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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.

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  • $\begingroup$ Two small points: 1) You should specify the ground field. 2) When you say "the Borel subgroup" you probably mean "some Borel subgroup" (I guess the same $B$ for $H_1$ and $H_2$?). $\endgroup$ – Jim Humphreys Dec 12 '13 at 21:04
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Check out the paper

Helminck, Aloysius G.(1-NCS); Schwarz, Gerald W.(1-BRND) Orbits and invariants associated with a pair of spherical varieties: some examples. (English summary) The 2000 Twente Conference on Lie Groups (Enschede). Acta Appl. Math. 73 (2002), no. 1-2, 103–113.

As commented by the authors, there seems to be some trouble to deal with the case of arbitrary spherical varieties.

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We answered this question in

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

It is open and dense.

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