Cohen-Macaulayness of modules of invariants for \(SL_ 2\) (Q1178919)

Let \(G\) be a reductive algebraic group over an algebraically closed field \(k\) of characteristic zero and let \(W\) be a finite-dimensional representation of \(G\). It is known that the ring \(k[W]^ G\) of invariant polynomials is Cohen-Macaulay. If \(U\) is another representation, the question of when \((U\otimes k[W])^ G\) is a Cohen-Macaulay \(k[W]^ G\)- module is studied. The group considered is \(Sl_ 2\) since the case of a torus is completely known through work of R. Stanley. The author determines some pairs of representations \(U\) and \(W\) which give Cohen- Macaulay modules. The methods used are those of geometric invariant theory. Some geometric assumption is made which restricts the possible set of representations which give Cohen-Macaulay modules. The author notes that a forthcoming paper will address this restriction.