According to a classical result of Shephard--Todd--Chevalley, finite linear
complex reflection groups are characterized by the property that their
algebra of polynomial invariants is free. If we consider these groups as
acting on the corresponding projective spaces (which are Hermitian symmetric
spaces of positive curvature), a natural infinite analogue of them are
cofinite discrete reflection groups in symmetric domains (Hermitian symmetric
spaces of negative curvature), the analogue of polynomials being automorphic
forms.
The only symmetric domains admitting reflections are complex balls
$B(n)=U(1,n)/(U(1)\times U(n))$ (of rank 1) and domains of Cartan type IV
$D(n)=O^+(2,n)/(SO(2)\times O(n))$ (of rank 2). Many examples of cofinite
discrete reflection groups in $B(n)$ and $D(n)$ for small n are known. For some
of them it is known that the algebra of automorphic forms is free. However,
due to a general result of Margulis for symmetric spaces of rank >1, any
cofinite discrete group in $D(n)$ containing a reflection, has a finite index
subgroup generated by reflections. This means that there are a lot of cofinite
discrete reflection groups in $D(n)$ for any $n$, and it is not likelyhood that
all of them share any good properties. To distinguish "good" reflection groups
$\Gamma\subset O(2,n)$, one can require that $dim H^2(\Gamma,\mathbb{Q})=1$. Under this
condition, there exists a semi-automorphic form (possibly, of fractional weight)
vanishing (with multiplicity 1) exactly at the mirrors of reflections contained
in $\Gamma$ (an analogue of the Vandermonde determinant). Hopefully, this will
permit to prove that, for such "good" reflection groups in $O(2,n)$, the algebra
of automorphic forms is free.
In particular, let $O_d$ is the ring of integers of the quadratic field $\mathbb{Q}(\sqrt d)$,
and $\sigma$ be its involution. The extended Hilbert modular group $\Gamma_d=
\langle PSL(2,O_d),\sigma\rangle$ is a cofinite discrete group in the domain $D(2)$
(which is the direct product of two copies of the hyperbolic plane). It is often
generated by reflections. One can try to calculate $H^2(\Gamma_d,\mathbb{Q})$, making use
of a presentation of $\Gamma_d$ obtained in a geometric way. This program was
realised for $d=2$ with the result that the group $\Gamma_2$ is "good" in the above
sense.
