The rational homotopical nilpotency of principal -bundles

Let Aut ( p ) denote the space of all self-fibre-homotopy equivalences of a principal U n ( C ) -bundle ξ : U n ( C ) → E → p X of simply connected CW complexes with E finite. We show that there exists an inequality: n − N ( p ) ⩽ Hnil Q ( Aut ( p ) 0 ) ⩽ n for any base space X, where N ( p ) = min { k | k = card ( i | χ 2 i ≠ 0 , i < n ) } with χ 2 i denoting the generalized characteristic class of the bundle p, and Hnil Q ( Aut ( p ) 0 ) = n − 1 when X = C P m with m ⩾ n − 2 and p is not rational trivial with mild restriction. We also show that Hnil Q ( Aut ( p ) 0 ) = n if p is a rational fibre-homotopy trivial bundle and X is finite.