Weight $q$-multiplicities for representations of the exceptional Lie algebra $mathfrak{g}_2$

Given a simple Lie algebra $mathfrak{g}$, Kostants weight $q$-multiplicity formula is an alternating sum over the Weyl group whose terms involve the $q$-analog of Kostants partition function. For $xi$ (a weight of $mathfrak{g}$), the $q$-analog of Kostants partition function is a polynomial-valued function defined by $wp_q(xi)=sum c_i q^i$ where $c_i$ is the number of ways $xi$ can be written as a sum of $i$ positive roots of $mathfrak{g}$. In this way, the evaluation of Kostants weight $q$-multiplicity formula at $q = 1$ recovers the multiplicity of a weight in a highest weight representation of $mathfrak{g}$. In this paper, we give closed formulas for computing weight $q$-multiplicities in a highest weight representation of the exceptional Lie algebra $mathfrak{g}_2$.