Let G be a connected reductive algebraic group over an algebraically closed field k. In a recent paper, Bate, Martin, Röhrle and Tange show that every (smooth) subgroup of G is separable provided that the characteristic of k is very good for G. Here separability of a subgroup means that its scheme-theoretic centralizer in G is smooth. Serre suggested extending this result to arbitrary, possibly non-smooth, subgroup schemes of G. The aim of this note is to prove this more general result. Moreover, we provide a condition on the characteristic of k that is necessary and sufficient for the smoothness of all centralizers in G. We finally relate this condition to other standard hypotheses on connected reductive groups.2010 Mathematics Subject Classification. 20G15.
For a field k, let G be a reductive k-group and V an affine k-variety on which G acts. Using the notion of cocharacter-closed G(k)-orbits in V , we prove a rational version of the celebrated Hilbert-Mumford Theorem from geometric invariant theory. We initiate a study of applications stemming from this rationality tool. A number of examples are discussed to illustrate the concept of cocharacter-closure and to highlight how it differs from the usual Zariski-closure.2010 Mathematics Subject Classification. 20G15 (14L24).Conversely, if G · v is closed then lim a→0 λ(a) · v lies in G · v for all λ such that the limit exists (cf. Section 2.4).A strengthening of the Hilbert-Mumford Theorem due to Hesselink [27], Kempf [29] and Rousseau [48] shows that if G · v is not closed, then there is a class of so-called "optimal" cocharacters λ opt such that the limit lim a→0 λ opt (a) · v exists in V but lies outside G · v. Each cocharacter λ opt enjoys some nice properties: for instance, if G is connected, then the parabolic subgroup P λopt associated to λ opt contains the stabiliser G v . Moreover, if G, V and the G-action are defined over a perfect subfield k 0 of k, k/k 0 is algebraic and v ∈ V (k 0 ), then λ opt is Gal(k/k 0 )-fixed and hence is defined over k 0 . The (strengthened) Hilbert-Mumford Theorem has become an indispensable tool in algebraic group theory and has numerous applications in geometric invariant theory and beyond [41]: e.g., geometric complexity theory [39], [40], nilpotent and unipotent elements of reductive groups [18], [44], [43], [2], moduli spaces of bundles [24], good quotients in geometric invariant theory [25], Hilbert schemes [42], moduli spaces of sheaves [28], the structure of the Horn cone [45], Kähler geometry [56], filtrations for representations of quivers [58], symplectic quotients [41, App. 2C], degenerations of modules [59] and G-complete reducibility [4], [9].Now suppose k is an arbitrary field, not necessarily algebraically closed. The orbit G · v is a union of G(k)-orbits. The structure of this set of G(k)-orbits can be very intricate. For instance, if w ∈ G·v and v, w are k-points then one can ask whether w is G(k)-conjugate to v. The answer is no in general; if k is perfect then this is controlled by the Galois 1-cohomology of G v (k s ) (see Remark 7.3(iv) or [10]). Things only get more complicated when one considers the G(k)-orbits that are contained in G · v. Orbits of actions of reductive groups over nonalgebraically closed fields have come under increasing attention, particularly from number theorists. For instance, suppose k is a global function field, let v ∈ V (k) and let C be the set of all w ∈ V (k) such that w is G(k ν )-conjugate to v for every completion k ν of k; then Conrad showed that C is a finite union of G(k)-orbits [19, Thm. 1.3.3]. Bremigan studied the strong topology of the orbits when k is a local field [17] (see also Remark 7.3(v) below). Now let V be the Lie algebra g with the adjoint action of G. When k is perfect, J. Levy proved that if x ∈ g, then any t...
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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