Lucas Fresse (University of Jerusalem)

On geometric properties of orbital varieties

To a nilpotent element x in a reductive Lie algebra, one can attach several algebraic varieties which play roles in geometric representation theory: its nilpotent orbit; the intersection of its nilpotent orbit with the nilradical of a Borel subalgebra (the irreducible components of this intersection are called orbital varieties); the fiber over x of the Springer resolution. There is a close relation between the Springer fiber over x and the orbital varieties attached to x. In this talk, we rely on this relation in order to study two properties of orbital varieties: the smoothness, and the property to contain a dense B-orbit. We concentrate on the type A. We provide several results which suggest that the two mentioned properties are related. This is a joint work with Anna Melnikov.

Chuying Fang (The Hong Kong University of Science and Technology)

Ad-nilpotent ideals and equivalence relations

Let $G$ be a complex simple algebraic group and $\mathfrak{g}$ its Lie algebra. In this talk, we will talk about the left equivalence relation of ad-nilpotent ideals of $\mathfrak{g}$ and its connection with affine Weyl groups, sign types and nilpotent orbits. In particular, if $G$ is simply laced, we prove that the equivalence relation corresponds to Lusztig's star operator.

Xu-Hua He (The Hong Kong University of Science and Technology )

Lusztig's G-stable pieces and nilpotent/unipotent varieties

Let $G$ be a connected simple algebraic group over an algebraically closed field. There is a partition of the wonderful compactification $¥bar G$ of $G$ into finitely many $G$-stable pieces, which was introduced by Lusztig in the study of character sheaves. Each piece is stable under the diagonal action of $G$ and in general, may contains infinitely many $G$-orbits. These pieces have found many interesting applications. In these 3 lectures, we will review some of the results relating to nilpotent/unipotent varieties. In the first talk, we recall the definition and discuss the closure relation of the $G$-stable pieces. There are some rich combinatorial structure behind it and we show that how to use the method of ``partial conjugation'' action to understand it. In the second talk, we introduce a Lagrangian subvariety $¥Lambda$ of the cotangent bundle of the wonderful compactification $¥bar G$, which is a generalization of the Steinberg variety. It is proved in a joint work with Lusztig that characteristic variety of character sheaf on $¥bar G$ is contained in $¥Lambda$. We then use $¥Lambda$ to study generalized Springer fiber for parabolic subgroups. In the third talk, we talk about the closure of the unipotent variety in $¥bar G$. It turns out that the boundary is a union of some $G$-stable pieces. We then discuss the relation to affine Deligne-Lusztig varieties of Iwahori level.

Jing-Song Huang (The Hong Kong University of Science and Technology )

Dirac Cohomology and Geometric Quantization

Let G be a connected semisimple Lie group having a finite center and a compact Cartan subgroup T with Lie(T)=t. We use Dirac cohomology to perform geometric quantization on G X t and obtain a natural correspondence from a discrete series to its Harish-Chandra parameter. We show this quantization satisfies the principle `quantization commutes with reduction' of Guillemin and Sternberg. We also obtain various models of discrete series representations. This is joint work with Meng-Kiat Chuah.

Toshiyuki Kobayashi (University of Tokyo)

Geometric quantization of coadjoint orbits, limits and restrictions

I plan to present some formulation with examples on
1. quantization commutes with restrictions
2. quantization commutes with limits
3. L2 models of minimal representations

Toshihiko Matsuki (Kyoto University)

An example of orthogonal triple flag variety of finite type

Let $G$ be the split special orthogonal group of degree $2n+1$ over a field $¥mathbb{F}$ of ${¥rm char}¥,¥mathbb{F}¥ne 2$. Then we describe $G$-orbits on the triple flag varieties $G/P¥times G/P¥times G/P$ and $G/P¥times G/P¥times G/B$ with respect to the diagonal action of $G$ where $P$ is a maximal parabolic subgroup of $G$ of the shape $(n,1,n) $ and $B$ is a Borel subgroup. As by-products, we also describe ${¥rm GL}_n$-orbits on $G/B$, $Q_{2n}$-orbits on the full flag variety of ${¥rm GL}_{2n}$ where $Q_{2n}$ is the fixed-point subgroup in ${¥rm Sp}_{2n}$ of a nonzero vector in $¥mathbb{F}^{2n}$ and $1¥times {¥rm Sp}_{2n}$-orbits on the full flag variety of ${¥rm GL}_{2n+1}$. In the same way, we can also solve the same problem for ${¥rm SO}_{2n}$ where the maximal parabolic subgroup $P$ is of the shape $(n,n)$. (c.f. arXiv: 1011.6468)

Hisayosi Matumoto (University of Tokyo)

Whittaker modules and vectors associated with the Jacobi parabolic subalgebras

The moment map for the generalized flag variety associated with a Jacobi parabolic subgroup of a complex symplectic group is not birational to its image. I would like to explain some counterexamples for the statement, which is correct in the birational case.

Shigeru Mukai (RIMS)

Simple Lie algebras and Legendre varieties

Yoshinori Namikawa (Kyoto University)

Slodowy slices and universal Poisson deformations

Brieskorn and Slodowy constructed the semi-universal deformation of an ADE surface singularity by using a slice (Slodowy slice) to a subregular nilpotent element of the corresponding Lie algebra. We will generalise this theorem to an arbitrary Slodowy slice. A Poisson deformation will play a crucial role in this generalisatio. This is a joint work with M. Lehn and C. Sorger.

Pavle Pandzic (University of Zagreb)

Translation principle for Dirac index

Unitarizable Harish-Chandra modules with nonzero Dirac cohomology should play an important role in the classification because they are extremal among all unitarizable modules in a certain precise sense. In the equal rank case, one can replace Dirac cohomology by its Euler characteristic, or Dirac index. The aim of this talk is to show how the index can vary in a translation family of virtual Harish-Chandra modules. Roughly speaking, if one member of the family has nonzero Dirac index, then so do almost all of the other members. This is joint work with Salah Mehdi.

Jiro Sekiguchi (Tokyo University of Agriculture and Technology)

Discriminants of polynomials and discriminants of real and complex reflection groups

Consider a polynomial $P(t)=t6+y_1t5+y_2t3+y_3t+¥frac{1}{20}y_22-¥frac{1}{4}y_1y_3$. Then there is a polynomial $f(y_1,y_2,y_3)$ such that the discriminant of $P(t)$ is $f(y)2$. Moreover $f(y)$ is regarded as the discriminant of the reflection group of type $H_3$. The speaker recognized this phenominon when he read the famous book of F. Klein on icosahedron. The purpose of this talk is to explain an idea of describing discriminants of real and complex reflection groups suggested by this phenominon.

Nobukazu Shimeno (Kwansei Gakuin University)

Boundary value problems on Riemannian symmetric spaces of the noncompact type

We characterize the image of the Poisson transform on each boundary component of a Riemannian symmetric space > of the noncompact type by a system of differential equations. The system corresponds to a generator system of a two sided ideals of an universal enveloping algebra, which are explicitly given by analogues of minimal polynomials of matrices. This talk is a joint work with Toshio Oshima.

Boris Sirola (University of Zagreb)

On certain nonsymmetric pairs of Lie algebras and some applications to representation theory

Let $(\mathfrak g,\mathfrak g_1)$ be a pair of (complex) Lie algebras with $\mathfrak g$ semisimple and $\mathfrak g_1\subseteq \mathfrak g$ reductive in $\mathfrak g$. Suppose also that the restriction of the Killing form of $\mathfrak g$ to $\mathfrak g_1$ is nondegenerate. We consider a particular subclass of such pairs that we call Cartan pairs. We show that in certain sense these pairs might be understood as a generlization of symmetric pairs. We also study some pairs $(\mathfrak g,\mathfrak g_1)$ where $\mathfrak g_1$ is self-normalizing in $\mathfrak g$. For pairs as above we have some useful results on geometry of nilpotent orbits and embedding of representations.

Kenji Taniguchi (Aoyama Gakuin University)

Composition series of the standard Whittaker (g,k)-modules

Let $G=KAN$ be an Iwasawa decomposition of a real reductive linear Lie group. Choose a non-degenerate unitary character $\eta$ of $N$. Let $M^{\eta}$ be the stabilizer of $\eta$ in $M:=Z_{K}(A)$. For an infinitesimal character $\Lambda$ and an irreducible representation $\sigma$ of $M^{\eta}$, define $I_{\eta,\Lambda,\sigma}$ to be the subspace of $C^{\infty}$-$\mathrm{Ind}_{M^{\eta}N}^{G}(\sigma \otimes \eta)$ consisting of those functions which (i) admit the infinitesimal character $\Lambda$, (ii) are left $K$-finite, (iii) grow moderately at the infinity. We call it the standard Whittaker $(\mathfrak{g},K)$-module. In my talk, I will talk about the $(\mathfrak{g},K)$-module structure of $I_{\eta,\Lambda,\sigma}$. First result is that this module is completely reducible if $\Lambda$ is generic, so its structure is completely determined by the results due to Kostant, Lynch, Matumoto, Wallach etc. For the non-generic case, the result that can be applied to general groups is hardly known. As an example, I will explain the case when $G$ is $U(n,1)$ and $\Lambda$ is regular integral.

Peter Trapa (University of Utah)

Spin representation of Weyl groups and nilpotent orbits

> Fix a semisimple Lie algebra g, Cartan subalgebra h, and let W denote the Weyl group of h in g. Since W acts by orthogonal transformations on the real span of roots of h in g, one can consider the preimage of W in an appropriate Pin group. Its irreducible genuine representations -- the so-called spin representations of W -- were classified in a case-by-case basis in the work of Schur, Morris, Reade and others. Recently, Dan Ciubotaru discovered a remarkable uniform classification of them (modulo a notion of equivalence) in terms of the nilpotent cone of the Lie algebra g. Ciubotaru's parametrization, which formally resembles Springer theory for W itself, fits perfectly with the theory of the p-adic Dirac operator (developed in joint work with Barbasch and Ciubotaru). This talk will present Ciubotaru's theorem and its consequences for unitary representation of split p-adic groups.

Akihito Wachi (Hokkaido University of Education)

Orbit graphs of symmetric pairs

Let $(G,K)$ be a complex symmetric pair, and $¥mathfrak{g} = ¥mathfrak{k} + ¥mathfrak{s}$ the complexified Cartan decomposition. For a nilpotent $G$-orbit $¥mathcal{O}$ on $¥mathfrak{g}$, the orbit graph associated to $¥mathcal{O}$ is given by the set of vertices consisting of the nilpotent $K$-orbits on $¥mathfrak{s}$ which generate the same $G$-orbit $¥mathcal{O}$. The edges are drawn between two $K$-orbits if the intersection of their closures contains a $K$-orbit of codimension one. We describe the structure and properties of orbit graphs for the symmetric pairs $(O_{p+q}, O_p ¥times o_q)$, $(Sp_{2n}, GL_n)$, $(Sp_{2p+2q}, Sp_{2p} ¥times Sp_{2q})$ and $(O_{2n}, GL_n)$.

Fuhai Zhu (Nankai University )

Dirac cohomology and branching laws

In this talk, we will first review the definition and basic facts about Dirac operator and Dirac cohomology of unitary representations and its relationship with Lie algebra cohomology. Then we will prove a $K$- character formula for irreducible unitary lowest weight modules and will show that the formula implies a generalized Littlewood restriction formula due to Enright and Willenbring.


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