Adjoint weights of a closed subgroup with a weight torus #
A torus map to a closed subgroup of GLₙ whose ambient map is diagonal with weights w
acts on tangent-matrix entry (i,j) through w i - w j. Testing the universal torus
point detects exactly the entries allowed in an adjoint weight space. The derivation
criterion needs no smoothness assumption; the comodule criterion assumes that the
augmentation cotangent module is finite projective.
The argument factors the matrix computations of
TauCeti.SpecialLinear.adDerivation_universalDiagonalTorus_eq_iff and its comodule
criterion through the injective closed-subgroup differential. This supplies the common
entrywise criterion used in classical pinnings.
References #
- J. S. Milne, Algebraic Groups (2017), §§21.1 and 24.6.
- B. Conrad, Reductive Group Schemes (2014), §5.1.
A weight-torus point scales each entry of the ambient tangent matrix of a closed subgroup by the difference of its two standard weights.
At the universal weight-torus point, a closed-subgroup tangent-matrix entry is multiplied by its integral character in the group-algebra basis.
A closed-subgroup tangent vector is an eigenvector at the universal weight-torus point exactly when entries of every other character vanish.
For a closed subgroup with a weight-torus factorization, membership in the actual adjoint comodule weight space is equivalent to vanishing of entries of every other weight.