The central action in differentiated reductive representations

Over an algebraically closed field of characteristic zero, the center of the Lie algebra of a reductive affine group acts diagonally in every rational representation. In particular each central tangent vector acts semisimply, including on representations of infinite dimension. This supplies the central-action condition in complete reducibility for reductive Lie algebras; arbitrary representations of a reductive Lie algebra need not satisfy that condition.

The center is the Lie algebra of the solvable radical, which is a torus. The comparison uses FiniteTypeCommHopfAlgCat.lieIdeal_solvableRadicalDefiningIdeal_eq_radical, the inclusion of the Lie center in the Lie radical, and reductiveCommHopfAlgProperty.torus_solvableRadical. The torus weight decomposition then diagonalizes its differentiated action.

References

• J. E. Humphreys, Linear Algebraic Groups, §13.
• J. S. Milne, Algebraic Groups (2017), Chapters 19 and 22.
• J. C. Jantzen, Representations of Algebraic Groups, I.2 and I.7.

theorem TauCeti.reductiveCommHopfAlgProperty.iSup_eigenspace_differential_eq_top_of_mem_center {k : Type u} [Field k] [IsAlgClosed k] [CharZero k] {H : FiniteTypeCommHopfAlgCat k} (hH : reductiveCommHopfAlgProperty k H) (V : Type w) [AddCommGroup V] [Module k V] [Comodule k (↑H.obj) V] (d : Derivation k (↑H.obj) (Bialgebra.CounitAlgebra k (↑H.obj) k)) (hd : d ∈ LieAlgebra.center k (Derivation k (↑H.obj) (Bialgebra.CounitAlgebra k (↑H.obj) k))) : ⨆ (μ : k), (Comodule.differential d).eigenspace μ = ⊤

In a rational representation of a reductive group in characteristic zero, the eigenspaces of any central tangent vector span the representation.

theorem TauCeti.reductiveCommHopfAlgProperty.isSemisimple_differential_of_mem_center {k : Type u} [Field k] [IsAlgClosed k] [CharZero k] {H : FiniteTypeCommHopfAlgCat k} (hH : reductiveCommHopfAlgProperty k H) (V : Type w) [AddCommGroup V] [Module k V] [Comodule k (↑H.obj) V] (d : Derivation k (↑H.obj) (Bialgebra.CounitAlgebra k (↑H.obj) k)) (hd : d ∈ LieAlgebra.center k (Derivation k (↑H.obj) (Bialgebra.CounitAlgebra k (↑H.obj) k))) : (Comodule.differential d).IsSemisimple

Every central element of the Lie algebra of a reductive group acts semisimply in every rational representation over an algebraically closed field of characteristic zero.