Representation Theory Seminar

Representation Theory Seminar
2008-2009
Fridays at 2pm in LCB 323

Last year: 2007-2008

Date	Speaker	Title
September 19	Hung Yean Loke (National University of Singapore)	Modular forms on double covers of split tori
September 26	Scott Crofts	Genuine Representations of ~Spin(n+1,n)
October 3	Matthew Ondrus (Weber State)	Whittaker modules for various Lie algebras and associative algebras
October 10	Ben Trahan	Iwahori-Hecke algebras
October 17		No meeting (Fall break)
October 31	Sarah Kitchen	Geometric Methods and Representations of Real Semi-Simple Lie Groups
November 7	Henryk Hecht	Characters of Harish-Chandra modules: an overview
November 14	Li Zhong (Idaho State)	Tensor products of discrete series representations
November 21	Dan Ciubotaru	Tempered modules in the exotic Deligne-Langlands correspondence
November 28		No meeting (Thanksgiving)
December 5	Kenyon Platt (BYU)	Classifying the Representation Type of the Infinitesimal Blocks of Parabolic Category O
January 30	Gordan Savin	Representation theoretic approach to Kohnen's plus space of half-integral weight modular forms I
February 6	Gordan Savin	Representation theoretic approach to Kohnen's plus space of half-integral weight modular forms II
February 13	Peter Trapa	Parameterizing nilpotent orbits
March 6	Thomas Haines (Maryland)	Shimura varieties with Γ_1(p)-level structure via Hecke algebra isomorphisms
March 13	Allen Moy (Hong Kong UST)	On a construction of elements in the Bernstein center
March 20		No meeting (Spring break)
March 27	Peter Trapa	Duality between GL(n,R) and the graded affine Hecke algebra for gl(n)

Maintained by Dan Ciubotaru.

September 19, 2008
Hung Yean Loke (National University of Singapore)
Title: Modular forms on double covers of split tori
Abstract: In this talk, I will describe certain modular forms on a split (non-abelian) torus of a non-linear double cover of a split Chevalley group over the rational adeles.
This is an on going project with Gordan Savin.

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September 26, 2008
Scott Crofts
Title: Genuine Representations of ~Spin(n+1,n)
Abstract: Let G=Spin(n+1,n) be the split real form of a simply connected complex Lie group of type B_n. Then G is connected and has a nonlinear double cover G^~= Spin^~(n+1,n).
In this talk we will discuss genuine representations of G^~ at certain half-integral infinitesimal characters. In particular, we prove a duality theorem for the multiplicities of formal characters of G^~.
Such "character multiplicity duality" is an important feature of linear groups, originally due to Vogan.
Software exhibiting this duality in certain cases will also be demonstrated.

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October 3, 2008
Matthew Ondrus (Weber State)
Title: Whittaker modules for various Lie algebras and associative algebras
Abstract: The notion of a Whittaker module for a Lie algebra was first studied in the setting of complex semisimple finite-dimensional Lie algebras. Since then, analogous modules have been defined and studied for various algebras. We will discuss some classical results on Whittaker modules and also some recent work on Whittaker modules for generalized Weyl algebras and for the Virasoro algebra.

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November 14, 2008
Li Zhong (Idaho State)
Title: Tensor products of discrete series representations
Abstract: The tensor product of discrete series representations has been studied by Repka in the case when both factors are either holomorphic or antiholomorphic discrete series. However discrete series other than holomorphic ones do exist and they are conjectured by Clozel to be associated to motivic objects. Therefore it is interesting to analyze tensor products of discrete series in more general cases. We follow Repka to discuss tensor product of discrete series from tensor products of principal series and propose a possible algorithm for the general case. We use explicit multiplicities of K-types for discrete series of Sp(4,R) to derive the decomposition of tensor product of discrete series representations.

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November 21, 2008
Dan Ciubotaru
Title: Tempered modules in the exotic Deligne-Langlands correspondence
Abstract: This is joint work with Syu Kato (RIMS). Recently, Kato defined "exotic" versions of the nilpotent cone and Steinberg variety for the symplectic group, which allowed him to construct geometrically the affine Hecke algebra of types B/C with arbitrary unequal parameters. I will explain how the tempered modules are classified in this picture, and the relations to Opdam-Solleveld recent classification, and to Lusztig's cuspidal local systems for spin groups. Finally, I will present some applications of this approach for the affine Hecke algebras that arise from (inner forms of) split p-adic groups in the unipotent category.

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December 5, 2008
Kenyon Platt (BYU)
Title: Classifying the Representation Type of the Infinitesimal Blocks of Parabolic Category O
Abstract: Parabolic category O is a generalization of Bernstein-Gelfand-Gelfand category O. This category decomposes into certain subcategories, called the infinitesimal blocks. Understanding these blocks is key to understanding the whole category. Some of these blocks are zero, so first we will determine when a given block is nonzero. This turns out to have an interesting answer, given in terms of nilpotent orbit theory. We will then consider the representation type of a nonzero block, discussing results in this direction, including further links to nilpotent orbits.

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March 6, 2009
Thomas Haines (Maryland)
Title: Shimura varieties with Γ_1(p)-level structure via Hecke algebra isomorphisms
Abstract: The Langlands-Kottwitz approach to the determination of the local zeta function of Shimura varieties at primes of good reduction consists in expressing the number of points modulo p as a product of a certain volume term, an orbital integral away from p, and a twisted orbital integral at p. It has been known for some time how to generalize this method to certain bad reduction cases coming from parahoric level structure. In this talk, I will explain how to generalize the method further to some deeper level situations. This is joint work with Michael Rapoport.

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March 13, 2009
Allen Moy (Hong Kong UST)
Title: On a construction of elements in the Bernstein center
Abstract: An indispensable tool in the representation theory of reductive Lie groups is the center of the universal enveloping algebra of the Lie algebra of a Lie group. In the 1980's, Bernstein introduced into the representation theory of reductive p-adic groups an analogue of the center of the universal enveloping algebra. We report on some joint work with Tadić concerning construction of elements in the Bernstein center for quasi-split groups.

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