The $λ$-Cosine Transforms, Differential Operators, and Funk Transforms on Stiefel and Grassmann Manifolds

Abstract: We introduce a new family of invariant differential operators associated with $\lambda$-cosine and Funk-Radon transforms on Stiefel and Grassmann manifolds. These operators reduce the order of the $\lambda$-cosine transforms and yield new inversion formulas. Intermediate Funk-cosine transforms corresponding to integration over matrices of lower rank are studied. The main tools are polar decomposition and Fourier analysis on matrix space.