The Strichartz conjecture for the Poisson transform on homogeneous line bundles over Noncompact Complex Grassmann manifolds

TL;DR

This paper proves a conjecture by Strichartz, characterizing the image of the Poisson transform on noncompact complex Grassmann manifolds for line bundles, establishing an isomorphism with a specific space of joint eigensections.

Contribution

It extends Strichartz's conjecture to nontrivial line bundles over complex Grassmann manifolds, providing an explicit image characterization of the Poisson transform.

Findings

The Poisson transform is an isomorphism for regular parameters.

Characterization of joint eigensections satisfying a growth condition.

Generalization of Strichartz's conjecture to nontrivial line bundles.

Abstract

Let \(X=G/K\) be a noncompact complex Grassmann manifold of rank \(r\). Let \(\tau_l\) be a character of \(K\), \(G\times_P{\C}\) and \(G\times_K{\C}\) the homogeneous line bundles associated with the representations \(\sigma_{\lambda,l}=\tau_l\otimes a^{\rho-i\lambda}\otimes 1\) of \(P=MAN\) and \(\tau_l\) of \(K\). We give an image characterization for the Poisson transform \(P_{\lambda,l}\) of \\\(L^2\)-sections of the unitary principal series representations of \(G\) parametrized by \(\sigma_{\lambda,l}\). More precisely for real and regular parameter \(\lambda\) in \(\mathfrak{a}^\ast\) we prove that \(P_{\lambda,l}\) is an isomorphism from \(L^2(K\times_M{\C})\) onto the space of joint eigensections \(F\) of the algebra of \(G\)-invariant differential operators on \(G\times_K{\C}\) that satisfy the following growth condition \begin{eqnarray*} \sup_{R>1}\frac{1}{R^r}\int_{B(R)}\mid F(g)\mid^2\, {\rm d}g<\infty. \end{eqnarray*} This generalizes a conjecture by Strichartz which corresponds to \(\tau_l\) trivial.