Quantum affine algebras, graded limits, and flags

TL;DR

This survey explores the recent links between quantum affine algebra representations and current algebra representations, emphasizing finite-dimensional cases, graded limits, and their connections to Macdonald polynomials and Demazure modules.

Contribution

It reviews recent developments in understanding the representation theory of quantum affine and current algebras, highlighting graded limits and Demazure modules.

Findings

Connections between quantum affine and current algebra representations clarified.

Demazure modules' structure and combinatorics are elucidated.

Relations to Macdonald polynomials are established.

Abstract

In this survey, we review some of the recent connections between the representation theory of (untwisted) quantum affine algebras and the representation theory of current algebras. We mainly focus on the finite-dimensional representations of these algebras. This connection arises via the notion of the graded and classical limit of finite-dimensional representations of quantum affine algebras. We explain how this study has led to interesting connections with Macdonald polynomials and discuss a BGG-type reciprocity result. We also discuss the role of Demazure modules in this theory and several recent results on the presentation, structure, and combinatorics of Demazure modules.