Representation Theory Seminar

Representation Theory Seminar
2011-2012
Fridays 4:10-5:10 in LCB 215

2010-2011: 2009-2010
2009-2010: 2009-2010
2008-2009: 2008-2009

Date	Speaker	Title
September 30	Nathan Geer (Utah State)	Generalized traces and modified dimensions
October 21	Matt Douglass (North Texas)	Something old, something new: Relating Coxeter arrangements and Solomon's descent algebra
October 28	Matthew Housley (BYU)	Skein Theoretic Constructions of Kazhdan-Lusztig Left Cell Representations
November 18	Cathy Kriloff (Idaho State)	Spectra of Cayley graphs of complex reflection groups
February 10	Eric Opdam (Amsterdam)	Tempered contractions of the Schneider-Stuhler resolutions
February 17	Dan Ciubotaru	Dirac index for graded affine Hecke algebras
February 24	Kay Magaard (Birmingham)	Maximal subgroups of classical groups
March 20 Tuesday 2pm in JWB 333	Stephen D. Miller (Rutgers)	Some applications of small representations to string theory
April 27	David Roe (PIMS)	The local Langlands correspondence and character sheaves

Maintained by Dan Ciubotaru.

September 30, 2011
Nathan Geer
Title: Generalized traces and modified dimensions
Abstract: In this talk I will discuss how to construct generalized traces on an ideal in certain module categories. As I will explain there are a several examples in representation theory where the usual trace and dimension are zero, but these generalized traces and modified dimensions are non-zero. Such examples include the representation theory of the Lie algebra sl(2) over a field of positive characteristic, Lie superalgebras over the complex numbers and quantum groups. In these examples the modified dimensions can be interpreted categorically and are closely related to some basic notions involving the representation theory. One motivation for this work is the construction of topological invariants. This is joint work with Jon Kujawa, Bertrand Patureau and Alexis Virelizier.

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October 21, 2011
Matt Douglass
Title: Something old, something new: Relating Coxeter arrangements and Solomon's descent algebra
Abstract: In 1986 Lehrer and Solomon conjectured that the representation of a finite Coxeter group W on the pth cohomology group of the complement of its arrangement is a sum of characters induced from linear characters of centralizers of elements of W. In this talk I'll describe some of the applications of the cohomology of the complement to solving other problems and report on recent progress toward a proof of the conjecture.

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October 28, 2011
Matt Housley
Title: Skein Theoretic Constructions of Kazhdan-Lusztig Left Cell Representations
Abstract: Kazhdan-Lusztig left cell representations of the symmetric group come equipped with canonical bases that have numerous interesting applications in the context of type A Lie groups. Unfortunately, these bases are at present only accessible through the computationally demanding Kazhdan-Lusztig algorithm, limiting direct empirical study to relatively small examples. In this talk, I will introduce a new way of constructing certain left cell representations via skein theory. Namely, left cell representations corresponding to two part partitions are computable by a graphical calculus. In addition, I will discuss recent collaborative work with Heather Russell and Julianna Tymoczko wherein we have developed a conjectural construction for [n,n,n] partitions which is supported by strong combinatorial evidence.

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November 18, 2011
Cathy Kriloff
Title: Spectra of Cayley graphs of complex reflection groups
Abstract: In a recent paper Renteln provided formulas and integrality results on distance spectra of Cayley graphs of finite real reflection groups utilizing reflection length and some representation theory. I will report on recent progress and conjectures in analyzing various spectra associated to Cayley graphs of complex reflection groups.

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February 4, 2012
Eric Opdam
Title: Tempered contractions of the Schneider-Stuhler resolutions
Abstract: A fundamental result of Ralf Meyer states that for a reductive p-adic group G one has Ext_H^i(V,W)=Ext_S^i(V,W) whenever V,W are tempered representations of G, H=H(G) is the Hecke algebra of G, and S=S(G) is the Harish-Chandra Schwartz algebra of G, an LF-space which is a topological completion of H(G). Meyer proved his result using the machinery of bornological vector spaces. We have tried to understand this result of Meyer from intuitively more elementary principles. Schneider and Stuhler have constructed a ``standard'' projective resolution P_*(V) of a smooth representation V of G by a complex of G-equivariant sheaves on the building B(G) of G. We show that for V tempered this resolution can be topologically completed while preserving the exactness. This yields an admissible projective resolution of S(G)-modules and leads to Meyer's results. The key construction is a contraction of the Schneider-Stuhler resolution which reflects the contractibility of the building B(G) as a topological space. The point is that this contraction extends continuously to the completed resolution.

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March 20, 2012
Stephen Miller
Title: Some applications of small representations to string theory
Abstract: I'll describe some recent joint work with string theorists Michael Green (Cambridge) and Pierre Vanhove (IHES) on some questions they encountered in studying graviton scattering amplitudes. The asymptotic expansion of these amplitudes in the low-energy limit is classically approximated through the Einstein-Hilber action. String theory posits additional terms which we found are Eisenstein series on exceptional groups. Because these Eisenstein series are automorphic realizations of small real representations, we are able to derive new properties about the amplitudes, such as characterizing their BPS orbit content in several limits. Additionally, we can verify that some of these automorphic forms are square-integrable. In the talk I'll mainly describe the mathematics involved, and how it relates to Arthur's conjectures and the work of the ATLAS project. If time permits I will describe some additional computations about the L^2 discrete spectrum of exceptional groups, and small representations they are related to.

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April 27, 2012
David Roe
Title: The local Langlands correspondence and character sheaves
Abstract: The local Langlands correspondence is a theorem for GL_n due to Harris, Taylor and Henniart, and recent work of Arthur makes good progress toward establishing it for all classical groups. In this talk I will describe a different approach to local Langlands, pioneered by DeBacker and Reeder, that works for arbitrary split reductive groups but imposes restrictions on the starting Galois representation. In particular I will focus on two stories: the extension of their method to tamely ramified unitary groups, and the beginnings of an effort to geometrize the method using character sheaves (joint with Clifton Cunningham).

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