A refinement of the Hodge stratification for connected reductive groups

For connected reductive groups G over a finite extension F of Q p and L the maximal unramified extension of F we study the setsH μ ̲ , N( G )of elements b ∈ G ( L ) with given Hodge points( b σ ) , ( b σ ) 2 , ... , ( b σ ) N . We explain the relationship to stratifications of some moduli scheme of abelian varieties defined by Goren and Oort respectively Andreatta and Goren. We show that for sufficiently large N the Newton point is constant on the setsH μ ̲ , N( G )and compute such N for certain classes of groups.