A Representation-Theoretic Approach to 𝑞𝑞-Characters

Abstract

We raise the question of whether (a slightly generalized notion of) 𝑞𝑞-characters can be constructed purely representation-theoretically. In the main example of the quantum toroidal 𝖌𝔩₁ algebra, geometric engineering of adjoint matter produces an explicit vertex operator 𝖱𝖱 which computes certain 𝑞𝑞-characters, namely Hirzebruch χy-genera, completely analogously to how the 𝚁-matrix 𝖱 computes 𝑞-characters. We give a geometric proof of the independence of preferred direction for the refined vertex in this and more general non-toric settings.