Representation Theory and Number Theory Seminar, 2017-2018

Friday, 3:30-4:30 PM, LCB 215



Fall 2017

August 25:
Title: Organizational meeting and student modularity seminar (Moss)

September 1:
Title: Student modularity seminar (Moss)

September 8:
Title: Student modularity seminar (Patrikis)

September 15:
Title: Student modularity seminar (Patrikis)

September 22:
Speaker: Anna Romanov, The University of Utah
Title: A Kazhdan-Lusztig algorithm for Whittaker modules
Abstract: The category of Whittaker modules for a complex semisimple Lie algebra generalizes the category of highest weight modules and displays similar structural properties. In particular, Whittaker modules have finite length composition series and all irreducible Whittaker modules appear as quotients of certain standard Whittaker modules which are generalizations of Verma modules. Using the localization theory of Beilinson-Bernstein, one obtains a beautiful geometric description of Whittaker modules as twisted sheaves of D-modules on the associated flag variety. I use this geometric setting to develop an analogue of Kazhdan-Lusztig algorithm for computing the multiplicities of irreducible Whittaker modules in the composition series of standard Whittaker modules.

September 29:
Speaker: Gil Moss, The University of Utah
Title: Toward a local Langlands correspondence in families
Abstract: In 2012 it was conjectured by Emerton and Helm that the local Langlands correspondence for GL(n) of a p-adic field (suitably normalized) should interpolate in \ell-adic families, where \ell is a prime different from p. Recently, Helm reformulated this conjecture in terms of the existence of a map from the integral Bernstein center to a Galois deformation ring. This connects both congruences and moduli spaces for objects on either side of the correspondence. In this talk we will present recent work (joint with David Helm) showing the existence of such a map and describing its image, and we will discuss a speculative generalization to split classical groups.

October 6:
Speaker: Erick Knight, University of Toronto
Title: Patching and the p-adic Jacquet-Langlands correspondence
Abstract: In this talk, I will explain how to use the Taylor-Wiles-Kisin patching method to study the p-adic Jacquet-Langlands correspondence. I will show that the two constructions of the p-adic Jacquet-Langlands correspondence due to myself and Scholze agree, and also determine the locally algebraic vectors inside the representations of the quaternion algebra. This is joint work with Przemyslaw Chojecki.

October 20:
Title: Student modularity seminar (Klevdal)

October 27:
Title: Student modularity seminar (Childers)

November 3:
Speaker: Sean McAfee, The University of Utah
Title: Twisted and Untwisted Cells for Real Reductive Lie Groups
Abstract: In classical Kazhdan-Lusztig theory, the action of the Hecke algebra H(W) on itself allows us to partition the Weyl group W into "cells" which carry information about representations of W and highest weight modules indexed by elements of W. Given an involution of W, we can define a notion of "twisted cells" of W as well. In this talk, I will describe the special relation between twisted and untwisted cells in the classical case, and show how recent work allows us to define twisted and untwisted cells for the set of Langlands parameters for a real reductive linear Lie group.

November 10:
Speaker: Allen Moy, HKUST
Title: An Euler-Poincaré formula for a depth zero Bernstein projector
Abstract: Work of Bezrukavnikov-Kazhdan-Varshavsky uses an equivariant system of trivial idempotents of Moy-Prasad groups to obtain an Euler-Poincaré formula for the r-depth Bernstein projector. We establish an Euler-Poincaré formula for the projector to an individual depth zero Bernstein component in terms of an equivariant system of Peter-Weyl idempotents of parahoric subgroups P associated to a block of the reductive quotient P. This work is joint with Dan Barbasch and Dan Ciubotaru.

November 17:
Speaker: Sean Taylor, LSU
Title: A Mixed Version of the Derived Category of Constructible Sheaves on Toric Varieties
Abstract: In "Koszul duality patterns in representation theory," Beilinson, Ginzburg, and Soergel provided the notion of a mixed abelian category. In this seminal paper, they used this theory to prove a parabolic-singular duality for BGG Category \mathcal{O}. For varieties over fields of characteristic p > 0, Deligne famously introduced the derived category of mixed constructible complexes. However, Deligne's category is not mixed in the above sense. The problem of producing a mixed structure was dealt with by Beilinson, Ginzburg, and Soergel in the abelian case and by Achar and Riche in the derived case. In this talk, we will explain recent results that accomplish producing a mixed structure on the derived category of constructible complexes over any toric variety. In the process, we prove that this also comes equipped with a mixed version of the category of perverse sheaves. It is then shown that some of functors commute with forgetting the mixed structure.

December 1:
Speaker: Preston Wake, UCLA
Title: The rank of Mazur's Eisenstein ideal
Abstract: In his landmark 1976 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. We use deformation theory of pseudorepresentations to study the corresponding Hecke algebra. We will discuss how this method can be used to refine Mazur's results, quantifying the number of Eisenstein congruences. Time permitting, we'll also discuss some partial results in the composite-level case. This is joint work with Carl Wang-Erickson.