scholarly journals Semisimplification for subgroups of reductive algebraic groups

2020 ◽  
Vol 8 ◽  
Author(s):  
Michael Bate ◽  
Benjamin Martin ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Invariant Theory ◽  
Algebraic Groups ◽  
Geometric Invariant ◽  
Reductive Algebraic Group ◽  
Complete Reducibility ◽  
Special Cases ◽  
Closed Fields ◽  
Completely Reducible ◽  
Algebraically Closed Fields

Let G be a reductive algebraic group—possibly non-connected—over a field k, and let H be a subgroup of G. If $G= {GL }_n$ , then there is a degeneration process for obtaining from H a completely reducible subgroup $H'$ of G; one takes a limit of H along a cocharacter of G in an appropriate sense. We generalise this idea to arbitrary reductive G using the notion of G-complete reducibility and results from geometric invariant theory over non-algebraically closed fields due to the authors and Herpel. Our construction produces a G-completely reducible subgroup $H'$ of G, unique up to $G(k)$ -conjugacy, which we call a k-semisimplification of H. This gives a single unifying construction that extends various special cases in the literature (in particular, it agrees with the usual notion for $G= GL _n$ and with Serre’s ‘G-analogue’ of semisimplification for subgroups of $G(k)$ from [19]). We also show that under some extra hypotheses, one can pick $H'$ in a more canonical way using the Tits Centre Conjecture for spherical buildings and/or the theory of optimal destabilising cocharacters introduced by Hesselink, Kempf, and Rousseau.

scholarly journals RELATIVE COMPLETE REDUCIBILITY AND NORMALIZED SUBGROUPS

2020 ◽  
Vol 8 ◽  
Author(s):  
MAIKE GRUCHOT ◽  
ALASTAIR LITTERICK ◽  
GERHARD RÖHRLE
Keyword(s):  
Automorphism Group ◽  
Algebraic Group ◽  
Lie Algebra ◽  
Reductive Algebraic Group ◽  
Identity Component ◽  
Complete Reducibility ◽  
Closed Fields ◽  
Completely Reducible ◽  
Reductive Subgroup ◽  
The Absolute

We study a relative variant of Serre’s notion of $G$ -complete reducibility for a reductive algebraic group $G$ . We let $K$ be a reductive subgroup of $G$ , and consider subgroups of $G$ that normalize the identity component $K^{\circ }$ . We show that such a subgroup is relatively $G$ -completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ }$ is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$ , as well as ‘rational’ versions over nonalgebraically closed fields.

Complete reducibility of subgroups of reductive algebraic groups over non-perfect fields IV: An 𝐹4 example

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Falk Bannuscher ◽  
Alastair Litterick ◽  
Tomohiro Uchiyama
Keyword(s):  
Algebraic Group ◽  
Algebraic Groups ◽  
Closed Field ◽  
Reductive Algebraic Group ◽  
Complete Reducibility ◽  
Completely Reducible ◽  
Rationality Problems ◽  
Reductive Algebraic Groups

Abstract Let 𝑘 be a non-perfect separably closed field. Let 𝐺 be a connected reductive algebraic group defined over 𝑘. We study rationality problems for Serre’s notion of complete reducibility of subgroups of 𝐺. In particular, we present the first example of a connected non-abelian 𝑘-subgroup 𝐻 of 𝐺 that is 𝐺-completely reducible but not 𝐺-completely reducible over 𝑘, and the first example of a connected non-abelian 𝑘-subgroup H ′ H^{\prime} of 𝐺 that is 𝐺-completely reducible over 𝑘 but not 𝐺-completely reducible. This is new: all previously known such examples are for finite (or non-connected) 𝐻 and H ′ H^{\prime} only.

scholarly journals Complete reducibility: variations on a theme of Serre

2021 ◽  
Author(s):  
Maike Gruchot ◽  
Alastair Litterick ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Spherical Building ◽  
Reductive Algebraic Group ◽  
Identity Component ◽  
General Properties ◽  
Complete Reducibility ◽  
Special Cases ◽  
Variations On A Theme

AbstractIn this note, we unify and extend various concepts in the area of G-complete reducibility, where G is a reductive algebraic group. By results of Serre and Bate–Martin–Röhrle, the usual notion of G-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of G. We show that other variations of this notion, such as relative complete reducibility and $$\sigma $$ σ -complete reducibility, can also be viewed as special cases of this building-theoretic definition, and hence a number of results from these areas are special cases of more general properties.

scholarly journals Nilpotent orbits of exceptional Lie algebras over algebraically closed fields of bad characteristic

1985 ◽  
Vol 38 (3) ◽  
pp. 330-350 ◽  
Author(s):  
D. F. Holt ◽  
N. Spaltenstein
Keyword(s):  
Finite Field ◽  
Lie Algebras ◽  
Closed Field ◽  
Nilpotent Orbits ◽  
Reductive Algebraic Group ◽  
Exceptional Lie Algebras ◽  
Closed Fields ◽  
Characteristic 3 ◽  
Algebraically Closed Fields

AbstractThe classification of the nilpotent orbits in the Lie algebra of a reductive algebraic group (over an algebraically closed field) is given in all the cases where it was not previously known (E7 and E8 in bad characteristic, F4 in characteristic 3). The paper exploits the tight relation with the corresponding situation over a finite field. A computer is used to study this case for suitable choices of the finite field.

scholarly journals Adelic descent theory

2017 ◽  
Vol 153 (8) ◽  
pp. 1706-1746
Author(s):  
Michael Groechenig
Keyword(s):  
Vector Bundles ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Reductive Algebraic Group ◽  
Perfect Complexes ◽  
Descent Theory ◽  
Closed Fields ◽  
Ring Of Continuous Functions ◽  
Reconstruction Theorem ◽  
Algebraically Closed Fields

A result of André Weil allows one to describe rank $n$ vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set $\text{GL}_{n}(\mathbb{A})$ of regular matrices over the ring of adèles (over algebraically closed fields, this result is also known to extend to $G$-torsors for a reductive algebraic group $G$). In the present paper we develop analogous adelic descriptions for vector and principal bundles on arbitrary Noetherian schemes, by proving an adelic descent theorem for perfect complexes. We show that for Beilinson’s co-simplicial ring of adèles $\mathbb{A}_{X}^{\bullet }$, we have an equivalence $\mathsf{Perf}(X)\simeq |\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })|$ between perfect complexes on $X$ and cartesian perfect complexes for $\mathbb{A}_{X}^{\bullet }$. Using the Tannakian formalism for symmetric monoidal $\infty$-categories, we conclude that a Noetherian scheme can be reconstructed from the co-simplicial ring of adèles. We view this statement as a scheme-theoretic analogue of Gelfand–Naimark’s reconstruction theorem for locally compact topological spaces from their ring of continuous functions. Several results for categories of perfect complexes over (a strong form of) flasque sheaves of algebras are established, which might be of independent interest.

Differentially algebraic group chunks

1990 ◽  
Vol 55 (3) ◽  
pp. 1138-1142 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Algebraic Group ◽  
Characteristic Zero ◽  
Quantifier Elimination ◽  
Closed Field ◽  
Zariski Topology ◽  
First Order ◽  
Differential Field ◽  
Closed Fields ◽  
Differential Fields ◽  
Algebraically Closed Fields

We point out that a group first order definable in a differentially closed field K of characteristic 0 can be definably equipped with the structure of a differentially algebraic group over K. This is a translation into the framework of differentially closed fields of what is known for groups definable in algebraically closed fields (Weil's theorem).I restrict myself here to showing (Theorem 20) how one can find a large “differentially algebraic group chunk” inside a group defined in a differentially closed field. The rest of the translation (Theorem 21) follows routinely, as in [B].What is, perhaps, of interest is that the proof proceeds at a completely general (soft) model theoretic level, once Facts 1–4 below are known.Fact 1. The theory of differentially closed fields of characteristic 0 is complete and has quantifier elimination in the language of differential fields (+, ·,0,1, −1,d).Fact 2. Affine n-space over a differentially closed field is a Noetherian space when equipped with the differential Zariski topology.Fact 3. If K is a differentially closed field, k ⊆ K a differential field, and a and are in k, then a is in the definable closure of k ◡ iff a ∈ ‹› (where k ‹› denotes the differential field generated by k and).Fact 4. The theory of differentially closed fields of characteristic zero is totally transcendental (in particular, stable).

On a Family of Distributions obtained from Orbits

1986 ◽  
Vol 38 (1) ◽  
pp. 179-214 ◽  
Author(s):  
James Arthur
Keyword(s):  
Conjugacy Class ◽  
Algebraic Group ◽  
Difficult Case ◽  
Reductive Algebraic Group ◽  
Section 8 ◽  
Special Cases ◽  
Image Position ◽  
The Right ◽  
Family Of Distributions ◽  
Opposite Extreme

Suppose that G is a reductive algebraic group defined over a number field F. The trace formula is an identityof distributions. The terms on the right are parametrized by “cuspidal automorphic data”, and are defined in terms of Eisenstein series. They have been evaluated rather explicitly in [3]. The terms on the left are parametrized by semisimple conjugacy classes and are defined in terms of related G(A) orbits. The object of this paper is to evaluate these terms.In previous papers we have already evaluated in two special cases. The easiest case occurs when corresponds to a regular semisimple conjugacy class in G(F). We showed in Section 8 of [1] that for such an , could be expressed as a weighted orbital integral over the conjugacy class of σ. (We actually assumed that was “unramified”, which is slightly more general.) The most difficult case is the opposite extreme, in which corresponds to {1}.

scholarly journals On the involution fixity of exceptional groups of Lie type

2018 ◽  
Vol 28 (03) ◽  
pp. 411-466 ◽  
Author(s):  
Timothy C. Burness ◽  
Adam R. Thomas
Keyword(s):  
Algebraic Geometry ◽  
Fixed Points ◽  
Algebraic Groups ◽  
Natural Analogue ◽  
Transitive Action ◽  
Exceptional Groups ◽  
Closed Fields ◽  
Suzuki Group ◽  
Algebraically Closed Fields ◽  
Groups Of Lie Type

The involution fixity [Formula: see text] of a permutation group [Formula: see text] of degree [Formula: see text] is the maximum number of fixed points of an involution. In this paper we study the involution fixity of primitive almost simple exceptional groups of Lie type. We show that if [Formula: see text] is the socle of such a group, then either [Formula: see text], or [Formula: see text] and [Formula: see text] is a Suzuki group in its natural [Formula: see text]-transitive action of degree [Formula: see text]. This bound is best possible and we present more detailed results for each family of exceptional groups, which allows us to determine the groups with [Formula: see text]. This extends recent work of Liebeck and Shalev, who established the bound [Formula: see text] for every almost simple primitive group of degree [Formula: see text] with socle [Formula: see text] (with a prescribed list of exceptions). Finally, by combining our results with the Lang–Weil estimates from algebraic geometry, we determine bounds on a natural analogue of involution fixity for primitive actions of exceptional algebraic groups over algebraically closed fields.

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