Pizzetti formula on the Grassmannian of 2-planes

Abstract

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 $${{\text {Gr}}}_o(m,2)$$ of 2-planes. This algebra is identified as the dual partner (in the sense of Howe duality) of the orthogonal group $${\text {SO}}(m)$$ 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.