Saturday, September 11, 2021 at Johns Hopkins University
David Ben-Zvi (University of Texas at Austin)
Title: Shearing and Geometric Arthur Parameters.
Abstract: Shearing is a name for the symmetry of the derived category of graded vector spaces which simultaneously shifts weights and cohomological degrees. I will explain some games one can play with shearing that play a role in my joint work with Yiannis Sakellaridis and Akshay Venkatesh. In particular I’ll explain the construction of the “spectral exponential sheaf”, a coherent analog of the exponential D-module or Artin-Schreier sheaf. Using this object we define a functor of “spectral Whittaker induction” on moduli of local systems on curves, and in particular a geometric form of Arthur parameters – local systems (Langlands parameters) which are sheared and twisted using a commuting action of SL_2.
Jayce Getz (Duke University)
Title: The Poisson summation conjecture for generalized Schubert varieties.
Abstract: (Joint with Y. Choie.) Conjectures of Braverman and Kazhdan, L. Lafforgue, Ngo and Sakellaridis state that spherical varieties admit Schwartz spaces, a Fourier transform or transforms, and corresponding Poisson summation formulae. I refer to this collection of conjectures as the Poisson summation conjecture. If the Poisson summation conjecture were known in general then the functional equation and meromorphic continuation of Langlands L-functions would follow. By converse theory, this in turn would imply much of Langlands functoriality.
I will define a family of generalized Schubert varieties and then outline the proof of the Poisson summation conjecture for these varieties. I will also explain how this allows one to prove a conjecture of Bump and Choie on the meromorphic continuation of Schubert Eisenstein series.
Ivan Loseu (Yale University)
Title: Unipotent Harish-Chandra bimodules.
Abstract: Unipotent representations of semisimple Lie groups is a very important and somewhat conjectural class of unitary representations. Some of these representations for complex groups (equivalently, Harish-Chandra bimodules) were defined in the seminal paper of Barbasch and Vogan from 1985 based on ideas of Arthur. From the beginning it was clear that the Barbasch-Vogan construction doesn’t cover all unipotent representations. The main construction of this talk is a geometric construction of Harish-Chandra bimodules that should exhaust all unipotent bimodules. A nontrivial result is that all unipotent bimodules in the sense of Barbasch and Vogan are also unipotent in our sense. The proof of this claim is based on the so called symplectic duality that in our case upgrades a classical duality for nilpotent orbits in the version of Barbasch and Vogan. Time permitting I will explain how this works. The talk is based on a joint work with Lucas Mason-Brown and Dmytro Matvieievskyi.
Mona Merling (University of Pennsylvania)
Title: Scissors congruence for manifolds via K-theory
Abstract: The classical scissors congruence problem asks whether given two polyhedra with the same volume, one can cut one into a finite number of smaller polyhedra and reassemble these to form the other. There is an analogous definition of an SK (German “schneiden und kleben,” cut and paste) relation for manifolds and classically defined scissors congruence (SK) groups for manifolds. Recent work of Jonathan Campbell and Inna Zakharevich has focused on building machinery for studying scissors congruence problems via algebraic K-theory, and applying these tools to studying the Grothendieck ring of varieties. I will talk about a new application of this framework: we will construct a K-theory spectrum of manifolds, which lifts the classical SK group, and a derived version of the Euler characteristic.
Carl Wang-Erickson (University of Pittsburgh)
Title: A fully faithful alternative to the Montreal functor.
Abstract: Let p be a prime number. The Montreal functor of P. Colmez sends p-adic representations of GL 2 (Q_p) to p-adic modules for Γ, where Γ denotes the absolute Galois group of Q_p . One result of V. Paškūnas’s study of the Montreal functor, which was a crucial step toward the p-adic Langlands correspondence for GL_2 (Q_p), is that its failure to be fully faithful boils down to the fact that it sends the trivial representation to zero. In this talk, for p greater than 3, we introduce a fully faithful alternative to the Montreal functor. It has a different target category: a derived category of modules over the stack of 2-dimensional p-adic representations of Γ. This is joint work with Christian Johansson and James Newton.