Lie Groups and Representation Theory Seminar 2013

List of speakers:
Soji Kaneyuki, Simon Goodwin, Nizar Demni #1, Nizar Demni #2, Yoshiki Oshima, Hisayosi Matumoto, Atsumu Sasaki, Michael Pevzner, Pierre Clare, Birgit Speh, Maarten van Pruijssen, Benjamin Harris, Takayuki Okuda, Yuichiro Tanaka, Adam Ehlers Nyholm Thomsen, (Representation Theory), Toshiyuki Kobayashi, Vaibhav Vaish, Yuichiro Tanaka, Pampa Paul, Dipendra Prasad, Dipendra Prasad, Ronald King, Simon Gindikin #1, Simon Gindikin #2, Koichi Kaizuka, Hisayosi Matumoto,
 Date: January 8 (Tue), 2013, 16:30-18:00 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Soji Kaneyuki (ss) (Sophia University, Professor Emeritus) Title: On the group of holomorphic and anti-holomorphic transformations of a compact Hermitian symmetric space and the G-structure) Abstract: [ pdf ] Let M be a compact irreducible Hermitian symmetric space. We determine the full group of holomorphic and anti-holomorphic transformations of M. Also we characterize that full group as the automorphism group of the G-structure on M, called a generalized conformal structure. Date: January 22 (Tue), 2013, 16:30-18:00 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Simon Goodwin (Birmingham University) Title: Representation theory of finite W-algebras Abstract: [ pdf ] There has been a great deal of recent research interest in finite W-algebras motivated by important connection with primitive ideals of universal enveloping algebras and applications in mathematical physics. There have been significant breakthroughs in the representation theory of finite W-algebras due to the research of a variety of mathematicians. In this talk, we will give an overview of the representation theory of finite W-algebras focusing on W-algebras associated to classical Lie algebras (joint with J. Brown) and W-algebras associated to general linear Lie superalgebras (joint with J. Brown and J. Brundan). Date: February 4 (Mon), 2013, 17:30-19:00 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Nizar Demni (Université de Rennes 1) Title: Dunkl processes assciated with dihedral systems, I Abstract: [ pdf ] I'll first give a brief and needed account on root systems and finite reflection groups. Then, I'll introduce Dunkl operators and give some properties. Once I'll do, I'll introduce Dunkl processes and their continuous components, so-called radial Dunkl processes. The latter generalize eigenvalues processes of some matrix-valued processes and reduces to reflected Brownian motion in Weyl chambers. Besides, Brownian motion in Weyl chambers corresponds to all multiplicity values equal one are constructed from a Brownian motion killed when it first hits the boundary of the Weyl chamber using the unique positive harmonic function (up to a constant) on the Weyl chamber. In the analytic side, determinantal formulas appear and are related to harmonic analysis on the Gelfand pair (Gl(n,C), U(n)). This is in agreement on the one side with the so-called reflection principle in stochastic processes theory and matches on the other side the so-called shift principle introduced by E. Opdam. Finally, I'll discuss the spectacular result of Biane-Bougerol-O'connell yielding to a Duistermaat-Heckman distribution for non crystallographic systems. Date: February 5 (Tue), 2013, 17:30-19:00 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Nizar Demni (Université de Rennes 1) Title: Dunkl processes assciated with dihedral systems, II Abstract: [ pdf ] I'll focus on dihedral systems and its semi group density. I'll show how one can write down this density using probabilistic techniques and give some interpretation using spherical harmonics. I'll also present some results attempting to get a close formula for the density: the main difficulty comes then from the inversion (in composition sense) of Tchebycheff polynomials of the first kind in some neighborhood. Finally, I'll display expressions through known special functions for even dihedral groups, and the unexplained connection between the obtained formulas and those of Ben Said-Kobayashi-Orsted. Date: April 2 (Tue), 2013, 16:30-18:00 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Yoshiki Oshima (哇F) (Kavli IPMU, the University of Tokyo) Title: Discrete branching laws of Zuckerman's derived functor modules (Zuckerman ֎Q̗UI) Abstract: [ pdf ] We consider the restriction of Zuckerman's derived functor modules with respect to symmetric pairs of real reductive groups. When they are discretely decomposable, explicit formulas for the branching laws are obtained by using a realization as D-module on the flag variety and the generalized BGG resolution. In this talk we would like to illustrate how to derive the formulas with a few examples. Date: April 3 (Wed), 2013, 16:30-18:00 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Hisayosi Matumoto ({v) (the University of Tokyo) Title: XJ[^ʉo}Q̊Ԃ̏^ɂ Abstract: [ pdf ] XJ[^ʉo}Q̊Ԃ̏^̕ނ͑Ήʉl̏ ϒ̊Ԃ sϐpf̕ނƓlł􉽂ȂǂƊ֘AĂB o}Q̊Ԃ̏^ɂĂVermaBernstein-Gelfand-GelfandɂL ʂ邪AXJ[^ʉo}Q̏ꍇɂRȈʉƂڂ ނɂ ̗\zlB̃Z~i[ł̖̗͂jUԂ̂\z  ǂ̂悤ȂƂĂ̂A^̏ꍇɂďڂB Date: April 9 (Tue), 2013, 16:30-18:00 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Atsumu Sasaki ( W) (Tokai University) Title: A characterization of non-tube type Hermitian symmetric spaces by visible actions (Japanese) Abstract: [ pdf ] We consider a non-symmetric complex Stein manifold D which is realized as a line bundle over the complexification of a non-compact irreducible Hermitian symmetric space G/K. In this talk, we will explain that the compact group action on D is strongly visible in the sense of Toshiyuki Kobayashi if and only if G/K is of non-tube type. In particular, we focus on our construction of slice which meets every orbit in D from the viewpoint of group theory, namely, we find an A-part of a generalized Cartan decomposition for homogeneous space D. We note that our choice of A-part is an abelian. Date: April 16 (Tue), 2013, 16:30-17:30 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Michael Pevzner (Reims University) Title: Non-standard models for small representations of GL(n,R) Abstract: [ pdf ] We shall present new models for some parabolically induced unitary representations of the real general linear group which involve Weyl symbolic calculus and furnish very efficient tools in studying branching laws for such representations. Date: April 16 (Tue), 2013, 17:30-18:30 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Pierre Clare (Penn. State University, USA) Title: Degenerate principal series of symplectic groups Abstract: [ pdf ] We will discuss properties of representations of symplectic groups induced from maximal parabolic subgroups of Heisenberg type, including K-types formulas, expressions of intertwining operators and the study of their spectrum. Speaker: Birgit Speh (Cornell University) Date: July 18 (Thu), 2013, 16:30-18:00 Place: Room 117, Graduate School of Mathematical Sciences, the University of Tokyo Title: Representations of reductive groups and L-functions. (I) Date: July 26 (Fri), 2013, 10:30-12:00 Place: Room 118, Graduate School of Mathematical Sciences, the University of Tokyo Title: Representations of reductive groups and L-functions. (II) Abstract: [ pdf ] This an introduction to the theory of L-functions and in particular of the local L-factors of representations in real and complex groups. Some familiarity with infinite dimensional representations would be very helpful, but I will not assume any knowledge of number theory. We will start in the first lecture by considering L-functions for Groessen characters and classical automorphic forms, in other words for automorphic representations of G(1) and GL(2). This will motivate the definition of the local L-factors of representations of GL(1,R) and GL(2,R). We will discuss Rankin convolutions and define the L-factors for infinite dimensional tempered representations of GL(n,R). In the second lecture we will quickly discuss Rankin Selberg integral approach to L-factors and then Shahidi's method of constructing L-functions by relating them to intertwining operators, leading to the definition of the the L-factors of tempered non degenerate representations. The lecture closes with a discussion of L-factors for nontempered representations. We note that our choice of A-part is an abelian. Speaker: Maarten van Pruijssen (Paderborn University) Date: September 4 (Wed), 2013, 18:00- Place: Room 470, Graduate School of Mathematical Sciences, the University of Tokyo Title: Reductive spherical pairs and orthogonal polynomials Speaker: Benjamin Harris (Louisiana State University, USA) Date: October 22 (Tue), 2013, 17:00-18:00 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Title: Representation Theory and Microlocal Analysis Abstract: [ pdf ] Suppose $H\subset K$ are compact, connected Lie groups, and suppose $\tau$ is an irreducible, unitary representation of $H$. In 1979, Kashiwara and Vergne proved a simple asymptotic formula for the decomposition of $Ind_H^K\tau$ by microlocally studying the regularity of vectors in this representation, thought of as vector valued functions on $K$. In 1998, Kobayashi proved a powerful criterion for the discrete decomposability of an irreducible, unitary representation $\pi$ of a reductive Lie group $G$ when restricted to a reductive subgroup $H$. One of his key ideas was to restrict $\pi$ to a representation of a maximal compact subgroup $K\subset G$, view $\pi$ as a subrepresentation of $L^2(K)$, and then use ideas similar to those developed by Kashiwara and Vergne. In a recent preprint the speaker wrote with Hongyu He and Gestur Olafsson, the authors consider the possibility of studying induction and restriction to a reductive Lie group $G$ by microlocally studying the regularity of the matrix coefficients of (possibly reducible) unitary representations of $G$, viewed as continuous functions on the (possibly noncompact) Lie group $G$. In this talk, we will outline the main results of this paper and give additional conjectures. Series lectures Speaker: Takayuki Okuda (cK) (Tohoku University) Date & place: October 23 (Wed), 2013, Room 128 October 24 (Thu), 2013, Room 123 October 30 (Wed), 2013, Room 128 October 31 (Thu), 2013, Room 370 Graduate School of Mathematical Sciences, the University of Tokyo Title: Dynkin's classification on semisimple subalgebras of semisimple Lie algebras Abstract: [ pdf ] fP[̕[̕ނɊւ E. B. Dynkin (Mat. Sbornik N.S. 1952, Amer. Math. Soc. Transl. 1957) ̌ʂЉ. {uł, ɗO^fP[̒ɖߍ܂ꂽfP[ ނ邽߂̃eNjbNɓIiĂb. E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Transl. 6 (1957), 111-244. Speaker: Yuichiro Tanaka (cYY) (the University of Tokyo) Date: October 29 (Tue), 2013, 16:30-18:00 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Title: Q̖d\̊􉽂ƉIȍp Abstract: [ pdf ] For a connected compact simple Lie group of type B or D, we find pairs $(V_{1},V_{2})$ of irreducible representations of G such that the tensor product representation $V_{1}\otimes V_{2}$ is multiplicity-free by a geometric consideration based on a notion of visible actions on complex manifolds, introduced by T. Kobayashi. The pairs we find exhaust all the multiplicity-free pairs by an earlier combinatorial classification due to Stembridge. Lecture Speaker: Adam Ehlers Nyholm Thomsen (Aarhus University and the University of Tokyo) Date: November 6 (Wed), 2013 Place: Room 128, Graduate School of Mathematical Sciences, the University of Tokyo Title: Calculating the global character of the metaplectic representation Abstract: [ pdf ] In this talk I will give a brief introduction to Metaplectic representation, also known as Segal-Shale-Weil representation, a unitary representation of the double cover of the symplectic group. This representation is intimately connected to the von-Neumann formulation of quantum mechanics, one way of realizing the representation is as intertwiners of the Schrödinger representation by the uniqueness in the Stone-von-Neumann theorem. The metaplectic representation is a very well studied representation and, among other things, serves as a key example in the study of minimal representations. The focus of this talk will be on calculating the global (or distributional or Harish-Chandra) character of the representation. JSPS-DST Asian Academic Seminar 2013 Discrete Mathematics & its Applications Sunday November 3 - Sunday November 10, 2013 (principal coordinators: M. Kotani, Bimal K. Roy) Representation Theory (organized by Toshiyuki Kobayashi) [ pdf ] Speaker: Toshiyuki Kobayashi (the University of Tokyo) Date: November 7 (Thu), 2013, 13:30-14:20 Title: Global Geometry and Analysis on Locally Pseudo-Riemannian Homogeneous Spaces Abstract: 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 of general signature, surprising little is known about global properties of the geometry even if we impose a locally homogeneous structure. Taking anti-de Sitter manifolds, which are locally modelled on AdS^n as an example, I plan to explain two programs: 1. (global shape) Existence problem of compact locally homogeneous spaces, and deformation theory. 2. (spectral analysis) Construction of the spectrum of the Laplacian, and its stability under the deformation of the geometric structure. Speaker: Vaibhav Vaish (the Institute of Mathematical Sciences) Date: November 7 (Thu), 2013, 14:30-15:20 Title: Weightless cohomology of algebraic varieties Abstract: Using Morel's weight truncations in categories of mixed sheaves, we attach to any variety defined over complex numbers, over finite fields or even over a number field, a series of groups called the weightless cohomology groups. These lie between the usual cohomology and the intersection cohomology, have a natural ring structure, satisfy Kunneth, and are functorial for certain morphisms. The construction is motivic and naturally arises in the context of Shimura Varieties where they capture the cohomology of Reductive Borel Serre compactification. The construction also yields invariants of singularities associated with the combinatorics of the boundary divisors in any resolution. Speaker: Yuichiro Tanaka (the University of Tokyo) Date: November 7 (Thu), 2013, 15:40-16:10 Title: Visible actions on generalized flag varieties — Geometry of multiplicity-free representations of $SO(N)$ Abstract: The subject of study is tensor product representations of irreducible representations of the orthogonal group, which are multiplicity-free. Here we say a group representation is multiplicity-free if any irreducible representation occurs at most once in its irreducible decomposition. The motivation is the theory of visible actions on complex manifolds, which was introduced by T. Kobayashi. In this theory, the main tool for proving the multiplicity-freeness property of group representations is the "propagation theorem of the multiplicity-freeness property". By using this theorem and Stembridge's classification result, we obtain the following: All the multiplicity-free tensor product representations of $SO(N)$ and $Spin(N)$ can be obtained from character, alternating tensor product and spin representations combined with visible actions on orthogonal generalized flag varieties. Speaker: Pampa Paul (Indian Statistical Institute, Kolkata) Date: November 7 (Thu), 2013, 16:10-16:40 Title: Holomorphic discrete series and Borel-de Siebenthal discrete series Abstract: Let $G_0$ be a simply connected non-compact real simple Lie group with maximal compact subgroup $K_0$. Let $T_0\subset K_0$ be a Cartan subgroup of $K_0$ as well as of $G_0$. So $G_0$ has discrete series representations. Denote by $\frak{g}, \frak{k},$ and $\frak{t}$, the complexifications of the Lie algebras $\frak{g}_0, \frak{k}_0$ and $\ frak{t}_0$ of $G_0, K_0$ and $T_0$ respectively. There exists a positive root system $\Delta^+$ of $\frak{g}$ with respect to $\frak{t}$, known as the Borel-de Siebenthal positive system for which there is exactly one non-compact simple root, denoted $\nu$. Let $\mu$ denote the highest root. If $G_0/K_0$ is Hermitian symmetric, then $\nu$ has coefficient $1$ in $\mu$ and one can define holomorphic discrete series representation of $G_0$ using $\Delta^+$. If $G_0/K_0$ is not Hermitian symmetric, the coefficient of $\nu$ in the highest root $\mu$ is $2$. In this case, Borel-de Siebenthal discrete series of $G_0$ is defined using $\Delta^+$ in a manner analogous to the holomorphic discrete series. Let $\nu^*$ be the fundamental weight corresponding to $\nu$ and $L_0$ be the centralizer in $K_0$ of the circle subgroup defined by $i\nu^*$. Note that $L_0 = K_0$, when $G_0/K_0$ is Hermitian symmetric. Otherwise, $L_0$ is a proper subgroup of $K_0$ and $K_0/L_0$ is an irreducible compact Hermitian symmetric space. Let $G$ be the simply connected Lie group with Lie algebra $\frak{g}$ and $K_0^* \subset G$ be the dual of $K_0$ with respect to $L_0$ (or, the image of $L_0$ in $G$). Then $K_0^*/L_0$ is an irreducible non-compact Hermitian symmetric space dual to $K_0/L_0$. In this talk, to each Borel-de Siebenthal discrete series of $G_0$, a holomorphic discrete series of $K_0^*$ will be associated and occurrence of common $L_0$-types in both the series will be discussed. Speaker: Dipendra Prasad (Tata Institute of Fundamental Research) Date: November 7 (Thu), 2013, 16:50-17:40 Title: Branching laws and the local Langlands correspondence Abstract: The decomposition of a representation of a group when restricted to a subgroup is an important problem well-studied for finite and compact Lie groups, and continues to be of much contemporary interest in the context of real and $p$-adic groups. We will survey some of the questions that have recently been considered drawing analogy with Compact Lie groups, and what it suggests in the context of real and $p$-adic groups via what is called the local Langlands correspondence. Colloquium at the Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Dipendra Prasad (Tata Institute of Fundamental Research) Date: November 8 (Fri), 2013, 16:30-17:30 Place: Room 123, Graduate School of Mathematical Sciences, the University of Tokyo Title: Ext Analogues of Branching laws Abstract: [ pdf ] The decomposition of a representation of a group when restricted to a subgroup is an important problem well-studied for finite and compact Lie groups, and continues to be of much contemporary interest in the context of real and $p$-adic groups. We will survey some of the questions that have recently been considered, and look at a variation of these questions involving concepts in homological algebra which gives rise to interesting newer questions. Speaker: Ronald King (the University of Southampton) Date: November 11 (Mon), 2013, 16:30-17:30 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Title: Alternating sign matrices, primed shifted tableaux and Tokuyama factorisation theorems Abstract: [ pdf ] Twenty years ago Okada established a remarkable set of identities relating weighted sums over half-turn alternating sign matrices (ASMs) to products taking the form of deformations of Weyl denominator formulae for Lie algebras B_n, C_n and D_n. Shortly afterwards Simpson added another such identity to the list. It will be shown that various classes of ASMs are in bijective correspondence with certain sets of shifted tableaux, and that statistics on these ASMs may be expressed in terms of the entries in corresponding compass point matrices (CPMs). This then enables the Okada and Simpson identities to be expressed in terms of weighted sums over primed shifted tableaux. This offers the possibility of extending each of these identities, that originally involved a single parameter and a single shifted tableau shape, to more general identities involving both sequences of parameters and shapes specified by arbitrary partitions. It is conjectured that in each case an appropriate multi-parameter weighted sum can be expressed as a product of a deformed Weyl denominator and group character of the type first proved in the A_n case by Tokuyma in 1988. The conjectured forms of the generalised Okada and Simpson identities will be given explicitly, along with an account of recent progress made in collaboration with Angèle Hamel in proving some of them. Speaker: Simon Gindikin (Rutgers University, USA) Date: November 19 (Tue), 2013, 16:30-17:30 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Title: Horospheres, wonderfull compactification and c-function Abstract: [ pdf ] I will discuss what is closures of horospheres at the wonderfull compactification and how does it connected with horospherical transforms, c-functions and product-formulas. (FMSP Lectures) Speaker: Simon Gindikin (Rutgers University, USA) Date & Place: (Lecture I) December 4 (Wed), 2013, 15:00-16:30, Room 128 (Lecture II) December 6 (Fri), 2013, 16:30-18:00, Room 118 Title: Horospheres: geometry and analysis Abstract: [ pdf ] About 50 years ago Gelfand suggested a conception of horospherical transform which, he hoped, must be an universal principle unifying non commutative harmonic analysis. I want to make several historic remarks (especially, since this year is the centennial anniversary of Gelfand). I want to discuss the evolution of this fundamental conception for 50 years and how Gelfand's dreams are looked today. I will discuss an elementary case of hyperboloids of any signature where there were not too much progress after old initial result of Gelfand-Graev. Speaker: Koichi Kaizuka (Lˌ) (University of Tsukuba) Date: December 17 (Tue), 2013, 16:30-17:30 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Title: A characterization of the $L^{2}$-range of the Poisson transform on symmetric spaces of noncompact type (RpNg^Ώ̋Ԃɂ|A\ϊ$L^{2}$-l̓ t) Abstract: [ pdf ] Characterizations of the joint eigenspaces of invariant differential operators in terms of the Poisson transform have been one of the central problems in harmonic analysis on symmetric spaces. From the point of view of spectral theory, Strichartz (J. Funct. Anal.(1989)) formulated a conjecture concerning a certain image characterization of the Poisson transform of the $L^{2}$-space on the boundary on symmetric spaces of noncompact type. In this talk, we employ techniques in scattering theory to present a positive answer to the Strichartz conjecture. (Lecture) Speaker: Hisayosi Matumoto ({v`) (the University of Tokyo) Date: December 18 (Wed), 2013, 15:00-16:30 Place: Room 122, Graduate School of Mathematical Sciences, the University of Tokyo Title: Whittaker vectors for Harish-Chandra modules Abstract: Let $G$ be a real reductive linear Lie group and let $G=KAN$ be its Iwasawa decomposition. We denote by ${\mathfrak g}$ the complexified Lie algebra corresponding to $G$. We consider a non-degenerate unitary character $\psi : N\rightarrow {\mathbb C}^\times$. Let $V$ be a Harish-Chandra $({\mathfrak g},K)$-module. Roughly speaking , a Whittaker model of $V$ is a realization of $V$ as a submodule of an induced representation $Ind_N^G(\psi)$. A Whittaker vector is an element in a "dual" of $V$ corresponding to a Whittaler model. In this talk, we discuss the following subjects. (1) We give a precise definition of Whittaker vectors, namely algebraic, real analytic, and $C^\infty$-Whittaker vectors. (2) We explain how to reduce the computation of the dimension of the spaces of Whittaker vectors for general Harish-Chandra modules to the special case of principal series representations. (3) We discuss the Whittaker vectors for principal series representations and explain why the dimension of the space of real analytic Whittaker vectors are greator than the dimension of the space of $C^\infty$- Whittaker vectors.