Differentiated actions of diagonalizable groups

Let π : H →ₐc[R] R[X] be the coordinate morphism of a homomorphism D(X) → G, and let V be a representation of G. A tangent vector d of D(X) acts on the x-weight space of V by the scalar d(x). Thus, over a field, every tangent vector coming from D(X) acts diagonally, and in particular semisimply, on V.

These results apply to the differentiated action of a split central torus on a rational representation. They require neither finite dimension nor characteristic zero. The weight formula works over any commutative ring and even for a commutative monoid X, without an antipode. In positive characteristic distinct characters can have the same differential; the weight-space inclusion in an eigenspace need not be an equality.

References

• J. C. Jantzen, Representations of Algebraic Groups, I.2 and I.7.
• J. S. Milne, Algebraic Groups (2017), Chapters 12 and 22.

theorem TauCeti.DiagonalizableGroup.differential_of_mem_weightSpace {R : Type u_1} {H : Type u_2} {X : Type u_3} (V : Type u_4) [CommRing R] [CommRing H] [Bialgebra R H] [CommMonoid X] [AddCommGroup V] [Module R V] [Comodule R H V] (π : H →ₐc[R] MonoidAlgebra R X) (d : Derivation R (MonoidAlgebra R X) (Bialgebra.CounitAlgebra R (MonoidAlgebra R X) R)) {x : X} {v : V} (hv : v ∈ weightSpace V π.toCoalgHom x) : (Comodule.differential ((derivationComp π) d)) v = (Bialgebra.CounitAlgebra.algEquivSelf R (MonoidAlgebra R X) R) (d (MonoidAlgebra.single x 1)) • v

A tangent vector from a diagonalizable group acts on a character space by the differential of that character.

theorem TauCeti.DiagonalizableGroup.weightSpace_le_eigenspace_differential {R : Type u_1} {H : Type u_2} {X : Type u_3} (V : Type u_4) [CommRing R] [CommRing H] [Bialgebra R H] [CommMonoid X] [AddCommGroup V] [Module R V] [Comodule R H V] (π : H →ₐc[R] MonoidAlgebra R X) (d : Derivation R (MonoidAlgebra R X) (Bialgebra.CounitAlgebra R (MonoidAlgebra R X) R)) (x : X) : weightSpace V π.toCoalgHom x ≤ (Comodule.differential ((derivationComp π) d)).eigenspace ((Bialgebra.CounitAlgebra.algEquivSelf R (MonoidAlgebra R X) R) (d (MonoidAlgebra.single x 1)))

Character spaces lie in the corresponding eigenspaces of the differentiated action. Different character spaces may have the same eigenvalue.

theorem TauCeti.DiagonalizableGroup.iSup_eigenspace_differential_eq_top {R : Type u_1} {H : Type u_2} {X : Type u_3} (V : Type u_4) [CommRing R] [CommRing H] [Bialgebra R H] [CommMonoid X] [AddCommGroup V] [Module R V] [Comodule R H V] (π : H →ₐc[R] MonoidAlgebra R X) (d : Derivation R (MonoidAlgebra R X) (Bialgebra.CounitAlgebra R (MonoidAlgebra R X) R)) : ⨆ (μ : R), (Comodule.differential ((derivationComp π) d)).eigenspace μ = ⊤

The eigenspaces of any tangent vector from a diagonalizable group span the representation. No finiteness or algebraic-closure assumption is needed.

theorem TauCeti.DiagonalizableGroup.isSemisimple_differential {X : Type u_3} (V : Type u_4) [CommMonoid X] [AddCommGroup V] {k : Type u_5} [Field k] {A : Type u_6} [CommRing A] [Bialgebra k A] [Module k V] [Comodule k A V] (π : A →ₐc[k] MonoidAlgebra k X) (d : Derivation k (MonoidAlgebra k X) (Bialgebra.CounitAlgebra k (MonoidAlgebra k X) k)) : (Comodule.differential ((derivationComp π) d)).IsSemisimple

Every tangent vector from a diagonalizable group acts semisimply in every rational representation, including representations of infinite dimension.