The behaviour at infinity of the Bruhat decomposition

For a connected reductive group G and a Borel subgroup B, we study the closures of double classes BBB in a (G x G)-equivariant "regular" compactification of G. We show that these closures (BgB) over bar intersect properly all (G x G)-orbits, with multiplicity one, and we describe the intersections. Moreover, we show that almost all (BgB) over bar are singular in codimension two exactly. We deduce this from more general results on B-orbits in a spherical homogeneous space G/H; they lead to formulas for homology classes of X-orbit closures in G/B, in terms of Schubert cycles.