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Radon transform on real, complex, and quaternionic Grassmannians. (English) Zbl 1125.26013

Author’s abstract: Let \(G_{n,k}(K)\) be the Grassmannian manifold of \(k\)-dimensional \(K\)-subspaces in \(K^n\), where \(K=R,C,H\) is the field of real, complex, or quaternionic numbers. For \(1\leq k <k'\leq n -1\), we define the Radon transform \((Rf)(\eta), \eta G_{n,k}'(K)\), for functions \(f(\xi)\) on \(G_{n,k}(K)\) as an integration over all \(\xi\eta\). When \(k+k'\leq n\), we give an inversion formula in terms of the Gørding-Gindikin fractional integration and the Cayley-type differential operator on the symmetric cone of positive \((k\times k)\)-matrices over \(K\). This generalizes the recent results of E. L. Grinberg and B. Rubin [Ann. of Math. (2) 159, No. 2, 783–817 (2004; Zbl 1071.44003)] for real Grassmannians.

MSC:

26A33 Fractional derivatives and integrals
57S15 Compact Lie groups of differentiable transformations
43A85 Harmonic analysis on homogeneous spaces

Citations:

Zbl 1071.44003
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References:

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