Lie Groups and Representation Theory Archives for Academic Year 2016


Families of Representations

When: Wed, September 7, 2016 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Jeffrey Adams (University of Maryland) - http://math.umd.edu/~jda
Abstract: The spherical prinicpal series of SL(2,R) is a family of representations I(nu) depending on a single complex parameter nu. Generically I(nu) is irreducible. For countably many nu I(nu) has finite composition series, but is not completely reducible. Question: is it possible to define a new family L(nu) so that L(nu) and I(nu) have the same composition factors for all nu, and L(nu) is completely reducible for all nu?

The Langlands-Kottwitz-Scholze Approach for Some Simple Shimura Varieties

When: Wed, September 21, 2016 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Patrick Daniels (UMD) -
Abstract: It is a fundamental problem in the Langlands program to express the zeta function of a Shimura variety as a product of automorphic L-functions. The work in this area originates with Eichler, Shimura, and others, who studied the Shimura varieties attached to GL2 and its inner forms. Many authors have contributed to the problem since, with the modern conjectural form due to Langlands, Kottwitz, and Rapoport.

In this talk we demonstrate Scholze's adaptation of the so-called Langlands-Kottwitz approach to studying the cohomology of Shimura varieties. Scholze first developed this technique in the case of the modular curve, and he later generalized those methods to compute the semi-simple zeta function of certain ``simple'' Shimura varieties. Our focus will be on this second case. In particular, we will explain the way in which Scholze is able to exploit the particularly nice geometry of the integral models of these Shimura varieties to adapt the approach to cases of bad reduction.


On the Canonical Representatives of a Finite Weyl Group

When: Wed, September 28, 2016 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Sean Rostami (Syracuse University) -
Abstract: Let G be a split reductive matrix group, T a split maximal torus in G,
and W its finite Weyl group. It is frequently impossible to find a
system of representatives for W in G which form a subgroup, but there
is a uniform method of defining representatives which, in some overall
sense, are as nice as possible. It is necessary for some questions to
quantify the failure of these representatives to form a subgroup. To
do this, it is necessary to understand certain sums which are
variations on a sum which appears frequently in Lie Theory. A question
involving the simple supercuspidals of Gross-Reeder and Reeder-Yu
requires such sums, and I present a new formula for these sums which
answers this question.

Topology of hyperplane arrangements and invariant theory

When: Wed, October 19, 2016 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Swarnava Mukhopadhyay (University of Maryland) - http://www2.math.umd.edu/~swarnava/
Abstract: V. Schechtman and A. Varchenko realize invariant theoretic objects in the cohomology of local systems on complements of hyperplane arrangements. We first consider a general problem of describing the image of compactly supported cohomology of a local system (defined on the complement of a hyperplane arrangement) in regular cohomology (this problem has some variants). Then, following Looijenga, we will connect this computation to invariant theory (Gauss-Manin realization of the KZ-connection). This is a joint work with Prakash Belkale and Patrick Brosnan.

Weight elimination in Serre-type conjectures

When: Wed, October 26, 2016 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Brandon Levin (U. Chicago) -
Abstract: I will discuss recent results towards the weight part of Serre's conjecture for GL_n as formulated by Herzig. The conjecture predicts the set of weights where an odd n-dimensional mod p Galois representation will appear in cohomology (modular weights) in terms of the restriction of the representation to the decomposition group at p. We show that the set of modular weights is always contained in the predicted set in generic situations. This is joint work with Daniel Le and Bao V. Le Hung.

PositiveGrassmannians and the geometry of Wilson loop diagrams

When: Wed, November 2, 2016 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Susama Agarwala (US Naval Academy) - https://sites.google.com/site/susamaagarwala/
Abstract: In this talk, I will introduce a family of diagrams, called Wilson loop diagrams, of physical significance in SYM N=4 theory. There is strong evidence that these diagrams should define a submanifold of the positive Grassmannian Gr_{\geq 0} (k, n), with k and n determined by the diagram. I present a map from these diagrams to the positive Grasmannians, and explore the geometry of the Wilson Loop diagrams.




On the Moy-Prasad filtration and supercuspidal representations

When: Mon, November 7, 2016 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Jessica Fintzen (U. Michigan) -
Abstract: Reeder and Yu gave recently a new construction of certain supercuspidal representations of p-adic reductive groups (called epipelagic representations). Their construction relies on the existence of stable vectors in the first Moy-Prasad filtration quotient under the action of a reductive quotient. We will explain these ingredients and present a theorem about the existence of such stable vectors for all primes p. This builds on a result of Reeder and Yu about the existence of stable vectors for large primes and generalizes the paper of the speaker and Romano, which treats the case of an absolutely simple split reductive group.
In addition, we will present a general set-up that allows us to compare the Moy-Prasad filtration representations for different primes p. This provides a tool to transfer results about the Moy-Prasad filtration from one prime to arbitrary primes and also yields a new description of the Moy-Prasad filtration representations.


On the average height of abelian varieties with complex multiplication

When: Wed, November 16, 2016 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Keerthi Madapusi Pera (U. Chicago) - http://math.uchicago.edu/~keerthi/
Abstract: In the 90s, generalizing the classical Chowla-Selberg formula, P. Colmez formulated a conjectural formula for the Faltings heights of abelian varieties with multiplication by the ring of integers in a CM field, which expresses them in terms of logarithmic derivatives at 1 of certain Artin L-functions. Using ideas of Gross, he also proved his conjecture for abelian CM extensions. In this talk, I will explain a proof of Colmez's conjecture in the average for an arbitrary CM field. This is joint work with F. Andreatta, E. Goren and B. Howard.

Automorphic L-functions, functoriality and the trace formula

When: Mon, November 21, 2016 - 2:00pm
Where: Kirwan Hall 1311
Speaker: James Arthur (U. Toronto) -
Abstract: Automorphic L-functions are Dirichlet series whose coefficients carry fundamental arithmetic information. The principle of functoriality is a cornerstone of the Langlands program. It postulates profound reciprocity laws that relate different automorphic L-functions, and hence the arithmetic data they contain. Finally, the trace formula is an identity that relates the data in automorphic L-functions with more concrete geometric information. We shall discuss these things, and the proposal of Langlands (known as Beyond Endoscopy) for bringing the trace formula to bear on the general principle of functoriality.


Some unipotent Arthur packets

When: Mon, December 5, 2016 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Jonathan Fernandes (University of Maryland) -


Towards a geometric understanding of the Mackey-Higson bijection

When: Wed, December 7, 2016 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Shilin Yu (Chinese University of Hong Kong) -
Abstract: Connes and Higson pointed out that the Baum-Connes conjecture in operator algebra suggests a surprising analogy between the tempered dual of a real reductive group and that of its Cartan motion group, which was first observed by Mackey in 1970's. Recently the Mackey-Higson bijection has been established by Afgoustidis. However, the reason for the existence of this bijection is not immediately clear. In this talk, I will show that this bijection follows naturally from deformation of Harish-Chandra D-modules and has a close relation with coadjoint orbit method. The construction leads to a new notion of `twisted' characteristic varieties of D-modules, which are Lagrangian subvarieties of the twisted cotangent bundle of the flag variety.

Vogan Duality

When: Wed, January 11, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Jeffrey Adams (University of Maryland) -
Abstract: This is an expository talk on Vogan duality, which plays an important role in representation theory of real reductive groups, and is an essential ingredient in the definition of Arthur packets for real groups.

Modularity of generating series of cycles on orthogonal Shimura varieties

When: Thu, January 19, 2017 - 2:00pm
Where: Kirwan Hall 1308
Speaker: Keerthi Madapusi Pera (U. Chicago) - http://math.uchicago.edu/~keerthi/
Abstract: Whenever one encounters a sequence of objects indexed by the integers, experience has taught us that it is sometimes unreasonably helpful to use a generating series to package them all together. On some occasions, one can even show that such a generating series is a modular form (in a suitable sense), and thus gains access to a whole raft of unforeseen relations between the terms involved via the rich arithmetic and analytic theory of modular forms.

Beginning with the work of Hirzebruch-Zagier and then Gross-Kohnen-Zagier, and continuing with that of S. Kudla, it has become clear that this procedure should work well when the objects involved are certain special cycles on modular curves and their higher dimensional analogues, Shimura varieties.

In this talk, I will explain a method of Borcherds that applies to codimension 1 cycles on orthogonal Shimura varieties, and sketch how it extends to cycles on their integral models. These modularity results have applications to computations of heights and arithmetic volumes on these Shimura varieties

This is joint with B. Howard.


Organizational Meeting

When: Wed, February 1, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Organizational Meeting () -


Schubert polynomials, theta polynomials, and raising operators I

When: Wed, February 22, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Harry Tamvakis (University of Maryland) - http://www.math.umd.edu/~harryt
Abstract: Schubert polynomials form a natural family of representatives for the Schubert classes on flag manifolds. We now have an understanding of these objects which is uniform for all the classical Lie groups. The Schur and theta polynomials, both defined
using raising operators, arise when we consider representatives for those Schubert
classes which come from Grassmannians. In these two talks, I will present an elementary
approach to this theory, focussing on the general linear and symplectic groups.

Schubert polynomials, theta polynomials, and raising operators II

When: Wed, March 1, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Harry Tamvakis (University of Maryland) - http://www.math.umd.edu/~harryt
Abstract: Schubert polynomials form a natural family of representatives for the Schubert classes on flag manifolds. We now have an understanding of these objects which is uniform for all the classical Lie groups. The Schur and theta polynomials, both defined
using raising operators, arise when we consider representatives for those Schubert
classes which come from Grassmannians. In these two talks, I will present an elementary
approach to this theory, focussing on the general linear and symplectic groups.

A generalization of the Springer resolution for GL_n(C)

When: Mon, March 6, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Martha Precup (Northwestern University) - http://www.math.northwestern.edu/~mprecup/
Abstract: The Springer correspondence relates irreducible representations of the symmetric group to a subset of simple perverse sheaves on the nilpotent cone of gl_n(C). The Springer resolution of the nilpotent cone and its fibers play an essential role in this result.

In the 1980's, Lusztig proved that each simple perverse sheaf on the nilpotent cone corresponds to an irreducible representation of a relative Weyl group. This series of results is known as the generalized Springer correspondence. The focus of this talk will be a map defined by Graham which plays a role in the generalized Springer correspondence analogous to that of the Springer resolution in the Springer correspondence. We will describe the fibers of this map using the combinatorics of standard tableaux. This talk is based on joint work with William Graham and Amber Russell.

Action of Weyl group on zero weight space

When: Mon, March 13, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Ilia Smilga (Yale University) -
Abstract: Consider a semisimple group G and a finite-dimensional irreducible representation rho of G on a space V. Assume that the zero weight space (in other terms, the set of vectors in V that are invariant by all elements of the Cartan subgroup A) is nontrivial. Then the Weyl group W = N(A)/Z(A) = N(A)/A has a well-defined action on this zero weight space. I am more specifically interested in the action of the "longest element" w0 of the group W, defined as the element that sends all positive roots to negative roots and vice-versa.

My goal is to classify all the representations for which the action of w0 on the zero weight space is nontrivial. (The motivation for that problem comes from dynamics of affine groups). Numerical experiments paint a very intriguing picture: this seems to be true for almost all representations, except those whose highest weights are multiples of fundamental weights, up to a certain threshold.

Quiver representations and generalizations of the pentagon identity

When: Wed, March 15, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Justin Allman (United States Naval Academy)
Abstract: The famous pentagon identity for quantum dilogarithms has a generalization for every Dynkin quiver (i.e. orientation of a simply-laced Dynkin diagram) due to Reineke. A more advanced generalization is associated with a pair of alternating Dynkin quivers, due to Keller, where the proof and description of the identities are described implicitly by cluster algebras and categories. In this talk, we present the Keller identities explicitly by a dimension counting argument. Namely, we calculate the Betti numbers of the equivariant "rapid decay" cohomology algebra of the quiver’s representation space in two different ways corresponding to two natural stratifications -- an approach suggested by Kontsevich and Soibelman in relation to the Cohomological Hall Algebra of quivers.

Arthur packets packets for p-adic groups and vanishing cycles of perverse sheaves

When: Tue, March 28, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Clifton Cunningham (University of Calgary) - http://math.ucalgary.ca/math_unitis/profiles/clifton-cunningham
Abstract: This talk will explain how to adapt the approach developed by Adams-Barbasch-Vogan to Arthur packets from Real groups to p-adic groups, and will illustrate this adaptation with several examples. We will also sketch a proof showing that Arthur packets are p-adic ABV packets for unipotent representations of p-adic special orthogonal groups.

Joint with: Bin Xu, Ahmed Moussaoui, Andrew Fiori, James Mracek

Local Arthur packets in the Archimedean case

When: Wed, March 29, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Nicolas Arancibia (Paris) -
Abstract: The objective of this talk is to give an introduction to the Local Arthur Conjecture (LAC). In a first time I will introduce the LAC for any local field of characteristic zero, then I will specialize to the Archimedean case and introduce the packets of irreducible unitary cohomological representations defined by J. Adams and J. Johnson in 1987 and the work of C. Moeglin and D. Renard on complex classical groups. If time allows me I will end the talk with a short exposition of the work of J. Adams, D. Barbasch and D. Vogan on the local Arthur conjecture.

Doubling constructions for covering groups and tensor product L-functions

When: Mon, April 10, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Yuangqing Cai (Boston University) -
Abstract: In the 1980's, Piatetski-Shapiro and Rallis discovered a family of Rankin-Selberg integrals for the classical groups that did not rely on Whittaker models. This is the so-called doubling method. In this work, we give a generalization of the doubling method. We present a family of integrals representing tensor product L-functions of classical groups with general linear groups. Our construction is uniform over all classical groups and their non-linear coverings, and is applicable to all cuspidal representations. These integrals remove the main obstruction to proving the existence of endoscopic lifts for all automorphic representations without using the trace formula. This is joint work with Friedberg, Ginzburg and Kaplan.

TBA

When: Mon, April 17, 2017 - 2:00pm
Where: Kirwan Hall 1310
Speaker: Jon Cohen (CMPS) -


On the Novikov higher signature conjecture: history and development

When: Wed, April 26, 2017 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Gennadi Kasparov (Vanderbilt)
Abstract: This is part of the Novikov celebration.
The Novikov higher signature conjecture played and continues to play an important role in the development of several areas of mathematics: topology, geometric group theory, K-theory of C*-algebras. I will give a brief exposition of the history and progress in research related with the Novikov conjecture up to the most recent results.

Transversally elliptic operators and K-theory

When: Fri, April 28, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Gennadi Kasparov (Vanderbilt)
Abstract: The class of (pseudo-) differential operators transversally elliptic with respect to a Lie group action on a manifold was introduced by M. Atiyah in the 70s. He also made an attempt to obtain a formula which calculates the index of such operators by topological means. This class of operators is interesting both from the point of view of geometry and analysis, but particularly by its relations with representation theory of Lie groups. In the 90s, N. Berline and M. Vergne have obtained a certain very complicated index formula. However, this did not stop further attempts to obtain something more reasonable and more useful in applications. I will try to explain the background of the theory and a different approach to an index formula.

Combinatorial evaluation of Hecke algebra traces.

When: Mon, May 1, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Mark Skandera (Lehigh University) - http://www.lehigh.edu/~mas906/
Abstract: The (type A) Hecke algebra $H_n(q)$ is a certain module over $\mathbb Z[q^{1/2},q^{-1/2}]$ which is a deformation
of the group algebra of the symmetric group. The $\mathbb Z[q^{1/2},q^{-1/2}]$-module of its trace functions
has rank equal to the number of integer partitions of $n$, and has bases which are natural deformations of
those of the symmetric group algebra trace module. While no known formulas give the evaluation of these traces at
the natural basis elements of $H_n(q)$, there are some nice combinatorial formulas for the evaulation of certain traces at certain Kazhdan-Lusztig basis elements. We will also discuss the open problem of evaluating these traces at other basis elements.

Gamma sheaves on reductive groups

When: Wed, May 3, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Tsao-Hsien Chen (University of Chicago) - https://sites.google.com/site/tsaohsienchen/
Abstract: Gamma sheaves on reductive groups, introduced by
Braverman and Kazhdan, are generalizations of Deligne’s Kloosterman sheaves. In the talk I will give an introduction of Braverman and Kazhdan’s
construction of gamma sheaves and then explain the motivation and a proof
of their acyclicity conjecture of gamma sheaves.

Some Unipotent Packets for Real Groups of Classical type

When: Wed, May 10, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Jonathan Fernandes (University of Maryland) -
Abstract: We will work in the framework where G is a classical real reductive group. Packets of unipotent representation of G are classified by real-forms of complex nilpotent orbits for the dual group. We will introduce the relevant invariants and provide an algorithm that computes the Langlands parameters in most of these Unipotent packets.

We will end the talk with a brief demonstration of the algorithm using the Atlas software.