The Lie–Luck conjecture for asymptotic representation ranks

Let $\mathsf{G}$ be a semisimple algebraic group over $\mathbb{C}$ and let $\Gamma$ be a finitely generated subgroup of $\mathsf{G}(\mathbb{C})$. Fix a central character $\chi$ and let $\mathsf{X}(\mathsf{T})_\chi$ be the dominant integral weights $\bm\lambda$ for which the center acts on $\mathsf{W}_{\bm\lambda}$ through $\chi$. For each such weight, let $\operatorname{rk}^{\Gamma}_{\mathsf{W}_{\bm\lambda}}$ be the associated Sylvester matrix rank function, and let $\operatorname{rk}^{\chi}_\Gamma$ be the twisted von Neumann rank. Lie–Luck conjecture. As rank functions on $\mathbb{C}[\Gamma]$,

$$\lim_{\min \lambda_i\to\infty}\operatorname{rk}^{\Gamma}_{\mathsf{W}_{\bm\lambda}}=\operatorname{rk}^{\chi}_\Gamma,$$

where the limit is taken over $\mathsf{X}(\mathsf{T})_\chi$. This is a rank-theoretic strengthening of the preceding asymptotic statements; the general qualitative behavior is conjectural, with the paper proving results only in particular settings.