Double cosets in F4

Abstract

In this paper we let G be the finite simple group F4(q), for q a prime power; we take P to be the maximal parabolic subgroup with Levi subgroup having derived group B3(q), and H to be the subgroup D4(q) generated by the long root elements of G. We consider the collection of (P,H)-double cosets in G; we obtain coset representatives and give the decomposition explicitly. As a consequence we are able to deduce that if q>2 then the action of F4(q) on the cosets of the maximal subgroup D4(q).S3 is not multiplicity-free, even if field automorphisms are applied.