Chevalley's restriction theorem for reductive symmetric superpairs

Abstract

Let ( g , k ) be a reductive symmetric superpair of even type, i.e. so that there exists an even Cartan subspace a ⊂ 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 | 2 q , C ) , endowed with a special involution. Here, the invariant algebra defines a singular algebraic curve.