Real and symmetric quasi-maps

Let G_R be a real reductive group and let X be the corresponding complex symmetric variety under the Cartan bijection. We construct a stratified homeomorphism between the based polynomial arc group of G_R and the based polynomial arc space of X. We also prove a multi-point version where we replace arcs by moduli spaces of quasi-maps from the projective line P¹ to G_R and X. The key ingredients in the proof include: (i) a multi-point generalization of the “Gram-Schmidt” factorization of loop groups, and (ii) a nodal degeneration of moduli spaces of quasi-maps. As an application, we show that for the closures of real spherical orbits in the real affine Grassmannian, their singularities near the base point are locally homeomorphic to complex algebraic varieties.