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

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