The Capelli identity and Radon transform for Grassmannians

TL;DR

This paper investigates invariant differential operators on Grassmannians, deriving their images under the Harish-Chandra homomorphism and providing a Radon inversion formula that generalizes recent results.

Contribution

It introduces a family of Capelli-type operators on Grassmannians, analyzes their properties, and extends Radon inversion formulas to a broader setting.

Findings

Determined the image of Capelli operators under Harish-Chandra homomorphism.

Extended Radon inversion formulas to non-compact duals of Grassmannians.

Established new invariant differential operators on Grassmannians.

Abstract

We study a family $C_{s,l}$ of Capelli-type invariant differential operators on the space of rectangular matrices over a real division algebra. The $C_{s,l}$ descend to invariant differential operators on the corresponding Grassmannian, which is a compact symmetric space, and we determine the image of the $C_{s,l}$ under the Harish-Chandra homomorphism. We also obtain analogous results for corresponding operators on the non-compact duals of the Grassmannians, and for line bundles. As an application we obtain a Radon inversion formula, which generalizes a recent result of B. Rubin for real Grassmannians.