  The concept of directed strongly regular graphs (DSRG) was introduced by
Duval in 1988 \cite{A}.In the present paper,we use representation theory of
finite groups in order to investigate the directed strongly regular Cayley
graphs.We first show that a Cayley graph $\mathcal{C}(G,S)$ is not a directed
strongly regular graph if $S$ is a union of some conjugate classes of $G$.This
generalizes an earlier result of Leif K.J{\o}rgensen \cite{J1} on abelian
groups.Secondly,by using induced representations,we have a look at the Cayley
graph $\mathcal{C}(N\rtimes_\theta H, N_1\times H_1)$ with $N_1\subseteq N$ and
$H_1\subseteq H$,determining its characteristic polynomial and its minimal
polynomial.Based on this result,we generalize the semidirect product method of
Art M. Duval and Dmitri Iourinski in \cite{D} and obtain a larger family of
directed strongly regular graphs.Finally,we construct some directed strongly
regular Cayley graphs on dihedral groups,which partially generalize the earlier
results of Mikhail Klin,Akihiro Munemasa,Mikhail Muzychuk,and Paul Hermann
Zieschang in \cite{K1}.By using character theory,we also give the
characterization of directed strongly regular Cayley graphs
$\mathcal{C}(D_n,X\cup Xa)$ with $X\cap X^{(-1)}=\emptyset$.
