Graded limits of minimal affinizations and Beyond: the multiplicty free case for type E6

TL;DR

This paper derives a graded character formula for specific modules over the current algebra of type E6, confirming conjectures about their structure and connecting them to minimal affinizations and projective modules.

Contribution

It provides a new graded character formula for modules in type E6, extending understanding of minimal affinizations and their projective properties under certain conditions.

Findings

Derived a graded character formula for modules in type E6

Confirmed isomorphism with classical limits of minimal affinizations under restrictions

Applied formula to projective modules in a subcategory of graded modules

Abstract

We obtain a graded character formula for certain graded modules for the current algebra over a simple Lie algebra of type E6. For certain values of their highest weight, these modules were conjectured to be isomorphic to the classical limit of the corresponding minimal affinizations of the associated quantum group. We prove that this is the case under further restrictions on the highest weight. Under another set of conditions on the highest weight, Chari and Greenstein have recently proved that they are projective objects of a full subcategory of the category of graded modules for the current algebra. Our formula applies to all of these projective modules.