Radon transforms on higher rank Grassmannians (Q1071208)

We define a Radon transform R from functions on Gr(k,n), the Grassmannian of projective k-planes in \({\mathbb{C}}P^ n\) to functions on Gr(l,n). If \(f\in C^{\infty}(Gr(k,n))\) and \(L\in Gr(k,n)\), then Rf(L) is the integral of f(H) over all k-planes H which lie in L. If \(R^ t\) is the dual transform, we show under suitable assumptions on k and l that \(R^ t R\) is invertible by a polynomial in the Casimir operators of \(U(n+1)\), the group of isometries of \({\mathbb{C}}P^ n\). We also treat the real and quaternionic cases. Finally, we indicate some possible variations and generalizations to flag manifolds.