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 $U\subset G$ 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.