THE BEHAVIOR 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'' c ompactification of G. We show that these closures (BgB) over bar inter sect properly all (G x G)-orbits, with multiplicity one, and we descri be the intersections. Moreover, we show that almost all (BgB) over bar are singular in codimension two exactly. We deduce this from more gen eral results on B-orbits in a spherical homogeneous space G/H; they le ad to formulas for homology classes of X-orbit closures in G/B, in ter ms of Schubert cycles.