Lie Groups and Representation Theory Archives for Academic Year 2015

Organizational Meeting

When: Wed, September 3, 2014 - 2:00pm
Where: 1311.0
Speaker: () -

Cocenters and representations of affine Hecke algebras

When: Wed, September 17, 2014 - 2:00pm
Where: Math 1311
Speaker: Xuhua He (UMD) -
Abstract: It is known that the number of conjugacy classes of a finite group equals the number of irreducible representations (over complex numbers). The conjugacy classes of a finite group give a natural basis of the cocenter of its group algebra. Thus the above equality can be reformulated as a duality between the cocenter of the group algebra and the Grothendieck group of its finite dimensional representations.

For affine Hecke algebras, the situation is much more complicated. First, the cocenter of affine Hecke algebras is harder to understand than the cocenter of group algebras. Second, for an affine Hecke algebra, the dimension of its cocenter is countably infinite and the number of irreducible representations is uncountable infinite. However, the ``cocenter-representation duality'' is still valid. This is what I am going to explain in this talk. It is based joint works with S. Nie, and joint work with D. Ciubotaru.

Computing Twisted Kazhdan-Lusztig-Vogan polynomials

When: Wed, October 1, 2014 - 2:00pm
Where: MATH 1311
Speaker: Jeffrey Adams (University of Maryland) -

Families of Harish Chandra modules connecting compact and noncompact Lie groups

When: Wed, October 15, 2014 - 2:00pm
Where: Math 1311
Speaker: Eyal Subag (Tel Aviv University) -

Abstract: Families of representations naturally appear in representation theory of real reductive Lie groups. In my talk I will demonstrate how the Lie groups themselves come in families and how families of representations (of non-isomorphic groups) play a significant role in representation theory. I’ll be focusing on the groups SU(1,1), SU(2) and their Cartan motion group. Furthermore, I will show that there exists an algebraic family of Harish Chandra pairs that is associated with these groups. We shall see how families of Harish Chandra modules relate representations of SU(1,1), SU(2) and their Cartan motion group. These families of HC modules will finally be used to provide some insights into the theory of contraction of representations and the Mackey bijection.

This talk is based on a joint work with Joseph Bernstein and Nigel Higson.

Geometric Satake Isomorphism, I

When: Wed, October 22, 2014 - 2:00pm
Where: Math 1311
Speaker: Tom Haines (UMCP) -
Abstract: We begin a series of talks meant to explain Timo Richarz' proof of the geometric Satake isomorphism.

Geometric Satake Isomorphism, II

When: Wed, October 29, 2014 - 2:00pm
Where: Math 1311
Speaker: Tom Haines (UMCP) -
Abstract: We continue the series of talks on Timo Richarz' proof of the geometric Satake isomorphism.

Geometric Satake Isomorphism, III

When: Wed, November 12, 2014 - 2:00pm
Where: Math 1311
Speaker: Tom Haines (UMCP) -
Abstract: We will continue the series of talks on Timo Richarz' proof of the geometric Satake isomorphism.

Rank-Level duality and Conformal Blocks divisors

When: Wed, November 19, 2014 - 2:00pm
Where: Math 1311
Speaker: Swarnava Mukhopadhyay (UMCP) -
Abstract: Conformal blocks are vector bundles on moduli space of curves with marked points that arise naturally in rational conformal field theory. They also give rise to a very interesting family of nef divisors and hence relate to questions on nef cone of moduli space of genus zero curves with n-marked points. Rank-level duality connects a conformal block associated to one Lie algebra to a conformal block for a different Lie algebra. In this talk we discuss relations among conformal blocks divisors that arise from rank-level duality.

Local systems of rank one

When: Wed, January 28, 2015 - 2:00pm
Where: Math 1311
Speaker: Nero Budur (KU Leuven) -
Abstract: We give a survey of recent results on the cohomology jump loci of
local systems of rank one, showing their connections with classical
singularity theory, Hodge theory, Bernstein-Sato polynomials, and
algebraic statistics. Joint work with Botong Wang.

Double theta polynomials and equivariant Giambelli formulas

When: Wed, February 11, 2015 - 2:00pm
Where: Math 1311
Speaker: Harry Tamvakis (UMD) -
Abstract: The double theta polynomials were defined in Elizabeth Wilson's 2010
PhD thesis, and were conjectured there to represent the equivariant
Schubert classes on symplectic Grassmannians. We now have several
different proofs of her conjecture, which gives an intrinsic algebraic model
for the equivariant cohomology ring of these Grassmannians. I will
discuss this theory and relate it to some previously known degeneracy
locus formulas.

Scaling of conformal blocks and generalized theta functions over the moduli space of curves

When: Wed, February 25, 2015 - 2:00pm
Where: Math 1311
Speaker: Angela Gibney (University of Georgia) -
Abstract: There is an identification, over smooth curves, of conformal blocks and generalized theta functions. I will talk about joint work with Prakash Belkale and Anna Kazanova, where we show that for conformal blocks in type A associated to projective varieties of minimal degree, if this interpretation extends to stable curves, certain identities between first Chern classes of vector bundles of conformal blocks will be satisfied. Examples show that sometimes the extension can fail, while other times it can hold.

An arithmetic transfer conjecture for ramified unitary groups

When: Mon, March 2, 2015 - 2:00pm
Where: Math 1311
Speaker: Brian Smithling (Johns Hopkins) -
Abstract: The arithmetic fundamental lemma is a conjectural relation proposed by W. Zhang in connection with a relative trace formula approach to the hermitian case of the arithmetic Gan-Gross-Prasad conjecture. It asserts a deep relation between the derivative of an orbital integral and an intersection number for cycles in a formal moduli space of p-divisible groups attached to an unramified unitary group over a p-adic field. I will report on an extension of the AFL conjecture to the setting of a ramified unitary group along with, for a unitary group in three variables, its proof. This is joint work with M. Rapoport and W. Zhang.

Schur Duality and Kazhdan-Lusztig Theory

When: Wed, April 1, 2015 - 2:00pm
Where: 1311.0
Speaker: Huanchen Baoi (University of Virginia) -
Abstract: The Kazhdan-Lusztig theory offered a powerful solution to the difficult problem of determining the irreducible characters in the category O of a simple Lie algebra. In type A, the Kazhdan-Lusztig theory can be reformulated in terms of Lusztig's canonical bases of quantum groups via the quantized Schur duality. In this talk, we will discuss a new theory of canonical bases arising from quantum symmetric pairs, initiated in joint work with Weiqiang Wang, which allows a new formulation of the Kazhdan-Lusztig theory in type B via a generalized Schur duality. We use such new canonical bases to formulate and establish the Kazhdan-Lusztig theory for the category O of the ortho-symplectic Lie superalgebras for the first time.

p-adic Deligne-Lusztig constructions and the local Langlands correspondence

When: Fri, April 10, 2015 - 2:00pm
Where: Math 1311
Speaker: Charlotte Chan (U. Michigan) -
Abstract: The representation theory of SL2(Fq) can be studied by studying the geometry of the Drinfeld curve. This is a special case of Deligne-Lusztig theory, which gives a beautiful geometric construction of the irreducible representations of finite reductive groups. I will discuss recent progress in studying Lusztig's conjectural construction of a p-adic analogue of this story. It turns out that for division algebras, the cohomology of the p-adic Deligne-Lusztig (ind-)scheme gives rise to supercuspidal representations of arbitrary depth and furthermore gives a geometric realization of the local Langlands and Jacquet-Langlands correspondences.

This talk is based on arXiv:1406.6122 and forthcoming work.

Twisted endoscopy from the sheaf-theoretic perspective

When: Wed, April 15, 2015 - 2:00pm
Where: 1310.0
Speaker: Paul Mezo (Carleton University) -
Abstract: Suppose G is a connected reductive algebraic group defined over the real numbers R. The Langlands correspondence partitions the set of irreducible representations of G(R) into finite sets of equivalence classes called L-packets. Shelstad associated the L-packets of G to the L-packets of endoscopic groups, which are in a sense smaller than G. The association is an identity involving distribution characters of the L-packets. Kottwitz and Shelstad later enriched the theory of endoscopy by introducing an automorphism of G. This "twisted" theory of endoscopy allows for more endoscopic groups and character identities. Meanwhile, an alternative perspective to the theory of endoscopy was developed by Adams, Barbasch and Vogan. They recast the theory into the framework of sheaves on a complex variety. I will sketch how the theory of twisted endoscopy may be incorporated into their framework.

Theta correspondence for dummies

When: Wed, April 29, 2015 - 2:00pm
Where: Math 1311
Speaker: Jeffrey Adams (University of Maryland) -
Abstract: Abstract: This is joint work with Dipendra Prasad and Gordan
Savin. The theta correspondence concerns quotients of the oscillator
representation omega restricted to a dual pair (G,G'). If pi is an
irreducible representation of G the representation Theta(pi)=Hom_G(omega,pi) of G' has a unique irreducible quotient
theta(pi). The theta, or Howe, correspondence is the bijection
pi->theta(pi). Instead of Hom it is natural to consider Ext_G(omega,pi) and the Euler-Poincare characteristic EP_G(omega,pi). Doing so makes many of the subtleties involved in the theta correspondence disappear. Results for EP are both simpler to state and to prove, and some naive expectations for the theta correspondence turn out to be true for the EP version. This gives a new perspective on the theta correspondence itself, including the structure of Theta(pi).

Strange duality of Verlinde spaces for G_2 and F_4

When: Mon, May 4, 2015 - 2:00pm
Where: Math 1311
Speaker: Swarnava Mukhopadhyay (UMCP) -
Abstract: We prove that the pull back of the canonical theta divisor for E_8-bundles at level one induces a strange duality between Verlinde spaces for G_2 and F_4 at level one on smooth curves of genus g. We also give parabolic generations and write down relations in the Picard group of \bar-M_{g,n}.

Local models for some unitary Shimura varieties

When: Wed, May 6, 2015 - 2:00pm
Where: Math 1311
Speaker: Sean Ballentine (UMCP) -