The spherical Paley-Wiener theorem on the complex Grassmann manifolds SU(𝑝+𝑞)/S(U_{𝑝}×U_{𝑞})

Abstract

We prove the Paley-Wiener theorem for the spherical transform on the complex Grassmann manifolds U / K = SU ( p + q ) / S ( U p × U q ) U/K=\mbox {SU}(p+q)/\mbox {S}(\mbox {U}_p\times \mbox {U}_q) . This theorem characterizes the K K -biinvariant smooth functions f f on the group U U that are supported in the K K -invariant ball of radius R R , with R R less than the injectivity radius of U / K U/K , in terms of holomorphic extendability, exponential growth, and Weyl invariance properties of the spherical Fourier transforms f ^ \hat {f} , originally defined on the discrete set Λ s p h \Lambda _{sph} of highest restricted spherical weights.