Closed orbits and uniform $S$-instability in geometric invariant theory

Let G be a reductive affine algebraic group and let X be an affine algebraic G-variety. We establish a (poly)stability criterion for points x ∈ X in terms of intrinsically defined closed subgroups Hx of G and relate it with the numerical criterion of Mumford and with Richardson and Bate-Martin-Röhrle criteria, in the case X = G N . Our criterion builds on a close analogue of a theorem of Mundet and Schmitt on polystability and allows the generalization to the algebraic group setting of results of Johnson-Millson and Sikora about complex representation varieties of finitely presented groups. By well established results, it also provides a restatement of the non-abelian Hodge theorem in terms of stability notions. the image of the representation, x(Γ) ⊂ G, is irreducible or completely reducible as a subgroup of G.In this paper, we generalize these relationships to a bigger class of affine G-varieties, where G is an affine reductive group, not necessarily irreducible, defined over an algebraically closed field k, of characteristic zero.Besides its intrinsic relevance as stability criteria, the constructions examined here perfectly agree with other known results for certain specific classes of G-varieties. For example, the non-abelian Hodge theorem (see [9] or [32]), in the case of complex reductive groups, can be restated as a correspondence between stable points in distinct (however homeomorphic) varieties. Also, from our set up, one can recover the Mundet-Schmitt criterion for polystability [18].To describe our main results, let Y (G) denote the set of one parameter subgroups (1PS for short) of G, that is, homomorphisms λ from the multiplicative group k × to G. Given a 1PS λ ∈ Y (G) and g ∈ G, the morphism t → λ(t)gλ(t) −1 may or may not extend to k (as a morphism of affine varieties). In case there is such an extension, we say that lim t→0 λ(t)gλ(t) −1 exists. It is known that P (λ) = g ∈ G : lim t→0 λ(t)gλ(t) −1

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“…This result motivates the following definition introduced by Levy (see [15] and also [3,Definition 3.8]). …”

Section: Ps Polystabilitymentioning

confidence: 72%

Stability of Affine G-Varieties and Irreducibility in Reductive Groups

Florentino

Casimiro

2012

Int. J. Math.

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“…The following theorem answers a question of Serre: [4,Ex. 5.12], we showed that Theorem 1.1 holds when G = GL(V ).…”

Section: Introductionmentioning

confidence: 94%

“…Tits [4] utilizes methods from geometric invariant theory and the concept of optimal destabilizing parabolic subgroups.…”

Section: The Centre Conjecture For Spherical Buildingsmentioning

confidence: 99%

“…In this sense, Serre's notion generalizes the usual concept of complete reducibility in representation theory. For more details and further results on this notion, see [12,13,1,3], and [4].…”

Section: Introductionmentioning

confidence: 99%

“…Then Σ has an Aut Δ k -centre.Remark 2.4.Suppose G is semisimple and k is a perfect field. It follows from[15, 5.7.2] that Aut G is an algebraic group also defined over k. In[4, Thm. 5.31], we give a uniform proof of the following special case of the Centre Conjecture: Let H be a…”

mentioning

confidence: 99%

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Maximal subalgebras of Cartan type in the exceptional Lie algebras

Herpel

Stewart

2015

Sel. Math. New Ser.

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Abstract. In this paper we initiate the study of the maximal subalgebras of exceptional simple classical Lie algebras g over algebraically closed fields k of positive characteristic p, such that the prime characteristic is good for g. In this paper we deal with what is surely the most unnatural case; that is, where the maximal subalgebra in question is a simple subalgebra of non-classical type. We show that only the first Witt algebra can occur as a subalgebra of g and give an explicit classification of when it is maximal in g.

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Closed orbits and uniform $S$-instability in geometric invariant theory

Cited by 39 publications

References 36 publications

Stability of Affine G-Varieties and Irreducibility in Reductive Groups

Stability of Affine G-Varieties and Irreducibility in Reductive Groups

Complete reducibility and separable field extensions

Maximal subalgebras of Cartan type in the exceptional Lie algebras

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