In this paper, we study the center and representation theory of finite $W$-algebra $\mathcal{T}(\mathfrak{g},e)$ over an algebraically closed field $\mathds{k}$ of characteristic $p\gg0$. We obtain an analogy of Veldkamp's theorem on the center, and then decide the structure of the reduced center of the associated reduced $W$-algebra $\mathcal{T}_\chi(\mathfrak{g},e)$ via that of the reduced enveloping algebra $U_\chi(\mathfrak{g})$. For the maximal spectrum $\text{Specm}(Z)$ of the center $Z=Z(\mathcal{T}(\mathfrak{g},e))$, we show that its Azumaya locus coincides with its smooth locus of smooth points, the former of which reflects irreducible representations of the maximal dimension for $\mathcal{T}(\mathfrak{g},e)$. 查看全文>>