## Lie Groups and Representation Theory Seminar 2011

List of speakers:
Pierre Clare, Hiroshi Oda, Gang Liu, Job Kuit, Taro Yoshino, Jun-ichi Mukuno, Hirotake Kurihara, Masahiko Kanai, Yoshiki Oshima, Laurant Demonet, Takayuki Okuda, Daniel Sternheimer, Masaki Kashiwara, Hung Yean Loke, Toshiyuki Kobayashi,
 Date: January 18 (Tue), 2011, 17:00-18:00 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Pierre Clare (Université d'Orleans & the University of Tokyo) Title: Connections between Noncommutative Geometry and Lie groups representations Abstract: [ pdf ] One of the principles of Noncommutative Geometry is to study singular spaces that the tools of classical analysis like algebras of continuous functions fail to describe, replacing them by more general C*-algebras. After recalling fundamental facts about C*-algebras, Hilbert modules and group C*-algebras, we will present constructions and results aiming to understand principal series representations and Knapp-Stein theory in the noncommutative geometrical framework. Eventually we will explain the relationship between the analysis of reduced group C*-algebras and the computation of the Connes-Kasparov isomorphisms. iWuj Date: January 17 (Mon) - 21 (Fri), 2011, 14:40-16:40 Place: Room 123, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Hiroshi Oda (Dc ) (Takushoku University) Title: Chevalley 藝Ƃ̗lXȊg Abstract: [ pdf ] Date: March 2 (Wed), 2011 Speaker: Gang Liu (Université de Poitiers) Title: Duflo's conjecture Date: March 2 (Wed), 2011 Speaker: Job Kuit (Utrecht University) Title: Radon transformation on reductive symmetric spaces: support theorems (Joint with Tuesday Seminar on Topology) Date: April 26 (Tue), 2011, 16:30-18:00 Place: Room 056, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Taro Yoshino (g쑾Y) (the University of Tokyo) Title: Topological Blow-up Abstract: [ pdf ] Suppose that a Lie group $G$ acts on a manifold $M$. The quotient space $X:=G\backslash M$ is locally compact, but not Hausdorff in general. Our aim is to understand such a non-Hausdorff space $X$. The space $X$ has the crack $S$. Rougly speaking, $S$ is the causal subset of non-Hausdorffness of $X$, and especially $X\setminus S$ is Hausdorff. We introduce the concept of topological blow-up' as a repair' of the crack. The repaired' space $\tilde{X}$ is locally compact and Hausdorff space containing $X\setminus S$ as its open subset. Moreover, the original space $X$ can be recovered from the pair of $(\tilde{X}, S)$. Date: May 24 (Tue), 2011, 16:30-18:00 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Jun-ichi Mukuno (쏃) (Nagoya University) Title: Properly discontinuous isometric group actions on inhomogeneous Lorentzian manifolds Abstract: [ pdf ] If a homogeneous space $G/H$ is acted properly discontinuously upon by a subgroup $\Gamma$ of $G$ via the left action, the quotient space $\Gamma \backslash G/H$ is called a Clifford--Klein form. In 1962, E. Calabi and L. Markus proved that there is no infinite subgroup of the Lorentz group $O(n+1, 1)$ whose left action on the de Sitter space $O(n+1, 1)/O(n, 1)$ is properly discontinuous. It follows that a compact Clifford--Klein form of the de Sitter space never exists. In this talk, we present a new extension of the theorem of E. Calabi and L. Markus to a certain class of Lorentzian manifolds that are not necessarily homogeneous. Date: May 31 (Tue), 2011, 16:30-17:30 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Hirotake Kurihara (I啐) (Tohoku University) Title: On character tables of association schemes based on attenuated spaces Abstract: [ pdf ] An association scheme is a pair of a finite set $X$ and a set of relations $\{R_i\}_{0\le i\le d}$ on $X$ which satisfies several axioms of regularity. The notion of association schemes is viewed as some axiomatized properties of transitive permutation groups in terms of combinatorics, and also the notion of association schemes is regarded as a generalization of the subring of the group ring spanned by the conjugacy classes of finite groups. Thus, the theory of association schemes had been developed in the study of finite permutation groups and representation theory. To determine the character tables of association schemes is an important first step to a systematic study of association schemes, and is helpful toward the classification of those schemes. In this talk, we determine the character tables of association schemes based on attenuated spaces. These association schemes are obtained from subspaces of a given dimension in attenuated spaces. (Joint with Topology Seminar) Date: June 7 (Tue), 2011, 16:30-18:00 Place: Room 056, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Masahiko Kanai (F) (The University of Tokyo) Title: Rigidity of group actions via invariant geometric structures Abstract: [ pdf ] It is a homomorphism into a FINITE dimensional Lie group that is concerned with in the classical RIGIDITY theorems such as those of Mostow and Margulis. In the meantime, differentiable GROUP ACTIONS for which we ask rigidity problems is a homomorphism into a diffeomorphism group, which is a typical example of INFINITE dimensional Lie groups. The purpose of the present talk is exhibiting several rigidity theorems for group actions in which I have been involved for years. Although quite a few fields of mathematics, such as ergodic theory, the theory of smooth dynamical systems, representation theory and so on, have made remarkable contributions to rigidity problems, I would rather emphasis geometric aspects: I would focus on those rigidity phenomenon for group actions that are observed by showing that the actions have INVARIANT GEOMETRIC STRUCTURES. Date: October 25 (Tue), 2011, 16:30-18:00 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Yoshiki Oshima (哇F) (The University of Tokyo) Title: Localization of Cohomological Induction Abstract: [ pdf ] Cohomological induction is defined for (g,K)-modules in an algebraic way and construct important representations such as (Harish-Chandra modules of) discrete series representations, principal series representations and Zuckerman's modules of semisimple Lie groups. Hecht, Milicic, Schmid, and Wolf proved that modules induced from one-dimensional representations of Borel subalgebra can be realized as D-modules on the flag variety. In this talk, we show a similar result for modules induced from more general representations. Date: November 15 (Tue), 2011, 16:30-18:00 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Laurant Demonet (Nagoya University) Title: Categorification of cluster algebras arising from unipotent subgroups of non-simply laced Lie groups Abstract: [ pdf ] We introduce an abstract framework to categorify some antisymetrizable cluster algebras by using actions of finite groups on stably 2-Calabi-Yau exact categories. We introduce the notion of the equivariant category and, with similar technics as in [K], [CK], [GLS1], [GLS2], [DK], [FK], [P], we construct some examples of such categorifications. For example, if we let Z/2Z act on the category of representations of the preprojective algebra of type A2n-1 via the only non trivial action on the diagram, we obtain the cluster structure on the coordinate ring of the maximal unipotent subgroup of the semi-simple Lie group of type Bn [D]. Hence, we can get relations between the cluster algebras categorified by some exact subcategories of these two categories. We also prove by the same methods as in [FK] a conjecture of Fomin and Zelevinsky stating that the cluster monomials are linearly independent. References [CK] P. Caldero, B. Keller, From triangulated categories to cluster algebras, Invent. Math. 172 (2008), no. 1, 169-211. [DK] R. Dehy, B. Keller, On the combinatorics of rigid objects in 2-Calabi-Yau categories, arXiv: 0709.0882. [D] L. Demonet, Cluster algebras and preprojective algebras: the non simply-laced case, C. R. Acad. Sci. Paris, Ser. I 346 (2008), 379-384. [FK] C. Fu, B. Keller, On cluster algebras with coefficients and 2-Calabi-Yau categories, arXiv: 0710.3152. [GLS1] C. Geiss, B. Leclerc, J. Schröer, Rigid modules over preprojective algebras, Invent. Math. 165 (2006), no. 3, 589-632. [GLS2] C. Geiss, B. Leclerc, J. Schröer, Cluster algebra structures and semicanoncial bases for unipotent groups, arXiv: math/0703039. [K] B. Keller, Categorification of acyclic cluster algebras: an introduction, arXiv: 0801.3103. [P] Y. Palu, Cluster characters for triangulated 2-Calabi-Yau categories, arXiv: math/0703540. Date: November 22 (Tue), 2011, 16:30-18:00 Place: Room 002, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Takayuki Okuda (cK) (The University of Tokyo) Title: Smallest complex nilpotent orbit with real points Abstract: [ pdf ] Let $\mathfrak{g}$ be a non-compact simple Lie algebra with no complex structures. In this talk, we show that there exists a complex nilpotent orbit $\mathcal{O}^{G_\mathbb{C}}_{\text{min},\mathfrak{g}}$ in $\mathfrak{g}_\mathbb{C}$ ($:=\mathfrak{g} \otimes \mathbb{C}$) containing all of real nilpotent orbits in $\mathfrak{g}$ of minimal positive dimension. For many $\mathfrak{g}$, the orbit $\mathcal{O}^{G_\mathbb{C}}_{\text{min},\mathfrak{g}}$ is just the complex minimal nilpotent orbit in $\mathfrak{g}_\mathbb{C}$. However, for the cases where $\mathfrak{g}$ is isomorphic to $\mathfrak{su}^*(2k)$, $\mathfrak{so}(n-1,1)$, $\mathfrak{sp}(p,q)$, $\mathfrak{e}_{6(-26)}$ or $\mathfrak{f}_{4(-20)}$, the orbit $\mathcal{O}^{G_\mathbb{C}}_{\text{min},\mathfrak{g}}$ is not the complex minimal nilpotent orbit in $\mathfrak{g}_\mathbb{C}$. We also determine $\mathcal{O}^{G_\mathbb{C}}_{\text{min},\mathfrak{g}}$ by describing the weighted Dynkin diagrams of these for such cases. Date: November 29 (Tue), 2011, 16:30-18:00 Place: Room 002, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Daniel Sternheimer (Rikkyo Univertiry and Université de Bourgogne) Title: Symmetries, (their) deformations, and physics: some perspectives and open problems from half a century of personal experience Abstract: [ pdf ] This is a flexible general framework, based on quite a number of papers, some of which are reviewed in: MR2285047 (2008c:53079) Sternheimer, Daniel. The geometry of space-time and its deformations from a physical perspective. From geometry to quantum mechanics, 287-301, Progr. Math., 252, Birkhäuser Boston, Boston, MA, 2007 http://monge.u-bourgogne.fr/d.sternh/papers/PiMOmori-DS.pdf iWu`j Date: December 5 (Mon) - 9 (Fri), 2011, 14:40-16:40 Speaker: Masaki Kashiwara () (RIMS, Kyoto University) Title: Khovanov-Lauda-Rouquier㐔Categorification Abstract: [ pdf ] Date: December 13 (Tue), 2011, 16:30-18:00 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Hung Yean Loke (National University of Singapore) Title: Local Theta lifts of unitary lowest weight modules to the indefinite orthogonal groups Abstract: [ pdf ] In this talk, I will discuss the local theta lifts of unitary lowest weight modules of $Sp(2p,R)$ to the indefinite orthogonal group $O(n,m)$. In a previous paper, Nishiyama and Zhu computed the associated cycles when the dual pair $Sp(2p,R) \times O(m,n)$ lies in the stable range, ie. $2p \leq \min(m,n)$. In this talk, I will report on a joint work with Jiajun Ma and U-Liang Tang at NUS where we extend the computation beyond the stable range. Our approach is to analyze the coherent sheaves generated by the graded modules. We will also need the Kobayashi's projection formula for discretely decomposable restrictions. Our study produces some interesting formulas on the $K$-types of the representations. In particular for some of these representations, the $K$-types formulas agree those in a conjecture of Vogan on the unipotent representations. IPMU Colloquium Date: December 14 (Wed), 2011, 15:30-17:00 Place: Lecture Hall, IPMU, the University of Tokyo Speaker: Toshiyuki Kobayashi (яrs) (The University of Tokyo) Title: Global Geometry and Analysis on Locally Homogeneous Spaces Abstract: [ pdf ] The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry. In contrast, in areas such as Lorentz geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry, surprising little is known about global properties of the geometry even if we impose a locally homogeneous structure. Further, almost nothing is known on global analysis on such spaces. Taking anti-de Sitter manifolds, which are locally modelled on AdS^n as an example, I plan to explain two programs: 1. (global shape) Is a locally homogeneous space closed? 2. (spectral analysis) Does spectrum of the Laplacian vary when we deform the geometric structure?