Pizzetti formula on the Grassmannian of 2-planes

This paper is devoted to the role played by the Higgs algebra H 3 in the generalisation of classical harmonic analysis from the sphere S m - 1 to the (oriented) Grassmann manifold Gr o ( m , 2 ) of 2-planes. This algebra is identified as the dual partner (in the sense of Howe duality) of the orthogonal group SO ( m ) acting on functions on the Grassmannian. This is then used to obtain a Pizzetti formula for integration over this manifold. The resulting formulas are finally compared to formulas obtained earlier for the Pizzetti integration over Stiefel manifolds, using an argument involving symmetry reduction.