Chevalley's restriction theorem for reductive symmetric superpairs

What is it about?

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^k(p∗)→S^W(a∗) 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. Here, the invariant algebra defines a singular algebraic curve.

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