Adjoint weight spaces of the symplectic group #
The paired diagonal torus acts on the (i,j) entry of a symplectic tangent matrix
through diagonalTorusWeight i - diagonalTorusWeight j, where the standard weight
is εᵢ on the first block and -εᵢ on the second. A cotangent-dual vector has
adjoint weight α exactly when its matrix entries of every other weight
vanish. This criterion uses the actual torus coaction and distinguishes characters
even in characteristic two and over rings with nilpotents. It provides the matrix
criterion for identifying the root lines and normalizing a symplectic pinning.
The universal-point argument follows
TauCeti.SpecialLinear.mem_adjointWeightSpace_iff and uses the symplectic
tangent Lie equivalence, quotient differential equivariance, and the general-linear
matrix conjugation formula. No field or reducedness assumption is needed.
References #
- J. S. Milne, Algebraic Groups (2017), §§21.1 and 24.6.
- B. Conrad, Reductive Group Schemes (2014), §5.1 (root spaces and pinnings).
A symplectic tangent vector transforms by α at the universal torus point exactly
when every entry of a different character vanishes.
A cotangent-dual vector has adjoint weight α exactly when every entry of a
different weight in its paired symplectic tangent matrix vanishes.
A nonzero entry of a symplectic adjoint weight vector determines its character. The assertion holds over every commutative base ring, including rings with zero divisors.