Polynomial invariants of infinite reflection groups

It is a famous theorem of Shepard-Todd that the ring of invariant polynomials of a finite complex reflection group W acting on a complex vector space V is actually itself a polynomial ring. In other words, if P(V) is the polynomial ring on V, then P(V)W is a (finitely generated) polynomial ring.

I would like to know if any work has been done to generalize this to the case of infinite complex reflection groups (or perhaps at the very least for complexified (infinite) Coxeter groups). To this end, after reading through modern proofs of the above theorem for finite complex reflection groups, it seems to me like most of the arguments factor through to the infinite case, except perhaps the key fact that the ring of invariant polynomials for a finite linear group is a finitely generated algebra. So, this question more or less boils down to the following:

If V is a finite dimensional complex vector space, and W is a (not necessarily finite) discrete subgroup of GL(V) which is finitely generated by pseudo-reflections, is the ring of polynomial invariants P(V)W finitely generated?


Source : Link , Question Author : Alexander Goldman , Answer Author : Community

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