Theorem Borel Subgroups Are Conjugate and Control Parabolics
theorem admitted

Borel Subgroups Are Conjugate and Control Parabolics

Let \(G\) be a connected linear algebraic group over an algebraically closed field.

Every Borel subgroup of \(G\) is parabolic, any two Borel subgroups are conjugate by an element of \(G\), and a closed subgroup \(P\subseteq G\) is parabolic if and only if it contains a Borel subgroup.

Proof

Let \(B\) be a Borel subgroup. The quotient \(G/B\) is complete by the standard projective embedding attached to a maximal connected solvable subgroup, hence \(B\) is parabolic.

If \(B_1\) and \(B_2\) are Borel subgroups, let \(B_1\) act on the complete variety \(G/B_2\) by left translation. The Borel fixed point theorem gives a fixed coset \(gB_2\), which means \(B_1\subseteq gB_2g^{-1}\). Maximality among connected solvable subgroups forces equality, so the two Borel subgroups are conjugate.

If \(P\) is parabolic, then \(G/P\) is complete. A Borel subgroup \(B\) acting on \(G/P\) fixes some point, so \(B\) is contained in a conjugate of \(P\); after conjugating, \(P\) contains a Borel. Conversely, any closed subgroup containing a Borel has quotient dominated by the complete variety \(G/B\), hence is parabolic.

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