I would like a finite dimensional irreducible representation of ${PSL}\_2(\mathbb{Z})$ which contains a proper sub-representation when restricted to the congruence subgroup $\Gamma(2)$, the kernel of $\mathrm{mod}\_2:{PSL}\_2(\mathbb{Z})\to {PSL}\_2(\mathbb{Z}\_2)$.


This would be used in constructing exact sequences of differential calculi on group algebras, and examining what relations exist between the differential invariants and the group structure. A seriously infinite noncommutative example would be really useful.


I have read the paper on the irreps of $B\_3$ and ${PSL}\_2(\mathbb{Z})$ by Imre Tuba and Hans Wenzl, but after calculations on some (I have not tried all of them!) of their examples, I cannot get a reducible rep for $\Gamma(2)$ which was not already reducible for ${PSL}\_2(\mathbb{Z})$.



Representations of $PSL_2(Z)$ and $\Gamma(2)$.
(Sorry: the earlier statement had to be modified a little; instead, I give a different construction). 


Take the standard representation $\rho$ of $PSL(2,\mathbb{Z}$ on $\mathbb{C}^3$ and tensor it with a representation $\theta$ (of dimenstion not one) of the finite group $PSL\_2(\mathbb{Z}/2\mathbb{Z})$ (such a representation exists since the group $PSL\_2(\mathbb{Z}/2\mathbb{Z})$ is not abelian). Take $\pi =\rho \otimes \theta$; restriced to $\Gamma (2)$, this is a sum of copies of $\rho$. 


The connected component of the Zariski closure of $\Gamma $ and $\Gamma (2)$ are the same both for $\rho$ and for $\pi$; hence for $\rho$ the Zariski closure of $\Gamma (2)$ is $G=PSL(2,\mathbb{C})\simeq SO(3,\mathbb{C})$.Thus the Zariski closure for $\Gamma $ in the larger representation $\pi$ is $G\times PSL(2,\mathbb{Z}/2\mathbb{Z})$, which shows that $\pi$ is irreducible since the latter group acts irreducibly on $\rho \otimes \theta\simeq \pi$.


