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This research aims to analyze the properties of spherical varieties defined over large fields and their equivariant forms.

March 29, 2026Open Access

Spherical Varieties over Large Fields

Key Points

This research aims to analyze the properties of spherical varieties defined over large fields and their equivariant forms.
Investigated spherical subgroups within connected reductive groups over various fields.
Analyzed conditions for a subgroup to be conjugate to one defined over the base field.
Provided Galois-cohomological conditions for the existence of equivariant forms.
Demonstrated that under certain conditions, a spherical subgroup can be conjugate to a subgroup over the base field.
Identified necessary and sufficient Galois-cohomological conditions for G-equivariant forms when quasi-split assumptions are relaxed.

Abstract

Let k 0 be a field of characteristic 0, k its algebraic closure, G a connected reductive group defined over k .Let H G be a spherical subgroup.We assume that k 0 is a large field, for example, k 0 is either the field 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 = 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 .

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