Adjoint groups, regular unipotent elements and discrete series characters

It is shown that if G is a finite Chevalley group or twisted type over a field of characteristic p and U is a maximal p-subgroup of G then any nonlinear irreducible character of U vanishes on regular elements. For groups of adjoint type the linear content of the restriction to U of a discrete series character J of G is calculated and it is deduced that J takes the value 0 or ${( - 1)^s}$ on regular elements of U $(s = {\text{rank}}\;G)$.