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Let  G  be a connected semi-simple algebraic group of adjoint type over an algebraically closed field and letbe the wonderful compactification of  G . For a fixed pair ( B , B  − ) of opposite Borel subgroups of  G , we look at intersections of Lusztig’s  G - stable pieces  and the  B  − × B -orbits in, as well as intersections of  B × B -orbits and  B  − × B  − -orbits in. We give explicit conditions for such intersections to be nonempty, and in each case, we show that every nonempty intersection is smooth and irreducible, that the closure of the intersection is equal to the intersection of the closures, and that the nonempty intersections form a strongly admissible partition of  Ḡ .

On Intersections of Certain Partitions of a Group Compactification