Restriction theorems and root systems for symmetric superspaces

In this paper we consider those involutions θ of a finite-dimensional Kac–Moody Lie superalgebra g, with associated decomposition g=k⊕p, for which a Cartan subspace a in p_0̄ is self-centralizing in p. For such θ the restriction map C_θ from p to a is injective on the algebra P(p)^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(p)^k by giving a complete set of generators for the image of C_θ. We also determine precisely when the restriction map R_θ from P(g)^g to P(p)^k is surjective. Finally we introduce the notion of a generalized restricted root system, and show that in the present setting the a-roots Δ(a,g) always form such a system.