A Convexity property for the SO(2,C)-double coset decomposition of SL(2,C) and applications to spherical functions

We make a fine study of the SO(2,C)-double coset decomposition in SL(2,C) and give a full description of the intersection of the various cells with the complex crown Xi of SL(2,R)/SO(2). A non-linear convexity theorem is proved and applications to analytically continued spherical functions are given.