Lie Groups and Representation Theory Archives for Academic Year 2018

Organizational meeting

When: Wed, August 30, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: () -

Characters of Finite Dimensional Representations

When: Wed, September 6, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Jeffrey Adams (University of Maryland) -
Abstract: The main objects of study of the Atlas of Lie Groups and
Representations are infinite dimensional representations. However
there are quite a few interesting open questions about finite
dimensional representations. One is: what can one say about the
signature of the invariant (Hermitian or bilinear) form on an
irreducible finite dimensional representation? How does this depend on
the real form of the group? Another one is: if g in G represents the
Coxeter element of the Weyl group, its trace in any finite dimensional
representation is 0,\pm 1 (Kostant). What do these values mean, and
are there other conjugacy classes like this?

Real representation of finite simple groups

When: Mon, September 18, 2017 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Ryan Vinroot (William and Mary ) -
Abstract: It has been conjectured that if G is a finite simple group, then every complex irreducible representation of G may be realized over the real numbers if and only if every element of G is the product of two involutions of G. This follows for most families of finite simple groups from work of various people over the past several decades, but not for the cases that G is either $p(2n,F_q) with q even or the simple orthogonal group Omega^{\pm}(4m,F_q) with q even. We will discuss the proof that this statement indeed holds for these symplectic groups, and what modifications must be made to the proof for the same method to apply to the simple orthogonal groups.

Unipotent Packets for Real Groups

When: Wed, October 11, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Jonathan Fernandes (University of Maryland) -

The pseudo-identity operator on the space of automorphic functions

When: Mon, October 16, 2017 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Jonathan Wang (IAS) -
Abstract: Let F be a global field and G a reductive group over F. We define a "strange" operator on the space of automorphic functions on G(A)/G(F). We discuss how this "pseudo-identity operator" relates to pseudo-Eisenstein series and inversion of the standard intertwining operator. We show that this operator is natural from the viewpoint of the geometric Langlands program via the functions-sheaves dictionary. The operator is also connected to Deligne-Lusztig duality and cohomological duality of representations over a local field.

Serre-Tate theory for Shimura varieties

When: Mon, October 23, 2017 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Rong Zhou (IAS) -
Abstract: We study the $\mu$ ordinary locus in the special fiber of Hodge-type Shimura varieties. This is the group theoretic analogue of the ordinary locus in the moduli space of abelian varieties. We show that each point on this locus has a canonical special point which lifts it and that the completed local ring at each point has a group-like structure, in analogy with classical Serre-Tate theory for ordinary abelian varieties. If we have time we will also talk about some generalizations and applications of these results. This is joint work with A. Shankar.

Relative Local Langlands Correspondence and Geometry of Parameter Spaces

When: Wed, October 25, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Dipendra Prasad (Tata Institute and University of Maryland) -

Supercuspidal L-packets

When: Wed, November 8, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Tasho Kaletha (University of Michigan) -
Abstract: Harish-Chandra has given a simple and explicit classification of the discrete series representations of reductive groups over the real numbers. We will describe a very similar classification that holds for a large proportion of the supercuspidal representations of reductive groups over non-archimedean local fields (which we may call regular). The analogy runs deeper: there is a remarkable parallel between the characters of regular supercuspidal representations and the characters of discrete series representations of real reductive groups. This leads to an explicit construction of the local Langlands correspondence for discrete Langlands parameters with trivial monodromy, under mild conditions on the residual characteristic.

Generalizations of Springer fibers associated to ad nilpotent ideals

When: Wed, November 15, 2017 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Ke Xue (University of Maryland) -
Abstract: We state that certain varieties arising as natural generalization of Springer fibers are paved by affines, i.e. possessing a cell decomposition into affine spaces. The proof will be presented in a sketch after introducing several motivations. The chief techniques of the proof are the Bala-Carter Theorem on nilpotent orbits and those employed in the 1988 paper by de Concini, Lusztig and Procesi concerning similar property of Springer fibers.

Transfer operators between relative trace formulas in rank one

When: Mon, November 27, 2017 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Yiannis Sakellaridis (Rutgers Newark and IAS) -
Abstract: I will introduce a new paradigm for comparing relative trace formulas,
in order to prove instances of (relative) functoriality and relations
between periods of automorphic forms.

More precisely, for a spherical variety X=H\G of rank one, I will prove
that there is an explicit "transfer operator" which transforms the
orbital integrals of the relative trace formula for X x X/G to the
orbital integrals of the Kuznetsov formula for GL(2) or SL(2), equipped
with suitable non-standard test functions. The operator is determined by
the L-value associated to the square of the H-period integral, and the
proof uses a deep theory of Friedrich Knop on the cotangent bundles of
spherical varieties. This is part of an ongoing joint project with
Daniel Johnstone and Rahul Krishna, who are proving instances of the
fundamental lemma. Globally, this transfer will induce an identity of
relative trace formulas and global relative characters, translating to
an Ichino–Ikeda type formula that relates the square of the H-period to
the said L-value.

This can be viewed as part of the program of relative functoriality, a
generalization of the Langlands functoriality conjecture, predicting
relations between the automorphic spectra of two spherical varieties
when there is a map between their dual groups. The case under
consideration here is the simplest non-abelian case of this, when the
dual groups are equal and of rank one. If time permits, I will discuss
how the transfer operator here and in a few examples of higher rank
where it is known is a "deformation" of an abelian transfer operator
obtained by replacing the spherical variety by its asymptotic cone (or
boundary degeneration).

Constructing Supercupsidal Representations

When: Wed, December 6, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Jeff Hakim (American University) -
Abstract: We discuss a revision of Jiu-Kang Yu’s construction of supercuspidal representations and connections with induction and the theory of distinguished representations.

Relative Local Langlands Correspondence II

When: Mon, December 11, 2017 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Dipendra Prasad (Tata Institute) -

Test functions for parahoric local models, I

When: Wed, January 31, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Tom Haines (UMCP) -
Abstract: Timo Richarz and I recently proved the test function conjecture of Kottwitz and myself which describes the trace of Frobenius on the nearby cycles sheaf for local models of Shimura varieties with parahoric level structure (and their equal characteristic counterparts). In this talk, I will discuss the statement of the result, along with history and examples, and its expected applications.

Test functions for parahoric local models, II

When: Wed, February 7, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Tom Haines (UMCP ) -
Abstract: In this second talk, I will discuss some of the ingredients of the proof of the test function conjecture for parahoric local models.

The q-Schur dualities of finite type

When: Wed, February 14, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Weiqiang Wang (University of Virginia)
Abstract: Schur algebra and Schur duality are usually referred to type A. In this talk we explain a generalization (in the quantized setting) which makes sense for any finite type such as type G_2 or E_8. We explain both algebraic and geometric constructions of the q-Schur algebras and q-Schur dualities; We show the q-Schur algebras have canonical bases with positivity and are closely related to the BGG category O. This is joint work with Li LUO (Shanghai), and it is related to earlier work with Huanchen Bao and others.

Categorification of Young idempotents

When: Wed, February 21, 2018 - 2:00pm
Where: Kirwan 1311
Speaker: Matthew Hogancamp (University of Southern California),

Abstract: I will discuss joint work with Ben Elias in which we construct complexes of Soergel bimodules which categorify the Young idempotents in the Hecke algebras of type A_n. The construction is an application our theory of categorical diagonalization to the case of the full twist Rouqiuer complex acting on Soergel bimodules. Our categorified Young idempotents are key elements in the Gorsky-Negut-Rasmussen conjectures, which relate categories of Soergel bimodules to Hilbert schemes of points in the plane. I will explain this connection, and give several examples.

Modular Koszul duality for the Hecke category

When: Mon, March 5, 2018 - 2:00pm
Where: Kirwan 1311
Speaker: Shotaro makisumi (Columbia University)
Abstract: The Hecke category, a categorification of the Hecke algebra, plays an important role in geometric representation theory. I will discuss a monoidal Koszul duality for the Hecke category (at least in cases arising from geometry), categorifying a certain involution of the Hecke algebra. This result has consequences for the modular representation theory of reductive groups. I will focus on the SL_2 case and explain the key constructions using a combinatorial/algebraic incarnation of the Hecke category (Soergel bimodules). (Joint with P.N. Achar, S. Riche, and G. Williamson.)

Quantum linear systems and representations of 2-groups

When: Wed, March 28, 2018 - 2:00pm
Where: Kirwan Hall 1311

Speaker: Carl A. Miller (University of Maryland) -
Abstract: The concept of a quantum linear system (Cleve-Mittal 2014) is a convergence point for a number of different topics in mathematics and physics. On one hand, they are relevant for cryptography and fundamental tests of quantum physics. On the other hand, their analysis so far has touched on representation theory, graph theory, and computational decidability, among other topics. In this talk I will discuss research in which we identify quantum linear systems that are rigid (i.e., those that have unique solutions). This is joint work with Amir Kalev and Aaron Ostrander.

The pro-p-Iwahori-Hecke algebra of a reductive p-adic group (candidacy exam)

When: Tue, April 17, 2018 - 2:00pm
Where: MTH 1310
Speaker: Yijie Gao

Representation ring of Levi subgroups versus cohomology ring of flag varieties

When: Wed, April 18, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Shrawan Kumar (UNC Chapel Hill) -
Abstract: See PDF.

Representations of p-adic groups

When: Wed, May 2, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Jessica Fintzen (IAS)
Abstract: The building blocks for complex representations of p-adic groups are called supercuspidal representations. I will survey what is known about the construction of supercuspidal representations, mention questions that remain mysterious until today, and explain some recent developments.