Let $(G,H\_1)$ and $(G,H\_2)$ be spherical pairs (i.e. $G$ is a reductive group, $H\_i$ are its closed subgroups and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H\_i$).


What can be said about the set of all elements $x \in G$ such that the conormal space at $x$ to its double coset $H\_1xH\_2$ does not intersect the nilpotent cone $\mathcal{N}$ in the dual Lie algebra of $G$? Is it open? Is it dense? Was it ever considered? 


We know that if $H\_1=H\_2$ is a symmetric subgroup then this is the set of regular semisimple elements, which is open and dense.



regular semisimple elements on spherical varieties
We answered this question in 


<http://arxiv.org/abs/1501.01479>.


It is open and dense.


