ON ALPERIN CONJECTURE FOR FINITE REDUCTIVE GROUPS

A recent theorem of the first author gives a count (with signs) of the classes of regular semi-simple elements of a finite reductive group. In this Note, we show that this formula also provides a proof of Alperin's conjecture for such a group in natural characteristic. This proof is entirely different from the original proof of Cabanes [2]. It involves identification of the orbit space of a reductive group on its spherical building.