The Koszul complex and a certain induced module for a quantum group

We give a description of a certain induced module for a quantum group of type $A$. Together with our previous results this gives a proof of Lusztig’s conjectural multiplicity formula for non-restricted modules over the De Concini-Kac type quantized enveloping algebra of type $A_n$ at the $\mathrm{\ell }$-th root of unity, where $\mathrm{\ell }$ is an odd integer satisfying $(\mathrm{\ell },n+1)=1$ and $\mathrm{\ell }>n+1$.