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

TL;DR

This paper describes a specific induced module for a quantum group of type A and proves Lusztig's conjectural multiplicity formula for certain non-restricted modules at roots of unity.

Contribution

It provides a new description of an induced module and completes the proof of Lusztig's conjecture in a specific quantum group setting.

Findings

Proof of Lusztig's multiplicity formula for non-restricted modules

Description of a certain induced module for quantum groups of type A

Validation of conjectural multiplicities at roots of unity

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 $\ell> n+1$.