Solvable Groups

A connected solvable algebraic group is one with a finite descending series of normal closed subgroups whose successive quotients are commutative. Borel subgroups (= maximal connected solvable subgroups) sit at the heart of reductive group theory.

Definition

A linear algebraic group \(H\) is solvable if there is a chain of closed subgroups \[ H = H_0 \;\supseteq\; H_1 \;\supseteq\; \cdots \;\supseteq\; H_n = 1 \] with each \(H_{i+1}\) normal in \(H_i\) and each quotient \(H_i / H_{i+1}\) commutative.

Equivalently, the derived series \(H, [H, H], [[H, H], [H, H]], \dots\) terminates at \(1\).

Lie–Kolchin theorem

Theorem (Lie–Kolchin). Let \(G\) be a connected solvable algebraic group over an algebraically closed field \(k\) acting linearly on a finite-dimensional vector space \(V\). Then there is a basis of \(V\) in which \(G\) acts by upper-triangular matrices.

Equivalently: every connected solvable subgroup of \(\mathrm{GL}_n\) is conjugate to a subgroup of the upper-triangular Borel.

Structure of connected solvable groups

For a connected solvable algebraic group \(B\) over an algebraically closed field \(k\):

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

where:

\(U = R_u(B)\) is the unipotent radical (a connected unipotent normal subgroup);
\(T\) is a maximal torus of \(B\) (any two are \(U\)-conjugate);
The product is semidirect.

This is the prototype of the general Levi decomposition.

Examples

The standard Borel of \(\mathrm{GL}_n\)

The upper-triangular subgroup \(B \subseteq \mathrm{GL}_n\): \[ T = \text{diagonal}, \qquad U = \text{strict upper-triangular}, \] \[ B \;=\; T \ltimes U. \]

\(B\) is the prototypical connected solvable group.

Tori

Every torus is commutative, hence solvable. The unipotent radical of a torus is trivial.

\(\mathbb{G}_a \rtimes \mathbb{G}_m\)

The "\(ax + b\)" group: \(\mathbb{G}_m\) acts on \(\mathbb{G}_a\) by multiplication. Connected, solvable, non-abelian, with maximal torus \(\mathbb{G}_m\) and unipotent radical \(\mathbb{G}_a\).

Borel subgroups

In a connected reductive group \(G\), a Borel subgroup is a maximal connected solvable subgroup. All Borel subgroups are conjugate over \(\bar k\); their fixed-point variety \(G/B\) is the flag variety, projective and smooth.

The Borel structure (Lie–Kolchin in disguise) is the geometric source of the entire Bruhat decomposition and root datum theory of \(G\).

Solvable groups over non-algebraically-closed fields

Over a general field \(k\), a connected solvable group may not have a maximal torus defined over \(k\); the obstruction is measured by \(H^1(k, T)\). For \(k\) a local or global field, this is governed by Galois cohomology and class field theory.

Examples:

Norm one tori \(R^{(1)}_{K/k}\mathbb{G}_m\) are connected solvable (in fact commutative) but not split unless \(K = k\).
Inner forms of Borel subgroups appear in the study of stable conjugacy and endoscopy.

Role in reductive structure and Langlands

Borel subgroups as maximal connected solvable subgroups define the flag variety \(G/B\) and the geometric setup of the Bruhat decomposition.
Parabolic subgroups are exactly those containing a Borel, generalising the solvable case.
Eisenstein series are constructed from solvable parabolics in the Langlands–Eisenstein theory; the constant term along a Borel recovers the Whittaker function and the Casselman–Shalika formula.

References

Borel, Linear Algebraic Groups (1991), §10.
Springer, Linear Algebraic Groups (1998), §6.
Humphreys, Linear Algebraic Groups (1975), §17.