On the number of real classes in the finite projective linear and unitary groups

Abstract

We show that for any $n$ and $q$, the number of real conjugacy classes in $\mathrm{PGL}(n,{\mathbb{F}}_q)$ is equal to the number of real conjugacy classes of $\mathrm{GL}(n,{\mathbb{F}}_q)$ which are contained in $\mathrm{SL}(n,{\mathbb{F}}_q)$, refining a result of Lehrer, and extending the result of Gill and Singh that this holds when $n$ is odd or $q$ is even. Further, we show that this quantity is equal to the number of real conjugacy classes in $\mathrm{PGU}(n,{\mathbb{F}}_q)$, and equal to the number of real conjugacy classes of $\mathrm{U}(n,{\mathbb{F}}_q)$ which are contained in $\mathrm{SU}(n,{\mathbb{F}}_q)$, refining results of Gow and Macdonald. We also give a generating function for this common quantity.