Centralizers of Nilpotent Elements in Basic Classical Lie Superalgebras in Good Characteristic

Abstract

AbstractLet $$\mathfrak {g}=\mathfrak {g}_{\bar{0}}\oplus \mathfrak {g}_{\bar{1}}$$ g = g 0 ¯ ⊕ g 1 ¯ be a basic classical Lie superalgebra over an algebraically closed field $$\mathbb {K}$$ K whose characteristic $$p>0$$ p > 0 is a good prime for $$\mathfrak {g}$$ g . Let $$G_{\bar{0}}$$ G 0 ¯ be the reductive algebraic group over $$\mathbb {K}$$ K such that $$\textrm{Lie}(G_{\bar{0}})=\mathfrak {g}_{\bar{0}}$$ Lie ( G 0 ¯ ) = g 0 ¯ . Suppose $$e\in \mathfrak {g}_{\bar{0}}$$ e ∈ g 0 ¯ is nilpotent. Write $$\mathfrak {g}^{e}$$ g e for the centralizer of e in $$\mathfrak {g}$$ g and $$\mathfrak {z}(\mathfrak {g}^{e})$$ z ( g e ) for the centre of $$\mathfrak {g}^{e}$$ g e . We calculate a basis for $$\mathfrak {g}^{e}$$ g e and $$\mathfrak {z}(\mathfrak {g}^{e})$$ z ( g e ) by using associated cocharacters $$\tau :\mathbb {K}^{\times }\rightarrow G_{\bar{0}}$$ τ : K × → G 0 ¯ of e. In addition, we give the classification of e which are reachable, strongly reachable or satisfy the Panyushev property for exceptional Lie superalgebras $$D(2,1;\alpha )$$ D ( 2 , 1 ; α ) , G(3) and F(4).