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

Abstract

We study the generating functions for Lusztig’s t-analog of weight multiplicities associated to integrable highest weight representations of the simplest affine Lie algebra \(A_1^{(1)}\). These generating series, termed t-string functions, are t-analogs of the string functions of \(A_1^{(1)}\). String functions are well studied for all affine Lie algebras and have important modularity properties. However, they are completely known in closed form only for the Lie algebra \(A_1^{(1)}\); in this case, Kac and Peterson showed that the string functions can be obtained in terms of certain indefinite modular forms of Hecke. In this paper, we generalize this description to the t-string functions of \(A_1^{(1)}\).