
Journal of Lie Theory 20 (2010), No. 3, 409436 Copyright Heldermann Verlag 2010 Conical Distributions on the Space of Flat Horocycles Fulton B. Gonzalez Dept. of Mathematics, Tufts University, Medford, MA 02155, U.S.A. fulton.gonzalez@tufts.edu [Abstractpdf] \def\sdir#1{\hbox{$\mathrel\times{\hskip 4.3pt %% {\vrule height 3.8 pt depth .01 pt}}\hskip 2pt_{#1}$}} \def\a{{\frak a}} \def\p{{\frak p}} \def\q{{\frak q}} Let $G_0=\p\sdir{}K$ be the Cartan motion group associated with a noncompact semisimple Riemannian symmetric pair $(G,\,K)$. Let $\a$ be a maximal abelian subspace of $\p$ 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 ${\Bbb D}(\Xi_0)$ 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 ${\Bbb D} (\Xi_0)$ is onedimensional, and we classify the set of all conical distributions on $\Xi_0$ when $G/K$ has rank one. We also consider the question of the irreducibility of the natural representation of $G_0$ on the eigenspaces of ${\Bbb D}(\Xi_0)$. Keywords: Conical distributions, Cartan motion group, horocycle Radon transform. MSC: 43A85; 22E46, 44A12 [ Fulltextpdf (290 KB)] for subscribers only. 