Tits' work on unipotent groups in nonzero characteristic

AbstractFor a Lie algebra $L$ over an algebraically closed field $F$ of nonzero characteristic, every finite dimensional $L$-module can be decomposed into a direct sum of submodules such that all composition factors of a summand have the same character. Using the concept of a character cluster, this result is generalised to fields which are not algebraically closed. Also, it is shown that if the soluble Lie algebra $L$ is in the saturated formation $\mathfrak{F}$ and if $V, W$ are irreducible $L$-modules with the same cluster and the $p$-operation vanishes on the centre of the $p$-envelope used, then $V, W$ are either both $\mathfrak{F}$-central or both $\mathfrak{F}$-eccentric. Clusters are used to generalise the construction of induced modules.

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Unipotent Groups

Introduction to Affine Group Schemes - Graduate Texts in Mathematics ◽

10.1007/978-1-4612-6217-6_8 ◽

1979 ◽

pp. 62-67

Author(s):

William C. Waterhouse

Keyword(s):

Unipotent Groups

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Linear Actions of Unipotent Groups

Algebraic Theory of Locally Nilpotent Derivations - Encyclopaedia of Mathematical Sciences ◽

10.1007/978-3-662-55350-3_6 ◽

2017 ◽

pp. 167-191

Author(s):

Gene Freudenburg

Keyword(s):

Unipotent Groups

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Shintani descent for algebraic groups and almost characters of unipotent groups

Compositio Mathematica ◽

10.1112/s0010437x16007429 ◽

2016 ◽

Vol 152 (8) ◽

pp. 1697-1724 ◽

Cited By ~ 1

Author(s):

Tanmay Deshpande

Keyword(s):

Finite Field ◽

Algebraic Group ◽

Algebraic Groups ◽

Unipotent Group ◽

Connected Component ◽

Unipotent Groups ◽

Character Sheaves ◽

Treat All ◽

Trace Of Frobenius

In this paper, we extend the notion of Shintani descent to general (possibly disconnected) algebraic groups defined over a finite field $\mathbb{F}_{q}$. For this, it is essential to treat all the pure inner $\mathbb{F}_{q}$-rational forms of the algebraic group at the same time. We prove that the notion of almost characters (introduced by Shoji using Shintani descent) is well defined for any neutrally unipotent algebraic group, i.e. an algebraic group whose neutral connected component is a unipotent group. We also prove that these almost characters coincide with the ‘trace of Frobenius’ functions associated with Frobenius-stable character sheaves on neutrally unipotent groups. In the course of the proof, we also prove that the modular categories that arise from Boyarchenko and Drinfeld’s theory of character sheaves on neutrally unipotent groups are in fact positive integral, confirming a conjecture due to Drinfeld.

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