Representation Theory Seminar

Representation Theory Seminar
2007-2008
Fridays at 4pm in LCB 323

Date	Speaker	Title
August 31	S. Crofts, C. Peterson, B. Trahan (student seminar)	Representations of Coxeter Groups and Hecke Algebras (after Kazhdan-Lusztig), Part I
September 7	S. Crofts, C. Peterson, B. Trahan (student seminar)	Representations of Coxeter Groups and Hecke Algebras (after Kazhdan-Lusztig), Part II
September 14	Dan Ciubotaru	Nilpotent Orbits in Graded Lie Algebras
September 21	Peter Trapa	Nilpotent Orbits and Representation Theory
September 28	Roger Zierau (Oklahoma State)	Conformally Invariant Differential Operators
October 5	Student Seminar (Crofts, Peterson, Trahan)	Degenerate Principal Series
October 12		No meeting (Fall Break)
October 19	Stephen Debacker (Michigan)	"L-packets" done incorrectly
October 26	Leticia Barchini (Oklahoma State)	Remarks on the geometry of the Springer fiber and applications
November 2	Student Seminar	Degenerate Principal Series II
November 16	Peter Trapa joint with the (almost?) Commutative Algebra seminar	Primitive Ideals in the Enveloping Algebra of a Complex Semisimple Lie Algebra
November 23		No meeting (Thanksgiving)
December 7	Alessandra Pantano (UC Irvine)	Unitary representations of real split groups
January 25	Dragan Milicic	Local Langlands correspondence for complex groups
February 1	Marketa Havlickova (MIT)	Boundaries of K-types in discrete series
February 15	Pramod Achar (Louisiana State)	Staggered Sheaves the seminar will meet in LCB 219 today
February 22	Syu Kato (RIMS and MSRI)	An exotic Deligne-Langlands correspondence for symplectic groups
March 7	Arun Ram (Wisconsin)	Path Models
March 21		No meeting (Springer Break)
March 28	Special double-header! 3pm, Pavle Pandzic (Zagreb/Cornell) 4pm, Monty McGovern (Washington)	Dirac cohomology of Harish-Chandra modules Singular sensations in the KGB picture
April 4	Eric Sommers (UMass Amherst)	A duality for nilpotent orbits
April 11	Marty Weissman (UC Santa Cruz)	Metaplectic tori

Maintained by Peter Trapa.

October 19, 2007
Stephen DeBacker (Michigan)
Title: "L-packets" done incorrectly
Abstract: In this talk, I will do things exactly incorrectly. In
trying to establish an "example" of the Local Langlands
Correspondence, the correct approach is to start with a geometric
object, called a Langlands parameter, and naturally associate to
this object a set of representations, called an L-packet, on the group.
One should then verify that the elements of the L-packet satisfy a
wide variety of conjectured properties. This has recently been done
for certain Langlands parameters by Mark Reeder and myself. In this
talk, I will try to provide a more intuitive understanding of why
our "L-packet"s are what they are. This intuition is based on the fact
that the representations that arise from the Langlands parameters we
consider are naturally associated to the nicest possible types of
tori in a p-adic group. These tori may be parameterized via
Bruhat-Tits theory, and this leads to a nice way to visualize the L-packets
we end up constructing. There will be lots of models and pictures.

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October 16, 2007
Leticia Barchini (Oklahoma State)
Title: Remarks on the geometry of the Springer fiber and applications
Abstract: available in pdf format

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December 7, 2007
Allesandra Pantano (UC Irvine)
Title: Unitary representations of real split groups
Abstract: The theory of unitary representations has important applications to
abstract harmonic analysis, and originated in the past century as a
natural continuation of classical Fourier analysis. In spite of the
extremely significant contributions made by Langlands,
Harish-Chandra and many other mathematicians, the problem of finding
the unitary dual of real reductive groups is still open: to this
day, a complete answer is only known for a very limited class of
groups.

In this talk I will describe some of the challenges in the
determination of the unitary dual of real split groups, and
introduce some directions of current research that can help us move
forward. In particular, I will discuss a method for comparing
intertwining operators for different groups, and relating the
unitarity of the corresponding representations.

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February 1, 2008
Marketa Havlickova (MIT)
Title: Boundaries of K-types in discrete series
Abstract: A fundamental problem about irreducible representations of a
reductive Lie group G is understanding their restriction to a
maximal compact subgroup K. In case of discrete series, the Blattner
character formula gives the multiplicity of any given irreducible
K-representation (or K-type) as an alternating sum. It is not
immediately clear from this formula which K-types, indexed by their
highest weights, have non-zero multiplicity. Evidence suggests that
the collection is very close to a set of lattice points in a convex
polyhedron. I shall describe a recursive algorithm for finding the
boundary facets of this polyhedron.

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February 15, 2008
Pramod Achar (Louisiana State)
Title: Staggered Sheaves
Abstract: Perverse sheaves, which form an abelian subcategory of the derived
category of constructible sheaves, have played an important role in
representation theory since their introduction, nearly 30 years ago.
Derived categories of coherent sheaves, on the other hand, are
relatively recent arrivals. I will describe some of the machinery
available in this setting, including, in particular, the
Deligne-Bezrukavnikov theory of "perverse coherent sheaves," and a
new construction, the category of "staggered sheaves." I will also
discuss some known and conjectural applications of these categories
to representation-theoretic questions.

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February 22, 2008
Syu Kato (RIMS and MSRI)
Title: An exotic Deligne-Langlands correspondence for symplectic groups
Abstract:
An affine Hecke algebra associated to a root datum is a q-analogue of its
affine Wely group. In general, it admits several (up to three) parameters.
The classification of irreducible representations of affine Hecke algebras
with equal-parameters is given by Kazhdan-Lusztig (and Ginzburg) as a
modification of the Deligne-Langlands conjecture. Their approach is based
on the geometry of the nilpotent cone of the corresponding Lie algebra
over complex numbers. This approach is later deepened by Lusztig in order
to deduce similar results for (various) integrally-weighted one-parameter
cases.

In this talk, we present a geometric realization of an affine Hecke
algebra H of type C with three parameters by replacing the nilpotent cone
of the Lie algebra with a certain Hilbert nilcone of a symplectic group in
the Kazhdan-Lusztig construction. This enables us to present a
Deligne-Langlands type classification of simple modules of H when the
parameters are sufficiently good. (The title Deligne-Langlands means that
our classification looks unmodified when compared with the Kazhdan-Lusztig
theorem.)

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March 7, 2008
Arun Ram (Wisconsin)
Title: Path Models
Abstract: This talk will be a survey of applications of path models: The
Weyl character formula, Schubert calculus, spherical functions, normal
forms in Chevalley groups, and indexing of points in affine flag varieties
and Mirkovic-Vilonen cycles.

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March 28, 2008
Pavle Pandzic (Zagreb/Cornell)

Title: Dirac cohomology of Harish-Chandra modules
Abstract: In the 1970's, Parthasarathy introduced a version of the Dirac
operator D attached to a real reductive group, and used it to construct
the discrete series representations. He also obtained a useful necessary
condition, Dirac operator inequality, for unitarizability of an irreducible
Harish-Chandra module. In 1997 Vogan studied a purely algebraic version
of D and used it to attach an invariant, called Dirac cohomology, to a Harish-Chandra
module X. He conjectured that Dirac cohomology, if nonzero, determines the
infinitesimal character of X. This conjecture was proved by Huang and myself in
2002. Subsequent generalizations to other settings were obtained by Kostant,
Kumar, Alekseev-Meinrenken and Kac-Frajria-Papi. Further results on Dirac
cohomology of Harish-Chandra modules included a relationship to n-cohomology
in some special cases (joint with Huang and Renard). In this talk I will give a
brief overview of the definitions and the above mentioned results. I will then
describe some further ideas and open questions. The topics I plan to mention
are algebraic Dirac induction, p^+ cohomology of unitary highest weight modules,
and sharpening the Dirac inequality.

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March 28, 2008
Monty McGovern (Washington)
Title: Singular sensations in the KGB picture
Abstract: The flag variety of a complex reductive group admits a well-known
decomposition into (Schubert) cells, each homeomorphic to $\Bbb C^m$
for some $m$. The closures of these cells, called Schubert varieties,
exhibit interesting singularities, which play an important role in the
representation theory of the group. The flag variety can be
decomposed in a different way, via orbits of a reductive rather than
solvable subgroup, and the closures of these orbits exhibit
singularities which are important in the representation theory of real
groups. I will apply a mixture of techniques from combinatorics and
representation theory to characterize these singularities in a number
of special cases.

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April 4, 2008
Eric Sommers (UMass Amherst)
Title: A duality for nilpotent orbits
Abstract: The talk concerns primitive ideals in the universal enveloping
algebra of a simple Lie algebra. For a nilpotent orbit O, the goal of the
talk is to locate, among all primitive ideals with associated variety O,
the one whose infinitesimal character is of shortest length. After
explaining the approach to computing the characters of shortest length
(which can be done purely in terms of root system data), we will explain a
connection to a duality map defined by Achar for nilpotent orbits.

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April 11, 2008
Marty Weissman (UC Santa Cruz)
Title: Metapletic Tori
Abstract: In 1968, Langlands generalized class field theory, from the
multiplicative group to arbitrary tori over local and global fields.
His observations at the time were crucial in forming the conjectures
of the Langlands program for arbitrary reductive groups over local and
global fields. Many authors have attempted to incorporate
"metaplectic groups" -- certain non-algebraic central extensions of
reductive groups -- into the Langlands program, typically by relating
representations of metaplectic groups to representations of related
reductive groups.

In this talk, I will try to follow the historical path of Langlands,
but allowing metaplectic groups throughout. Specifically, I will
discuss algebraic tori and their metaplectic covers. Next, I will
explain how to generalize the 1968 result of Langlands to parameterize
representations of metaplectic tori (over local fields). Finally, I
will describe what these observations for tori suggest for the future
development of a metaplectic Langlands program.

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