Radon, cosine and sine transforms on Grassmannian manifolds.
Journal article, 2009
Let G n,r (double-struck capital K) = G/K be the Grassmannian manifold of k-dimensional (double-struck capital K)-subspaces in (double-struck capital K) n, where (double-struck capital K) = ℝ,ℂ,ℍ is the field of real, complex, or quaternionic numbers. The Radon, cosine, and sine transforms, ℛ r′ ,r , (script capital C) r′ ,r, and (script capital S) r′ ,r , from the L 2 space L 2(G n,r (double-struck capital K)) to the space L 2(G n,r′ (double-struck capital K)), for r, r′ ≤ n - 1 are defined as integral operators in terms of inclusion relations of and angles between the subspaces. We compute the spectral symbols of the transforms and characterize their images under the decomposition of L 2 spaces into irreducible subspaces of G. For that purpose we prove two Bernstein-Sato-type formulas on general root systems of type BC for the sine- and cosine-type functions. It is observed further that the Knapp-Stein intertwining operator for certain induced representations is given by the sine transform and as a consequence we give the unitary structure of the Stein's complementary series in the compact picture.