Unipotent elements and generalized exponential maps

Sobaje, Paul

doi:10.1016/j.aim.2018.05.015

Cited by 3 publications

(4 citation statements)

References 27 publications

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“…This has been observed by Sobaje in [Sob15]. Furthermore, the non-separably good characteristics case is addressed in [Sob18, § 7]. The author explains there why Springer isomorphisms fail to exist without this assumption.…”

Section: Introductionmentioning

confidence: 76%

“…When is simple, Sobaje reminds the reader of the existence of Springer isomorphisms in separably good characteristics (see [Sob15, Theorem 1.1 and Remark 2]). Moreover, in [Sob18, § 7] the author investigates the non-separably good characteristic case. He also emphasises that in separably good characteristics one can always find an isomorphism that restricts to an isomorphism of reduced schemes for any Borel subgroup .…”

Section: Springer Isomorphisms and -Formalismmentioning

confidence: 99%

“…Note that here is endowed with the group law induced by the Baker–Campbell–Hausdorff law (which is well-defined, see the preamble of [Sei00, § 5]). When , Sobaje explains in [Sob18, Theorem 6.0.2] that there exists a unique Springer isomorphism that restricts to , whose tangent map is the identity and that is compatible with the -power. Note that when , maps satisfying the three aforementioned requirements still exist, but this time, there are many of such maps.…”

Section: Springer Isomorphisms and -Formalismmentioning

confidence: 99%

“…Note that when , maps satisfying the three aforementioned requirements still exist, but this time, there are many of such maps. Sobaje study and classify in [Sob18, Theorem 6.0.2] a specific class of such maps, the so-called generalised exponential maps.…”

Section: Springer Isomorphisms and -Formalismmentioning

confidence: 99%

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Integration questions in separably good characteristics

Jeannin¹

2023

Compositio Math.

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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.

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Section: Introductionmentioning

confidence: 76%

Section: Springer Isomorphisms and -Formalismmentioning

confidence: 99%

Section: Springer Isomorphisms and -Formalismmentioning

confidence: 99%

Section: Springer Isomorphisms and -Formalismmentioning

confidence: 99%

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Integration questions in separably good characteristics

Jeannin¹

2023

Compositio Math.

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Jordan blocks of nilpotent elements in some irreducible representations of classical groups in good characteristic

Korhonen

2021

Journal of Pure and Applied Algebra

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Let G be a classical group with natural module V and Lie algebra g over an algebraically closed field K of good characteristic. For rational irreducible representations f : G → GL(W ) occurring as composition factors of V ⊗ V * , ∧ 2 (V ), and S 2 (V ), we describe the Jordan normal form of df (e) for all nilpotent elements e ∈ g. The description is given in terms of the Jordan block sizes of the action of e on V ⊗ V * , ∧ 2 (V ), and S 2 (V ), for which recursive formulae are known. Our results are in analogue to earlier work (Proc. Amer. Math. Soc., 2019), where we considered these same representations and described the Jordan normal form of f (u) for every unipotent element u ∈ G.

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