Commentarii Mathematici Helvetici


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Volume 73, Issue 1, 1998, pp. 137–174
DOI: 10.1007/s000140050049

The behaviour at infinity of the Bruhat decomposition

Michel Brion (1)

(1) Institut Fourier, Université Grenoble I, B.P. 74, F-38402, SAINT MARTIN D'HERES CEDEX, FRANCE

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.

Keywords: Bruhat decomposition, regular embedding, Chow ring