This paper is concerned with
the study of certain irreducible representations, over the field of complex numbers, of
finite groups of Lie type, and especially with the characters afforded by these
representations. The methods used are based on the theory of blocks with
cyclic defect groups for certain prime different from the characteristic, called
special primes, relative to which the groups have cyclic Sylow subgroups.
Character values are obtained on certain regular semisimple classes, and
all Deligne-Lusztig virtual characters relative to certain maximal tori are
decomposed.