Generating functions and statistics on spaces of maximal tori in classical Lie groups

Abstract

In this paper we use generating function methods to obtain new asymptotic results about spaces of $F$-stable maximal tori in $GL_n(\overline{F_q})$, $Sp_{2n}(\overline{F_q})$, and $SO_{2n+1}(\overline{F_q})$. We recover stability results of Church--Ellenberg--Farb and Jim\'enez Rolland--Wilson for "polynomial" statistics on these spaces, and we compute explicit formulas for their stable values. We derive a double generating function for the characters of the cohomology of flag varieties in type B/C, which we use to obtain analogs in type B/C of results of Chen: we recover "twisted homological stability" for the spaces of maximal tori in $Sp_{2n}(\mathbb{C})$ and $SO_{2n+1}(\mathbb{C})$, and we compute a generating function for their "stable twisted Betti numbers". We also give a new proof of a result of Lehrer using symmetric function theory.