Conical Distributions on the Space of Flat Horocycles

Abstract

Let $G_0=K⋉\mathfrak{p}$ be the Cartan motion group associated with a noncompact semisimple Riemannian symmetric pair $(G,K)$. Let $\mathfrak{a}$ be a maximal abelian subspace of $\mathfrak{p}$ and let $\p=\a+\q$ be the corresponding orthogonal decomposition. A flat horocycle in $\p$ is a $G_0$-translate of $\q$. A conical distribution on the space ${\Xi }_0$ of flat horocycles is an eigendistribution of the algebra $\mathbb{D}({\Xi }_0)$ of $G_0$-invariant differential operators on ${\Xi }_0$ which is invariant under the left action of the isotropy subgroup of $G_0$ fixing $\q$. We prove that the space of conical distributions belonging to each generic eigenspace of $\mathbb{D}({\Xi }_0)$ is one-dimensional, and we classify the set of all conical distributions on ${\Xi }_0$ when $G/K$ has rank one.