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

Abstract

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 $\ell >n+1$.

In the remark about the application of our main theorem to the representation theory, a correction is made on the order of the root of 1