Modular finite $W$-algebras

Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $G$ be a connected reductive algebraic group over $k$. Under some standard hypothesis on $G$, we give a direct approach to the finite $W$-algebra $U(mathfrak g,e)$ associated to a nilpotent element $e in mathfrak g = operatorname{Lie} G$. We prove a PBW theorem and deduce a number of consequences, then move on to define and study the $p$-centre of $U(mathfrak g,e)$, which allows us to define reduced finite $W$-algebras $U_eta(mathfrak g,e)$ and we verify that they coincide with those previously appearing in the work of Premet. Finally, we prove a modular version of Skryabins equivalence of categories, generalizing recent work of the second author.