Brion, Michel The behavior at infinity of the Bruhat decomposition. (English) Zbl 0935.14029 Comment. Math. Helv. 73, No. 1, 137-174 (1998). Summary: 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. Cited in 4 ReviewsCited in 45 Documents MSC: 14L30 Group actions on varieties or schemes (quotients) 14M17 Homogeneous spaces and generalizations 14M15 Grassmannians, Schubert varieties, flag manifolds Keywords:Bruhat decomposition; equivariant compactification; regular embedding; spherical homogeneous space; Schubert cycles PDFBibTeX XMLCite \textit{M. Brion}, Comment. Math. Helv. 73, No. 1, 137--174 (1998; Zbl 0935.14029) Full Text: DOI