In this paper, we study the center Z of the finite W-algebra $${\mathcal{T}}({\mathfrak{g}},e)$$ associated with a semi-simple Lie algebra $$\mathfrak{g}$$ over an algebraically closed field $$\mathbb{k}$$ of characteristic p≫0, and an arbitrarily given nilpotent element $$e \in{\mathfrak{g}} $$ We obtain an analogue of Veldkamp’s theorem on the center. For the maximal spectrum Specm(Z), we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $${\mathcal{T}}({\mathfrak{g}},e)$$ .

在本文中，我们研究了与半简单李代数 $$\mathfrak{相关的有限W代数 $${\mathcal{T}}({\mathfrak{g}},e)$$ 的中心Z g}$$ 在特征p ≫0的代数闭域 $$\mathbb{k}$$ 和任意给定的幂零元素 $$e \in{\mathfrak{g}} $$ 上我们得到 Veldkamp 的类比中心定理。对于最大谱 Specm( Z )，我们证明了它的 Azumaya 轨迹与其平滑点的平滑轨迹重合。 前一个轨迹反映了$${\mathcal{T}}({\mathfrak{g}},e)$$ 的最大维数的不可约表示。