 1. The Tor and Ext functors for smooth representations of real algebraic groups
Abstract Inspired by the recent work of Dipendra Prasad in the $p$-adic setting, we define the Tor and Ext functors for an appropriate category of smooth representations of a real algebraic group $G$, and give some applications. This is joint work with Birgit Speh.
Title 2. Generalized and degenerate Whittaker models associated to nilpotent orbits
Abstract In this talk, we examine the relation between the different spaces of Whittaker models that can be attached to a nilpotent orbit. We will also explore their relation to other nilpotent invariants (like the wave front set) and show some examples and applications. This is joint work with Dmitry Gourevitch and Siddhartha Sahi.
Title 3. Local Theta lifting of generalized Whittaker models
Abstract In this talk, we describe the behavior of the space of generalized Whittaker models attached to a nilpotent orbit under the local theta correspondence. This description is a generalization of a result of Moeglin in the p-adic setting. This is joint work with Chengbo Zhu.
Organizers: T. Kobayashi, T. Kubo, H. Matumoto, H. Sekiguchi
Date: January 27 (Tue), 2015, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Hironori Oya (大矢浩徳) (The University of Tokyo)
Title: Representations of quantized function algebras and the transition matrices from Canonical bases to PBW bases
Abstract:
[ pdf ]
Let $G$ be a connected simply connected simple complex algebraic group of type $ADE$ and $\mathfrak{g}$ the corresponding simple Lie algebra. In this talk, I will explain our new algebraic proof of the positivity of the transition matrices from the canonical basis to the PBW bases of $U_q(\mathfrak{n}^+)$. Here, $U_q(\mathfrak{n}^+)$ denotes the positive part of the quantized enveloping algebra $U_q(\ mathfrak{g})$. (This positivity, which is a generalization of Lusztig's result, was originally proved by Kato (Duke Math. J. 163 (2014)).) We use the relation between $U_q(\mathfrak{n}^+)$ and the specific irreducible representations of the quantized function algebra $\mathbb{Q} _q[G]$. This relation has recently been pointed out by Kuniba, Okado and Yamada (SIGMA. 9 (2013)). Firstly, we study it taking into account the right $U_q(\mathfrak{g})$-algebra structure of $\mathbb{Q}_q[G]$. Next, we calculate the transition matrices from the canonical basis to the PBW bases using the result obtained in the first step.

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