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Pizzetti formula on the Grassmannian of 2-planes

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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.

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Acknowledgements

This research was supported by the Fund for Scientific Research-Flanders (FWO-Vlaanderen), Project ‘Construction of algebra realisations using Dirac operators’, Grant G.0116.13N. The second author was partially supported by JSPS KAKENHI Grant Number JP19K03480.

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Authors and Affiliations

  1. University of Antwerp, Antwerpen, Belgium

    D. Eelbode

  2. Waseda University, Tokyo, Japan

    Y. Homma

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  1. D. Eelbode
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  2. Y. Homma
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Correspondence to D. Eelbode.

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Eelbode, D., Homma, Y. Pizzetti formula on the Grassmannian of 2-planes. Ann Glob Anal Geom 58, 325–350 (2020). https://doi.org/10.1007/s10455-020-09731-8

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  • DOI: https://doi.org/10.1007/s10455-020-09731-8

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