## 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.

### MSC:

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