Nilpotent centralizers and good filtrations (Q6620730)

Let \(G\) be a connected reductive algebraic group over an algebraically closed field of positive characteristic \(p\). For a connected reductive subgroup \(H\) of \(G\), the pair \((G,H)\) is called a \textit{Donkin pair} (or \textit{good filtration pair}) if each \(G\)-module with a good filtration restricts to an \(H\)-module with a good filtration.\N\NIf the characteristic \(p\) is good for \(G\) then, whenever \(x\) is a nilpotent element of the Lie algebra of \(G\), the centraliser \(G^{x}\) admits a decomposition \(G^{x} = G_{\mathrm{red}}^{x} \ltimes G_{\mathrm{unip}}^x\), where \(G_{\mathrm{unip}}^x\) is a connected unipotent group and \((G_{\mathrm{red}}^{x})^{\circ}\) is reductive.\N\NThe main result of the paper states: If \(p\) is good for \(G\), then \((G,(G_\mathrm{red}^{x})^{\circ})\) is a Donkin pair, for all nilpotent \(x\) in the Lie algebra of \(G\). In fact the same holds slightly more generally: If \(p\) is \textit{pretty good} in the sense of \textit{S. Herpel} [Trans. Am. Math. Soc. 365, No. 7, 3753--3774 (2013; Zbl 1298.20057)], then the component group of \(G_{\mathrm{red}}^{x}\) has order coprime to \(p\), the category of finite-dimensional \(G_{\mathrm{red}}^{x}\) modules is a highest weight category and Corollary 1.2 of the present paper states: In this situation, \((G,G_{\mathrm{red}}^{x}\)) is a Donkin pair for all nilpotent \(x\).\N\NThe proof proceeds by reducing to the case that \(G\) is quasi-simple and then proceeding case-by-case on the Lie type of \(G\). This involves collecting together previously-known examples of Donkin pairs from prior work of various authors; exhibiting some more families Donkin pairs; and then making use of explicit knowledge of nilpotent element centralisers in these groups, given for instance by \textit{M. W. Liebeck} and \textit{G. M. Seitz} [Unipotent and nilpotent classes in simple algebraic groups and Lie algebras. Providence, RI: American Mathematical Society (AMS) (2012; Zbl 1251.20001)].