Radon, Cosine and Sine Transforms on Grassmannian Manifolds

LetGn,r(K) be the Grassmannian manifold of k-dimensionalK-subspaces in K where K = R,C,H is the field of real, complex or quaternionic numbers. We consider the Radon, cosine and sine transforms, Rr′,r, Cr′,r and Sr′,r, from the L space L2(Gn,r(K)) to the space L 2(Gn,r′(K)), for r, r ′ ≤ n − 1. The L spaces are decomposed into irreducible representations of G with multiplicity free. We compute the spectral symbols of the transforms under the decomposition. For that purpose we prove two Bernstein-Sato type formulas on general root systems of type BC for the sine and cosine type functions on the compact torus R/2πQ generalizing our recent results for the hyperbolic sine and cosine functions on the non-compact space R. We find then also a characterization of the images of the transforms. Our results generalize those of Alesker-Bernstein and Grinberg. We prove further that the Knapp-Stein intertwining operator for certain induced representations is given by the sine transform and we give the unitary structure of the Stein’s complementary series in the compact picture.