Do we already know the classification of the finite-dimensional irreducible unitary representations of the modular group $PSL(2,\mathbb{Z})=\mathbb{Z}/2*\mathbb{Z}/3$? I'm particularly interested in the following question.
Q: Is there a maximal dimension for the finite-dimensional irreducible unitary representations of $PSL(2,\mathbb{Z})$?
The answer is No.
There is a mod-$p$ map $f:PSL(2,\mathbb Z) \rightarrow PSL(2,\mathbb F_p)$.
The permutation representation of $PSL(2,\mathbb F_p)$ on the projective line on $\mathbb F_p$ with $p+1$ points splits as a $1$-dimensional and a $p$-dimensional irreducible representation, which can be checked by the character table.
Thus, there is a $p$-dimensional irreducible representation for $PSL(2,\mathbb Z)$. As $p$ can be arbitrarily large, there is no maximal dimension for the finite-dimensional irreducible unitary representations of $PSL(2,\mathbb Z) $.
