Intertwining operators and geometry

Explicit Hilbert spaces for the unitary dual of rank one orthogonal groups

21 janv. 2025, 17:15

40m

Amphithéâtre Hermite (Institut Henri Poincaré)

Orateur

Christian Arends (Aahrus University)

Description

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 obtained from the non-compact picture on a maximal unipotent subgroup $N\cong {\mathbb{R}}^n$ by applying the Euclidean Fourier transform. As an application, we describe the space of Whittaker vectors on all irreducible Casselman--Wallach representations. Moreover, the new realizations of the irreducible unitary representations immediately reveal their decomposition into irreducible representations of a parabolic subgroup, thus providing a simple proof of a recent result of Liu--Oshima--Yu. This is joint work with Frederik Bang-Jensen and Jan Frahm.