Restriction Theorems and Root Systems for Symmetric Superspaces

In this paper we consider those involutions $\theta$ of a finite-dimensional Kac-Moody Lie superalgebra $\mathfrak{g}$, with associated decomposition $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$, for which a Cartan subspace $\mathfrak{a}$ in ${\mathfrak{p}}_{\bar{0}}$ is self-centralizing in $\mathfrak{p}$. For such $\theta$ the restriction map $C_{\theta }$ from $\mathfrak{p}$ to $\mathfrak{a}$ is injective on the algebra $P(\mathfrak{p}{)}^{\mathfrak{k}}$ of $\mathfrak{k}$-invariant polynomials on $\mathfrak{p}$. There are five infinite families and five exceptional cases of such involutions, and for each case we explicitly determine the structure of $P(\mathfrak{p}{)}^{\mathfrak{k}}$ by giving a complete set of generators for the image of $C_{\theta }$. We also determine precisely when the restriction map $R_{\theta }$ from $P(\mathfrak{g}{)}^{\mathfrak{g}}$ to $P(\mathfrak{p}{)}^{\mathfrak{k}}$ is surjective. Finally we introduce the notion of a generalized restricted root system, and show that in the present setting the $\mathfrak{a}$-roots $\mathrm{\Delta }(\mathfrak{a},\mathfrak{g})$ always form such a system.