Rationality conjecture for the branching generating function of finite subgroups of $\mathbf{SL}_n\mathbb{C}$

Let $n\geq 2$ and let $\Gamma$ be a subgroup of $\mathbf{SL}_n\mathbb{C}$. Let $\{\gamma_0,\ldots,\gamma_l\}$ be the equivalence classes of irreducible finite-dimensional complex representations of $\Gamma$, with $\gamma_0$ trivial. For the irreducible $\mathfrak{sl}_n\mathbb{C}$-module of highest weight $p_1\varpi_1+\cdots+p_{n-1}\varpi_{n-1}$, write its restriction to $\Gamma$ as

$$\left.\pi_{\mathbf p}\right|_{\Gamma}=\bigoplus_{i=0}^l m_i(\mathbf p)\gamma_i,$$

where $\mathbf p=(p_1,\ldots,p_{n-1})\in\mathbb N^{n-1}$. Set $\mathbf t^{\mathbf p}=t_1^{p_1}\cdots t_{n-1}^{p_{n-1}}$, let $\mathcal E=(e_0,\ldots,e_l)$ be the canonical basis of $\mathbb C^{l+1}$, and define

$$v_{\mathbf p}:=\sum_{i=0}^l m_i(\mathbf p)e_i,\qquad P_\Gamma(\mathbf t):=\sum_{\mathbf p\in\mathbb N^{n-1}}v_{\mathbf p}\mathbf t^{\mathbf p}.$$

Write $P_\Gamma(\mathbf t)_i$ for the $i$-th coordinate. Rationality conjecture. The coefficients of the vector $P_\Gamma(\mathbf t)$ are rational functions in $\mathbf t$: for every $i\in\{0,\ldots,l\}$, there are polynomials $N_\Gamma(\mathbf t)_i,D_\Gamma(\mathbf t)\in\mathbb Q[\mathbf t]$ such that

$$P_\Gamma(\mathbf t)_i=\frac{N_\Gamma(\mathbf t)_i}{D_\Gamma(\mathbf t)}.$$

This conjecture generalizes the known results of Kostant for $\mathbf{SL}_2\mathbb C$ and the authors' results for $\mathbf{SL}_3\mathbb C$; it asserts rationality of the multivariable branching generating functions for the restrictions of irreducible representations to $\Gamma$.