Questions on invariant operators of finite group representations

Question

1) Is there an equivalent of the Casimir operator for an irreducible representation of a finite group? 2) Given an invariant operator of a certain group, can I check if it is invariant under only that group (and it's subgroups)? Alternatively, given a group, is there a way I can construct an operator invariant under only that group and nothing bigger?

Basically I am trying to construct Hamiltonians that are invariant under specific finite groups. This was my original question on Physics stack exchange.

Answer 1

1) You can take any inner product and sum over the group to get an invariant inner product: $$ g(v,w) = \frac1{|G|}\sum_{x\in G} \langle x.v, x.w\rangle. $$ Then the Laplacian (the usual formula with respect to an orthonormal basis) is something like the Kasimir. But it is invariant under the larger group $O(V,g)$.

2) Each invariant polynomial is a polynomial in the finitely many basic invariant polynomial: Think of the statement: any symmetric polynomial is a polynomial in the elementary symmetric polynomials. You need a generating system for the algebra of invariant polynomials in order separate $G$-orbits. If you only allow for second order polynomials (you ask for an operator!), you will not be able to separate orbits, thus this will be invariant under larger groups in general, as for the Kasimir above.