Lie Groups and Representation Theory Archives for Academic Year 2016

On the Schwartz space of a smooth semi-algebraic stack

When: Fri, October 2, 2015 - 10:00am
Speaker: Yiannis Sakellaridis (Rutgers) -
Abstract: There are many instances in the field of automorphic forms where nice classification results do not exist for a single group or space, but they do exist if one considers some "pure inner forms" at the same time. It has been suggested by Joseph Bernstein that this phenomenon should be explained by replacing the spaces and groups by appropriate quotient stacks.

In this talk I will explain how to extend the notions of Schwartz functions (or rather measures) from smooth, semi-algebraic ("Nash") manifolds, to the corresponding category of stacks.

I will also describe an approach to orbital integrals (in this setting considered as "evaluation maps" on global Schwartz spaces) which does not use truncation.

The Real/Quaternionic Indicator

When: Wed, October 28, 2015 - 2:00pm
Where: 1311.0
Speaker: Ran Cui (Maryland) -

Hecke algebras and $p$-adic groups, II

When: Wed, November 18, 2015 - 2:00pm
Where: Math 1311
Speaker: Xuhua He (University of Maryland) -
Abstract: In a 1957 paper, Tits explained the analogy between the symmetric group $S_n$ and the general linear group over a finite field $\mathbb F_q$ and indicated that $S_n$ should be regarded as the general linear group over $\mathbb F_1$, the field of one element.

Following Tits' philosophy, we may informally regard the affine Weyl groups as the reductive group over $\mathbb Q_1$, the $1$-adic field. Although it might be premature to develop the theory of $1$-adic field at the current stage, we do have a fairly good understanding on the conjugacy classes of the affine Weyl groups, together with the length function on it, and such knowledge allows us to reveal a great part of the structure of the conjugacy classes of $p$-adic groups. In the first talk, we will explain how such knowledge is used to understand the Frobenius-twisted conjugacy classes of loop groups and discuss some further applications to Shimura varieties. In the second talk, we will explain how such knowledge is used in the study of cocenters and representations of affine Hecke algebras and discuss some further application to the representations of $p$-adic groups.

Schur duality, canonical bases, and quantum symmetric pairs of affine type

When: Wed, December 2, 2015 - 2:00pm
Where: Math 1311
Speaker: Chun-Ju Lai (University of Virginia) -
Abstract: Recently, generalizing the work of Beilinson, Lusztig, and MacPherson of finite type A, Bao, Kujawa, Li, and Wang constructed the quantum algebras arising from partial flag varieties of finite type B/C, altogether with their canonical bases. They also provided a geometric realization of a Schur-type duality between these algebras and the Hecke algebras of type B/C acting on a tensor space. These quantum algebras are coideal subalgebras of quantum gl(n), which also form quantum symmetric pairs, with quantum gl(n). The canonical bases arising from quantum symmetric pairs was used by Bao-Wang to formulate Kazhdan-Lusztig theory for BGG category O. The above can be reformulated within the framework of Hecke algebras without geometry.

In this talk I will explain the affinization of above in two (Hecke algebra theoretic and geometric) approaches. This is a joint work with Zhaobing Fan, Yiqiang Li, Li Luo, and Weiqiang Wang.

Canonical bases arising from quantum symmetric pairs

When: Wed, December 9, 2015 - 2:00pm
Where: 1311
Speaker: Huanchen Bao (Maryland) -
Abstract: A quantum symmetric pair consists of a quantum group and its coideal subalgebra. The coideal subalgebra is a quantum analog of the fixed point subalgebra of the enveloping algebra of the underline Lie algebra with respect to certain involution. Recently, we initiated a theory of canonical bases arising from quantum symmetric pairs. Such canonical bases have been used to formulate the Kazhdan-Lusztig theory for ortho-symplectic Lie superalgebras for the first time. In this talk, we shall generalize the construction of canonical bases to all quantum symmetric pairs of finite types, as well as many Kac-Moody cases. We further construct the canonical bases on the modifed coideal subalgebras, generalizing Lusztig's construction of the canonical bases on the modified quantum groups. This is joint work with Weiqiang Wang.

Schur Indicator for Split Groups

When: Wed, December 16, 2015 - 1:00pm
Where: 1311.0
Speaker: Rob McLean (Maryland) -

Frobenius semisimplicity

When: Wed, February 24, 2016 - 2:00pm
Where: Math 1311
Speaker: Tom Haines (UMCP) -
Abstract: I will explain part of my joint work with Mark de Cataldo and Li Li aimed at proving special cases of the general conjecture that Frobenius acts semisimply on the cohomology of any variety over a finite field. The paper in question is arXiv:1602.00645.

Feynman periods: numbers and geometry

When: Wed, March 9, 2016 - 2:00pm
Where: Math 1311
Speaker: Dmitry Doryn (Center for Geometry and Physics, Pohang, South Korea) -
Abstract: I will speak on the Feynman periods, the values of Feynman integrals in (massless, scalar) phi^4 theory, from the number-theoretical perspective. Then I define a closely related geometrical object, the graph hypersurface. One can try to study the geometry of these hypersurfaces (cohomology, Grothendieck ring, number of rational points over finite fields) and to relate it to the periods. The most interesting results come out from the study of the c_2 invariant (on the arithmetical side).

Level structures beyond the Drinfeld case

When: Wed, March 23, 2016 - 2:00pm
Where: Math 1311
Speaker: Preston Wake (UCLA) -
Abstract: Drinfeld level structures are a key concept in the arithmetic study of the moduli of elliptic curves. They also play an important role in the moduli of 1 dimensional p-divisible groups, and related Shimura varieties studied by Harris and Taylor. I'll explain why Drinfeld level structures (and the related "full set of sections" defined by Katz and Mazur) are not adequate for studying more general Shimura varieties. I'll discuss two examples of a satisfying theory of level structure outside the Drinfeld case:
i) full level structures on the group \mu_p x \mu_p;
ii) Gamma_1(p^r)-type level structures on an arbitrary p-divisible group (joint work with R. Kottwitz).

Plancherel measure and Shahidi local coefficients for coverings of p-adic SL(2,F)

When: Wed, March 30, 2016 - 2:00pm
Where: Math 1311
Speaker: Dani Szpruch (Howard University)

Abstract: Plancherel measure is a certain meromorphic function associated with a parabolic induction on a quasi-split reductive group defined over a local field. Among other applications, it determines the reducibility points on the unitary axis. It was conjectured by Langlands that this invariant is a ratio of certain L-functions and hence of arithmetic significance. Under some mild assumptions, this conjecture was proven for generic inducing data by Shahidi, utilizing the uniqueness of Whittaker model. In this talk we shall discuss the computation of the Plancherel measure for genuine principal series representations of non-algebraic coverings of p-adic SL(2), although uniqueness of Whittaker model fails. Along the way we define a higher dimensional analog for Shahidi local coefficients and relate it to Tate gamma factor. This is a joint work with David Goldberg.

The Real-Quaternionic Indicator of Irreducible Self-Conjugate Representations of Real Reductive Groups (thesis defense)

When: Mon, April 4, 2016 - 3:00pm
Where: MTH 1313
Speaker: Ran Cui (UMD)

The solution to the sphere packing problem in 24 dimensions via modular forms

When: Wed, April 6, 2016 - 2:00pm
Where: Math 1311
Speaker: Stephen Miller (Rutgers) -
Abstract: Maryna Viazovska recently made a stunning breakthrough on
sphere packing by showing the E8 root lattice gives the densest packing
of spheres in 8 dimensional space [arxiv:1603.04246]. This is the first
result of its kind for dimensions > 3, and follows an approach suggested
by Cohn-Elkies from 1999 through harmonic analysis. The talk will
describe this method, as well as new joint work with Viazovska, Cohn,
Kumar, and Radchenko that completely solves the sphere packing problem
in dimension 24 [arxiv:1603.06518]: the Leech lattice gives the densest
packing of spheres in 24 dimensions, and no other periodic packing is as
dense as it.

Subtleties of H^3 and twisted K-Theory for compact Lie groups

When: Mon, April 11, 2016 - 3:00pm
Where: MTH 1313
Speaker: Jonathan Rosenberg (UMD) -
Abstract: We explicitly compute the map on H^3 induced by a covering of compact simple Lie groups. The result is complicated and quite surprising. We also delve further into the twisted K-theory of compact Lie groups, previously studied by Moore-Maldacena-Seiberg, Hopkins, Braun, and Douglas, and the connection between Langlands duality and T-duality, studied by Daenzer-Van Erp and Bunke-Nikolaus. This is joint work with Mathai Varghese.

This is a cross-listing from the Geometry Seminar

The wonderful compactification for quantum groups

When: Wed, April 13, 2016 - 2:00pm
Where: MATH1311
Speaker: Jordan Ganev (University of Texas, Austin) -
Abstract: The wonderful compactification of a semisimple group features in
several areas of geometric representation theory and in the theory of
spherical varieties. It was first introduced by De Concini and Procesi,
and can be realized as a quotient of the Vinberg semigroup. The wonderful compactification links the geometry of the group to the geometry of its partial flag varieties, and encodes the asymptotic of matrix coefficients for the group. We review several constructions of the wonderful compactification and its basic properties. We then introduce quantum group versions of both the wonderful compactification and the Winberg semigroup, describe quantum D-modules on these spaces, and discuss conjectural applications of these newly-defined objects.

Dual Bernstein Center

When: Wed, April 20, 2016 - 2:00pm
Where: Math 1311
Speaker: Ahmed Moussaoui (U. Calgary) -
Abstract: The main purpose of this talk is to construct a Bernstein decomposition of the set of (enhanced) Langlands parameters.

To do that we will present a conjecture on the (enhanced) Langlands parameters of supercuspidal representations of a split p-adic group. In the case of split classical groups we will explain how to define a cuspidal support map. We obtain then an algebra which we call the dual Bernstein centre and which is defined just in term of enhanced Langlands parameters. In particular this is independent of the validity of the local Langlands correspondence. This algebra contains the stable Bernstein centre defined by Thomas Haines which was also construct only in terms of Langlands parameters. At the end, we will be able to show that the LLC establish an isomorphism between the Bernstein centre and the dual Bernstein centre for split classical groups.

Transfer of representations and the Bernstein center for inner forms of GL(n)

When: Wed, April 27, 2016 - 2:00pm
Where: Math 1311
Speaker: Jon Cohen (UMCP) -
Abstract: I will discuss a construction and characterization of the Local Langlands Correspondence for Inner Forms of GL_n, and its relation to a method for constructing explicit matching functions via a transference of Bernstein centers.

Classical Invariants of Principal Series (thesis defense --- note special date/time)

When: Thu, May 5, 2016 - 1:00pm
Where: Math 1310
Speaker: Rob McLean (UMCP)