The behavior at infinity of the Bruhat decomposition. (English) Zbl 0935.14029

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.


14L30 Group actions on varieties or schemes (quotients)
14M17 Homogeneous spaces and generalizations
14M15 Grassmannians, Schubert varieties, flag manifolds
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