TITLE: Classification of finite complex reflection groups
QUESTION [6 upvotes]: 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?

REPLY [3 votes]: Answering the first question, if the field has characteristic zero then the classification will be reduced to Shephard-Todd. By this I mean that every finite reflection group will be on Shephard-Todd list. In the opposite direction, finite Weyl groups will appear over every field and the rest of them will need some algebraic integers to be present in your field... 
In positive characteristic, the life gets tough as pseudoreflections can be unipotent. I do not know whether classification is known but I can say that the list will get much longer...