The $t$-analogs of string functions for $A_1^{(1)}$ and Hecke indefinite modular forms

TL;DR

This paper explores the $t$-analogs of string functions for the affine Lie algebra $A_1^{(1)}$, revealing their connection to Hecke indefinite modular forms and providing a new description of these functions as radial averages.

Contribution

It introduces a comprehensive description of the $t$-string functions for $A_1^{(1)}$, extending the classical string functions and linking them to modular forms.

Findings

$t$-string functions generalize classical string functions.

Values of $t$-string functions are radial averages of extended Hecke modular forms.

Provides a new perspective on the structure of affine Lie algebra representations.

Abstract

We study generating functions for Lusztig's $t$-analog of weight multiplicities associated to integrable highest weight representations of the simplest affine Lie algebra $A_1^{(1)}$. At $t=1$, these reduce to the {\em string functions} of $A_1^{(1)}$, which were shown by Kac and Peterson to be related to certain Hecke indefinite modular forms. Using their methods, we obtain a description of the general $t$-string function; we show that its values can be realized as radial averages of a certain extension of the Hecke indefinite modular form.