STRUCTURE OF INTERNAL MODULES AND A FORMULA FOR THE SPHERICAL VECTOR OF MINIMAL REPRESENTATIONS

Let G be a simply connected Chevalley group, and P =MN a maximal parabolic subgroup of G. Let n be the Lie algebra of N. A choice of Chevalley basis defines a Z-structure on n, The structure ofM orbits over Z on irreducible subquotients of n could be highly non-trivial, and very interesting as Bhargava [B] shows. In the first part of this paper we deal with thi