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Journal of Lie Theory 20 (2010), No. 3, 409--436
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.


\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 one-dimensional, 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

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