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Fixing a closed $K$-orbit in $\\B$, we associate to every $K$-orbit on $\\B$ some subsets of the Weyl group of $G$, and we study them as invariants of the $K$-orbits. When ${\\bf k} = {\\mathbb C}$, these invariants are used to determine when an orbit of a real form of $G$ and an orbit of a Borel subgroup of $G$ have non-empty intersection in $\\B$. 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