Real and symmetric quasi-maps

TL;DR

This paper constructs a stratified homeomorphism linking polynomial arc groups of real reductive groups and symmetric varieties, extending to moduli spaces of quasi-maps, revealing local homeomorphism of orbit closures to complex varieties.

Contribution

It introduces a multi-point homeomorphism between arc spaces and moduli spaces of quasi-maps, generalizing loop group factorizations and degeneration techniques.

Findings

Stratified homeomorphism between arc groups and symmetric varieties.

Extension to moduli spaces of quasi-maps from $P^1$.

Orbit closures have singularities locally homeomorphic to complex varieties.

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.