Pizzetti formula on the Grassmannian of 2-planes
2020
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.
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