Chevalley's restriction theorem for reductive symmetric superpairs

TL;DR

This paper generalizes Chevalley's restriction theorem to reductive symmetric superpairs, explicitly determines the image of the restriction map, and explores new invariant behaviors in specific superalgebra examples.

Contribution

It extends Chevalley's restriction theorem to symmetric superpairs, providing explicit descriptions and new insights into invariant structures in superalgebra contexts.

Findings

The restriction map is injective and its image is explicitly characterized.

The theorem applies to symmetric superpairs of group type, generalizing previous results.

In certain superalgebras, invariants form a singular algebraic curve.

Abstract

Let (g,k) be a reductive symmetric superpair of even type, i.e. so that there exists an even Cartan subspace a in p. The restriction map S(p^*)^k->S(a^*)^W where W=W(g_0:a) is the Weyl group, is injective. We determine its image explicitly. In particular, our theorem applies to the case of a symmetric superpair of group type, i.e. (k+k,k) with the flip involution where k is a classical Lie superalgebra with a non-degenerate invariant even form (equivalently, a finite-dimensional contragredient Lie superalgebra). Thus, we obtain a new proof of the generalisation of Chevalley's restriction theorem due to Sergeev and Kac, Gorelik. For general symmetric superpairs, the invariants exhibit a new and surprising behaviour. We illustrate this phenomenon by a detailed discussion in the example g=C(q+1)=osp(2|2q,C), endowed with a special involution. In this case, the invariant algebra defines a singular algebraic curve.