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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'sconjectural multiplicity formula for non-restricted modules over the DeConcini-Kac type quantized enveloping algebra of type $A_n$ at the $\ell$-throot of unity, where $\ell$ is an odd integer satisfying $(\ell,n+1)=1$ and$\ell> n+1$.

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