Radon, cosine and sine transforms on Grassmannian manifolds

TL;DR

This paper analyzes Radon, cosine, and sine transforms on Grassmannian manifolds over real, complex, and quaternionic fields, computing their spectral symbols, characterizing their images, and connecting them to representation theory and intertwining operators.

Contribution

It introduces explicit spectral symbols for these transforms on Grassmannians and generalizes Bernstein-Sato formulas to root systems of type BC, linking to representation theory.

Findings

Spectral symbols of the transforms are explicitly computed.

Characterization of the transforms' images is provided.

The sine transform is identified as the Knapp-Stein intertwining operator.

Abstract

Let $G_{n,r}(\bbK)$ be the Grassmannian manifold of $k$-dimensional $\bbK$-subspaces in $\bbK^n$ 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, $\mathcal R_{r^\prime, r}$, $\mathcal C_{r^\prime, r}$ and $\mathcal S_{r^\prime, r}$, from the $L^2$ space $L^2(G_{n,r}(\bbK))$ to the space $L^2(G_{n,r^\prime}(\bbK))$, for $r, r^\prime \le n-1$. The $L^2$ 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 $\mathbb R^r/{2\pi Q^\vee}$ 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.