Let \(G\) be a connected reductive algebraic group and \(U\) an irreducible finite dimensional representation of \(G\) and \(S\) the simple roots of \(G\). Then \(G\) acts on the polynomial ring \(R=SU\) associated to \(U\), and \(R^G\) is Cohen-Macaulay by the Hochster-Robert theorem. A conjecture of Stanley [proved partially by the author: \textit{M. van den Bergh}, Invent. Math. 106, 389-409 (1991; Zbl 0761.13004)] gives sufficient conditions for \((U\otimes_{\mathbb C}R)^G\) to be Cohen-Macaulay. Let \((R^G)^+\) be the positive part of \(R^G\). The purpose of this paper is to compute the local cohomology modules \(H^i_{(R^G)^+}((U\otimes_{\mathbb C}R)^G)\) and so to give weaker sufficient conditions for \((U\otimes_{\mathbb C}R)^G\) to be Cohen-Macaulay.