On the Coadjoint Representation of Z2-contractions of Reductive Lie Algebras

The ground field k is algebraically closed and of characteristic zero. Let g be a reductive algebraic Lie algebra. Classical results of Kostant [7] give a fairly complete invarianttheoretic picture of the (co)adjoint representation of g. Let σ ∈ Aut(g) be an involution and g = g0 ⊕ g1 the corresponding Z2-grading. Associated to this decomposition, there is a non-reductive Lie algebra k = g0 ⋉ g1, the semi-direct product of the Lie algebra g0 and g0-module g1. LetK denote a connected group with Lie algebra k. A remarkable property of the Lie algebra contraction g ; k is that it preserves the transcendence degree of the algebras of invariants for both adjoint and coadjoint representations of k; i.e., trdeg k[k] = trdeg k[k] = rk g. The latter equality also shows that ind k = rk g. In [14], we proved that many good properties of Kostant’s picture for (g, ad ) carry over to (k, ad ). In particular, k[k] is a polynomial algebra and the quotient mapping πk : k → k/K = Spec (k[k]) is equidimensional. The goal of this article is to study the invariants of (k, ad). Motivated by several examples, we come up with the following