The rational homotopical nilpotency of principal Un(C)-bundles

Let Aut(p) denote the space of all self-fibre-homotopy equivalences of a principal Un(C)-bundle ξ:Un(C)→E→pX of simply connected CW complexes with E finite. We show that there exists an inequality:n−N(p)⩽HnilQ(Aut(p)0)⩽n for any base space X, where N(p)=min{k|k=card(i|χ2i≠0,i