Chevalley's restriction theorem for reductive symmetric superpairs

Details Chevalley's restriction theorem for reductive symmetric superpairs Chevalley's restriction theorem for reductive symmetric superpairs

2008-12-18

arxiv.org/abs/0812.3530v4

J. Algebra 323 (2010) no. 4 1159-1185

Mathematics

Representation Theory

35 pages; v4: revised submission to J.Alg., accepted for publication under the proviso of revision

Scientific paper

10.1016/j.jalgebra.2009.11.014

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.

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