Background:

Let $K$ be a field and let $V$ be a finite-dimensional $K$-vector space. A pseudoreflection (or usually imprecisely just reflection) in $V$ is an element $1 \neq s \in \mathrm{GL}(V)$ fixing a hyperplane. A reflection representation of a group $W$ over $K$ is a $K$-linear representation $\rho:W \rightarrow \mathrm{GL}(V)$, such that $\rho(W)$ is generated by reflections. A group $W$ is called a reflection group over $K$ if it admits a reflection representation over $K$.

Shephard-Todd classified (see below) the finite irreducible reflection groups over $\mathbb{C}$ (i.e. those finite groups admitting an irreducible reflection representation over $\mathbb{C}$).

Question:

Is there also a classification of the finite irreducible reflection representations over $\mathbb{C}$?

Edit: This question is very imprecise as indicated in the comments below. I should say what "classification of representations" means, and I have to admit: I don't know. A few ideas in this direction are:

• determine the isomorphism classes of finite irreducible reflection representations over $\mathbb{C}$, where an isomorphism between two reflection representations $\rho:W \rightarrow \mathrm{GL}(V)$, $\rho':W' \rightarrow \mathrm{GL}(V')$ is a vector space isomorphism $f:V \rightarrow V'$ such that $f \rho(G) f^{-1} = \rho'(G)$. (I think the Shephard-Todd classification is a classification relative to this notion!?)

• the same as above but an isomorphism is a vector space isomorphism $f:V \rightarrow V'$ and a group isomorphism $\varphi:W \rightarrow W'$ such that $f \rho(g) f^{-1} = \rho'( \varphi(g) )$ for all $g \in W$.

• consider pairs $(W,T)$ consisting of a finite irreducible reflection group over $\mathbb{C}$ and a subset $T$ which are generating reflections of some irreducible reflection representation of $W$ and then determine isomorphism classes of such pairs.

• [Insert your idea here].

My motivation for this question is something like this: A Cherednik-Algebra is defined for any finite irreducible reflection representation over $\mathbb{C}$. In what sense does the algebra depend on the group alone and not on the choice of a particular reflection representation? This problem does not exist for the cyclotomic Hecke algebras.