A biunimodular function on a cyclic group of prime order is a function with constant modulus whose Fourier transform also has constant modulus. For instance gaussian functions are biunimodular. According to a theorem of Haagerup there are only finitely many biunimodular functions up to scalar. In this talk we will construct new biunimodular functions for all prime $p>5$. The proof relies on...
Tempered representations of a group are the unitary representations that are weakly contained in the regular representation. The standard way to say that a representation is close to being tempered is probably to use Fell's topology which involves convergence of matrix coefficients. I will discuss another (in general much stronger) notion of closeness, that originates from random matrix theory...
Given a group $G$ and a real number $p$, it is natural to study representations of G by linear isometries on $L^p$-spaces. Of course, the case where $p$ is equal to 2 corresponds to the familiar and much studied case of unitary representations of $G$. For a semisimple Lie group $G$, we will give a complete classification of all its irreducible $L^p$-representations, for $p\neq2$.
Crossed product algebras are fundamental objects that describe actions of a Lie group G on a Fréchet algebra A. In this talk we will consider the convolution algebra of compactly supported smooth functions on G with values in A. Using geometrical arguments, we will canonically identify the periodic cyclic homology of this crossed product (up to a dimension shift) with the homology of the...
Let $G$ be a connected real semisimple Lie group, and $K
Dirac cohomology of a discrete series representation is similar to the highest weight vectors of a highest weight module. We discuss how to formulate the character lifting of discrete series by employing Dirac cohomology. This lifting is closely related to the Langlands functoriality and the Howe dual pair correspondence.
In 1990, Kashiwara and Lusztig independently discovered the theory of crystal bases for complex semisimple Lie algebras. This theory says that if we deform the universal enveloping algebra by the Drinfled-Jimbo quantisation procedure, and let the quantisation parameter go to infinity, then the structure theory of the finite-dimensional representations becomes extremely simple. This allows an...
I will outline a puzzle relating to group actions in noncommutative geometry and its resolution by way of conformal geometry. I will discuss the discrepancy between Kasparov's bounded picture of KK-theory and the unbounded picture of KK-theory, including the spectral triples so important to noncommutative geometry, particularly with respect to group equivariance. The discrepancy occurs already...
Let $G$ be a connected linear real reductive group with a maximal compact subgroup $K$. In this talk, we will explain an approach to study the large-time behavior of the heat kernel on the corresponding homogeneous space $G/K$ using Bismut’s formula. We will discuss how Bismut’s formula provides a natural link between the index theory and Vogan's minimal $K$-type theory. In particular, we...
I will talk about how Delorme's techniques used in his proof of the Paley-Wiener theorem can be applied to the Casselman-Wallach Schwartz algebra of rapidly decreasing smooth functions on a real reductive group, and how they can be used to give a sort of structure theorem strong enough to prove a refined version of the Connes-Kasparov isomorphism. If there's time, I will discuss how the...
The goal of this talk is a simple, concrete description of the topological K-theory of the reduced C*-algebra of a reductive p-adic group $G$, in terms coming from representation theory.
We will discuss the structure of several group algebras of $G$, and how the K-theory can be computed from the space of irreducible tempered $G$-representations. The final description is closely related to...
The category of smooth representations of a $p$-adic group $G$ admits a decomposition into Bernstein blocks. With Paul Baum, Roger Plymen and Maarten Solleveld, we have formulated a conjecture which relates these blocks to (possibly twisted) extended quotients of complex algebraic varieties by finite groups.
In this talk, I will first introduce $p$-adic groups and some of their...
Recently, Nigel Higson and Alexandre Afgoustidis made precise an analogy proposed by George Mackey between some unitary representations of a semisimple Lie group and unitary representations of its associated semidirect product group. In this talk, I will show a construction of an embedding of the reduced group C-algebra of the Cartan motion group into the reduced group C-algebra of the...
