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Let $\overline{GR}(n,k)$ be the manifold of all affine k-dimensional subspaces in $R^n$, and let $R:C^{\infty}_c(\overline{GR}(n,k))\to C^{\infty}_c(\overline{GR}(n,l))$, $0\le k<l\le n-1$, be the Radon transform defined in a standard way as follows: $$ (Rf)(E)=\int\limits_{\{L\in \overline{GR}(n,k)):L\subset E\}}f(L)dL. $$ Here the integration is with respect to the Haar measure on $\{L\in \overline{GR}(n,k)):L\subset E\}=\overline{GR}(k,l)$. QUESTION. Is the inversion formula for the Radon transform $R$ known?

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Yes, it is known. See Boris Rubin, Radon transforms on affine Grassmannians, 2004, in particular theorem 4.2. A free full text pdf is available at the linked page. The reconstruction formula is not very simple, so I will not reproduce it here.

Radon transforms on Grassmannians are also discussed in Helgason's book Radon Transform, section II.4.F.

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