  We introduce the Primitive Eulerian polynomial $P_{\cal A}(z)$ of a central
hyperplane arrangement ${\cal A}$. It is a reparametrization of its
cocharacteristic polynomial. Previous work of the first author implicitly show
that, for simplicial arrangements, $P_{\cal A}(z)$ has nonnegative
coefficients. For reflection arrangements of type A and B, the same work
interprets the coefficients of $P_{\cal A}(z)$ using the (flag)excedance
statistic on (signed) permutations. The main result of this article is to
provide an interpretation of the coefficients of $P_{\cal A}(z)$ for all
simplicial arrangements only using the geometry and combinatorics of ${\cal
A}$.
  This new interpretation sheds more light to the case of reflection
arrangements and, for the first time, gives combinatorial meaning to the
coefficients of the Primitive Eulerian polynomial of the reflection arrangement
of type D. In type B, we find a connection between the Primitive Eulerian
polynomial and the $1/2$-Eulerian polynomial of Savage and Viswanathan (2012).
We present some real-rootedness results and conjectures for $P_{\cal A}(z)$.
