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Integration questions in separably good characteristics
- Part of
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- Journal:
- Compositio Mathematica / Volume 159 / Issue 5 / May 2023
- Published online by Cambridge University Press:
- 24 April 2023, pp. 890-932
- Print publication:
- May 2023
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- Article
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Let
$G$ be a reductive group over an algebraically closed field 
$k$ of separably good characteristic 
$p>0$ for 
$G$. Under these assumptions, a Springer isomorphism 
$\phi : \mathcal {N}_{\mathrm {red}}(\mathfrak {g}) \rightarrow \mathcal {V}_{\mathrm {red}}(G)$ from the nilpotent scheme of 
$\mathfrak {g}$ to the unipotent scheme of 
$G$ always exists and allows one to integrate any 
$p$-nilpotent element of 
$\mathfrak {g}$ into a unipotent element of 
$G$. One should wonder whether such a punctual integration can lead to an integration of restricted 
$p$-nil 
$p$-subalgebras of 
$\mathfrak {g}= \operatorname {Lie}(G)$. We provide a counter-example of the existence of such an integration in general, as well as criteria to integrate some restricted 
$p$-nil 
$p$-subalgebras of 
$\mathfrak {g}$ (that are maximal in a certain sense). This requires the generalisation of the notion of infinitesimal saturation first introduced by Deligne and the extension of one of his theorems on infinitesimally saturated subgroups of 
$G$ to the previously mentioned framework.
