The Plancherel formula for L^2(GL_n(F)\GL_n(E)) and applications to the Ichino-Ikeda and formal degree conjectures for unitary groups

Raphael Beuzart-Plessis
CNRS
March 6, 2018
Abstract : Let $E/F$ be a quadratic extension of local fields of characteristic zero. In this talk, I will explain two ways to compute the Plancherel decomposition of $L^2(GL_n(F)\backslash GL_n(E))$. In both cases, the result involves the image of base change from unitary groups to $GL_n(E)$ and is in accordance with a general conjecture of Sakellaridis-Venkatesh on the spectral decomposition of spherical varieties. We will also give applications of our formulas to the so-called Ichino-Ikeda and formal degree conjectures for unitary groups.

Restriction problem for non-generic representation of Arthur type

Wee Teck Gan
National University of Singapore
March 6, 2018
Abstract: The Gross-Prasad conjecture considers a branching problem for generic Arthur packets of classical groups. In this talk, we will describe progress towards extending this conjecture to nongeneric Arthur packets (this is joint work with Gross and Prasad). For GL(n), we describe some recent progress towards this conjecture by Max Gurevich.

Relative character asymptotics and applications

Paul Nelson
ETH Zurich
March 6, 2018
Abstract: Relative (or spherical) characters describe the restriction of a representation to a subgroup. They arise naturally in the study of periods of automorphic forms, e.g., in the setting of conjectures of Gan-Gross-Prasad and Ichino-Ikeda. I will discuss the problem of their asymptotic estimation, emphasizing some known results, open problems and applications.

Representations of p-adic groups

Jessica Fintzen
University of Michigan; Member, School of Mathematics
March 5, 2018

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.

Supercuspidal L-packets

Tasho Kaletha
University of Michigan
March 5, 2018
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.

Local eigenvalue statistics of random band matrices

Tatyana Shcherbina
Princeton University
February 28, 2018

Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems, since they interpolate between mean-field type Wigner matrices and random Schrodinger operators. In particular, RBM can be used to model the Anderson metal-insulator phase transition (crossover) even in 1d. In this talk we will discuss some recent progress in application of the supersymmetric method (SUSY) and transfer matrix approach to the analysis of local spectral characteristics of some specific types of RBM.

Local to global relations of periods (continued)

Erez Lapid
Weizmann Institute of Science; Member, School of Mathematics
February 27, 2018

Rankin-Selberg integrals provide factorization of certain period integrals into local counterparts. Other, more elusive, periods can be studied in principle by the relative trace formula and other methods.
 
Following Waldspurger, Ichino-Ikeda formulated a local to global conjecture about the Gross-Prasad periods. A more general setup was subsequently considered by Sakellaridis-Venkatesh.
 
I will discuss some of these principles as well as a result on Whittaker coefficients joint with Zhengyu Mao.