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

Abstract: A result of Lehrer describes a beautiful relationship between topological and combinatorial data on certain families of varieties with actions of finite reflection groups. His formula relates the cohomology of complex varieties to point counts on associated varieties over finite fields. Church, Ellenberg, and Farb use their representation stability results on the cohomology of flag manifolds, together with classical results on the cohomology rings, to prove asymptotic stability for "polynomial" statistics on associated varieties over finite fields. In this paper we investigate the underlying algebraic structure of these families' cohomology rings that makes the formulas convergent. We prove that asymptotic stability holds in general for subquotients of FI$\mathcal{W}$-algebras finitely generated in degree at most one, a result that is in a sense sharp. As a consequence, we obtain convergence results for polynomial 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 new proof of the stability theorem for invariant maximal tori in $\mathrm{GL}_n(\overline{F_q})$ due to Church-Ellenberg-Farb.