Integral weight decompositions of a symmetric algebra

Let f be an endomorphism of a vector space M over a field K whose eigenspaces at the integers span M, so M = ⨁ⱼ M(j) with f acting on M(j) by j. The derivation D_f of the symmetric algebra S(M) extending f (SymmetricAlgebra.derivationOfEnd) is then diagonalizable with integer eigenvalues as well: a monomial in eigenvectors of weights j₁, …, jₙ is an eigenvector of D_f of weight j₁ + ⋯ + jₙ, by the Leibniz rule.

These weights are the characters of the torus action that the grading induces. If g is an endomorphism of M acting on each M(j) by cʲ for a unit c, then the algebra endomorphism SymmetricAlgebra.map g of S(M) acts on the weight-w eigenspace of D_f by cʷ. In characteristic zero the integer eigenspaces of D_f are independent, so this holds for every eigenvector of D_f of weight w, not merely for the monomials. In particular every element of S(M) killed by D_f is fixed by map g. This is the multiplicative counterpart of SymmetricAlgebra.map_exp_apply_of_derivationOfEnd_eq_zero, where f is nilpotent and g is its exponential; here f is semisimple and g ranges over the cocharacter it integrates to, and no exponential is available over a general field.

For the coadjoint action of an integral semisimple element h of a Lie algebra L on the polynomial functions S(L*), this shows that the polynomial functions killed by h are invariant under the grading automorphisms of L defined by h.

Main results

• SymmetricAlgebra.iSup_eigenspace_derivationOfEnd_eq_top: if the integer eigenspaces of f span M, then the integer eigenspaces of D_f span S(M).
• SymmetricAlgebra.map_apply_of_derivationOfEnd_eq_smul: if moreover g acts on M(j) by cʲ, then map g acts on every eigenvector of D_f of integer weight w by cʷ.

References

• J. E. Humphreys, Linear Algebraic Groups, GTM 21, §16, for the weight decomposition of a torus representation and the induced decomposition of its symmetric algebra.

theorem SymmetricAlgebra.iSup_eigenspace_derivationOfEnd_eq_top {K : Type u} {M : Type v} [Field K] [AddCommGroup M] [Module K M] {f : Module.End K M} (hf : ⨆ (j : ℤ), f.eigenspace ↑j = ⊤) : ⨆ (w : ℤ), Module.End.eigenspace ↑((derivationOfEnd K M) f) ↑w = ⊤

The induced derivation of an integrally diagonalizable endomorphism is integrally diagonalizable. If the eigenspaces of f at the integers span M, then the eigenspaces at the integers of the derivation D_f of S(M) extending f span S(M).

theorem SymmetricAlgebra.map_apply_of_derivationOfEnd_eq_smul {K : Type u} {M : Type v} [Field K] [AddCommGroup M] [Module K M] [CharZero K] {f g : Module.End K M} (hf : ⨆ (j : ℤ), f.eigenspace ↑j = ⊤) {c : Kˣ} (hg : ∀ (j : ℤ), ∀ m ∈ f.eigenspace ↑j, g m = ↑(c ^ j) • m) {w : ℤ} {p : SymmetricAlgebra K M} (hp : ((derivationOfEnd K M) f) p = ↑w • p) : (map K g) p = ↑(c ^ w) • p

The torus attached to an integral weight decomposition acts on weight vectors by characters. Suppose the eigenspaces M(j) of f at the integers span M, and g acts on each M(j) by c ^ j for a unit c. Then the algebra map map g of S(M) acts by c ^ w on every eigenvector of weight w of the derivation D_f extending f.