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The Chevalley–Shephard–Todd theorem states that given a finite-dimensional faithful representation of a finite group $G$ on a vector space $V$ over a field $k$ whose characteristic does not divide the order of $G$, the action of $G$ is generated by pseudoreflections just if the invariant functions $k[V]^G$ form a polynomial ring.

  1. Is there any known generalization (perhaps somewhat weakened) to the case $V$ is infinite-dimensional and $G$ is only assumed to be discrete (or some other finiteness condition)?

  2. If not, is there an obvious reason there can be no such generalization?

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  • $\begingroup$ I'm not sure how "pseudo-reflection" would be defined in case $V$ is infinite dimensional. On the other hand, there may be some possibilities when the action of $G$ is assumed to be locally finite dimensional. I'm not sure what the constraints should be on $G$ and $V$ to get a meaningful generalization. $\endgroup$
    – Jim Humphreys
    Commented Jun 2, 2017 at 22:03
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    $\begingroup$ The motivating example for C-S-T is the theory of symmetric polynomials. An infinite-dimensional analogue would have to involve infinite sums, so it's no longer "just" a question of polynomial algebra. [Presumably a "pseudoreflection" is a linear transformation that acts as $1$ on some hyperplane (codimension-$1$ subspace) and a nontrivial root of unity on some $1$-dimensional complementary subspace.] $\endgroup$
    – Noam D. Elkies
    Commented Jun 2, 2017 at 22:06
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    $\begingroup$ Consider the infinite symmetric group $S_{\infty}$ acting on $k[x_1, x_2, \dots ]$. The ring of invariants is just $k$, which is certainly a polynomial ring. But $S_{\infty}$ doesn't seem to me to be generated by pseudoreflections in any sense. One problem here is that away from the finite case we no longer have the ability to average over the group to produce invariants, so there's no reason there should be any. $\endgroup$
    – Qiaochu Yuan
    Commented Jun 2, 2017 at 23:12
  • $\begingroup$ True, and averaging is why we make the order restriction. $\endgroup$
    – jdc
    Commented Jun 3, 2017 at 1:35

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