  Let $G$ be a digraph with adjacency matrix $A(G)$ and outdegrees diagonal
matrix $D(G)$. For any real $\alpha\in[0,1]$, the $A_\alpha$ matrix
$A_\alpha(G)$ of a digraph $G$ is defined as $A_\alpha(G)=\alpha
D(G)+(1-\alpha)A(G)$. The eigenvalue of $A_\alpha(G)$ with the largest modulus
is called the $A_\alpha$ spectral radius of $G$. In this paper, we first give
some upper bounds for the $A_\alpha$ spectral radius of a digraph and we also
characterize the extremal digraphs attaining these bounds. Moreover, we define
the $A_\alpha$ energy of a digraph $G$ as
$E^{A_\alpha}(G)=\sum\limits_{i=1}^n(\lambda^\alpha_i(G))^2$, where $n$ is the
number of vertices and $\lambda^\alpha_i(G)$ $(i=1,2,\ldots,n)$ are the
eigenvalues of $A_\alpha(G)$. We obtain a formula for $E^{A_\alpha}(G)$, and
give a lower and upper bounds for $E^{A_\alpha}(G)$ and characterize the
extremal digraphs that attain the lower and upper bounds. Finally, we
characterize the extremal digraphs with maximum and minimum $A_\alpha$ energy
among all directed trees and unicyclic digraphs, respectively.
