Weight Levi subgroups are reductive #
The Levi subgroup of GL_N preserving the weight spaces of an integer weight is reductive over
every field. Its coordinate algebra is smooth and geometrically connected, and over an algebraic
closure its normal smooth unipotent closed subgroups are trivial. The latter statement follows
from the faithful, completely reducible standard comodule, including when weights repeat.
The result is stated for the finite-type coordinate Hopf algebra and allows rank zero and arbitrary characteristic.
References #
- J. S. Milne, Algebraic Groups (2017), Chapters 13 and 19.
- T. A. Springer, Linear Algebraic Groups, §§2.2 and 2.4.
- Formal source:
TauCeti.Algebra.AlgebraicGroup.GeneralLinear.Reductive.
theorem
TauCeti.GeneralLinear.reductiveCommHopfAlgProperty_weightLeviFiniteTypeCoordinateHopfAlgebra
(k : Type u)
[Field k]
{N : ℕ}
(w : Fin N → ℤ)
:
The block-diagonal Levi subgroup of GL_N attached to any integer weight is reductive over
every field, including in rank zero and when weight blocks repeat.