On spaces of morphisms of curves in algebraic homogeneous spaces

We prove here the following result. Let X be an affine curve and G/H an affine algebraic homogeneous space over C. Assume that either X is affine or that G and H are semisimple modulo their unipotent radicals. Let C(X,G/H) denote the space of continuous maps of X in G/H (both spaces given their natural Hausdorff topologies) with the compact open topology. Let M(X,G/H) be the C points of the ind-variety of morphisms of X in G/H with the inductive limit Hausdorff topology. Then the inclusion M(X,G/H)→C(X,G/H) is a homotopy equivalence.