Canonical representations on the two-sheeted hyperboloid
One of the sources for getting canonical representations on symmetric spaces \({\mathcal L}=G/H\) consists of the following. We take some group \(\widetilde G\) (overgroup) containing \(G\), take series representations \(\widetilde R\) of \(\widetilde G\) induced by the character of some parabolic subgroup \(\widetilde P\) associated with \(G/H\), and restrict these representations to \(G\). In the paper under review the author carries out this program for a crucial example: the two-sheeted hyperboloid \(\mathcal L\) in \({\mathbb R}^3\). The hyperboloid \(\mathcal L\) can be identified with the Riemann sphere \(\overline{\mathbb C}\) without the equator \(S: | z| =1\). Canonical representations of the group \(G=\text{SU}(1,1)\) are defined as the restriction of spherical representations of the overgroup \(\widetilde G=\text{SL}(2,{\mathbb C})\). The author decomposes these canonical representations into irreducible constituents and decomposes the Berezin form considered on the corresponding spaces.
scientific article; zbMATH DE number 5078037