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 $\ell $-th root of unity, where $\ell $ is an odd integer satisfying $(\ell ,n+1)=1$ and $\ell> n+1$.