Vladimir Molchanov

Canonical representations for hyperboloids: an interaction with an overalgebra

Let $\cal X$ be a homogeneous space $G/H$ where $G={\rm SO}_0 (p,q)$, $H={\rm SO}_0 (p,q-1)$, a hyperboloid in ${\Bbb R}^n$, $n=p+q$. Canonical representations of $G$ on the hyperboloid $\cal X$ are defined as restrictions to $G$ of maximal degenerate series representations of an overgroup $\widetilde G={\rm SL}(n,\Bbb R)$. We write explicitly an interaction of Poisson and Fourier transforms for canonical representations with Lie operators of $\widetilde G$. For elements of the Lie algebra
$\widetilde {\mathfrak g}$ of $\widetilde G$ not belonging to the Lie algebra ${\mathfrak g}$ of $G$, this interaction contains differential operators of the fourth, second and zero orders.