On covariants of reductive algebraic groups. (Q1866461)

Let \(G\) be a reductive algebraic group over an algebraically closed field of characteristic zero. Assume \(G\) acts on an affine variety \(X\). Let \(x\in X\) be a point whose orbit closure \(Y=\overline{G\cdot x}\) is normal. Assume the orbit \(G\cdot x\) has a complement of codimension at least two in \(Y\). Let \(M\) be a finite dimensional \(G\)-module. Let \(M^{G_x}\) be the submodule of invariants under the stabiliser group \(G_x\). There is an evaluation map \(\varepsilon_x\) which associates to any \(G\)-equivariant polynomial morphism \(\phi: X\to M\) its value \(\phi(x)\) in \(M^{G_x}\). The main result is that \(\varepsilon_x\) is onto. Several applications are given.