表現論セミナー

Date

February 16 (Tue), 10:30-12:30, 2010

Room

Dept. of Math. Building no. 3 Room 108

Speaker

Prof. Dan Ciubotaru (Utah)

Title

On formal degrees for discrete series of classical affine Hecke algebras.

Abstract

  The talk is based on joint work with Syu Kato. The expected stability of L-packets of discrete series for p-adic groups implies that the formal degrees of the discrete series in the same L-packet have to be proportional. In Lusztig's category of representations with unipotent cuspidal support, this problem can be translated to one for affine Hecke algebras with unequal parameters. Following Reeder, Opdam, and Solleveld, the formal degree of a discrete series for affine Hecke algebras are known up to a rational constant (depending on the discrete series). Reeder conjectured a precise form for this constant, and verified this for the Hecke algebras arising for split exceptional groups. We compute the missing constants for the affine Hecke algebras of classical types with unequal parameters. The method of calculation is a consequence of a new algorithm for the W-structure of tempered modules for these Hecke algebras, based on Kato's exotic geometry.

Organizers S.Ariki, Syu Kato, Shin-ichi Kato

Date

January 14 (Thu), 10:30-12:30

Room

Dept. of Math. Building no. 3 Room 552

Speaker

功刀直子氏 (東京理科大)

Title

Principal blocks of general linear groups with non-abelian Sylow subgroups in non-defining characteristic

Abstract

  有限一般線型群の非定義体に関するモジュラー表現を考える。 Chuang-Rouquier により, 一般線型群に関するブルエの可換不足群予想は解決され, その結果として可換シロー部分群を持つ場合, 定義体に関する条件が同じ2つの主ブロックは 森田同値になることが得られている。 森田同値に関して同様のことを,非可換シロー部分群 を持つ場合に考察する。

Organizers S.Ariki, Syu Kato, Shin-ichi Kato

Date

December 10 (Wednesday), 10:30-12:30

Room

Dept. of Math. Building no. 3 Room 552

Speaker

佐垣大輔氏 (筑波大)

Title

Path model for extremal weight modules over quantum affine algebras of infinite rank.

Abstract

  Littelmann は, 1994年と1995年の論文において, Lakshmibai-Seshadri (LS) パスの概念を導入した: 対称化可能な Kac-Moody リー環 g の整ウェイト \lambda が 与えられたとき, 型 \lambda の LS パスとは, 閉区間 [0,1] から h_R^* (Cartan 部分代数 h の実形 h_R の双対空間) への 区分的に線形で連続な写像であって, \lambda を通る Weyl 群 軌道に関するある組合せ論的な条件を満たすもののことである. さらに Littelmann はルート作用素と呼ばれる写像を定義し, それを用いて型 \lambda の LS パス全体の集合 B(\lambda) に クリスタルの構造を与えた. その後, Kashiwara と Joseph に より, \lambda が支配的整ウェイトである場合は B(\lambda) は 最高ウェイト \lambda の既約最高ウェイト U_{q}(g)-加群の 結晶基底に同型であることが証明された.

今回のセミナーでは, LS パスの定義や基本的な結果について 簡単に解説した後, B_{\infty}, C_{\infty}, D_{\infty} 型の "infinite rank affine Lie algebra" (Kac の教科書の7.11節 参照) に対する LS パスについて考察し, 次の定理を述べる:

[定理] g を B_{\infty}, C_{\infty}, D_{\infty} の infinite rank affine Lie algebra とし, \lambda を g の (支配的とは限ら ない) 整ウェイトとする. このとき, 型 \lambda の LS パスの なすクリスタル B(\lambda) は, extremal ウェイト \lambda の extremal ウェイト U_{q}(g) 加群の結晶基底と同型である.

それから, LS パスのなすクリスタルのテンソル積 B(\lambda) \otimes B(\mu) の (連結成分への) 分解則 についても議論しようと思う.

Organizers S.Ariki, Syu Kato, Shin-ichi Kato

Date

11月20日 (金) 14:00-16:00,
25日 (水), 26日 (木) 10:30-12:30

Room

京大・理学部3号館305号室
日程・講演時間・部屋がいつもと違いますのでご注意ください

Speaker

Cedric Bonnafe氏 (Besancon)

Title

Geometry of Deligne-Lusztig varieties and representations

Abstract

  In 1976, the fundamental paper by Deligne and Lusztig lead to many significant advances in the representation theory of finite reductive groups. Their theory is build on varieties (the so-called Deligne-Lusztig varieties) on which the finite reductive group acts: one can then recover many representations by studying their l-adic cohomology. This method applies as well to ordinary representations (character theory) as to modular representations (blocks, decomposition matrices, Broue's abelian defect conjecture). Our series of lectures will roughly follow the following plan:

First lecture - geometry of Deligne-Lusztig varieties
Second lecture - Ordinary representations
Third lecture - Modular representations

Organizers S.Ariki, Syu Kato, Shin-ichi Kato

Date

November 4 (Wednesday), 10:30-12:30

Room

Dept. of Math. Building no. 3 Room 552

Speaker

谷崎俊之氏 (阪市大)

Title

Variations on a theme of Bezrukavnikov-Mirkovic-Rumynin

Abstract

  Bezrukavnikov-Mirkovic-Rumynin gave a correspondence between representations of simple Lie algebras in positive characteristics and $D$-modules on the corresponding flag manifold. The aim of the present talk is to give its analogue for quantized enveloping algebras at roots of 1.

More precisely, we establish a derived equivalence between the category of certain modules over the (De Concini-Kac type) quantized enveloping algebras at roots of 1 and that of (crystalline) $D$-modules on the quantized flag manifold.

At roots of 1 we can associate a sheaf of rings $\tilde{D}$ on the ordinary flag manifold over the complex number field, so that the category of $D$-modules is equivalent to that of $\tilde{D}$-modules.

Let $Z$ be the center of $\tilde{D}$. We can show that $\tilde{D}$ is an Azumaya algebra on $Spec Z$. We can also show that restrictions of $\tilde{D}$ to certain closed subsets are split Azumaya algebras.

By those results we obtain a correspondence between representations of quantized enveloping algebras at roots of 1 and $O$-modules on the Springer fibers. This implies, for example, Lusztig's conjecture on the number of irreducible representations of quantized enveloping algebras with specified central character.

A closely related result using a different definition of $D$-modules is also given by Backelin-Kremnizer.

Organizers S.Ariki, Syu Kato, Shin-ichi Kato

Date

October 29 (Thursday), 10:30-12:30

Room

Dept. of Math. Building no. 3 Room 552

Speaker

Prof. Tomoyuki Arakawa (Nara WU)

Title

Varieties of nilpotent orbits, modular invariant representations of Kac-Moody algebras, and lisse representation of affine W-algebras

Abstract

  In my talk I will discuss the relationship among the following three topics:
(1) Some class of varieties of nilpotent orbits such as {x in g; (adx)^n=0}, where g is a simple Lie algebra and n is an even integer;
(2) Modular invariant representations (Kac-Wakimoto admissible representations) of Kac-Moody algebras;
(3) Lisse (or C_2-cofinite) representations of affine W-algebras, which can be considered as quantization of infinite jet schemes of special transversal slices (Slodowy slices).

As a consequence I will prove the C_2 cofintiness of all (non- principal) exceptional W-algebras recently discovered by Kac-Wakimoto. In fact I will show there are more C_2 -cofinite W-algebras. This gives a new, huge examples of C_2 cofinite vertex operator algebras.

Organizers S.Ariki, Syu Kato, Shin-ichi Kato

Date

10月21日 (水), 22日(木), 28日 (水), 10:30-12:30

Room

京大・理学部3号館305号室

Speaker

Changchang Xi氏 (RIMS/北京師範大学)

Title

Constructions of derived equivalences and stable equivalences

Abstract

  This is a series of three lectures on constructions of derived equivalences and stable equivalences of Morita type. We will start with basic definitions and facts, survey some fundamental results, and show new methods of how to construct these equivalences from given ones. During the course, we will also consider when a derived equivalence implies a stable equivalence of Morita type. This leads to a generalization of a result by Rickard for self-injective algebras.

Organizers S.Ariki, Syu Kato, Shin-ichi Kato

Date

10月15日 (木), 10:30-12:30

Room

京大・理学部3号館552号室

Speaker

Michel Duflo氏 (Paris VII)

Title

Weyl's functional calculus and equivariant differential forms

Abstract

  Let A1,A2,...,Ad be d Hermitian matrices of size n. Weyl's functional calculus is a compactly supported distribution W on $R^d$ which associates to a smooth function f of d variables a matrix W(f) := f(A1,...,An). Forty years ago, Edward Nelson gave a formula for W, explicitly describing it as the derivative of a probability measure on Rd supported on the joint numerical range of the Ai. We show how this formula fits in the setting of Hamiltonian geometry and equivariant differential forms.

Organizers S.Ariki, Syu Kato, Shin-ichi Kato

Date

10月8日 (木), 10:30-12:30

Room

京大・理学部3号館552号室

Speaker

清水健一氏 (筑波大・数学)

Title

有限次元ホップ代数の表現のなすモノイダル圏の不変量について

Abstract

  モノイダル圏(またはテンソル圏)とは、対象の間に結合的な二項演算が定義されて いるような圏であり、群や量子群などの表現論に限らず、トポロジーなどにおいて も盛んに研究されている。本講演では、モノイダル圏において定義される組みひも 構造や中心構成などについて、ホップ代数の表現論の立場から解説を行うとともに、 それらを用いて構成される有限次元ホップ代数の表現のなすモノイダル圏に対する 不変量を紹介する。この不変量は、特に有限群の群環に対しては簡単に計算でき、 また実際に表現環の同型な多くの半単純ホップ代数を区別できる。時間が許せば、 これらの構成の幾何学的背景に関しても解説を行いたい。

Organizers S.Ariki, Syu Kato, Shin-ichi Kato

Date

August 3 (Mon), 14:00-15:00, 15:30-16:30

Room

Dept.Math. (Building 3) Room 552

Speaker

Milind Sohoni (IIT Bombay)

Title

(talk 1) Geometric Complexity Theory
(talk 2) The quantum deformation of the restriction of GL_{mn}-modules to GL_m \times GL_n

Abstract

  (talk 1)
Let X be an m-by-m matrix and consider the (i) the form det(X), the determinant,and, (ii) perm(X), the permanent (i.e., the determinant without the signs).
There are efficient algorithms to compute the determinant, while no such algorithm is known for the permanent. P v. NP and other questions in theoretical computer science ask for such proofs.

In our approach, we convert the non-existence of an algorithm to the existence of "obstructions" or "witnesses", which are constructs from geometric invariant theory and ultimately representation theory. We show that the GL(X) orbits, and their closures, of det(X), perm(X) have a lot to do with the question. We show that both det(X) and perm(X) are GIT-stable. Since both the forms have a very distinctive stabilizers H in GL(X), this immediately brings us to the Peter-Weyl modules, i.e. GL(X)-modules with an H-invariant vector. The witnesses or obstructions are from such modules.

We also survey another approach to the problem and analyse its GIT content. Finally, we pose the group restriction problem, i.e., the presence of H-invariant vectors in a G-module, as the main problem. We outline why quantum algebras should be the key tool for the problem.

This is joint work with Ketan Mulmuley.


(talk 2)
We consider the problem of classification of P-W modules for a pair (H,G), as posed earlier. We quickly survey the key results in the (G, G \times G), which is now well-understood in its algebraic and combinatorial facets. We hope for a similar insight for the stabilizer of the determinant, i.e., (GL_m \times GL_m , GL_{m^2 }), where the embedding is
(A,B)(X) \rightarrow AXB^{-1}.

We consider the (slightly more general) embedding
GL_m \times GL_n \rightarrow GL_{mn}.

A quantization of the above embedding is not known. However, for any Weyl module $V_{\lambda }(\C^mn)$ of $GL_{mn}$ we construct a $U_q (gl_m ) \otimes U_q (gl_n )$-structure, which at q=1 matches the classical embedding.

This is done in two steps: (i) A faithful embedding of $U_q (gl_m ) \otimes U_q (gl_n ) \rightarrow U_q (gl_{mn})$ for the action on $\wedge^k (\C^mn)$. (ii) a straightening law which helps construct the general $V_{\lambda }(\C^mn)$. For the $\wedge^k (\C^mn)$ we also exhibit the bi-crystal structure, thereby constructing the algebra behind the combinatorial results of Danilov et al.

The general problem is to construct a bi-crystal structure on semistandard tableaus with entries in [mn]. Our earlier result proves that such a structure exists. We demonstrate a candidate U_q (gl_m ) crystal structure for n=2 and ask if this is what arises from the above quantization.

This is ongoing joint work with Bharat Adsul and K. V. Subrahmanyam.

Organizers S.Ariki, Shin-ichi Kato, Syu Kato,

Date

June 18 (Thu), 10:30-12:30

Room

RIMS room 102 (いつもと部屋が違いますのでご注意下さい)

Speaker

Prof. Hiraku Nakajima (RIMS)

Title

Quiver varieties and cluster algebras

Abstract

  Motivated by a recent conjecture by Hernandez and Leclerc, we embed a Fomin-Zelevinsky cluster algebra into the Grothendieck ring R of the category of representations of quantum loop algebras U_q(Lg) of a symmetric Kac-Moody Lie algebra, studied earlier by the author via perverse sheaves on graded quiver varieties. Graded quiver varieties controlling the image can be identified with varieties which Lusztig used to define the canonical base. The cluster monomials form a subset of the base given by the classes of simple modules in R, or Lusztig's dual canonical base. The positivity and linearly independence (and probably many other) conjectures of cluster monomials follow as consequences, when there is a seed with a bipartite quiver. Simple modules corresponding to cluster monomials factorize into tensor products of `prime' simple ones according to the cluster expansion.

Organizers S.Ariki, Shin-ichi Kato, Syu Kato,

Date

June 11 (Thu), 10:30-12:30

Room

Dept.Math. (Building 3) Room 552

Speaker

Prof. Mikhail Khovanov (Columbia University)

Title

Categorification of the Iwahori-Hecke algebra and quantum groups

Abstract

  We will review the diagramatic description of Soergel's categorification of the Iwahori-Hecke algebra and explain the diagrammatics for the categorification of quantum groups. The talk is based on joint works with Ben Elias and Aaron Lauda.

Organizers S.Ariki, Shin-ichi Kato, Syu Kato,

Date

5/18(月) 115号室 15:00--17:00
5/19(火) 115号室 14:00--16:00
5/20(水) 202号室 15:00--17:00
5/21(木) 202号室 15:00--17:00
5/22(金) 202号室 13:30--15:30
(曜日によって部屋、時間が異なりますのでご注意ください)

Room

京大・数理解析研究所

Speaker

Shrawan Kumar氏 (UNC)

Title

Eigenvalue problem for reductive groups

Abstract

  こちらをご参照ください。 [pdf]

Organizers S.Ariki, Syu Kato, Shin-ichi Kato

Date

May 7 (Thu), 10:30--12:30

Room

京大・理学部3号館552号室

Speaker

橋本光靖氏 (名大多元数理)

Title

良いフィルター付けと不変式環

Abstract

  多項式環に簡約群が作用すると、不変式環は標数0では有理特異点を持ちます。 特に Cohen-Macaulay になるのですが、正標数ではそうならない例があります。 しかしながら、正標数でも、重要な例で不変式環が Cohen-Macaulay になるものは 多いです。本講演では、正標数の体上の多項式環に簡約群が線型に 作用し、多項式環が表現として S. Donkin の意味で良いフィルター付 けを持つならば、不変式環が強F正則であることを示したのでそのことについて 話します。 強F正則性は密着閉包の概念とともに M. Hochster と C.Huneke によって考えられた 正標数の可換環論の概念で、不変式論とのなじみがいいことがだん だん明らかになってきています。強F正則ならば Cohen-Macaulay です。 定理の証明には Steinberg module の標準的な性質や、O.Mathieu に よるテンサー積定理など、Jantzen の教科書に出てくるような簡約群の表現論が 用いられます。

Organizers S.Ariki, Syu Kato, Shin-ichi Kato

Date

April 23 (Thu), 10:30-12:30

Room

Dept.Math. (Building 3) Room 552

Speaker

庄司俊明氏 (名大・多元)

Title

A geometric realization of Kostka functions associated to complex reflection groups

Abstract

  Kostka 多項式は一般線形群の巾零軌道の閉包の上の交叉コホモロジーにより記述できることが Lusztig により知られている。 Kostka多項式は分割によってラベル付けされるが、それをr個の分割の組で置き換えた関数が構成でき、それを複素鏡映群に付随した Kostka関数という。Achar-Henderson はr = 2 の場合にこの Kostka 関数がある種の軌道から得られる交叉コホモロジーで記述できることを示した。この講演では、その拡張として一般のrに対して、Kostka関数が交叉コホモロジーにより記述できることを示す。

Organizers S.Ariki, Syu Kato, Shin-ichi Kato

Date

April 16 (Thu), 10:30-12:30

Room

Dept.Math. (Building 3) Room 552

Speaker

浅芝秀人氏 (静岡大・理)

Title

Covering theory of categories without free action assumption and a 2-categorical generalization of Cohen-Montgomery duality

Abstract

  群Gの作用する(自由作用とは仮定しない)小圏とそれらの間の"弱 G同変"関手,お よびそれらの間の射のなす2圏をG-Catとおき,G次数付き小圏とそれらの間の"弱同次 "関手,およびそれらの間の射のなす2圏をG-GrCatとおくと,

(1) 軌道圏をとる操作は,2関手(-)/G : G-Cat → G- GrCat を導く;
(2) スマッシュ積をとる操作は,2関手 (-)#G : G-GrCat → G-Cat を導く;
(3) これらは互いに"弱2擬逆"になる.

以上は,Cibils-Marcosの定理およびCohen-Montgomery dualityの一般化を与える. さらに自然な関手 C →C/G および B#G →B を典型的な例とするG被覆関手 F : C → Bについて,

(4) Fの制限関手 F. は,圏同値 Mod B → Mod^G A(Mod B はB加群の圏,Mod^G A は,不変A加群からなるMod Aの充満圏)を導く;
(5) F. の左随伴関手(Fの左Kan拡大で定義される関手) は,圏同値 Mod A → Mod_G B(Mod_G B は,G次数B加群とそれらの間の次数を保つ射の圏)を導く.

時間があればこれらの導来同値などへの応用について述べる.

Organizers S.Ariki, Syu Kato, Shin-ichi Kato

Date

April 9 (Thu), 10:30-12:30

Room

Dept.Math. (Building 3) Room 552

Speaker

Prof. Susumu Ariki (RIMS)

Title

Graded q-Schur algebras

Abstract

  In this talk, we show that we may grade the Dipper-James' q-Schur algebra and that the graded decomposition numbers of the algebra is given by the plus version of the Leclerc-Thibon canonical basis in the deformed Fock space. This is the graded analogue of the work by Varagnolo and Vasserot, which we may obtain by specializing our result at v=1.

Organizers S.Ariki, Syu Kato, Shin-ichi Kato

Date

January 20 (Tue), 11:00-12:00 & 13:30--14:30

Room

RIMS room 102

Speaker

Prof. Patrik Delorme (Marseille)

Title

A Paley-Wiener theorem for Whittaker functions on a reductive $p$-adic group.

Abstract

  We define a Fourier transform for functions on a reductive $p$-adic group, which transform by a nondegenerate character of a maximal unipotent subgroup, and with compact support modulo this unipotent. This transformation, as well as wave packets are studied using a theory of the constant term. Then, a result of Heiermann is used to characterize the image of this Fourier transform.

In the second part some details will be given on the theory of the constant term.
Also I will try to explain (partial) analogous results for $p$-adic symmetric spaces (rational continuation of Eisenstein integrals, constant term).

Organizers S. Ariki, S.Kato

Date

January 19 (Mon), 11:00-12:00 & 13:30--14:30

Room

RIMS room 202

Speaker

Prof. Alexsander Samokhin (Oregon)

Title

Tilting bundles via the Frobenius morphism.

Abstract

  In these lectures we will discuss an approach to construct tilting bundles on algebraic varieties with the emphasis on homogeneous spaces and smooth toric varieties, and its relations to the D-affinity of flag varieties in positive characteristic. The proposed approach uses positive characteristic methods, notably the Frobenius morphism. We will start with a review of previously known methods to obtain tilting bundles, such as strong exceptional collections in derived categories of coherent sheaves. We will give examples of varieties that have strong exceptional collections and recall well-known results of Kapranov's as well as some more recent results of Kuznetsov's and of ours. We will then talk about inspirational works by Bezrukavnikov-Mirkovic-Rumynin on localization of modules over Lie algebras in positive characteristic and by Bezrukavnikov-Kaledin on the McKay correspondence in the symplectic case. Next we will talk about our own results on tilting bundles via the Frobenius morphism and work out in detail several examples of varieties when a tilting bundle can be obtained by taking the pushforward of a line bundle under the Frobenius. We will discuss implications of these results for the D-affinity of homogeneous spaces in positive characteristic and state a conjecture about such pushforwards.

Organizers S. Ariki, S.Kato

Date

January 13 (Tue), 10:30-12:00
January 16 (Fri), 16:30-18:00
January 21 (Wed), 10:30-12:00
January 22 (Thu), 10:30-12:00

Room

RIMS room 102 (Tue) and RIMS room 202 (Fri)
Room 009 for the last two talks.

Speaker

Aaron Lauda 氏(Columbia)

Title

(1) Categorification of quantum sl(2)
(2) Introduction to rings R(v)
(3) Rings R(v) and a categorification of quantum sl(n)
(4) Cyclotomic quotients of rings R(v)

Abstract

  Crane and Frenkel proposed that 4-dimensional TQFTs could be obtained by categorifying quantum groups at root of unity using their canonical bases. In my lectures I will explain joint work with Mikhail Khovanov which makes some steps towards this goal. We will see how various diagrammatically defined algebraic structures produce categorifications of quantum groups.

In my first lecture I will begin with the graphical calculus that categorifies the quantum enveloping algebra of sl(2) at generic q. We will see how the definition arises naturally by considering a semilinear form on the quantum enveloping algebra. The second lecture introduces the diagrammatically defined graded algebra R(v) that categorifies the positive part of the quantum enveloping algebra for any Kac-Moody algebra. In the third lecture we combine these two ideas to obtain a conjectural categegorification of the whole quantum enveloping algebra for any Kac-Moody algebra. We can prove this conjecture for sl(n). In the final lecture I will discuss cyclotomic quotients of the rings R(v). These are quotients of R(v) that are conjectured to categorify irreducible highest weight representations of quantum Kac-Moody algebras. We will also discuss recent work of Brundan and Kleshchev that proves this conjecture for type A. This work can be used to introduce a new Z-grading on blocks of the symmetric group and the associated Hecke algebra.

Organizers S. Ariki, S.Kato

Date

December 16 (Tue), 11:00-12:00, 13:30-14:30,

Room

数理解析研究所102号室

Speaker

西山享氏 (京大)

Title

多重旗多様体上の軌道の有限性について

Abstract

多重旗多様体とは部分旗多様体の直積を意味する。多重旗多様体 $ G/P_1 \times G/P_2 \times \dots \times G/P_k $ への $ G $ の対角的作用を考えた とき、その軌道が有限になる場合がMagyar-Weyman-Zelevinsky によって分類さ れている ($G$ が古典群の場合)。典型的な場合は $ Flag(\C^n) \times Flag(\C^n) \times P(\C^n) $ への $ GL_n $ が作用している場合であり、この 場合は mirabolic case と呼ばれて、モーメント写像を介した Steinberg 多様 体や Springer ファイバーの構造などが詳細に研究されている (Travkin, Finkelberg, Ginzburg)。この場合、モーメント写像の像の一部には自然に Achar-Henderson の enhanced nilpotent cone が現れる。
講演では、以上のような多重旗多様体上の軌道の有限性を巡る理論を対称対 $(G, K)$ の場合に拡張することを論じる。
Magyar-Weyman-Zelevinsky の分類によれば多重旗多様体上の $ G $ 軌道が有限 になるのは $k \leq 3$ (旗多様体の個数が3個以下)の場合に限られるが、それ を用いた自然な設定により、$K$ 軌道の有限性や、重複度自由な作用との関係、 またモーメント写像とその像としてあらわれる冪零多様体についていままでわ かってきたことを述べたい。
この研究は名大・多元数理の落合啓之氏との共同研究である。 

Organizers S. Ariki, S.Kato

Date

December 9 (Tue), 11:00-12:00, 13:00-14:00,

Room

数理解析研究所102号室

Speaker

並河良典氏 (京大)

Title

Induced nilpotent orbits and birational geometry

Abstract

複素単純リー環のべき零軌道の閉包(の正規化)は 複素シンプレクティック特異点になる。べき零軌道に関して、誘導軌道と いう概念がある。この概念は、双有理幾何のほうでは、複素シンプレクティック特異 点の Q-factorial terminalization と呼ばれる良い(部分的)特異点解消 に対応する。このことを説明するのが本講演の目的である。 

Organizers S. Ariki, S.Kato

Date

November 25 (Tue), 11:00-12:00, 13:00-14:00

Room

数理解析研究所102号室

Speaker

山川大亮氏 (京大)

Title

Geometry of multiplicative preprojective algebra

Abstract

Multiplicative preprojective algebraとは,Crawley-Boevey--Shawが Deligne-Simpson問題へのquiverの表現論からのアプローチとして 導入した(deformed) preprojective algebraの類似物です. 本講演ではこの代数の表現のモジュライ空間, すなわち乗法的箙多様体についての講演者の結果を紹介します. 講演の前半では乗法的箙多様体の定義や基本的性質, また背景となっているDeligne-Simpson問題について, 後半では箙多様体と比較してどういう事が分かるか, といった事について話をする予定です.

Organizers S. Ariki, S.Kato

Date

November 18 (Tue), 11:00-12:00, 13:00-14:00

Room

数理解析研究所102号室

Speaker

松本詔氏 (名古屋大)

Title

ジャック測度とランダム置換の最長増加部分列

Abstract

1999年、Baik-Deift-Johansson により対称群のプランシェレル測度に対 するある極限定理が得られた。 それによると、プランシェレル測度に従うランダム分割の、最大成分の分布は、分割 の重さが大きくなるとき にある極限分布$F$を持つ。またRSK対応を通じて見ると、それはランダム置換の最長 増加部分列の極限分布も同 時に与えている。一方でその極限分布$F$は、GUEランダム行列の最大固有値の極限分 布(Tracy-Widom分布)でもあった。 今回の講演では、プランシェレル測度の拡張であるジャック測度に従うランダム分割 を取り扱う。 長さに制限の入ったランダム分割に対する極限定理を与える。またその系として、あ る条件をもつランダム置換の最長増加部 分列の長さの極限分布が、トレースが0のGSEランダム行列の最大固有値の分布に一致 することを示す。

Organizers S. Ariki, S.Kato

Date

November 6 (Thu), 13:00-15:00,

Room

数理解析研究所102号室

Speaker

Joachim Hilgert氏 (Paderborn)

Title

Chevalley's restriction theorem for super-symmetric Riemannian symmetric spaces

Abstract

We start by explaining the concept of a super-symmetric Riemannian symmetric spaces and present the examples studied by Zirnbauer in the context of universality classes of random matrices. For these classes we then show how to formulate and approach an analog of Chevalley's restriction theorem for radial super-functions. It turns out that in the presence of even Cartan spaces radial functions are always even and have Weyl group invariant restictions to the Cartan spaces. The restriction map turns out to be injective and in general not surjective. Functions in the image have to satisfy additional regularity conditions coming from the odd restricted roots. The proof of the conjectured characterization of the image is not complete yet. We explain the method of proof and the problems in completing it. This is joint work in progress with A. Alldridge (Paderborn) and M.~Zirnbauer (Cologne)

Organizers S. Ariki, S.Kato

Date

October 16 (Thu), 13:00-15:00

Room

数理解析研究所102号室

Speaker

三町勝久氏 (東工大)

Title

セルバーグ型積分に付随するねじれチェインと量子群

Abstract

In 1991, Felder and Wieczerkowski discussed the action of the quantum group $U_q(sl_2)$ on the family of the paths in the homology groups, of which coefficents are given by the local system associated with a Selberg type integral. The present talk is to reformulate it from the viewpoint of the recent progress of the study of the twisted homology group.

Organizers S. Ariki, S.Kato

Date

October 3 (Fri), 11:00-12:00
October 6 (Mon) 11:00-12:00, 13:30-14:30

Room

RIMS room 402 (Friday) and room 202 (Monday)

Speaker

Pramod Achar (Louisiana State U.)

Title

Introduction to staggered sheaves

Abstract

"Staggered sheaves" are certain complexes of coherent sheaves with many remarkable properties resembling those of perverse sheaves. In this series of three talks, I will give an introduction to the theory of staggered sheaves, starting from the definition, and aiming to cover all the main results of the theory:description of simple objects; purity and decomposition phenomena; and projective and standard objects. Throughout, I will try to illustrate features of the theory with elementary examples. (I will not assume any familiarity with perverse sheaves.) Parts of this work are joint with D. Treumann and with D. Sage. A tentative outline for the three talks is as follows:
1. Overview; comparison with perverse sheaves; staggered sheaves on a G-orbit.
2. Staggered IC functor; filtrations of the derived category; purity theorem.
3. Ext-groups; decomposition theorem; projective and standard objects.

Organizers S. Ariki, S.Kato

Date

8月28日(木), 13:00-15:00

Room

京都大学数理解析研究所 102号室

Speaker

Shona Yu (The University of Sydney/Technische Universiteit Eindhoven)

Title

The Cyclotomic Birman-Murakami-Wenzl (BMW) Algebras

Abstract

The Birman-Murakami-Wenzl (BMW) algebras are closely tied with the Artin braid group of type A, the Iwahori-Hecke algebras of type A, and may be thought of as a deformation of the Brauer algebras.

Its algebraic definition was originally motivated by the Kauffman link invariant and, geometrically, it is isomorphic to the Kauffman tangle algebra. These algebras also feature in the theory of quantum groups, statistical mechanics, and even topological quantum field theory.

Motivated by type B knot theory and the cyclotomic Hecke algebras of type G(k,1,n) (aka the Ariki-Koike algebras), Häring-Oldenburg defined the cyclotomic BMW algebras. In this talk, we investigate the structure of these algebras and show they have a diagrammatic interpretation as a certain cylindrical analogue of the Kauffman Tangle algebras. In particular, we provide a basis which may be explicitly described both algebraically and diagrammatically in terms of "cylindrical" tangles. This basis turns out to be cellular, in the sense of Graham and Lehrer.

This talk is a presentation of the results in my Ph.D. thesis, completed end of 2007 at the University of Sydney, Australia.

Organizers S. Ariki, S.Kato

Date

7月29日(火), 11:00-12:00, 13:30-14:30

Room

京都大学数理解析研究所 202号室 (普段とは違う部屋となります)

Speaker

榎本 直也 氏(京都大学)

Title

対称結晶の幾何学的構成について

Abstract

A型アフィンヘッケ環のLascoux-Leclerc-Thibon-Ariki理論は、ある種の表現 の組成重複度や分岐則を量子群の結晶基底や大域基底を用いて記述する理論であ る。この理論では、アフィンヘッケ環の幾何学的表現論と量子群の幾何学的表現 論、特に、$U_v^-$とその大域基底をquiverの表現論を用いて幾何学的に構成す るG. Lusztigの理論が大きな役割を果たした。
 最近、講演者と柏原正樹氏は、量子群とそのDynkin対合を用いて、「対称結 晶」と呼ばれる概念を導入し、B型アフィンヘッケ環に対してもLLTA型予想を定 式化した。
 本講演では、Dynkin対合付きのquiverの表現論を用いて、対称結晶とその大域 基底を幾何学的に構成するという結果を紹介する。これは、上記のLusztig理論 の類似物である。そのため、本講演の前半では、Lusztig理論の概要を説明す る。これは、quiverの表現のモジュライ空間から得られるある種の圏の Grothendieck群として$U_v^-$を実現し、その大域基底を単純偏屈層を用いて記 述するというものである。後半では、Dynkin対合付きのquiverの表現のモジュラ イ空間を導入し、前半の議論とパラレルな形で、対称結晶の幾何学的構成につい て説明する。

Organizers S. Ariki, S.Kato

Date

7月15日(火), 10:30-12:00, 13:00-14:30

Room

京都大学数理解析研究所 102号室

Speaker

阿部 紀行 氏(東京大学)

Title

On the existence of homomorphisms between principal series of complex semisimple Lie groups

Abstract

In this talk, I give the condition for the existence of non-zero homomorphisms between principal series of complex semisimple Lie groups. I also give the condition for the existence of non-zero homomorphisms between twisted Verma modules, which is an extension of a result of Verma and Bernstein-Gelfand-Gelfand.

Organizers S. Ariki, S.Kato

Date

6月24日(火), 11:00-12:00, 13:00-14:00

Room

京都大学数理解析研究所 102号室

Speaker

山田 裕史 氏 (岡山大学)

Title

Compound basis for the space of symmetric functions

Abstract

The aim of this talk is to introduce a compound basis for the space of symmetric functions.
Our basis consists of products of Schur functions and $Q$-functions. The basis elements are indexed by the partitions.
It is well known that the Schur functions form an orthonormal basis for our space.
A natural question arises. How are these two bases connected?

In this talk we present some numerical results of the transition matrix for these bases.
In particular we will see that the determinant of the transition matrix is a power of 2.
This is not a surprising fact.
However the explicit formula involves an interesting combinatorial feature.

Organizers S. Ariki, S.Kato

Date

6月17日(火), 11:00-12:00, 13:00-14:00

Room

京都大学数理解析研究所 102号室

Speaker

内藤 聡 氏 (筑波大学)

Title

Mirkovic-Vilonen polytopes of Demazure crystals extremal Mirkovic-Vilonen polytopes

Abstract

Mirkovic-Vilonen (MV for short) polytopes are the images of MV cycles in the affine Grassmannian under the moment map, and these polytopes provide a realization of the crystal basis (highest weight crystal) for the irreducible highest weight module over a quantized universal enveloping algebra (of the Langlands dual Lie algebra).

In this talk, I will give an explicit description of the subset of MV polytopes corresponding to a Demazure (sub-) crystal of a highest weight crystal. Also, I will give a polytopal expression for MV polytopes corresponding to extremal elements in a highest weight crystal.

This is a joint work with D. Sagaki.

Organizers S. Ariki, S.Kato

Date

6月12日(木), 13:00-15:00

Room

京都大学数理解析研究所 102号室

Speaker

笠谷 昌弘 氏(京都大学)

Title

Polynomial solutions of the qKZ equation, and relative topics

Abstract

In this talk, I explain a construction of polynomial solutions of the quantum Knizhnik-Zamolodchikov (qKZ) equation in terms of non-symmetric Macdonald polynomials. This is the joint work with Y. Takeyama. I will also explain "positivity conjecture" proposed by V. Pasquier and me. The conjecture claims that the evaluation of components of certain polynomial solutions at $z_i=1$ ($z_i$'s are variables) is in $\mathbb{N}[\tau]$, where $\tau$ is a parameter of the solutions.

Organizers S. Ariki, S.Kato

Date

6月10日(火), 11:00-12:00, 13:00-14:00

Room

京都大学数理解析研究所 102号室

Speaker

辻井 健修 氏 (大阪市立大学)

Title

A simple proof of Pommerening's theorem

Abstract

Let $G$ be a connected reductive algebraic group over an algebraically closed field of characteristic $p$. Suppose that $p$ is good for the root system of $G$. Pommerening's theorem says that any distinguished nilpotent element in $Lie(G)$ is a Richardson element for a distinguished parabolic subgroup of $G$. This theorem implies the Bala-Carter theorem in good characteristic. In this talk, we will give a short and direct proof of Pommerening's theorem by using the Kempf-Rousseau theory, which was also used in Premet's proof.

Organizers S. Ariki, S.Kato

Date

5月22日(木), 午後1時〜3時 (時間変更しました)

Room

京都大学数理解析研究所 102号室

Speaker

越谷 重夫 氏(千葉大学)

Title

Blocks of finite groups with metacyclic defect groups

Abstract

Richard Brauer (1901--77),the almost unique pioneer of modular representation theory of finite groups, posed many interesting and important problems, conjectures, questions... early 1960's. Many of them are still open. Essentially and originally due to the Brauer's philosophy, we have had three important conjectures given by Jon Alperin, Everett Dade and Michel Broue, which were announced during late 1980's and early 1990's. In the talk, we will be discussing a kind of analogue of Broue's Abelian defect group conjecture, especially blocks of finite groups with non-abelian but metacyclic defect groups. Hopefully, something new and interesting would show up.

Organizers S. Ariki, S.Kato

Date

4月15日(火), 午前10時〜11時, 午後11時10分〜12時10分

Room

京都大学数理解析研究所 102号室

Speaker

前野 俊昭 氏(京大・工)

Title

Recent topics on quantum Schubert calculus

Abstract

The quantum Schubert calculus on flag varieties has been developed after Givental and Kim gave the presentation of the quantum cohomology ring of the flag variety of type A. In this talk I will survey recent topics of the quantum Schubert calculus. In the first part, I will summerize some results and conjectures on the quantum K-theory invented by Givental and Lee. In the second part, I will talk on Peterson's isomorphism between the equivariant homology of the affine Grassmannian and the quantum cohomology ring of the corresponding flag variety (which has been proved by Lam and Shimozono).

Organizers S. Ariki, S.Kato

Date

2月19日(火), 午前11時〜12時, 午後1時〜2時

Room

京都大学数理解析研究所 102号室

Speaker

山根宏之 氏(大阪大)

Title

Coxeter groupoids has solvable word problem

Abstract

The notion of the Weyl groupoids arises naturally in studying (generalization of Kac-Moody) Lie superalgebras and Nichols algebras. We show that the Weyl groupoids are defined only by Coxeter-type relations and that Matsumoto-type theorem holds for them (so the word problem of them is solvable), this is joint work with I.Heckenberger [arXiv:QA/0610823, to appear in Math.Z. (the electronic version has already appeared)]. This gives a possible answer to a problem posed by V.Serganova. Having motivation toward to physical application, we give a Drinfeld realization of the affine quantum superalgebra $U_q D^{(1)}(2,1;x)$ using Lusztig and Beck's argument and the corresponding (extended affine) Weyl groupoids, this is joint work with I.Heckenberger, F.Spill, A.Torrielli [arXiv:0705.1071, to appear in RIMS Kokyuroku Bessatsu]. In this talk, we also introduce Hecke algebroids associated with the Weyl groupoids and discuss their representation theory.

Organizers S. Ariki, S.Kato

Date

1月22日(火), 午前11時〜12時, 午後1時〜2時

Room

京都大学数理解析研究所 102号室

Speaker

宇野 勝博 氏(大阪教育大・教育)

Title

Fusion systems of finite groups and correspondences of characters

Abstract

A fusion system is a category whose objects are subgroups of a fixed p-group, where p is a prime, and morphisms are abstract generalization of conjugation maps. A few decades ago, fusion systems were considered for structure theorems of finite groups, but recently, they are studied from a topological point of view, since they are related to classifying spaces. In the talk, we give the definition of fusion systems and state recent developments. Moreover, the relationship with Broue's perfect isometries between character rings is presented.

Organizers S. Ariki, S.Kato

Date

1月15日(火), 午前11時〜12時, 午後1時〜2時

Room

京都大学数理解析研究所 102号室

Speaker

Michael Pevzner氏 (University of Reims, 東大数理)

Title

Generalized Rankin-Cohen brackets

Abstract

The particular geometric structure of causal symmetric spaces allows the definition of a covariant quantization of these homogeneous manifolds. We will discuss how do the composition formulae (#-products) of quantized operators give rise to a new interpretation of Rankin-Cohen brackets and permit to connect them with the branching laws of tensor products of holomorphic discrete series representations.

Organizers S. Ariki, S.Kato

Date

12月18日(火), 午前11時〜12時, 午後1時〜2時

Room

京都大学数理解析研究所 102号室

Speaker

兼田 正治 氏 (大阪市立大)

Title

On complete exceptional sequences of coherent sheaves on homogeneous projective varieties

Abstract

The existence of a complete exceptional sequence of coherent sheaves on a complex projective variety supports Kontsevich's homological mirror conjecture.
Assume the variety is homogeneous and write it as G/P with G a reductive group and P a parabolic subgroup of G.
For G of rank at most 2 we have recently found a Karoubian complete strongly exceptional poset of locally free sheaves of finite rank on G/P, parametrized by W/W_P, W (resp. W_P) the Weyl group of G (resp. P), verifying a conjecture of Catanese.
They are constructed by examining the Frobenius direct image of the structure sheaf on G/P in positive characteristic.

Organizers S. Ariki, S.Kato

Date

10月23日(火), 午前11時〜12時, 午後1時〜2時

Room

京都大学数理解析研究所 102号室

Speaker

阿部 健 氏 (京大・数理研)

Title

シンプレクティックバンドルに対するstrange duality写像の退化について

Abstract

アフィンリー環のconformal blockの空間と代数曲線上の 主G束のモジュライの上の一般テータの空間は同型であることが知られています。 代数曲線が特異点を持つものに退化する時、 conformal blockの空間は(いろいろな)conformal block の直和になります。これをfactorization theorem と呼びます。
セミナーの前半では、このfactorization theoremが 一般テータの側ではどのように理解されるか、について、 $SL(2)$バンドルのときに見てみます。
セミナーの後半では、一般テータの空間の間の strange duality (rank-level duality とも呼ばれる)と言う現象 を紹介します。 strange dualityは、二つの然るべき一般テータの空間 はdualの関係にある、と言うものです。 セミナーでは、シンプレクティック版のstrange duality について、代数曲線が特異点を持つものに退化するとき、 duality mapがどのように分解するか、について話したいと思います。

Comments [pdf]
Organizers S. Ariki, S.Kato

Date

10月23日(火), 午前11時〜12時

Room

京都大学数理解析研究所 102号室

Speaker

Pablo Ramacher 氏 (Goettingen大)

Title

Representation theory of real algebraic groups on affine G-varieties

Abstract

We consider a smooth real affine algebraic variety M, together with a real linear algebraic group acting regularly on M. We then study the regular representation of G on the Banach space of functions on M vanishing at infinity by introducing a certain dense subspace of analytic vectors. If G is reductive, and K a maximal compact subgroup, the considered subspace constitutes a (g,K)-module in the sense of Harish-Chandra and Lepowsky, and by taking suitable subquotients, we construct admissible (g,K)-modules as well as K-finite representations.

Comments [pdf]
Organizers S. Ariki, S.Kato

Date

10月18日(木), 午前11時〜12時, 午後1時〜2時

Room

京都大学数理解析研究所 102号室 / 402号室(午後1時〜2時)

Speaker

桑原 敏郎氏 (京大・理)

Title

Rational Cherednik algebras and quiver varieties for $\mathbb{Z}/l\mathbb{Z}$ case varieties

Abstract

Rational Cherednik algebra は Dunkl operator などによって 生成される非可換代数で Ariki-Koike algebra の表現論と深く 関係します。 今回扱うのは比較的簡単な場合で、巡回群 ${\mathbb Z} / l {\mathbb Z}$ に関する場合です。このとき、rational Cherednik algebra は Crawley-Boevey と Holland によって定義されたA型 の deformed preprojective algebra $T_\lambda$と同型になり ます。 $T_\lambda$は $(l-1)$-次元のパラメタを持ちます。
$T_\lambda$に対して表現の圏にパラメタをシフトする関手
\[S_\lambda^\theta : T_\lambda \longrightarrow T_{\lambda+\theta}\]
$\theta \in {\mathbb Z}^{l-1}$ごとに定義できます。さらにこの関手 が圏同値であるときに、$T_\lambda$のフィルター付き表現の圏 から箙多様体上の連接層を構成する方法が知られています。
こういった構成は$T_\lambda$の表現論やそこでの関手 $S_\lambda^\theta$の振る舞いが箙多様体の幾何と深く関係して いるということを示唆しています。$T_\lambda$自身を表現と してみたとき、対応する連接層が箙多様体の tautological bundle になる事が中心的結果ですが、他に箙多様体の twisting sheaf の $({\mathbb C}^*)^2$-固定点での stalk の情報から同型
\[S_\lambda^\theta(\Delta_\lambda(i)) \longrightarrow \Delta_{\lambda+\theta}(i)\]
を具体的に構成することができます。ここで$\Delta_\lambda(i)$ は$T_\lambda$の標準加群です。こういったことが具体的に 構成できることは${\mathbb Z} / l {\mathbb Z}$の場合に特有 の事情 ですが、一般の場合でも幾何の情報から表現論の構造がある程度 わかるかもしれないと考えられています。

Comments [pdf]
Organizers S. Ariki, S.Kato

Date

10月9日(火), 午前11時〜12時, 午後1時〜2時

Room

京都大学数理解析研究所 102号室

Speaker

原下 秀士 氏 (東大・数理)

Title

Ekedahl-Oort strata contained in the supersingular locus and Deligne-Lusztig varieties

Abstract

この講演では、超特異アーベル多様体のモジュライ空間に含まれる Ekedahl-Oort strata を Deligne-Lusztig 多様体で記述するというお話をします。 アーベル多様体とそのモジュライ空間の構造(Newton polygon strata と Ekedahl-O ort strata の基本性質)とDeligne-Lusztig 多様体についての復習からはじめて主 定理の紹介をします。

Organizers S. Ariki

Date

10月4日(木), 午後2時〜4時

Room

京都大学数理解析研究所 402号室

Speaker

加藤 周 氏 (数理研)

Title

Quivers attached to exotic Deligne-Langlands correspondence

Abstract

3パラメタのC型のaffine Hekce代数は表現論的には他の古典型affine Hecke代数 の表現論のかなりの部分を支配するという意味で際立った代数である。
また、そのような状況に対応してこの代数の表現論は通常のaffine Hecke代数の いわゆるDeligne-Langlands-Lusztig予想に類似はするが異なる幾何学的記述を 持つ事が知られている。(exotic Deligne-Langlands対応 cf. math.RT/0601155)
今回の講演ではこの記述は(パラメタが十分に良い場合は)いわゆるA型のquiverの 表現空間の変種であり、いくつかの面ではそのようなquiverの類似物であると 見なせるという事を解説する。さらにこの事を用いてどのquiverの表現もどきに どのような既約表現が対応するかということについても述べる。

Organizers S. Ariki

Date

10月2日(火), 午前11時〜12時, 午後1時〜2時

Room

京都大学数理解析研究所 102号室

Speaker

伊山 修 氏 (名大・多元数理)

Title

Cluster tilting objects associated with preprojective algebras

Abstract

A. Buan, I. Reiten, J, Scottとの共同研究。
non-Dynkin型のpreprojective algebra \Lambda上の加群圏を考察する。 Coxeter群の元wに対して、\Lambdaの傾イデアルI_wが構成され、 対応して2-Calabi-Yau Frobenius圏Sub(\Lambda/I_w)が得られる。 wの最短表示毎にSub(\Lambda/I_w)のクラスター傾対象が得られる。

Organizers S. Ariki

Room

Room 115, RIMS, Kyoto University

Speaker

加藤 周 氏 (数理研)

Date & Title

2007年4月2日(月), 15:00-16:30
Deformations of nilpotent cones and Springer correspondences

2007年4月3日(火), 15:00-16:30
An exotic Deligne-Langlands correspondence for symplectic groups

                       (註)講演は日本語のみにて行われます。

Comments  
Organizers S. Ariki

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