Borel Subgroups

A Borel subgroup of a connected reductive (or more generally, connected smooth affine) algebraic group \(G\) is a maximal connected solvable closed subgroup. Borel subgroups are the foundational building block for the structure theory of reductive groups.

Definition

A closed subgroup \(B \subseteq G\) is a Borel subgroup if:

\(B\) is connected.
\(B\) is solvable.
\(B\) is maximal among connected solvable closed subgroups.

Equivalently (over an algebraically closed field), \(B\) is a Borel iff the quotient \(G/B\) is a complete (projective) variety. The flag variety \(G/B\) is the home of much of the geometric representation theory of \(G\).

Borel's fixed-point theorem

Theorem (Borel). Every connected solvable algebraic group acting on a complete variety has a fixed point.

Consequence: any two Borel subgroups of \(G\) are conjugate (over an algebraically closed field). The Borel subgroups form a single conjugacy class.

Borel subgroup of \(\operatorname{GL}_n\)

The standard Borel \(B \subseteq \operatorname{GL}_n\) is the subgroup of upper-triangular invertible matrices. It has dimension \(\binom{n+1}{2}\) and the structure

\[ B \;\cong\; T \ltimes U, \]

where \(T\) is the diagonal torus and \(U\) is the unipotent radical (strictly upper-triangular matrices, an iterated extension of \(\mathbb{G}_a\)'s).

The flag variety \(\operatorname{GL}_n / B\) is the variety of complete flags \(0 \subset V_1 \subset V_2 \subset \dots \subset V_n = k^n\) with \(\dim V_i = i\), of dimension \(\binom{n}{2}\).

Borel pairs and maximal tori

A Borel pair is a pair \((B, T)\) where \(B\) is a Borel subgroup and \(T \subseteq B\) is a maximal torus. Every Borel subgroup contains a maximal torus (a result of Chevalley); the Borel pairs are all conjugate under \(G(\bar{k})\).

A Borel pair determines:

a basis of the root system (the simple roots are the characters of \(T\) appearing in \(\operatorname{Lie}(B/T)\)),
a positive system of roots (those appearing in \(\operatorname{Lie}(B)\)),
a based root datum (see based root data).

Bruhat decomposition

For a connected reductive group \(G\) with Borel \(B\), maximal torus \(T \subseteq B\), and Weyl group \(W = N_G(T)/T\):

\[ G \;=\; \bigsqcup_{w \in W} B w B \]

as a disjoint union of double cosets. This is the Bruhat decomposition; the cells \(BwB\) are locally closed subvarieties of \(G\), and the closure relations are governed by the Bruhat order on \(W\).

Parabolic subgroups

A parabolic subgroup of \(G\) is a closed subgroup containing some Borel. Equivalently, a closed subgroup \(P\) such that \(G/P\) is a complete variety. For \(G\) split, parabolics containing a fixed Borel \(B\) are in bijection with subsets of the simple roots.

For \(\operatorname{GL}_n\), parabolic subgroups are stabilisers of partial flags.

\(k\)-rational vs.\ geometric Borel subgroups

Over a non-algebraically closed field \(k\):

The set of Borel subgroups of \(G_{\bar{k}}\) carries a Galois action.
\(G\) is quasi-split iff there is a Galois-fixed (equivalently, \(k\)-rational) Borel subgroup.
The classification of forms via Galois cohomology is closely tied to the existence and conjugacy of \(k\)-Borels.