So, a question I was thinking about for Cherednik algebras (more on that later) ended up reducing to the following invariant theory statement I wasn’t able to prove (or at least have been too lazy to try).
So, let G be a reductive (or if you like real groups better, compact) group acting on a complex vector space V. Consider the polynomial functions as a module over invariant functions . For each irreducible representation W of G, we have a W-isotypic component . (Or, if you like, we can take the eigenspace decomposition for the action of class functions on G by convolution, if G is compact). Of course, is an module in an obvious way.
Question. Is finitely generated as an module?
Note: this is well known for a finite group, since itself is a finitely generated module in this case. The question above obviously fails for all of when is not finite, but the isotypic components are more “the right size.”