Restriction theorems and root systems for symmetric superspaces

Abstract

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