Conjugacy of Borel subgroups

Any two Borel subgroups of a finite-type affine group over an algebraically closed field are conjugate by a rational point. The ambient group may be disconnected or nonreduced, and the result holds in every characteristic.

For a smooth connected group, induction on Lie dimension proves the theorem. A solvable group is its own unique Borel subgroup. Otherwise the closed-projective-orbit criterion supplies a proper closed subgroup containing conjugates of every Borel candidate. Taking its reduction and identity component retains these containments and produces a smooth connected group of strictly smaller dimension. Conjugacy there transfers to the original group. Finally the reduced identity component handles an arbitrary ambient group.

The proof combines the proper-subgroup construction in Borel.ProjectiveOrbit, HopfIdeal.finrank_quotientLie_lt, and the closed-subgroup transfers in Borel.Conjugacy.Basic. The reduction is realized using HopfIdeal.vanishingIdeal and rational point separation.

References

• A. Borel, Linear Algebraic Groups, 2nd ed. (1991), §11.1.
• J. S. Milne, Algebraic Groups (2017), Theorem 17.6.

theorem TauCeti.HopfIdeal.IsBorelOverAlgClosed.exists_conjugate_eq {k : Type u} [Field k] {H : FiniteTypeCommHopfAlgCat k} {I J : HopfIdeal k ↑H.obj} (hI : IsBorelOverAlgClosed k H I) (hJ : IsBorelOverAlgClosed k H J) : ∃ (g : WithConv (↑H.obj →ₐ[k] k)), I.conjugate g = J

Borel conjugacy. Any two Borel subgroups of a finite-type affine group over an algebraically closed field are conjugate by a rational point. The ambient group need not be smooth, reduced, or connected.