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

摘要

In this paper we use generating function methods to obtain new asymptotic results about spaces of F-stable maximal tori in GLn(overline{Fq}), Sp2n(overline{Fq}), and SO2n+1(overline{Fq}). We recover stability results of Church-Ellenberg-Farb and Jiménez 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 Sp2n(C) and SO2n+1(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.