The normalizer modulo the centralizer is finite étale

Let D(X) be a closed diagonalizable subgroup of a finite-type affine group over an algebraically closed field. If its scheme normalizer is reduced, the represented quotient of the normalizer by the scheme centralizer is finite étale. In characteristic zero the reducedness hypothesis follows from Cartier's theorem.

The quotient coordinate algebra is the coinvariant algebra for the centralizer ideal inside the normalizer. The result describes the full quotient scheme, including when the diagonalizable subgroup is nonreduced.

For a maximal torus in a reductive group, identifying the centralizer with the torus turns this quotient into the Weyl group scheme. That identification is a separate result.

References

• J. S. Milne, Algebraic Groups (2017), §§4.a and 21.1.
• T. A. Springer, Linear Algebraic Groups, §7.1.

theorem BialgHom.moduleFinite_coinvariants_centralizer_normalizer {k : Type u} [Field k] [IsAlgClosed k] {H : TauCeti.FiniteTypeCommHopfAlgCat k} {X : Type u} [CommGroup X] (π : ↑H.obj →ₐc[k] MonoidAlgebra k X) (hπ : Function.Surjective ⇑π) [IsReduced ↑(H.quotient (π.normalizerHopfIdeal hπ)).obj] : Module.Finite k ↥(π.centralizerHopfIdeal.map (Bialgebra.Quotient.mkBialgHom (π.normalizerHopfIdeal hπ).toIdeal)).coinvariants

The coordinate algebra of the scheme normalizer modulo the scheme centralizer is finite-dimensional. No reducedness or smoothness of the ambient group is required.

theorem BialgHom.algebraEtale_coinvariants_centralizer_normalizer {k : Type u} [Field k] [IsAlgClosed k] {H : TauCeti.FiniteTypeCommHopfAlgCat k} {X : Type u} [CommGroup X] (π : ↑H.obj →ₐc[k] MonoidAlgebra k X) (hπ : Function.Surjective ⇑π) [IsReduced ↑(H.quotient (π.normalizerHopfIdeal hπ)).obj] : Algebra.Etale k ↥(π.centralizerHopfIdeal.map (Bialgebra.Quotient.mkBialgHom (π.normalizerHopfIdeal hπ).toIdeal)).coinvariants

The represented quotient of the scheme normalizer by the scheme centralizer is étale, retaining the full subgroup schemes even when D(X) is nonreduced.