Tambara Workshop 2007
Geometry and Representations in Lie Theory

ABSTRACTS

• David Vogan (MIT) / "The character table for E₈(R)"

  A group of about twenty mathematicians working on a project called "Atlas of Lie groups and representations" recently completed computation of character tables for all the real forms of the exceptional Lie groups, using the algorithms introduced by Kazhdan and Lusztig almost thirty years ago. In the case of the split real form of E₈, the character table (in a very compressed form) occupies about fifty gigabytes of disk space. I will talk about two (closely related) questions:

  • What assurance is there that these enormous tables are correct?
  • How can one extract from them information that a human can understand and find interesting?

• David Vogan (MIT) / "The orbit method and D-modules"

  Perhaps the most difficult aspect of the (still unsolved) problem of classifying unitary representations is the construction of "unusual" unitary representations of simple Lie groups: those (like the trivial representation, or the metaplectic representation) that are not part of an infinite family.

  The orbit method of Kirillov and Kostant suggests that such representations ought to be "attached" to nilpotent orbits in the dual of the Lie algebra, but it does not say how these representations ought to be constructed.

  I will recall how the theory of D-modules can sometimes be used to carry out such constructions, and what still needs to be done to prove that the resulting representations are unitary.

• Tomoyuki Arakawa (Womens Univ. Nara) / "Hightest weight categories and representations of W-algebras"

  Affine W-algebras are very interesting vertex algebras, which can be considered as a chiralization of Kostant-Lynch theory. In the previous works we have determined their irreducible characters in the case of principal and minimal nilpotent orbits.

  In this talk we discuss the generalization of this results to other nilpotent orbits.

• Noriyuki Abe (Tokyo) / "On a generalization of Jacquet modules of degenerate principal series representations"

  The notion of Jacquet modules was introduced by Casselman. He also suggested some generalization of the Jacquet modules. This generalization of Jacquet modules is related to the Whittaker models while the original Jacquet modules is related to homomorphisms to the principal series representations.

  In this talk we introduce some filtration of this generalized Jacquet modules of degenerate principal series representations and investigate this filtration.

• Toshihiko Matsuki (Kyoto) / "Generalized Schubert cells and the complex crown"

  S. Gindikin and I conjectured in 2001 that "almost all" the domains in the complexification of a symmetric space arising from the duality of orbits on flag manifolds would coincide with the "complex crown". This conjecture was solved affirmatively in 2004 with many people's contributions from 1999 through 2004.

  The problem was simple in principle: We have only to compute xS \cap T for x \in G_C, K_C-B double cosets S and G_R-B double cosets T. I would like to explain some technical parts to solve the conjecture by computing some typical simple examples.

• Dan Ciubotaru (Utah) / "Matching of Kazhdan-Lusztig polynomials for real and p-adic groups" (tentative)

  The Grothendieck groups of categories of admissible representations of a reductive algebraic group $G$ over a local field $F$ have two distinguished bases: one formed by the standard modules, and the other by the irreducible modules.

  The fundamental question of finding the corresponding (unitriangular) change of basis matrix is answered by Kazhdan-Lusztig polynomials.

  From a geometric point of view, these polynomials give a measure for the singularities of closures of certain orbits. In joint work (in progress) with P. Trapa, we study correspondences between the geometries and Kazhdan-Lusztig polynomials for $G(\mathbb R)$ and $G(\mathbb Q_p).$ In particular, we construct a functor for $GL(n,\mathbb R)$, similar to the Arakawa and Suzuki functor from category $\mathcal O$ to the category of modules for the affine graded Hecke algebra of $GL(n,\mathbb Q_p).$

• Toshio Oshima (Tokyo) / "Subsystems of a root system (E₈)" (tentative)

  We give a combinatorial description of isomorphic classes of the homomorphisms between root systems. By this description we classify subststems of a root system and explain the result for E₈.

• Kyo Nishiyama (Kyoto) / "Degenerate principal series and the asymptotic cone of semisimple orbits"

  The talk is based on the joint work with Peter Trapa.

  In the study of dual pair correspondence (Howe duality, or theta correspondence), S.T.Lee and C.Zhu investigated the Howe's maximal quotient lifted from the trivial representation outside the stable range. It turns out to be isomorphic to a degenerate principal series induced from a maximal parabolic subgroup of Siegel type in their situation.

  In this talk, we study the geometric picture of the above phenomenon.

  We discuss the orbit structure of the null fiber of the related quotient map (moment map), the resolution of the singularlities of the irreducible components of the null fiber. The quotient of the null fiber is precisely the associated variety of the degenerate principal series in question.

  On the other hand, the quotient of the resolution is isomorphic to the conormal bundle of a closed K_C-orbit on the partial flag variety, which produces the maximal irreducible subquotients of the degenerate principal series via moment map.

• Nobukazu Shimeno (Okayama Univ. Science) / "Heckman-Opdam hypergeometric functions and their specializations"

  We consider solutions of Heckman-Opdam's hypergeometric system associated with a root system. We discuss confluence of the Heckman-Opdam hypergeometric function to the Whittaker function, restrictions to singular sets, and solutions for other real forms. This talk is based on the joint work with Toshio Oshima.

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