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In this paper we study the nilpotency of certain groups of self homotopy equivalences. Our main goal is to extend, to localized homotopy groups and/or homotopy groups with coefficients, the general principle of Dror and Zabrodsky by which a group of self homotopy equivalences of a finite complex which acts nilpotently on the homotopy groups is itself nilpotent.
Let ξ = (E, p, B, F) be a Hurewicz fibration. In this paper we study the space LG(ξ) consisting of fibre homotopy self equivalences of ξ inducing by restriction to the fibre a self homotopy equivalence of F belonging to the group G. We give in particular conditions implying that π1(LG(ξ)) is finitely generated or that L1(ξ) has the same rational homotopy type as aut1(F).
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