TITLE: Is there a topological Chevalley-Shephard-Todd Theorem?
QUESTION [6 upvotes]: Is the following true:
For a representation of a finite group $G$ on $\mathbb{C}^n$, the quotient $\mathbb{C}^n/G$ is a topological manifold if and only if $G$ is generated by pseudo-reflections.
( A pseudo-reflection is an element of $G$ whose fixed set is of codimension $\leq 1$)

REPLY [3 votes]: To elaborate on Jason's great answer.
The answer is no and the counter-example is the binary icosahedral group together with the sum of an irreducible representation of degree $2$ and a unit representation, over $\mathbb{C}$.
Let $\Gamma$ be the binary icosahedral group--the unique perfect and fixed-point-free finite group, and consider one of it irreducible representations of degree  $2$ over complex numbers (it is fixed-point free), for example the one resulting from embedding of $\Gamma$ in $SL_2(\mathbb{C})$. Then the quotient $\mathbb{C}^2/\Gamma$ is called the Poincare Homology Sphere.  Now by the affirmative answer to Milnor's double suspension problem, the double suspension of a homology sphere is a true sphere.  Now the image of the origin in $(\mathbb{C}^2\times \mathbb{C})/\Gamma$, where the second factor comes with a trivial $\Gamma$-action, looks locally like the vertex of the cone over double suspension of the Poincare Homology Sphere, hence the quotient space is locally Euclidean at the image of the origin.  But then, given the scaling action, it is locally Euclidean everywhere.
EDIT  That said, I claim the following sufficient condition:
Suppose no sub-quotient of a finite group $\Gamma$ is isomorphic to the binary icosahedral group.  Then $\Gamma$ satisfies the topological Chevalley-Shephard-Todd Theorem:  
Given a linear representation of $\Gamma$, finite degree over $\mathbb{C}$, the quotient is a topological manifold iff $\Gamma$ is generated by pseudo-reflections.