NORTh 8
The 8th Workshop on Nilpotent Orbits and Representation Theory

ABSTRACTS

• Abe, Noriyuki (Univ. Tokyo, Japan) / "On a generalization of the BGG category"

  Using a formulation of Fiebig, we can attache ``the BGG category O'' for a Coxeter system. If a Coxeter system is the Weyl group of a semisimple Lie algebra, this category is similar to the integral regular block of the BGG category. We show some properties of this category.

• Huang, Jing-Song (HKUST, HongKong) / "Invariant Differential Operators and Eigenspace Representations"

  We show how to use the Taylor expansion to relate the eigenfunctions on general semisimple symmetric spaces to Riemannian symmetric spaces.

• Ikeda, Takeshi (Okayama University of Science, Japan) / "Equivariant K-theory of isotropic Grassmannians"

  Let $X$ be the Grassmannian of maximal isotropic subspaces in a complex vector space equipped with a non-degenerate symmetric or skew-symmetric bilinear form. We introduce a distinguished family of polynomials that represent the structure sheaves of the Schubert varieties in the torus equivariant $K$-ring of coherent sheaves on $X$. These polynomials are $K$-theoretical deformation of the so-called factorial Q- or P-functions introduced by V. Ivanov. This talk is based on joint work with H. Naruse.

• Kato, Syu (RIMS, Japan) / "On exotic standard modules of affine Hecke algebras of type B/C"

  We prove that the total homology groups of (1-)exotic Springer fibers are generated by its top-term. This confirms a question in [Achar-Henderson, Adv. Math. (2008)]. As a consequence, we see that exotic standard modules of affine Hecke algebras of type B/C are spanned by a simple module, provided if the parameter configuration is sufficiently nice. If time allows, we discuss asymptotic version of affine Hecke algebras of type B/C with unequal parameters.

• Kobayashi, Toshiyuki (Univ. Tokyo, Japan) / "Fundamental Groups of Locally Complex Symmetric Spaces --- an application of symmetries of nilpotent orbits"

  NA

• Kroetz, Bernhard (MPI, Germany) / "Analytic representation theory of real reductive groups"

  In this talk we will give a notion of analytic representation of a Lie group. Then we will show that the category of admissible representations of a real reductive group is equivalent to the category of Harish-Chandra modules, i.o.w. every Harish-Chandra module admits a unique analytic globalization.

• Losev, Ivan (MIT, USA) / "Fedosov quantization and W-algebras"

  W-algebras (of finite type) are certain associative algebras arising in the representation theory of universal enveloping algebras of semisimple Lie algebras. In full generality they were defined by Premet in the beginning of this decade but trace back to the work of Kostant of late 70-'s. In this talk I am going to explain how Fedosov deformation quantization applies to the study of W-algebras.

• Matumoto, Hisayosi (Univ. Tokyo, Japan) / "On homomorphisms between scalar generalized Verma modules"

  Let ${\mathfrak g}$ be a complex semisimple Lie algebra. We call a parabolic subalgebra ${\mathfrak p}$ of ${\mathfrak g}$ normal, if any parabolic subalgebra which has a common Levi part with ${\mathfrak p}$ is conjugate to ${\mathfrak p}$ under an inner automorphism of ${\mathfrak g}$. For a normal parabolic subalgebra, we have a good notion of the restricted root system or the little Weyl group. We have a comparison result on the Bruhat order on the Weyl group for ${\mathfrak g}$ and the little Weyl group. We apply this result to the existence problem of the homomorphisms between scalar generalized Verma modules.

• Ochiai, Hiroyuki (Nagoya Univ., Japan) / "Orbits on products of flag varieties"

  NA

• Oda, Hiroshi (Takushoku Univ., Japan) / "Functors connecting graded Hecke algebras and real reductive Lie groups"

  One knows many notions and results are common both in the representation theory of a real reductive Lie group G and in the representation theory of the graded Hecke algebra H associated with an Iwasawa decomposition G = KAN. In this talk we construct a simple functor Ξ from the category FD of finite-dimensional H-modules to the category of certain kinds of Harish-Chandra modules for G. The left adjoint functor of Ξ is also constructed. Although some properties of these functors are not so desirable and further modification may be needed in their construction, Ξ has several good natures. For example, the multiplicity of a single-petaled K-type V in Ξ(X) (X ∈ Ob(FD) coincides with the intertwining number between X_W (W = W(G,A) ⊂ H, the Weyl group) and a W-module corresponding to V, and parabolic inductions for H are compatible with those for G under Ξ.

• Trapa, Peter (Univ. Utah, USA) / "Parametrizing real nilpotent orbits."

  The classification of nilpotent adjoint orbits in the Lie algebra of a real semisimple Lie group G is essentially complete. It can be made explicit in any particular case, but often extensive case-by-case analysis is needed (for instance, if G is disconnected). This talk is about a canonical way to parametrize certain real orbits, namely those which are contained in the special piece corresponding to an even complex adjoint orbit. (The parametrization is in terms of the geometry of partial flag varieties.) This is based on joint work with Ciubotaru and Nishiyama and, most recently, Barbasch.

• Wachi, Akihito (Hokkaido Inst. Tech., Japan)/ "The number of nilpotent orbits for symmetric pairs"

  We show a formula for the number of the nilpotent K-orbits on p, where K is a complex Lie group, and p is a complex Lie algebra determined by the Cartan decomposition g_R = k_R + p_R. We also introduce an equivalence relation of orbits, and study their equivalent classes. This talk is based on joint work with Kyo Nishiyama and Yasuhide Numata.

• Zhu, Chengbo (NUS, Singapore) / "Uniqueness of certain mixed models: a new descent method"

  I will discuss a recent work (joint with D. Jiang and B. Sun) on the uniqueness of a certain exceptional model for $\GL(6)$ representations, in which we introduce a new descent argument based on two geometric notions attached to submanifolds, called metrical properness and unipotent $\chi$-incompatibility.

• Zierau, Roger (Oklahoma State Univ., USA) / "Smoothness of a family of components of Springer fibers for classical groups"

  Some geometric properties of components of Springer fibers associated to closed K-orbits will be discussed. In particular these components are shown to be iterated bundles, and are therefore smooth. Some topological properties of these components will be give.

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