We generalize Kostant’s
convexity theorem for the Iwasawa decomposition of a real semisimple Lie group G to
the following situation. Let τ be an involution of G, and H = (Gτ)0. Then there
exists an Iwasawa decomposition G = KApN with certain compatibility properties,
e.g. τ(K) = K, τ(Ap) = Ap. Let ap=Lie(Ap), H : G → ap the projection according
to the Iwasawa decomposition and Epq the projection of ap onto the −1 eigenspace
apq of dτ(e). Let X ∈ apq. Then the main result of this paper describes the
image of the map H → apq, h → Epq∘ H(exp(X) ⋅ h) as the vector sum of a
closed convex polyhedral cone and the convex hull of a Weyl group orbit
through X. For τ a Cartan involution it gives precisely Kostant’s description of
H(exp(X) ⋅ K).
