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 this question in the case when G is simply laced and N is abelian. In a sense, this is the most banal case. Our results can be described as follows. Let Mss be the “semi-simple” part of M . It is more natural to work with Mss. Starting with the highest root β one can, in a canonical fashion, define a maximal sequence of orthogonal roots β, β1, . . . , βr−1 in the Lie algebra n. Let eβ, . . . , eβr−1 be the corresponding Chevalley basis elements in n. Then every M ss(Z)-orbit in n contains an element