Bruhat decomposition

2020 Mathematics Subject Classification: Primary: 20G [MSN][ZBL]

The Bruhat decomposition is a representation of a connected split algebraic reductive group $G$, as the union of double cosets of a Borel subgroup, parametrized by the Weyl group of $G$. More exactly, let $B,B^-$ be opposite Borel subgroups of a reductive group $G$; let $U,U^-$ be the respective unipotent parts of $B,B^-$ (cf. Linear algebraic group) and let $W$ be the Weyl group of $G$. In what follows $w$ denotes both an element of $W$ and its representative in the normalizer of the torus $B\cap B^-$, since the construction presented below is independent of the representative chosen. The group $U_w^- = U\cap wU^-w^{-1}$ will then be considered for each $w\in W$. The group $G$ is then representable as the union of the non-intersecting double cosets $BwB$ ($w\in W$), and the morphism $U_w^-\times B \to BwB$ defined by $(x,y)\mapsto xwy$ is an isomorphism of algebraic varieties. An even more precise formulation of the Bruhat decomposition will yield a cellular decomposition of the projective variety $G/B$. Namely, if $x_0$ is a fixed (with respect to the left shifts by elements from $B$) point of $G/B$ (such a point always exists, cf. Borel fixed-point theorem), $G/B$ will be the union of non-intersecting $U$-orbits of the type $U(w(x_0))$, $w\in W$ (cf. Algebraic group of transformations), and the morphism $U_w^- \to Uw(x_0)$ ($u\mapsto u(w(x_0))$) is an isomorphism of algebraic varieties. All groups $U_w^-$, being varieties, are isomorphic to an affine space; if the ground field is the field of complex numbers, then each of the above $U$-orbits is a cell in the sense of algebraic topology so that the homology of $G/B$ 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 [Ch]. A. Borel and J. Tits generalized the construction of Bruhat decompositions to the non split groups $G_k$ of $k$-points of a $k$-defined algebraic group [BoTi], the role of Borel subgroups being played by minimal parabolic $k$-subgroups, the role of the groups $U$ by their unipotent radicals; the Weyl $k$-group $W_k$ or the relative Weyl group was considered instead of $W$.

References

[Bo]	A. Borel, "Linear algebraic groups", Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[BoTi]	A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES, 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402
[Ch]	Séminaire C. Chevalley, "Classification des groupes de Lie algébriques", 2, Ecole Norm. Sup. (1956-1958) MR0106966 Zbl 0092.26301