Bruhat decomposition

A representation of a connected algebraic reductive group as the union of double cosets of a Borel subgroup, parametrized by the Weyl group of . More exactly, let be opposite Borel subgroups of a reductive group ; let be the respective unipotent parts of (cf. Linear algebraic group) and let be the Weyl group of . In what follows denotes both an element of and its representative in the normalizer of the torus , since the construction presented below is independent of the representative chosen. The group will then be considered for each . The group is then representable as the union of the non-intersecting double cosets (), and the morphism is an isomorphism of algebraic varieties. An even more precise formulation of the Bruhat decomposition will yield a cellular decomposition of the projective variety . Namely, if is a fixed (with respect to the left shifts by elements from ) point of (such a point always exists, cf. Borel fixed-point theorem), will be the union of non-intersecting -orbits of the type , (cf. Algebraic group of transformations), and the morphism is an isomorphism of algebraic varieties. All groups , being varieties, are isomorphic to an affine space; if the ground field is the field of complex numbers, then each of the above -orbits is a cell in the sense of algebraic topology so that the homology of can be calculated. The existence of a Bruhat decomposition for a number of classical groups was established in 1956 by F. Bruhat, and was proved in the general case by C. Chevalley [3]. A. Borel and J. Tits generalized the construction of Bruhat decompositions to the groups of -points of a -defined algebraic group [2], the role of Borel subgroups being played by minimal parabolic -subgroups, the role of the groups by their unipotent radicals; the Weyl -group or the relative Weyl group was considered instead of .

References

[1]	A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2]	A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES , 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402
[3]	C. Chevalley, "Classification des groupes de Lie algébriques" , 2 , Ecole Norm. Sup. (1958) Zbl 0092.26301