Weyl group symmetry of q-characters

TL;DR

This paper establishes that the ring of q-characters of quantum affine algebras is exactly the subring of Weyl group invariants in a certain completion, resolving a longstanding question and linking to representation theory and cluster algebra categorification.

Contribution

It proves that the Weyl group invariants in a completed ring correspond precisely to q-characters, clarifying their structure and relation to representation categories.

Findings

Weyl group action on the ring Y is defined and analyzed.

The subring of W-invariants is shown to be isomorphic to the ring of q-characters.

Screening operators are identified as limits of simple reflections.

Abstract

We define an action of the Weyl group W of a simple Lie algebra g on a completion of the ring Y, which is the codomain of the q-character homomorphism of the corresponding quantum affine algebra U_q(g^). We prove that the subring of W-invariants of Y is precisely the ring of q-characters, which is isomorphic to the Grothendieck ring of the category of finite-dimensional representations of U_q(g^). This resolves an old puzzle in the theory of q-characters. We also identify the screening operators, which were previously used to describe the ring of q-characters, as the subleading terms of simple reflections from W in a certain limit. Our results have already found applications to the study of the category O of representations of the Borel subalgebra of U_q(g^) in arXiv:2312.13256 and to the categorification of cluster algebras in arXiv:2401.04616.