Let $G$ be a compact simple lie group of rank $n$. Then the Poincaré series of $G$ is given by $$P(G,q)=\prod_{i=1}^n (1+q^{2d_i-1}),$$ where the integers $d_1\leq d_2\leq \cdots \leq d_n$ are the degrees of the fundamental polynomial invariants of the Weyl group. More generally, these degrees exist for any finite subgroup of the unitary group $U(n)$ generated by pseudoreflections. (A pseudoreflection $\varphi$ satisfies $\dim\ker(\varphi-1)=n-1$.)
Even though there is no Lie group corresponding to a general finite reflection group, one can still consider the formula for the Poincaré series. The symmetry group of the icosahedron (type $H_3$) has degrees $2,6,10$ and so its "Poincaré series" is given by $$(1+q^3)(1+q^{11})(1+q^{19})=1+q^3+q^{11}+q^{14}+q^{19}+q^{22}+q^{30}+q^{33}.$$ There is no Lie group corresponding to this Poincaré series, and the usual reason given is that the group $H_3$ does not stabilize a lattice.
My question is this: Is there any topological reason why the "Poincaré series" of type $H_3$ cannot be realized as a Lie group? What hope is there of finding a $33$-dimensional "pseudo-Lie group" of type $H_3$? More generally, is there a topological explanation for the fact that Weyl groups are special among reflection groups?
