Appears in collection : 2025 - T1 - WS1 - Intertwining operators and geometry

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.