Degeneration of horospheres in spherical homogeneous spaces
Horospheres in a certain class (containing all simply connected symmetric spaces) of spherical homogeneous spaces \(G/H\) for semisimple algebraic groups \(G\) are studied. Here, the stationary subgroups \(H\) are supposed to be excellent (this is a notion introduced in the article). Generic horospheres are ``generic'' orbits of a maximal unipotent subgroup \(U \subset G\). But there are some other horospheres, which are not generally orbits of \(U\). They are called here degenerate horospheres and they are formed by \(U\)-orbits of lower dimensions. The main result of the article is a description of such degenerate horospheres. Two examples for \(G=\mathrm{SL}_n({\mathbf C}), H=\mathrm{SO}_n(\mathbf C)\) and for \(G=\mathrm{SL}_2({\mathbf C}) \times\mathrm{SL}_2({\mathbf C}) \times\mathrm{SL}_2({\mathbf C})\), \(H=\mathrm{SL}_2(\mathbf C)\) (diagonally embedded into \(G\); here the subgroup \(H\) is not excellent) are considered.
scientific article; zbMATH DE number 6951616