  In this paper we prove a conjecture by De Concini, Kac and Procesi \cite{CP} (Corollary \ref{conj}):   The dimension of any $M\in U_q-\mood^\chi$ is divisible by $l^{codim_\mathcal{B}\mathcal{B}_\chi}$. 
Proof of the De Concini-Kac-Procesi conjecture