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We study the Radon transform Rf of functions on Stiefel and Grassmann manifolds. We establish a connection between Rf and Gårding-Gindikin fractional integrals associated to the cone of positive definite matrices. By using this connection, we obtain Abel-type representations and explicit inversion formulae for Rf and the corresponding dual Radon transform. We work with the space of continuous functions and also with L p spaces.

We define a Radon transform R from functions Gr(&, A?), the Grassmannian of projective A:-planes in CP" to functions on Gr(/, n). If / e C°°(Gr(A:, n)) and L e Gr(/,n), then Rf(L) is the integral of /(//) over all Λ>planes Ή which lie in L. If R' is the dual transform, we show under suitable assumptions on k and / that R'R is invertible by a polynomial in the Casimir operators of U(n 4-1), the group of isometries CP". We also treat the real and quaternionic cases. Finally, we indicate some possible variations and generalizations to flag manifolds.

Abstract. The Radon transform R on CP" associates to a point function/(jc) the hyperplane function Rf(H) by integration over the hyperplane H. Il R' is the dual transform, we can invert R'R by a polynomial in the Laplace-Beltrami operator, and verify the formula of Helgason [7] with very simple computations.We view the Radon transform as a G-invariant map between representations of the group of isometries G = U(n + 1) on function spaces attached to CP". Pulling back to a sphere via a suitable Hopf fibration and using the theory of spherical harmonics, we can decompose these representations into irreducibles. The scalar by which R acts on each irreducible is given by a simple integral. Thus we obtain an explicit formula for R. The action of R'R is immediately related to the spectrum of CP". This shows that R'R can be inverted by a polynomial in the Laplace-Beltrami operator. Similar procedures give corresponding results for the other compact 2-point homogeneous spaces: RT"1, HP", OP", as well as spheres.0. Introduction. In 1917 Radon showed that a function in Euclidean space could be recovered from its integrals over hyperplanes and thereby defined the now celebrated Radon transform. It associates to a point function fix) the hyperplane function RfiH) = JHf. This type of transform arises naturally in many geometric situations-whenever a nice family of submanifolds (such as hyperplanes) is present. Thus Helgason considered the Radon transform in the setting of homogeneous spaces [8], and Gelfand in a quite general topological setting [3]. In particular, Helgason studied the cases of compact two-point homogeneous spaces using the theory of antipodal manifolds. Since (except for spheres) these spaces are projective spaces, they can also be studied using projective geometry and algebra.In this paper we consider the Radon tranform on compact two-point homogeneous spaces from the point of view of group representations. The Peter-Weyl theory for compact Lie groups makes the analysis very simple. Thus we can derive important properties of the Radon transform with only a few computations, and these computations bring out the nature of the geometry involved and the behaviour of the Radon transform under it. In particular, we can recover the inversion formulae of Helgason involving the Laplace-Beltrami operator [7]. This was already done by Guillemin for the sphere S2 [5], Therefore, we skip the cases of spheres and also real projective spaces (which are essentially the same). In fact, the analysis for