Homogeneous spaces of Hilbert type

TL;DR

This paper extends the class of algebraic quotient spaces known to be of Hilbert type by including certain non-connected subgroups, especially abelian subgroups, of simply connected groups over number fields.

Contribution

It proves that quotients of simply connected groups by abelian subgroups are of Hilbert type, generalizing previous results to non-connected subgroups.

Findings

G/H is of Hilbert type when H is a smooth connected subgroup.

G/H is of Hilbert type for abelian, not necessarily connected, subgroups of simply connected groups.

Extension of Hilbert type results to non-connected subgroups over number fields.

Abstract

Let k be a global field. Let G be a connected linear algebraic k-group, assumed reductive when k is a function field. It follows from a result of a preprint by Bary-Soroker, Fehm and Petersen that when H is a smooth connected k-subgroup of G, the quotient space G/H is of Hilbert type. We prove a similar result for certain non-connected k-subgroups H of G. In particular, we prove that if G is a simply connected k-group over a number field k, and H is an abelian k-subgroup of G, not necessarily connected, then G/H is of Hilbert type.