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A result of Lehrer describes a beautiful relationship between topological andcombinatorial data on certain families of varieties with actions of finitereflection groups. His formula relates the cohomology of complex varieties topoint counts on associated varieties over finite fields. Church, Ellenberg, andFarb use their representation stability results on the cohomology of flagmanifolds, together with classical results on the cohomology rings, to proveasymptotic stability for "polynomial" statistics on associated varieties overfinite fields. In this paper we investigate the underlying algebraic structureof these families' cohomology rings that makes the formulas convergent. Weprove that asymptotic stability holds in general for subquotients ofFI$_\mathcal{W}$-algebras finitely generated in degree at most one, a resultthat is in a sense sharp. As a consequence, we obtain convergence results forpolynomial statistics on the set of maximal tori in$\mathrm{Sp}_{2n}(\overline{F_q})$ and $\mathrm{SO}_{2n+1}(\overline{F_q})$that are invariant under the Frobenius morphism. Our results also give a newproof of the stability theorem for invariant maximal tori in$\mathrm{GL}_n(\overline{F_q})$ due to Church-Ellenberg-Farb.

Convergence criteria for FI$_\mathcal{W}$-algebras and polynomial statistics on maximal tori in type B/C