}$ with this highest weight. Given an arbitrary finite-dimensional (complex) $G$-representation $V$, we can always decompose it into irreducible sub-representations $
  V \cong \bigoplus_{\lambda \in \Lambda^*_{G,+}} m_{G,V}(\lambda) \, V_{G,\lambda}
$. We shall call the function $m_{G,V}$ thus defined the *highest weight multiplicity function*.

If we restrict the representation to the maximal torus, we can similarly decompose into irreducible representations. Since $T_G$ is a compact Abelian group, we can always jointly diagonalize its action, and it follows that the irreducible representations are one-dimensional. The joint eigenvalues can be encoded as a weight $\beta \in \Lambda^*_G$, and we will denote the corresponding irreducible representation of $T_G$ by ${\mathbb C}_\beta$. The decomposition $
  V \cong \bigoplus_{\beta \in \Lambda^*_G} m_{T_G,V}(\beta) \, {\mathbb C}_\beta
$ then defines the *weight multiplicity function* $m_{T_G,V}$. We also set $[k] = \{ 1,\ldots,k \}$, and write $f \sim g$ for the asymptotic equivalence $\lim_{k \rightarrow \infty} f(k)/g(k) = 1$.

An equivalent way of encoding weight multiplicities is in terms of the (formal) *character*, $$\operatorname{ch}V = \sum_\beta m_{T_G,V}(\beta) \, e^\beta,$$ which can be understood as the generating function of $m_{T_G,V}$. Formally, $\operatorname{ch}V$ is an element of the group ring ${\mathbb Z}[\Lambda^*_G]$, which consists of (finite) linear combinations of basis elements $e^\beta$ subject to the relation $e^\beta e^{\beta'} = e^{\beta + \beta'}$. The character of an irreducible representation $V_{G,\lambda}$ is given by the *Weyl character formula* [@knapp02 p. 319], $$\label{weyl character formula}
  \operatorname{ch}V_{G,\lambda} =
  \frac {\sum_{w \in W_G} \det(w) \, e^{w(\lambda + \rho)}} {e^\rho \prod_{\alpha \in R_{G,+}} \left( 1 - e^{-\alpha} \right)}.$$ Observe that we have $$\label{derivation kostant partition function}
  \begin{aligned}
  \frac 1 {\prod_{\alpha \in R_{G,+}} \left( 1 - e^{-\alpha} \right)}
  &= \prod_{\m