Geometric invariant theory and stretched Kostka quasi-polynomials

For $G$ a semisimple, simply-connected complex algebraic group and two dominant integral weights $\lambda ,\mu$, we consider the dimensions of weight spaces $V_{\lambda }(\mu )$ of weight $\mu$ in the irreducible, finite-dimensional highest weight $\lambda$ representation. For natural numbers $N$, the function $N\mapsto \dim V_{N\lambda }(N\mu )$ is a quasi-polynomial in $N$, the stretched Kostka quasi-polynomial. Using methods of geometric invariant theory (GIT), we realize the degree of this quasi-polynomial as the dimension of a certain GIT quotient. As a result, we resolve a conjecture of Gao and Gao on an explicit formula for this degree. We also discuss periods of this quasi-polynomial determined by the GIT approach, and give computational evidence supporting a geometric determination of the minimal period.