  We use the progenerator constructed in our previous paper to give a necessary
condition for a simple module of a finite reductive group to be cuspidal, or
more generally to obtain information on which Harish-Chandra series it can lie
in. As a first application we show the irreducibility of the smallest unipotent
character in any Harish-Chandra series. Secondly, we determine a unitriangular
approximation to part of the unipotent decomposition matrix of finite
orthogonal groups and prove a gap result on certain Brauer character degrees.
