Radon, cosine and sine transforms on Grassmannian manifolds

2008/10/29 by Genkai Zhang, Zhang, Genkai
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Geometry and complex manifolds #Mathematical Analysis and Transform Methods #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.0810.5257

openalex publication_date 2008/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G_n,r(\bbK) be the Grassmannian manifold of k-dimensional \bbK-subspaces in \bbKⁿ where \bbK=\mathbb R, \mathbb C, \mathbb H is the field of real, complex or quaternionic numbers. We consider the Radon, cosine and sine transforms, \mathcal R_{r^′, r}, \mathcal C_{r^′, r} and \mathcal S_{r^′, r}, from the L² space L²(G_n,r(\bbK)) to the space L²(G_n,r^′(\bbK)), for r, r^′ ≤ n-1. The L² spaces are decomposed into irreducible representations of G with multiplicity free. We compute the spectral symbols of the transforms under the decomposition. For that purpose we prove two Bernstein-Sato type formulas on general root systems of type BC for the sine and cosine type functions on the compact torus \mathbb R^r/2πQ^\vee generalizing our recent results for the hyperbolic sine and cosine functions on the non-compact space \mathbb R^r. We find then also a characterization of the images of the transforms. Our results generalize those of Alesker-Bernstein and Grinberg. We prove further that the Knapp-Stein intertwining operator for certain induced representations is given by the sine transform and we give the unitary structure of the Stein's complementary series in the compact picture.

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