
Journal of Lie Theory 30 (2020), No. 3, 653672 Copyright Heldermann Verlag 2020 Spherical Varieties over Large Fields Stephan Snegirov Faculty of Mathematics, Northwestern University, Evanston, IL 60208, U.S.A. stephansnegirov2024@u.northwestern.edu [Abstractpdf] Let $k_0$ be a field of characteristic 0, $k$ its algebraic closure, $G$ a connected reductive group defined over $k$. Let $H\subset G$ be a spherical subgroup. We assume that $k_0$ is a large field, for example, $k_0$ is either the field $\mathbb{R}$ of real numbers or a $p$adic field. Let $G_0$ be a quasisplit $k_0$form of $G$. We show that if $H$ has selfnormalizing normalizer, and $\Gamma = {\rm Gal}\,(k/k_0)$ preserves the combinatorial invariants of $G/H$, then $H$ is conjugate to a subgroup defined over $k_0$, and hence, the $G$variety $G/H$ admits a $G_0$equivariant $k_0$form. In the case when $G_0$ is not assumed to be quasisplit, we give a necessary and sufficient Galoiscohomological condition for the existence of a $G_0$equivariant $k_0$form of $G/H$. Keywords: Equivariant form, inner form, algebraic group, spherical homogeneous space. MSC: 20G15, 12G05, 14M17, 14G27, 14M27. [ Fulltextpdf (186 KB)] for subscribers only. 