Degeneration of Horospheres in Spherical Homogeneous Spaces

Horospheres for an action of a semisimple algebraic group G on an affine variety X are the generic orbits of a maximal unipotent subgroup U ? G or, equivalently, the generic fibers of the categorical quotient of the variety X by the action of U, which is defined by the values of the highest weight functions. The remaining fibers of this quotient (which we call degenerate horospheres) for a certain class of spherical G-varieties containing all simply connected symmetric spaces are studied.