Directed Strongly Regular Cayley Graphs on Dihedral groups

In this paper,\;we characterize some certain directed strongly regular Cayley graphs on Dihedral groups $D_{p^\alpha}$,\;where $p$ is a prime and $\alpha\geqslant 1$ is a positive integer.\;We achieve this result by using some tools from Representation Theory,\;Fourier Transformation and Algebraic Number Theory.\;