On an analytic description of the α-cosine transform on real Grassmannians

The goal of this paper is to describe the α-cosine transform on functions on real Grassmannian Gri(ℝn) in analytic terms as explicitly as possible. We show that for all but finitely many complex α the α-cosine transform is a composition of the (α + 2)-cosine transform with an explicitly written (though complicated) O(n)-invariant differential operator. For all exceptional values of α except one, we interpret the α-cosine transform explicitly as either the Radon transform or composition of two Radon transforms. Explicit interpretation of the transform corresponding to the last remaining value α, which is-(min{i,n-i} + 1), is still an open problem.