  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
$\ell> n+1$.
