The Invariant Polynomials on Simple Lie Superalgebras

Chevalley’s theorem states that for any simple finite dimensional Lie algebra g: (1) the restriction homomorphism of the algebra of polynomials S(g∗) −→ S(h∗) onto the Cartan subalgebra h induces an isomorphism S(g∗)g ∼= S(h∗)W , where W is the Weyl group of g; (2) each g-invariant polynomial is a linear combination of the polynomials trρ(x)k , where ρ is a finite dimensional representation of g. None of these facts is necessarily true for simple Lie superalgebras. We reformulate Chevalley’s theorem as formula (∗) below to include Lie superalgebras. Let h be the split Cartan subalgebra of g; let R = R+ ∪ R− be the set of nonzero roots of g, the union of positive and negative ones. Set R̃+ = {α ∈ R+ | −α ∈ R−}. For each root α ∈ R̃+ denote by g(α) the Lie superalgebra generated by h and the root superspaces gα and g−α. Let the image of S(g(α)∗)g(α) under the restriction homomorphism S(g(α)∗) −→ S(h∗) be denoted by Iα(h∗) and the image of S(g∗)g by I(h∗). Then I(h∗) = ⋂