Conjugation by a rational point in a representation #
Let g be a rational point of an affine group G and π an algebra-valued point. In a
representation V of G, the point π acts on g v as g acts on the result of letting the
conjugate point g⁻¹ π g act on v. In particular, if the conjugate point scales v, then π
scales g v by the same scalar. This is how rational points normalizing a subgroup permute its
weight spaces: the subgroup's universal point π scales a weight vector by its character, and the
conjugated character is read off from g⁻¹ π g.
No reducedness, finite type, or field hypothesis is needed, and the value algebra of π may be
nonreduced.
References #
- J. S. Milne, Algebraic Groups (2017), §21.1.
- J. C. Jantzen, Representations of Algebraic Groups, I.2.
theorem
TauCeti.Comodule.endOfPoint_one_tmul_basePointsRepresentation_of_conj
{R : Type u_1}
{H : Type u_2}
{V : Type u_3}
{C : Type u_4}
[CommSemiring R]
[CommSemiring H]
[HopfAlgebra R H]
[AddCommMonoid V]
[Module R V]
[Comodule R H V]
[CommSemiring C]
[Algebra R C]
(π : H →ₐ[R] C)
(g : WithConv (H →ₐ[R] R))
{v : V}
{y : C}
(h : (endOfPoint V (π.comp (HopfAlgebra.pointConjugationAlgHom g⁻¹))) (1 ⊗ₜ[R] v) = y ⊗ₜ[R] v)
:
(endOfPoint V π) (1 ⊗ₜ[R] ((basePointsRepresentation V) g) v) = y ⊗ₜ[R] ((basePointsRepresentation V) g) v
If the conjugate g⁻¹ π g of an algebra-valued point π by a rational point g scales v
by y, then π scales g v by y.