Real and symmetric quasi-maps

Abstract

Let $G_{\mathbb{R}}$ 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 $G_{\mathbb{R}}$ 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 ${\mathbb{P}}^1$ to $G_{\mathbb{R}}$ 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.

complete revision (including title) with some material moved to arXiv:2006.10279; 22 pages, 1 figure