Journal of Lie Theory 27 (2017), No. 3, 623--636
Copyright Heldermann Verlag 2017
On the Kernel of the Maximal Flat Radon Transform on Symmetric Spaces of Compact Type
Eric L. Grinberg
Dept. of Mathematics, University of Massachusetts, 100 Morrissey Boulevard, Boston, MA 02125, U.S.A.
eric.grinberg@umb.edu
Steven Glenn Jackson
Dept. of Mathematics, University of Massachusetts, 100 Morrissey Boulevard, Boston, MA 02125, U.S.A.
jackson@math.umb.edu
[Abstract-pdf]
Let $M$ be a Riemannian globally symmetric space of compact type, $M'$ its set of maximal flat totally geodesic tori, and Ad$(M)$ its adjoint space. We show that the kernel of the maximal flat Radon transform $\tau\colon L^2(M) \rightarrow L^2(M')$ is precisely the orthogonal complement of the image of the pullback map $L^2({\rm Ad}(M))\rightarrow L^2(M)$. In particular, we show that the maximal flat Radon transform is injective if and only if $M$ coincides with its adjoint space.
Keywords: Integral geometry, Radon transform, symmetric space.
MSC: 44A12; 22E30, 22E46, 43A85, 53C35, 53C65
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