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 BgB in a $ (G \times G) $-equivariant "regular" compactification of G. We show that these closures $ \overline {BgB} $ intersect properly all $ (G \times G) $-orbits, with multiplicity one, and we describe the intersections. Moreover, we show that almost all $ \overline {BgB} $ 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 H-orbit closures in G/B, in terms of Schubert cycles.