The Strichartz conjecture for the Poisson transform on homogeneous line bundles over Noncompact Complex Grassmann manifolds
Let \(X=G/K\) be a noncompact complex Grassmann manifold of rank \(r\). Let \(τ_l\) be a character of \(K\), \(G\times_P{\C}\) and \(G\times_K{\C}\) the homogeneous line bundles associated with the representations \(σ_{λ,l}=τ_l\otimes a^{ρ-iλ}\otimes 1\) of \(P=MAN\) and \(τ_l\) of \(K\). We give an image characterization for the Poisson transform \(P_{λ,l}\) of \\\(L^2\)-sections of the unitary principal series representations of \(G\) parametrized by \(σ_{λ,l}\). More precisely for real and regular parameter \(λ\) in \(\mathfrak{a}^\ast\) we prove that \(P_{λ,l}\) 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 \(τ_l\) trivial.