Polytopal invariance for Lusztig's $q$-weight multiplicities

Let $\mathrm{\Phi }$ be an irreducible root system normalized so that long roots have length $\sqrt{2}$, and let $\mu \le \lambda$ be dominant weights. We associate to $(\mathrm{\Phi },\mu ,\lambda )$ the convex polytope $$P_{\mu ,\lambda }^{\mathrm{\Phi }}=(\mu +C)\cap (\lambda -C)\cap (\bar{D}-\rho ),$$ where $C={\mathbb{R}}_{\ge 0}{\mathrm{\Phi }}^{+}$ and $\bar{D}$ is the closed dominant Weyl chamber. We prove that if $P_{\mu ,\lambda }^{\mathrm{\Phi }}$ and $P_{{\mu }^{\mathrm{\prime }},{\lambda }^{\mathrm{\prime }}}^{{\mathrm{\Phi }}^{\mathrm{\prime }}}$ are isometric, then $$K_{\lambda \mu }^{\mathrm{\Phi }}(q)=K_{{\lambda }^{\mathrm{\prime }}{\mu }^{\mathrm{\prime }}}^{{\mathrm{\Phi }}^{\mathrm{\prime }}}(q),$$ where $K_{\lambda \mu }^{\mathrm{\Phi }}(q)$ is Lusztig’s $q$-analogue of the $\mu$-weight multiplicity in the irreducible highest-weight representation $V(\lambda )$, often called the generalized Kostka-Foulkes polynomial. Moreover, we construct the polynomial directly from the metric shape of the polytope, and we classify when two such polytopes are isometric. Such coincidences are abundant and can occur in different Dynkin types, and in different Dynkin families. By the Lusztig-Kato theorem, the main theorem gives a metric analogue, in the spherical setting, of the combinatorial invariance conjecture for Kazhdan-Lusztig polynomials.