Lie Groups and Representation Theory Archives for Academic Year 2020

Geometrization of orbital integrals of spherical Hecke functions

When: Wed, September 4, 2019 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Jingren Chi (UMCP) -
Abstract: We will talk about certain algebraic varieties, first studied by Kottwitz-Viehmann, that encodes orbital integrals of spherical Hecke functions on a reductive group over equal characteristic non-archimedean local field. We report on joint work with A. Bouthier that proves a dimension formula for these varieties and hence provides a description of the asymptotics of the corresponding orbital integrals. We will also explain a conjectural description of the number of irreducible components.

Partial orders on the Weyl group and unipotent classes

When: Wed, September 18, 2019 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Jeffrey Adams (University of Maryland) -
Abstract: Lusztig has defined a surjective map from conjugacy classes in the Weyl group to unipotent conjugacy classes. He says that the fact the this works is a "miracle". We would like to understand some properties of this map. In this talk I'll discuss the behavior of the map with respect to natural order on conjugacy classes in W and the closure relations for unipotent classes. This is joint work with Xuhua He and Sian Nie.

On arithmetic transfer conjectures

When: Wed, September 25, 2019 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Michael Rapoport (UMCP) -
Abstract: Arithmetic Transfer conjectures are the analogues of Wei
Zhang's Arithmetic Fundamental Lemma conjecture in the presence of
ramification. Some such conjectures were given in two papers by
Smithling, Zhang and myself. I will present more such conjectures. This
is joint work in progress with S. Kudla, B. Smithling and W. Zhang.

nipotent Representations attached to Principal Nilpotent Orbits

When: Wed, October 23, 2019 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Lucas Mason-Brown (MIT) -

Abstract: Let G be a real reductive group. The classification of the irreducible unitary representations of G is one of the major unsolved problems in representation theory. There is evidence to suggest that every such representation can be constructed (through several types of induction) from a finite set of building blocks, called the unipotent representations. These representations are `attached' (in a certain mysterious sense) to the nilpotent orbits of G on the dual space of its Lie algebra. The theory of unipotent representations hands us a finite set of distinguished classes in the K-theory of the nilpotent cone. It would be extremely interesting to have a geometric description of these classes. In this talk, we provide one, in the special case of the principal nilpotent orbit. Time permitting, we will formulate a conjecture about the general case.

Equivalent definitions of Arthur-packets for real classical groups

When: Wed, October 30, 2019 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Paul Mezo (Carleton University) -

Abstract: In his most recent book, Arthur defines A(rthur)-packets for classical groups using techniques from harmonic analysis. For real groups and alternative definition of A-packets has been know since the early 90s. This approach, due to Adams-Barbasch-Vogan, relies on sheaf-theoretic techniques instead of harmonic analysis. We will report on work in progress, joint with N. Arancibia, in proving that these two different definitions for A-packets are equivalent for real classical groups.

L-functions, orbital integrals, and algebraic varieties in the Langlands Program

When: Fri, November 22, 2019 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Yihang Zhu (Columbia University) -
Abstract: The Langlands Program predicts a deep link between the world of analysis and representation theory and the world of number theory and algebraic geometry. The key objects that synthesize everything together are the L-functions. On the other hand, orbital integrals are concrete and fundamental objects in representation theory. We shall see how orbital integrals could be employed in the study of certain algebraic varieties and L-functions arising naturally in the Langlands program.

Around the Arithmetic Gan-Gross-Prasad Conjecture

When: Mon, December 2, 2019 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Brian Smithling (UMCP) -
Abstract: The famous theorem of Gross and Zagier relates the NĂ©ron-Tate
height of Heegner points on modular curves to the first central
derivative of an L-function. The Arithmetic Gan-Gross-Prasad (AGGP)
Conjecture is a vast generalization of this theorem to
higher-dimensional Shimura varieties attached to unitary and orthogonal
groups proposed by Gan-Gross-Prasad and, in a more precise form in the
unitary case, by W. Zhang. I will give an overview of my joint work
with M. Rapoport and W. Zhang in the context of various aspects of the
approach to the AGGP conjecture laid out by W. Zhang.


When: Mon, April 6, 2020 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Moshe Adrian (Queens College) -
Abstract: TBA