Self homotopy equivalences of classifying spaces of compact connected Lie groups

We describe, for any compact connected Lie group G and any prime p, the monoid of self maps $BG_{^p}$ → $BG_{^p}$ which are rational equivalences. Here, $BG_{^p}$ denotes the p-adic completion of the classifying space of G. Among other things, we show that two such maps are homotopic if and only if they induce the same homomorphism in rational cohomology, if and only if their restrictions to the classifying space of the maximal torus of G are homotopic.