In this paper we shall be concerned with the decomposition of the permutation characters of coset actions of a certain type. Specifically, we let G be a connected reductive algebraic group over an algebraically closed field K of characteristic p > 0, and σ be a Frobenius map of G: for convenience we set F = σ2, and write Gσ and GF for the groups of fixed points. The action with which this paper deals is then that of GF on cosets of Gσ: this covers actions such as G(q2) on G(q) and (where appropriate) G(q2) on 2G(q2). We find that actions of this sort are either almost or actually multiplicity-free (an action is said to be multiplicity-free if no irreducible constituent of the permutation character occurs more than once): moreover some which are not multiplicity-free can be made so by the application of outer automorphisms. We shall classify precisely which almost simple actions of this type are multiplicity-free in Theorems 4, 10 and 11: these results are used in. The material that follows is taken from the author's Ph.D. thesis: we also mention a recent preprint of Kawanaka which covers very similar ground.

Preliminaries

If H is a finite group, we shall denote by Ĥ the set of (complex) irreducible characters of H. We shall mainly use the standard notation for finite groups of Lie type as given in. However, in a few instances we employ a slight modification, necessitated by our use of more than one Frobenius map.