I plan to discuss an application of branching laws in spectral analysis on standard locally symmetric spaces, extending beyond the classical Riemannian setting. Recent advancements have overcome challenges in global analysis with indefinite metrics, thanks to developments in the branching theory of infinite-dimensional representations of reductive groups, which are based on geometries with...
The symmetry breaking of infinite-dimensional representations reveals several remarkable families of equivariant differential operators. We will demonstrate how their global properties can be captured using the concept of a generating operator and explore some applications to branching problems.
A Riemann-Roch type formula serves as the the cornerstone in establishing the Atiyah-Singer index theory via the K-theory method. The classical deformation-to-the-normal-cone approach offers a perspective from noncommutative geometry on formulating the analytic index. In this work, we propose a topological method that combines a Riemann-Roch theorem with deformation-to-the-normal-cone...
We describe the construction of a Fredholm module adapted to the proof of the Baum-Connes (or Connes-Kasparov) conjecture with coefficients for real rank one simple Lie groups (e.g. $Sp(n,1)$). The main ingredients are a BGG complex on the flag manifold associated to the Borel subgroup, and a suitable Poisson transform from the above complex to the space of L2-harmonic forms on the associated...
For unitary representations of a reductive Lie group, in addition to finding all equivalence classes, there is the problem of describing concrete models of individual representations. This is important for several applications, note the example of the metaplectic representation for the double cover of the symplectic group. For the covering group of three by three real matrices of determinant...
Let $G$ be a real reductive group. Let $\pi_1$ and $\pi_2$ be unitary irreducible representations of $G$. The decomposition of the tensor product $\pi_1\otimes\pi_2$ has been a long-standing problem in harmonic analysis. In this talk, we will discuss this problem for the case where $G=Spin(n, 1)$. It turns out that the decomposition of $\pi_1\otimes\pi_2$ in this case is closely related to the...
In this talk, I will present a way to construct differential symmetry breaking operators between principal series representations induced from the minimal parabolic for the pair (GL(n+1,R),GL(n,R)). The construction, based on the so-called the source operator method, leads to DSBO for some "generic" parameters. I will also discuss the non-generic case based on the n=2 example.
In this talk, we construct and classify all differential symmetry breaking operators between certain principal series representations of the groups $SO_0(4,1)$ and $SO_0(3,1)$. Moreover, we prove that for these representations, we obtain a localness theorem, namely, we have that any symmetry breaking operator is given by a differential (local) symmetry breaking operator.
The Paley-Wiener space for compactly supported smooth functions $C^\infty_c(G)$ on a semisimple Lie group $G$ is characterised by certain intertwining conditions, known as \textit{Delorme's intertwining conditions}, which are challenging to work with. Using the concept of Collingwood's boxes, we demonstrate how these relationships can be simplified and visualised in specific cases such as $G =...
We realize all irreducible unitary representations of the group $\mathrm{SO}_0(n+1,1)$ on explicit Hilbert spaces of vector-valued $L^2$-functions on $\mathbb{R}^n \setminus{\{0\}}$. The key ingredient in our construction is an explicit expression for the standard Knapp--Stein intertwining operators between arbitrary principal series representations in the so-called $F$-picture which is...
One major question in the representation theory of locally compact groups is how an irreducible representation of a group $G$ decomposes if restricted to a subgroup $H$. For $\pi$ and $\tau$ irreducible representations of $G$ and $H$, respectively, elements of $Hom_H(\pi\vert_{H}, \tau)$ are referred to as symmetry breaking operators, a term coined by Kobayashi. In a recent joint paper with...
Given a real irreducible dual pair there is an integral kernel operator which maps the distribution character of an irreducible admissible representation of the group with the smaller or equal rank to an invariant eigendistribution on the group with the larger or equal rank. If the pair is in the stable range and if the representation is unitary, then the resulting distribution is the...
In this talk I will discuss how the well-known explicit construction of the local theta correspondence by J.S. Li has a simple interpretation in terms of induced representations group $C^{*}$-algebras in the sense of M.A.Rieffel. This picture allows us deduce that in the standard cases where Li’s method works, local theta correspondence arises from a continuous functor. In special cases, the...
I will discuss the restriction of representations in the discrete spectrum of of the symmetric space GL(n,R)/GL(p,R)GL(n-p,R) to GL(n-1,R).
In this talk, I will discuss three Cartan subalgebras (or root systems) related to the branching problem of reductive Lie groups. One Cartan subalgebra describes complexity of an embedding of $G$-varieties. This is related to intertwining operators (symmetry breaking operators). The others are defined by the annihilators of $\mathfrak{g}$-modules or their non-zero vectors. They are related to...
In my talk, I will explain my PhD research project, which is about poles and zeros of the Harish-Chandra $\mu$-function. This function appears in the representation theory of $p$-adic groups, and is defined using intertwining operators between parabolically induced representations. It can be used to describe Bernstein blocks in the category of smooth representations of a reductive $p$-adic...
Equivariant quantum channels are completely positive trace-preserving maps intertwining representations of a group G. Lieb and Solovej (2014) studied traces of the functional calculus of equivariant quantum channels for SU(2) to establish a Wehrl-type inequality for integrals of convex functions of matrix coefficients. In particular, they showed that coherent highest weight states minimize the...
The $ (k, a) $-generalized Fourier transform $ \mathscr{F}_{k, a} $ introduced by Ben Saïd--Kobayashi--Ørsted is a deformation family of the classical Fourier transform with a Dunkl parameter $ k $ and a parameter $ a > 0 $ that interpolates minimal representations of two different simple Lie groups. In this session, we will talk about some new results when $ a $ is not positive. As a main...
In recent years, an analytical framework based on the “$(k,a)$ -generalized Fourier analysis” introduced by Ben Saîd--Kobayashi--Ørsted has been actively studied. This is a novel branch of harmonic analysis that deforms the traditional Fourier analysis theory using two parameters, $k$ and $a$, arising from Dunkl theory and the interpolation theory of minimal representations of Lie groups. In...
With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the notion of visible action for holomorphic actions of Lie groups on complex manifolds. His propagation theorem of the multiplicity-freeness property produces various kinds of multiplicity-free theorems for unitary representations realized in the space of holomorphic sections of an...
Let ${\mathrm{E}}_n({\mathbb{C}})$ denote the connected complex Lie group of type ${\mathrm{E}}_n$ for $n = 6, 7$. These two groups contain the following reductive pairs:
\begin{align}
T_1({\mathbb{C}}) \times {\mathrm{Spin}}(10,{\mathbb{C}}) & \subset {\mathrm{E}}_6({\mathbb{C}}), \cr
T_2({\mathbb{C}}) \times {\mathrm{Spin}}(8,{\mathbb{C}}) & \subset {\mathrm{E}}_6({\mathbb{C}}),...
Intertwining operators between parabolically induced representations play a fundamental role in the study of the tempered dual of reductive groups. Therefore it is not surprising to see related objects, such as R-groups, appear in the description of the reduced C*-algebra associated to these groups. The purpose of this talk will be to explain how various techniques of operator algebraic nature...
