Radon, cosine and sine transforms on Grassmannian manifolds

Let $G_{n,r}(bbK)$ be the Grassmannian manifold of $k$-dimensional $bbK$-subspaces in $bbK^n$ where $bbK=mathbb R, mathbb C, mathbb H$ is the field of real, complex or quaternionic numbers. We consider the Radon, cosine and sine transforms, $mathcal R_{r^prime, r}$, $mathcal C_{r^prime, r}$ and $mathcal S_{r^prime, r}$, from the $L^2$ space $L^2(G_{n,r}(bbK))$ to the space $L^2(G_{n,r^prime}(bbK))$, for $r, r^prime le n-1$. The $L^2$ 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 $mathbb R^r/{2pi Q^vee}$ generalizing our recent results for the hyperbolic sine and cosine functions on the non-compact space $mathbb R^r$. 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 Steins complementary series in the compact picture.