Journal of Lie Theory

no. 3
Spherical Varieties over Large Fields
Journal of Lie Theory, Volume 30 (2020) no. 3, pp. 653-672
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 quasi-split $k_0$-form of $G$. We show that if $H$ has self-normalizing 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 quasi-split, we give a necessary and sufficient Galois-cohomological condition for the existence of a $G_0$-equivariant $k_0$-form of $G/H$.
DOI: 10.5802/jolt.1133
Classification: 20G15, 12G05, 14M17, 14G27, 14M27
Keywords: Equivariant form, inner form, algebraic group, spherical homogeneous space 
@article{JOLT_2020_30_3_a2,
     author = {S. Snegirov},
     title = {Spherical {Varieties} over {Large} {Fields}},
     journal = {Journal of Lie Theory},
     pages = {653--672},
     year = {2020},
     volume = {30},
     number = {3},
     doi = {10.5802/jolt.1133},
     zbl = {1530.20164},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1133/}
}
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S. Snegirov. Spherical Varieties over Large Fields. Journal of Lie Theory, Volume 30 (2020) no. 3, pp. 653-672. doi: 10.5802/jolt.1133

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