mbda_z \\ \delta\end{smallmatrix}\right)} \}.
  \end{aligned}$$ After choosing a basis of the lattice $\Lambda^*_{G_{\operatorname{ss}}}$ we arrive at the asserted formula (with $s' = \dim T_{G_{\operatorname{ss}}}$ and $u = t + r_H$).

We stress that the proof of is constructive: The maps $\mathcal A$ and $\mathcal B$, whose existence is asserted by the theorem, are defined in in terms of $A$ and $B$, whose construction is described explicitly in [@billeyguilleminrassart04 Proof of Theorem 2.1] (for the case of $\mathfrak g = {\mathfrak{su}}(d)$) and in [@bliem08 §4] (for the general case). See for an illustration in the context of the Kronecker coefficients.

If one uses the Kostant multiplicity formula instead of in the proof of then one arrives at a similar formula for the multiplicities $m^\lambda_\mu$ involving an additional alternating sum over the Weyl group of $G$. After completion of this work, we have learned of [@heckman82 Lemma 3.1] which is derived in this spirit.

Piecewise Quasi-Polynomiality {#piecewise-quasi-polynomiality .unnumbered}
-----------------------------

Let us use the fundamental weight bases fixed above to identify $\Lambda^*_G \cong {\mathbb Z}^{r_G}$ and $\Lambda^*_H \cong {\mathbb Z}^{r_H}$. The group homomorphisms $\mathcal A$ and $\mathcal B$ correspond to matrices with integer entries, which we shall denote by the same symbols. Observe that the formula in in essence amounts to counting the number $n(y) := \# \left( \Delta_{\mathcal A, \mathcal B}(y) \cap {\mathbb Z}^{s+s'} \right)$ of integral points in certain rational convex polytopes of the form $$\label{weight restriction polytope}
  \Delta_{\mathcal A,\mathcal B}(y) := \{ x \in {\mathbb R}^{s+s'} : x_1, \ldots, x_s \geq 0, \mathcal A x = \mathcal B y \},$$ parametrized by $y \in {\mathbb Z}^{r_G+r_H}$. Explicitly, $$\label{concrete alternating sum}
  m^\lambda_\mu = \sum_{\gamma \in \Gamma_H} c_\gamma n(\lambda, \mu + \gamma).$$ It is well-known that $n(y)$ is a *piecewise quasi-polynomial* function in $y$ [@claussloechn