Workshop

" Topics in the Theory of Weyl Groups and Root Systems "
in honor of Professor Jiro Sekiguchi on his 60th birthday

ABSTRACTS

• Sekiguchi, Jiro (Tokyo Univ. Agriculture and Technology) / "Holonomic systems related with free divisors of rank three"

  In a joint work with M. Kato (Univ. Ryukyus), we obtained systems of uniformization equations with singularities along discriminant sets of irreducible real and complex reflection groups of rank three. Such systems are examples of holonomic systems of three variables and they are regarded as ordinary differential equations of one variable by fixing the remaining two variables. These ordinary differential equations have three singular points in the complex plane. In my talk I discuss the holonomic systems and ordinary differential equations thus obtained.

• Fukui, Tetsuo (Mukogawa Women's University) / "Configurations of eight lines on a real projective plane coming from 8LC sets"

  This study is joint work with Professor Jiro Sekiguchi. The Weyl group $W(E_8)$ of type $E_8$ acts on the configuration space of labelled eight lines with some conditions on a real projective plane. This configuration space is identified with an affine open subset $\mathcal{S}$ of ${\bf R}^8$. Let $\mathcal{P}_8$ be the totality of connected components of $\mathcal{S}$. Then $W(E_8)$ also acts on $\mathcal{P}_8$. On the other hand, we defined 8LC sets consisting of ten roots of the root system of type $E_8$. To each 8LC set there associates a diagram consisting of ten circles analogous to Dynkin diagram. Let $\mathcal{LC}_8$ be the totality of extended 8LC sets. It is an interesting problem to classify combinatorial equivalence classes of simple 2-arrangements on a real projective plane. The main result in this paper is that there are 135 number of combinatorial equivalence classes of simple 2-arrangements defined by the systems of labelled eight lines in the image of the $W(E_8)$-equivariant map of $\mathcal{LC}_8$ to $\mathcal{P}_8$.

• Henderson, Anthony (Univ. Sydney) / "The exotic Robinson-Schensted correspondence"

  The well-known Robinson-Schensted correspondence between the symmetric group $S_n$ and pairs of same-shape standard Young tableaux with $n$ boxes has a geometric interpretation due to Steinberg. $S_n$ parametrizes the orbits of $GL(V)$ in $Fl(V)\times Fl(V)$, where $Fl(V)$ is the variety of complete flags in the vector space $V$ of dimension $n$. The conormal bundle to an orbit $O_w$ consists of triples $(F_1,F_2,x)$ where $(F_1,F_2)$ is in $O_w$ and $x$ is a nilpotent endomorphism of $V$ which preserves both flags. The tableaux corresponding to $w$ record the action of $x$ on $F_1$ and $F_2$ for a generic triple in this conormal bundle. Various analogues of this correspondence are known. A very similar case arises from the orbits of $Sp(V)$ in $Fl(V)$ when $V$ has a symplectic form: here one gets a bijection between the set of fixed-point-free involutions in $S_n$ and the set of standard tableaux with $n$ boxes in which every column has even length. I will explain the analogous correspondence for the orbits of $Sp(V)$ in $V\times Fl(V)$, which is related to the exotic nilpotent cone for $Sp(V)$. This is joint work with Peter Trapa (University of Utah).

• Ishibe, Tadashi (Hiroshima Univ.) / "Monoids in the fundamental groups of the logarithmic free divisors in $ \mathbb{C}^3 $"

  We study monoids generated by certain Zariski-van Kampen generators in the 17 fundamental groups of the complement of logarithmic free divisors in
  $\mathbb{C}^3$ listed by Jiro Sekiguchi. They admit positive homogeneous presentations. Five of them are Artin monoids and eight of them are free abelian monoids. The remaining four monoids are not Gau\ss ian and, hence, are neither Garside nor Artin . However, we introduce the concept of fundamental elements for positive homogenously presented monoids, and show that all 17 monoids possess fundamental elements. As an application of the study of monoids, we solve some decision problems for the fundamental groups except three cases.

• Munemasa, Akihiro (Tohoku Univ.) / "Smallest eigenvalues of graphs and root systems of type A, D and E"

  During 1960's, Alan Hoffman investigated eigenvalues of the adjacency matrix of a graph. It became apparent that the class of line graphs formed a large part of graphs with smallest eigenvalue at least $-2$, but it was not realized until Cameron, Goethals, Seidel and Shult (1976) discovered that graphs with smallest eigenvalue at least $-2$ are precisely those which can be represented by a root system of type A, D or E. A complete description of such graphs was published in a monograph written by Cvetkovi\'c, Rowlinson and Simi\'c (2004). Although, unlike the theory of root systems, there is no theory for classifying configuration of vectors of squared length 3, I will show how to use root systems of type A, D or E to obtain some information on certain graphs with smallest eigenvalues at least $-3$. This is based on joint work with Hye Jin Jang, Jack Koolen and Tetsuji Taniguchi.

• Noël, Alfred (Univ. Massachusets Boston) / "Variations on a recent theorem of Kostant"

  In a 2007 paper entitled "On the Centralizer of $K$ in $U(\mathfrak{g})$", Bertram Kostant gave an algorithm for computing the generators of the centralizer of a maximal compact group $K$ in the enveloping algebra of a complex semisimple Lie algebra $U(\mathfrak{g})$. His motivation is to rescue an old program that sought to determine an irreducible Harish-Chandra module $ H $ of $U(\mathfrak{g})$ via the finite-dimensional action of the above centralizer on a primary $\mathfrak{k}$-component in $ H $. "This original approach of Harish-Chandra to a determination of all $H$ has largely been abandoned because one knows very little about generators of that centralizer". However, Kostant's approach produces an algorithm of very high time and space complexity. We present a different algorithm of a lower complexity and compute some previously unknown examples.
  This is joint work with Steven Jackson.

• Ochiai, Hiroyuki (Kyushu Univ.) / "On Sekiguchi correspondence"

  Kostant-Sekiguchi correspondence, so called, is the bijective correspondence between the set of nilpotent (co)adjoint orbits on a semisimple Lie algebra and the set of nilpotent $K_{\mathbb{C}}$-orbits on the complexified tangent space of the corresponding Riemannian symmetric space.
  From the beginning of the theory, Sekiguchi has the correspondence on the context of non-Riemannian semisimple symmetric spaces. I will review this Sekiguchi's work, as well as an application/ interpretation to the geometric invariants of representation theory.

• Okuda, Takayuki (Univ. Tokyo) / "Relation between nilpotent orbits and proper actions of $SL(2,\mathbb{R})$ (tentative)"

  Let $G$ be a linear reductive Lie group and $H$ a reductive subgroup of $G$. For a discrete subgroup $\Gamma$ of $G$, if the action of $\Gamma$ on the homogeneous space $G/H$ is properly discontinuous, then we call $\Gamma$ a discontinuous group for $G/H$ and $\Gamma \backslash G/H$ a Clifford-Klein form. Our main problem is that ``What discrete group $\Gamma$ can be arise as a discontinuous group for $G/H$?'' In this talk, we focus on the relationship among the following three topics:
  * Discontinuous group for $G/H$ which is isomorphic to a surface group of genus $g$.
  * Proper actions of $SL(2,\mathbb{R})$ on $G/H$.
  * Nilpotent orbits $\mathcal{O}$ in $\mathfrak{g}$ satisfying that: the hyperbolic orbit in $\mathfrak{g}$ associated with $\mathcal{O}$ (by the Jacobson--Morozov theorem) does not meets $\mathfrak{h}$ ($\mathfrak{g}$, $\mathfrak{h}$ is the Lie algebra of $G$, $H$, respectively).
  Furthermore, we give a classification of semisimple symmetric pair $(G,H)$ which admits a surface group as a discontinuous group for $G/H$.

• Sepanski, Mark R. (Baylor Univ.) / "Distinguished orbits and the L-S category of simply connected compact Lie groups"

  We show that the Lusternik-Schnirelmann category of a simple, simply connected, compact Lie group $G$ is bounded above by the sum of the relative categories of certain distinguished conjugacy classes in $G$ corresponding to the vertices of the fundamental alcove for the action of the affine Weyl group on the Lie algebra of a maximal torus of $G$. (this is joint work with M. Hunziker).

• Shimeno, Nobukazu (Kwansei Gakuin Univ.)/ "Restriction of the Heckman-Opdam hypergeometric system"

  In a joint work with T. Oshima, we observed that intereting classes of ordinary differential equations, such as generalized hypergeometric differential equations and equations in even family appear by restricting the Heckman-Opdam hypergeometric system to singular lines. I will talk about restrictions with emphasis on examples.

• Takayama, Nobuki (Kobe Univ., Japan) / "A-hypergeometric equations"

  This is a survey talk on A-hypergeometric equations, which was introduced by Gel'fand, Kapranov, Zelevinsky in the late 1980's. GKZ gave a new idea to study multi-variable hypergeometric functions. After their break-through, a lot of interesting results have been obtained by utilizing D-modules, commutative algebra, combinatorics and affine toric ideals. I have written a chapter on the A-hypergeometric equations for the so-called Askey-Bateman project and my talk will follow this article.

• Takeuchi, Kiyoshi (Univ. Tsukuba) / "On confluent A-hypergeometric functions "

  In 1990, Gelfand, Kapranov and Zelevinsky proved that their A-hypergeometric functions admit integral representations. In this talk, we introduce a generalization of this result to Adolphson's confluent A-hypergeometric functions. A method of toric compactifications will be used for this purpose. This is a joint work with A. Esterov.

• Tanabé, Susumu (Galatasaray Univ.) / "Period integrals for complete intersection varieties and monodromy groups related to them"

  In this talk, we will discuss about concrete expression of solutions to Gauss-Manin system or Picard-Fuchs equation associated to affine complete intersection varieties. Our main interest will be focused on several cases where the concrete monodromy representation of the solutions is available. As an example of our investigations on Horn type hypergeometric functions, we show the following. Let $Y$ be a Calabi-Yau complete intersection in a weighted projective space.The space of quadratic invariants of the hypergeometric group $H$ associated with the period integrals of the mirror CI variety to $Y$ is one-dimensional and spanned by the Gram matrix of a split-generator of the derived category of coherent sheaves on $Y$ with respect to the Euler form. It will be also shown that the hypergeometric group $H$ is generated by a set of pseudo-reflexions (not necessarily involutive reflexions).

• Terao, Hiroaki (Hokkaido Univ.) / "The Shi arrangements and the Bernoulli polynomials"

  The Shi arrangement was introduced by J.-Y. Shi in relation to the Kazhdan-Lustzig cells of affine Weyl groups. It is an affine deformation of the arrangement associated with the root system of the type $A$. One of its remarkable properties is the fact that the Poincare polynomial factors as $(t+\ell+1)^{\ell}$. In particular, the number of chambers is equal to $(\ell+2)^{\ell}$. Ch. Athanasiadis showed that the cone A of the Shi arrangement is a free arrangement, which explains why the Poincare polynomial factors. He used the addition theorem to show the freeness. In this talk, we give an explicit construction of a basis for the derivation module $D(A)$ in terms of the Bernoulli polynomials, which are proved to be inherent in the study of Shi arrangements. The cases of the other root systems and the relations with the flat generators will be also discussed.

• Trapa, Peter (Univ. Utah) / "Character formulas for discrete series of affine Hecke algebras"

  Motivated by techniques from the theory of Riemannian symmetric spaces, we give a character formula for the W-module structure of a discrete series representation of an affine Hecke algebra. It seems likely that a similar formula holds for Hecke algebras attached to noncrystallographic roots systems and complex reflection groups. This is joint work with Dan Ciubotaru.

• Zhu, Fuhai (Nankai Univ.) / "Dirac operators and dual pair correspondence (tentative)"

  It is well-known that nonzero Dirac cohomology of an irreducible unitary representation determines the infinitesimal character of such representation. But in most cases, untary representations have trivial Dirac cohomology. In this talk, I will use some algebra generated by the Dirac operator to understand the behavior of the Dirac operator in general cases and explain some dual pair correspondence therein.

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