  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.
