A remark on Gelfand-Graev characters for finite simple groups

The famous Gelfand-Graev character of a group of Lie type G is a multiplicity free character of shape nu (G) , where nu is a suitable degree 1 character of a Sylow p-subgroup and p is the defining characteristic of G. We show that, for an arbitrary non-abelian simple group G, if nu is a linear character of a Sylow p-subgroup of G such that nu (G) is multiplicity free, then G is isomorphic to either a group of Lie type in defining characteristic p, or to a group PSL(2, q), where either p = q + 1, or p = 2 and q + 1 or q - 1 is a 2-power.