## Lie Groups and Representation Theory Seminar 2012

List of speakers:
Yves Benoist, Toshihisa Kubo #1, Eric Opdam, Toshihisa Kubo #2, Takayuki Okuda, Oskar Hamlet #1, Oskar Hamlet #2, Ali Baklouti #1, Ali Baklouti #2, Toshihisa Kubo #3, Hiroshi Konno, Masaki Watanabe, Tatsuyuki Hikita,
 GCOE Lectures Date: May 29 (Tue), 2012, 16:30-18:00 June 5 (Tue), 2012, 16:30-18:00 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Yves Benoist (CNRS, Orsay) Title: Random walk on reductive groups I, II Abstract: [ pdf ] The asymptotic behavior of the sum of real numbers chosen independantly with same probability law is controled by many classical theorems: Law of Large Numbers, Central Limit Theorem, Law of Iterated Logarithm, Local Limit Theorem, Large deviation Principle, 0-1 Law,... In these introductory talks I will recall these classical results and explain their analogs for products of matrices chosen independantly with same probability law, when the action of the support of the law is semisimple. We will see that the dynamics of the corresponding action on the flag variety is a crucial tool for studying these non-commutative random walks. Date: June 12 (Tue), 2012, 16:30-18:00 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Toshihisa Kubo (vۗv) (the University of Tokyo) Title: Conformally invariant systems of differential operators of non-Heisenberg parabolic type Abstract: [ pdf ] Minkowski spacewave operatorconformally invariant operator̓T^IȗłB ߔNABarchini-Kable-Zierauɂ1differential operator conformal invariancesystemɈʉꂽ (conformally invariant systems)B ̃Z~i[łmaximal non-Heisenberg parabolicgāA ̗lsecond order differential operatorsystem肽B ܂conformally invariant systeḿAgeneralized Verma moduleԂ homomorphismU邪AԂ΂ɂĂqׂB Date: July 17 (Tue), 2012, 17:00-18:30 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Eric Opdam (University of Amsterdam) Title: Dirac induction for graded affine Hecke algebras Abstract: [ pdf ] In recent joint work with Dan Ciubotaru and Peter Trapa we constructed a model for the discrete series representations of graded affine Hecke algebras as the index of a Dirac operator. We discuss the K-theoretic meaning of this result, and the remarkable relation between elliptic character theory of a Weyl group and the ordinary character theory of its Pin cover. Date: July 24 (Tue), 2012, 16:30-18:00 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Toshihisa Kubo (vۗv) (the University of Tokyo) Title: The Dynkin index and conformally invariant systems of Heisenberg parabolic type Abstract: [ pdf ] Heisenberg parabolic subalgebra\zꂽconformally invariant systemɊւ̒萔ɂčl@B̒萔͂conformally invariant system\zۂɏdvȖʂA̎̔䗦 ƂĒ`ꂽŋ̓IȎȂǂ͕\ȂB{uł͂ ̓IlȎDynkin indexďЉB Date: November 6 (Tue), 2012, 16:30-18:00 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Takayuki Okuda (cK) (the University of Tokyo) Title: An explicit construction of spherical designs on S3 Abstract: [ pdf ] The existence of spherical t-designs on S^d for any t and d are proved by Seymour-Zaslavsky in 1984. However, explicit constructions of spherical designs were not known for d > 2 and large t. In this talk, for a given spherical t-design Y on S^2, we give an algorithm to make a spherical 2t-design X on S^3 which maps Y by a Hopf map. In particular, by combining with the results of Kuperberg in 2005, we have an explicit construction of spherical t-designs on S^3 for any t. Date: November 13 (Tue), 2012, 16:30-18:00 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Oskar Hamlet (Chalmers University) Title: Tight maps, a classification Abstract: [ pdf ] Tight maps/homomorphisms were introduced during the study of rigidity properties of surface groups in Hermitian Lie groups. In this talk I'll discuss the properties of tight maps, their connection to rigidity theory and my work classifying them. (Talk) Date: November 16 (Fri), 2012, 14:50-16:50 Place: Room 123, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Oskar Hamlet (Chalmers University) Title: Kaehler area of triangles in Hermitian symmetric spaces Abstract: [ pdf ] In this talk I will follow the paper "The Gromov norm of the Kaehler class of symmetric domains" by Domic & Toledo. In the paper a very concrete formula for calculating triangle areas is given and using this one shows that the Kaehler class is a bounded class. (Talk) Date: November 20 (Tue), 2012, 14:50-16:00 Place: Room 123, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Ali Baklouti (Sfax University) Title: On some aspects of deformation theory of discontinuous subgroups acting on exponential homogeneous spaces Abstract: [ pdf ] In this lecture, I will introduce the Kobayashi's new framework of deformation theory for general non-Riemannian homogeneous spaces, putting the focus on the context of exponential solvable settings. Date: November 20 (Tue), 2012, 16:30-17:30 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Ali Baklouti (Sfax University) Title: On the geometry of discontinuous subgroups acting on some homogeneous spaces Abstract: [ pdf ] Let $G$ be a Lie group, $H$ a closed subgroup of $G$ and $\ Gamma$ a discontinuous subgroup for the homogeneous space $G/H$. I first introduce the deformation space ${\mathcal{T}}^{K_o}(\Gamma, G, H)$ of the action of $\Gamma$ on $G/H$ in the sense of Kobayashi and some of its refined versions, namely the Clifford--Klein space of deformations of the form ${\mathcal{X}}=\Gamma \backslash G/H$. The deformation space ${\mathcal{T}}^{G_o}(\Gamma, G,H)$ of marked $(G,H)$- structures on ${\mathcal{X}}$ in the sense of Goldman is also introduced. As an important motivation, I will explain the connection between the spaces ${\mathcal{T}}^{K_o}(\Gamma, G, H)$ and ${\mathcal{T}}^{G_o}(\ Gamma, G, H)$ and study some of their topological features, namely the rigidity in the sense of Selberg--Weil--Kobayashi and the stability in the sense of Kobayashi--Nasrin. The latter appears to be of major interest to write down the connection explicitly. (Talk) Date: November 21 (Wed), 2012, 14:50-16:50 Place: Room 123, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Kubo Toshihisa (vۗv) (the University of Tokyo) Title: The Heisenberg ultrahyperbolic equation and the associated Kelvin transform Abstract: [ pdf ] In this talk we study the paper "The Heisenberg ultrahyperbolic equation: K-finite and polynomial solutions" by Anthony C. Kable. We mainly focus on the conformal invariance property of the Heisenberg ultra hyperbolic operator. If time permits then we are also going to discuss a result on the polynomial solutions of the operator. Date: November 27 (Tue), 2012, 16:30-18:00 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Hiroshi Konno (G) (the University of Tokyo) Title: Convergence of Kahler to real polarizations on flag manifolds (l̂̃P[[΋ɂ̎΋ɂւ̎) Abstract: [ pdf ] In this talk we will discuss geometric quantization of a flag manifold. In particular, we construct a family of complex structures on a flag manifold that converge 'at the quantum level' to the real polarization coming from the Gelfand-Cetlin integrable system. Our construction is based on a toric degeneration of flag varieties and a deformation of Kähler structure on toric varieties by symplectic potentials. This is a joint work with Mark Hamilton. Date: November 29 (Thu), 2012, 16:30-17:30 Place: Room 122, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Masaki Watanabe (n) (the University of Tokyo) Title: On a relation between certain character values of symmetric groups Abstract: [ pdf ] We present a relation of new kind between character values of symmetric groups which explains a curious phenomenon in character tables of symmetric groups. Similar relations for characters of Brauer and walled Brauer algebras and projective characters of symmetric groups are also presented. Date: December 11 (Tue), 2012, 17:00-17:30 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Tatsuyuki Hikita (DcCV) (Kyoto University) Title: Affine Springer fibers of type A and combinatorics of diagonal coinvariants Abstract: [ pdf ] We introduce certain filtrations on the homology of affine Springer fibers of type A and give combinatorial formulas for the bigraded Frobenius series of the associated graded modules. The results are essentially given by generalizations of the symmetric function introduced by Haglund, Haiman, Loehr, Remmel, and Ulyanov which is conjectured to coincide with the bigraded Frobenius series of the ring of diagonal coinvariants.