Centralizers of nilpotent elements in basic classical Lie superalgebras in good characteristic

TL;DR

This paper computes bases for the centralizer and its center of nilpotent elements in basic classical Lie superalgebras over fields with good characteristic, and classifies certain nilpotent elements in exceptional cases.

Contribution

It provides explicit bases for centralizers and their centers in basic classical Lie superalgebras and classifies special nilpotent elements in exceptional Lie superalgebras.

Findings

Explicit bases for  and 's center in classical Lie superalgebras.

Classification of reachable and Panyushev property-satisfying nilpotent elements.

Results applicable to Lie superalgebras over fields with good characteristic.

Abstract

Let \mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}} be a basic classical Lie superalgebra over an algebraically closed field \mathbb{K} whose characteristic p>0 is a good prime for \mathfrak{g}. Let G_{\bar{0}} be the reductive algebraic group over \mathbb{K} such that \mathrm{Lie}(G_{\bar{0}})=\mathfrak{g}_{\bar{0}}. Suppose e\in\mathfrak{g}_{\bar{0}} is nilpotent. Write \mathfrak{g}^{e} for the centralizer of e in \mathfrak{g} and \mathfrak{z}(\mathfrak{g}^{e}) for the centre of \mathfrak{g}^{e}. We calculate a basis for \mathfrak{g}^{e} and \mathfrak{z}(\mathfrak{g}^{e}) by using associated cocharacters \tau:\mathbb{K}^{\times}\rightarrow G_{\bar{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), G(3) and F(4).