Let $(G,H)$ be a spherical pair (i.e. $G$ is a reductive group, $H$ is a closed subgroup and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H$). Let $U$ be the unipotent radical of $B$.

What can be said about the set of all elements $x$ in $G/H$ such that the stabilizer of $x$ in $U$ is contained inside the derived group $[U,U]$?

Is it open? We know that it is a union of $B$-orbits. In the group case (i.e. $G=H \times H$ with $H$ diagonally embedded) we know that this is the open Bruhat cell. We do not know what it is even for general symmetric pairs.


This set is a special case of regular semisimple elements on spherical varieties.

In particular it is open and dense.

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