A Paley-Wiener theorem for the Theta-spherical transform: the even multiplicity case

The $\Theta$-spherical functions generalize the spherical functions on Riemannian symmetric spaces and the spherical functions on non-compactly causal symmetric spaces. In this article we consider the case of even multiplicity functions. We construct a differential shift operator $D_m$ with smooth coefficients which generates the $\Theta$-spherical functions from finite sums of exponential functions. We then use this fact to prove a Paley-Wiener theorem for the $\Theta$-spherical transfrom.