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).
Categories: math.RT