This talk is a review of aspects, old and new, of the connection between harmonic analysis on real reductive groups and the structure of certain $C^*$-algebras attached to them.
Topics will include the topology of the tempered dual, the Baum--Connes--Kasparov conjecture in K-theory, and the Mackey bijection between the tempered dual of a real reductive group and the unitary dual of its Cartan...
The first lecture will present a general formalism for theorems in harmonic analysis, which applies not only to spherical varieties but also to other spaces of polynomial growth (such as finite-volume quotients of semisimple Lie groups). We will discuss the variation of discrete series, asymptotic cones (boundary degenerations), and how to build the $L^2$ and Harish-Chandra Schwartz spaces out...
When a Lie group $G$ acts transitively on a manifold $X$, an irreducible subrepresentation of the unitary representation $L^2(X)$ is called a discrete series representation of $X$. The discrete series plays an important role in the study of harmonic analysis for symmetric spaces. In this talk, we would like to give sufficient conditions for the existence of discrete series for general...
I will discuss recent works with Bram Mesland and Magnus Goffeng in which we have shown that in many cases, theta correspondence, both local and global, lends itself well to a C*-algebraic formulation.
The first lecture will present a general formalism for theorems in harmonic analysis, which applies not only to spherical varieties but also to other spaces of polynomial growth (such as finite-volume quotients of semisimple Lie groups). We will discuss the variation of discrete series, asymptotic cones (boundary degenerations), and how to build the $L^2$ and Harish-Chandra Schwartz spaces out...
In these two lectures I will give an overview of the relative Langlands duality conjectures, as they are formulated in my work with Yiannis Sakellaridis and Akshay Venkatesh. The rough idea is that multiplicity-free harmonic analysis problems (spherical varieties and their variants) come in pairs associated to Langlands dual groups, so that representation theoretic questions on one side are...
In the context of the relative Langlands program, the notion of relative cuspidality was introduced by Kato and Takano based on the support of relative matrix coefficients. In this talk, I will explain how theta correspondence can be used to demonstrate relative cuspidality of non-supercuspidal distinguished representations, based on a relative character identity.
In these two lectures I will give an overview of the relative Langlands duality conjectures, as they are formulated in my work with Yiannis Sakellaridis and Akshay Venkatesh. The rough idea is that multiplicity-free harmonic analysis problems (spherical varieties and their variants) come in pairs associated to Langlands dual groups, so that representation theoretic questions on one side are...
