数理科学広域演習, Winter School 2015 on Representation Theory of Real Reductive Groups, Hironori Oya, Piotr Pragacz, Bent Ørsted #1, Yuichiro Tanaka, Ryosuke Nakahama, Bent Ørsted #2, Bent Ørsted #3, Anton Evseev, Takeyoshi Kogiso, Anatoly Vershik, Analytic Representation Theory of Lie Groups, Michael Pevzner, Paul Baum #1, Paul Baum #2, Toshiaki Hattori, Paul Baum #3, Fabian Januszewski,

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 GroupsGraduate School of Mathematical Sciences, the University of Tokyo January 24 (Sat)-26 (Mon), 2015 | |

January 24 (Sat) | 13:00-14:00, Room 126Peter Trapa"Unitary representations of reductive Lie groups I" |

14:30-15:30, Room 126Raul Gomez"The Tor and Ext functors for smooth representations of real algebraic groups" | |

16:30-17:30, Room 126Benjamin Harris"The Geometry of Tempered Characters" | |

January 25 (Sun) | 9:00-10:00, Room 126Raul Gomez"Generalized and degenerate Whittaker models associated to nilpotent orbits" |

10:30-11:30, Room 126Benjamin Harris"The Geometry of Harmonic Analysis" | |

12:00-13:00, Room 126Peter Trapa"Unitary representations of reductive Lie groups II" | |

January 26 (Mon) | 9:30-10:30, Room 122Benjamin Harris"The Geometry of Nontempered Characters" |

11:00-12:00, Room 122Raul Gomez"Local Theta lifting of generalized Whittaker models" | |

12:30-13:30, Room 128Peter 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. |

Date: | March 24 (Tue), 2015, 18:00-19:30 |

Place: | Room 126, Graduate School of Mathematical Sciences, the University of Tokyo |

Speaker: | Piotr Pragacz (Institute of Mathematics, Polish Academy of Sciences) |

Title: | A Gysin formula for Hall-Littlewood polynomials |

Abstract: [ pdf ] |
Schubert calculus on Grassmannians is governed by Schur S- functions, the one on Lagrangian Grassmannians by Schur Q-functions. There were several attempts to give a unifying approach to both situations. We propose to use Hall-Littlewood symmetric polynomials. They appeared implicitly in Hall's study of the combinatorial lattice structure of finite abelian p-groups and in Green's calculations of the characters of GL(n) over finite fields; they appeared explicitly in the work of Littlewood on some problems in representation theory. With the projection in a Grassmann bundle, there is associated its Gysin map, induced by pushing forward cycles (topologists call it "integration along fibers"). We state and prove a Gysin formula for HL-polynomials in these bundles. We discuss its two specializations, giving better insights to previously known formulas for Schur S- and P-functions. |

Date: | April 7 (Tue), 2015, 16:30-17:30 |

Place: | Room 122, Graduate School of Mathematical Sciences, the University of Tokyo |

Speaker: | Bent Ørsted (Aarhus University and the University of Tokyo) |

Title: | Branching laws and elliptic boundary value problems |

Abstract: [ pdf ] |
Classically the Poisson transform relates harmonic functions in the complex upper half plane to their boundary values on the real axis. In some recent work by Caffarelli et al. some new generalizations of this appears in connection with the fractional Laplacian. In this lecture we shall explain how the symmetry-breaking operators introduced by T. Kobayashi for studying branching laws may shed new light on the situation for elliptic boundary value problems. This is based on joint work with J. Möllers and G. Zhang. |

Date: | April 14 (Tue), 2015, 16:30-18:00 |

Place: | Room 122, Graduate School of Mathematical Sciences, the University of Tokyo |

Speaker: | Yuichiro Tanaka (田中雄一郎) (Institute of Mathematics for Industry, Kyushu University) |

Title: | Visible actions of compact Lie groups on complex spherical varieties (複素球多様体へのコンパクトリー群の可視的作用について) |

Abstract: [ pdf ] |
With the aim of uniform treatment of multiplicity-free representations
of Lie groups,
T. Kobayashi introduced the theory of visible actions on complex
manifolds. In this talk we consider visible actions of a compact real form U of a connected complex reductive algebraic group G on G-spherical varieties. Here a complex G-variety X is said to be spherical if a Borel subgroup of G has an open orbit on X. The sphericity implies the multiplicity-freeness property of the space of polynomials on X. Our main result gives an abstract proof for the visibility of U-actions. As a corollary, we obtain an alternative proof for the visibility of U- actions on linear multiplicity-free spaces, which was earlier proved by A. Sasaki (2009, 2011), and the visibility of U-actions on generalized flag varieties, earlier proved by Kobayashi (2007) and T- (2013, 2014). |

Date: | April 21 (Tue), 2015, 17:00-18:30 |

Place: | Room 122, Graduate School of Mathematical Sciences, the University of Tokyo |

Speaker: | Ryosuke Nakahama (中濱良祐) (The University of Tokyo) |

Title: | Norm computation and analytic continuation of vector valued holomorphic discrete series representations (ベクトル値正則離散系列表現のノルム計算と解析接続) |

Abstract: [ pdf ] |
The holomorphic discrete series representations is
realized on the space of vector-valued holomorphic functions
on the complex bounded symmetric domains.
When the parameter is sufficiently large, then its norm is given by
the converging integral,
but when the parameter becomes small, then the integral does not
converge.
However, if once we compute the norm explicitly,
then we can consider its analytic continuation,
and can discuss its properties, such as unitarizability.
In this talk we treat the results on explicit norm computation.
正則離散系列表現は，複素有界対称領域上のベクトル値正則関数空間上に実現さ れる． そのノルムはパラメータが十分大きい場合には収束する積分で表せるが， パラメータが小さくなるとその積分は収束しなくなる． しかし，ノルムを具体的に計算することによって， その小さいパラメータへの解析接続を考えることができ， そのユニタリ化可能性などの性質を論じることができる． 本講演ではノルムの具体的な計算に関する結果を扱う． |

(Colloquium at the University of Tokyo) | |

Date: | April 24 (Fri), 2015, 16:50-17:50 |

Place: | Room 123, Graduate School of Mathematical Sciences, the University of Tokyo |

Speaker: | Bent Ørsted (Aarhus University and the University of Tokyo) |

Title: | Rigidity of conformal functionals on spheres |

Abstract: [ pdf ] | On a compact smooth manifold one may construct a Riemannian metric in many different ways. Each metric gives rise to natural elliptic operators such as the Laplace-Beltrami operator and corresponding spectral invariants, e.g. the eigenvalues, the trace of the heat semigroup, and the zeta function. In this lecture we shall consider such functionals on the space of metrics on the sphere, combining conformal differential geometry and representation theory of semisimple Lie groups to obtain results about local extremal properties of special functionals. This is based on joint work with Niels Martin Möller. |

Date: | April 28 (Tue), 2015, 17:00-18:30 |

Place: | Room 122, Graduate School of Mathematical Sciences, the University of Tokyo |

Speaker: | Bent Ørsted (Aarhus University and the University of Tokyo) |

Title: | Restricting automorphic forms to geodesic cycles |

Abstract: [ pdf ] | We find estimates for the restriction of automorphic forms on hyperbolic manifolds to compact geodesic cycles in terms of their expansion into eigenfunctions of the Laplacian. Our method resembles earlier work on products of automorphic forms by Bernstein and Reznikov, and it uses Kobayashi's new symmetry-breaking kernels. This is joint work with Jan Möllers. |

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 ではないに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 は（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。） |

Date: | June 30 (Tue), 2015, 17:00-18:30 |

Place: | Room 122, Graduate School of Mathematical Sciences, the University of Tokyo |

Speaker: | Anatoly Vershik (St. Petersburg Department of Steklov Institute of Mathematics) |

Title: | Random subgroups and representation theory |

Abstract: [ pdf ] |
The following problem had been appeared independently in different
teams and various reason:
to describe the Borel measures on the lattice of all subgroups of
given group, which are invariant with respect to the action of the group
by conjugacy. The main interest of course represents nonatomic measures
which exist not for any group. I will explain how these measures connected with characters and representations of the group, and describe the complete list of such measures for infinite symmetric group. |

Analytic Representation Theory of Lie Groups | |

Date: | July 1 (Wed)-July 4 (Sat), 2015 |

Place: | Kavli IPMU, the University of Tokyo |

Intensive lectures (集中講義) | |

Date: | July 7 & 14, 2015, 15:00-16:00 |

Place: | Room 126, Graduate School of Mathematical Sciences, the University of Tokyo |

Speaker: | Michael Pevzner (Reims Univeristy) |

Title: | Different aspects of Rankin-Cohen brackets |

Abstract: |
Rankin-Cohen brackets form an infinite family of bi-differential
operators having a surprisingly rich internal structure. We shall
illustrate some of its
manifestations in the first lecture and give some explanations in the
second one. Lecture 1. First examples: differential operations on modular forms and quantization of the one-sheeted hyperboloid. Lecture 2. Rankin-Cohen brackets as symmetry breaking operators: efficiency of the F-method. |

Intensive lectures (集中講義) | |

Date: | July 15-17, 23 & 24, 2015, 15:00-16:30 |

Place: | Room 122 (July 15, 16, 23); Room 126 (July 24); Room 128 (July 17), Graduate School of Mathematical Sciences, the University of Tokyo |

Speaker: | Paul Baum (Penn State Univeristy) |

Title: | INDEX THEORY AND K-HOMOLOGY |

Abstract: |
This series of five lectures will prove the Atiyah-Singer index theorem
as a corollary of Bott periodicity, and then give an
exposition of some further developments.
Lecture 1 . Dirac Operator.
Lecture 2. Atiyah-Singer Revisited.
Lecture 3. What is K-homology?
Lecture 4. Beyond Ellipticity.
Lecture 5. The Riemann-Roch Theorem. |

Date: | July 14 (Tue), 2015, 17:00-18:30 |

Place: | Room 122, Graduate School of Mathematical Sciences, the University of Tokyo |

Speaker: | Paul Baum (Penn State Univeristy) |

Title: | MORITA EQUIVALENCE REVISITED |

Abstract: [ pdf ] |
Let X be a complex affine variety and k its coordinate algebra. A k-
algebra is an algebra A over the complex numbers which is a k-module
(with an evident compatibility between the algebra structure of A and
the k-module structure of A). A is not required to have a unit.
A k-algebra A is of finite type if as a k-module A is finitely
generated. This talk will review Morita equivalence for k-algebras and
will then
introduce --- for finite type k-algebras ---a weakening of Morita
equivalence called geometric equivalence. The new equivalence relation
preserves the primitive ideal space (i.e. the set of isomorphism classes
of irreducible A-modules) and the periodic cyclic homology of A.
However, the new equivalence relation permits a tearing apart of strata
in the primitive ideal space which is not allowed by Morita equivalence. Let G be a connected split reductive p-adic group, The ABPS (Aubert- Baum-Plymen-Solleveld) conjecture states that the finite type algebra which Bernstein assigns to any given Bernstein component in the smooth dual of G, is geometrically equivalent to the coordinate algebra of the associated extended quotient. The second talk will give an exposition of the ABPS conjecture. |

Date: | July 21 (Tue), 2015, 15:30-16:30 |

Place: | Room 122, Graduate School of Mathematical Sciences, the University of Tokyo |

Speaker: | Toshiaki Hattori (服部俊昭) (Tokyo Institute of Technology) |

Title: | Shimizu's lemma for SL(3,R)/SO(3) (清水の補題のSL(3,R)/SO(3)の場合への拡張について) |

Abstract: [ pdf ] |
We find a generalization of Shimizu's lemma in the case
of the symmetric space SL(3, R)/SO(3) of noncompact type of rank 2.
We also find a relation between this lemma and displacement of
horoballs by isometries. PSL(2,C)の部分群の離散性に関する必要条件である清水の補題， Jorgensenの不等式を双曲空間から他の階数１の対称空間の場合 に拡張しようという研究が現在進行中であるが, 高階の対称空間 についてそのような結果はまだないようである。階数が２の対称 空間で最も簡単なSL(3,R)/SO(3)の場合に清水の補題を拡張する 試みについてお話しする。 |

Date: | July 21 (Tue), 2015, 17:00-18:30 |

Place: | Room 122, Graduate School of Mathematical Sciences, the University of Tokyo |

Speaker: | Paul Baum (Penn State Univeristy) |

Title: | GEOMETRIC STRUCTURE IN SMOOTH DUAL |

Abstract: [ pdf ] |
Let G be a connected split reductive p-adic group. Examples are GL(n,
F) , SL(n, F) , SO(n, F) , Sp(2n, F) , PGL(n, F)
where n can be any positive integer and F can be any finite extension of
the field Q_p of p-adic numbers. The smooth
(or admissible) dual of G is the set of equivalence classes of smooth
irreducible representations of G. This talk will first
review the theory of the Bernstein center. According to this theory, the
smooth dual of G is the disjoint union of subsets
known as the Bernstein components. The talk will then explain the ABPS
(Aubert-Baum-Plymen-Solleveld) conjecture
which states that each Bernstein component is a complex affine variety.
Each of these complex affine varieties is explicitly
identified as the extended quotient associated to the given Bernstein
component. The ABPS conjecture has been proved for GL(n, F), SO(n, F), and Sp(2n, F). |

Date: | July 28 (Tue), 2015, 17:00-18:30 |

Place: | Room 122, Graduate School of Mathematical Sciences, the University of Tokyo |

Speaker: | Fabian Januszewski (Karlsruhe Institute of Technology (KIT)) |

Title: | On (g,K)-modules over arbitrary fields and applications to special values of L-functions |

Abstract: [ pdf ] | I will introduce g,K-modules over arbitrary fields of characteristic 0 and discuss their fundamental properties and constructions, including Zuckerman functors. This may be applied to produce models of certain standard modules over number fields, which has applications to special values of automorphic L-functions, and also furnishes the space of regular algebraic cusp forms of GL(n) with a natural global Q-structure. |

© Toshiyuki Kobayashi