The invariant polynomials on simple Lie superalgebras

Chevalley's theorem states that for any simple finite dimensional Lie algebra ${\mathfrak{g}}$: (1) the restriction homomorphism of the algebra of polynomials $S({\mathfrak{g}}^*)\longrightarrow S({\mathfrak{h}}^*)$ onto the Cartan subalgebra ${\mathfrak{h}}$ induces an isomorphism $S({\mathfrak{g}}^*)^{{\mathfrak{g}}}\cong S({\mathfrak{h}}^*)^{W}$, where $W$ is the Weyl group of ${\mathfrak{g}}$; (2) each ${\mathfrak{g}}$-invariant polynomial is a linear combination of the polynomials $\tr \rho(x)^k$, where $\rho$ is a finite dimensional representation of ${\mathfrak{g}}$.

None of these facts is necessarily true for simple Lie superalgebras. We reformulate Chevalley's theorem as formula $(*)$ below to include Lie superalgebras. Let ${\mathfrak{h}}$ be the split Cartan subalgebra of ${\mathfrak{g}}$; let $R=R_+\cup R_-$ be the set of nonzero roots of ${\mathfrak{g}}$, the union of positive and negative ones. Set $\tilde R_+=\{\alpha \in R_+\mid -\alpha \in R_-\}$. For each root $\alpha \in \tilde R_+$ denote by ${\mathfrak{g}}(\alpha)$ the Lie superalgebra generated by ${\mathfrak{h}}$ and the root superspaces ${\mathfrak{g}}_\alpha$ and ${\mathfrak{g}}_{-\alpha}$. Let the image of $S({\mathfrak{g}}(\alpha)^*)^{{\mathfrak{g}}(\alpha)}$ under the restriction homomorphism $S({\mathfrak{g}}(\alpha)^*)\longrightarrow S({\mathfrak{h}}^*)$ be denoted by $I^{\alpha}({\mathfrak{h}}^*)$ and the image of $S({\mathfrak{g}}^*)^{{\mathfrak{g}}}$ by $I({\mathfrak{h}}^*)$. Then

Chevalley's theorem for anti-invariant polynomials is also presented.