Pizzetti formula on the Grassmannian of 2-planes
The authors study the Higgs algebras \(H_3\) in the harmonic analysis of the Grassmann manifolds \(\mathrm{Gr}_0(m,2):=\mathrm{SO}(m)/(\mathrm{SO}(m-2)\times \mathrm{SO}(2))\). These algebras appear in the Howe duality where \(\mathrm{SO}(m)\) acts on \(\mathrm{Gr}_0(m,2)\). In particular, an orthogonal decomposition of the vector space of all poynomials under the joint action of \(\mathrm{SO}(m)\times H_3\) is given. This decomposition is then used to derive a Pizetti formula for integrals of the form \(\int_{\mathrm{Gr}_0(m,2)} f\> d\mu\) with the uniform distribution \(\mu\) on \(\mathrm{Gr}_0(m,2)\). The authors also present a connection to a Pizetti formula for Stiefel manifolds by K.~Coulembier and M.~Kieburg.
scientific article; zbMATH DE number 7259251
scientific article