Let G be a connected reductive linear algebraic group. The aim of this note
is to settle a question of J-P. Serre concerning the behaviour of his notion of
G-complete reducibility under separable field extensions. Part of our proof
relies on the recently established Tits Centre Conjecture for the spherical
building of the reductive group G.Comment: 5 pages; to appear in Comptes rendus Mathematiqu

Bate, Michael

Martin, Benjamin

Roehrle, Gerhard

English

arXiv

Röhrle, Gerhard

Max Planck Society eDoc Server

Complete reducibility and separable field extensions

Michael Bate

Benjamin Martin

Gerhard Röhrle

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 348 (2010) 495–497Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comGroup Theory/GeometryComplete reducibility and separable field extensionsComplète réductibilité et extensions de corps séparablesMichael Bate a, Benjamin Martin b, Gerhard Röhrle ca Department of Mathematics, University of York, York YO10 5DD, United Kingdomb Mathematics and Statistics Department, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealandc Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germanya r t i c l e i n f o a b s t r a c tArticle history:Received 27 February 2010Accepted after revision 6 April 2010Available online 24 April 2010Presented by Jean-Pierre SerreLet G be a connected reductive linear algebraic group. The aim of this note is to settlea question of J-P. Serre concerning the behaviour of his notion of G-complete reducibilityunder separable field extensions. Part of our proof relies on the recently established TitsCentre Conjecture for the spherical building of the reductive group G .© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éSoit G un groupe algébrique linéaire réductif connexe. Le but de cette Note est derépondre à une question de J-P. Serre concernant le comportement par extensions decorps séparables, de la notion de G-réductibilité complète qu’il a introduite. Une partiede nos arguments repose sur la solution récente de la conjecture du centre de Tits pourles immeubles sphériques du groupe réductif G .© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionThroughout, G denotes a connected reductive linear algebraic group defined over a field k. Following Serre, [13], a sub-group H of G is called G-completely reducible over k (G-cr over k) if whenever H is contained in a k-defined parabolicsubgroup P of G , there exists a k-defined Levi subgroup of P containing H . In case V is a finite dimensional k-vector spaceand G = GL(V ), a subgroup H of G is G-completely reducible over k precisely when V is a semisimple H-module, [13, 1.3,3.2.2]. In this sense, Serre’s notion generalizes the usual concept of complete reducibility in representation theory. For moredetails and further results on this notion, see [12,13,1,3], and [4].The following theorem answers a question of Serre:Theorem 1.1. Suppose k1/k is a separable extension of fields. Let G be a reductive group defined over k, and let H be a k-definedsubgroup of G. Then H is G-completely reducible over k if and only if H is G-completely reducible over k1 .The reverse implication in Theorem 1.1 is proved in [4, Thm. 5.11]. The proof of [4, Thm. 5.11] rests on a generalrationality result, [4, Thm. 3.1], concerning G-orbits in an affine variety. We present a proof of the forward direction of thestatement in Section 3 based on the recently established Tits Centre Conjecture, Theorem 2.3.E-mail addresses: meb505@york.ac.uk (M. Bate), B.Martin@math.canterbury.ac.nz (B. Martin), gerhard.roehrle@rub.de (G. Röhrle).1631-073X/$ – see front matter © 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2010.04.013496 M. Bate et al. / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 495–497Remarks 1.2. (i) In [4, Ex. 5.12], we showed that Theorem 1.1 holds when G = GL(V ).(ii) Theorem 1.1 was proved in [1, Thm. 5.8] for k perfect, by passing back and forth between k and its algebraic closurek and between k1 and k. In general this approach fails, because the extension k/k need not be separable.(iii) There are examples showing that each implication in Theorem 1.1 fails without the separability assumption on theextension k1/k; see [1, Ex. 5.11] and [3, Ex. 7.22].2. The Centre Conjecture for spherical buildingsLet Δk denote the spherical building of G over k, [15, Sec. 5]: the simplices of Δk correspond to k-defined parabolicsubgroups of G . Given a k-defined parabolic subgroup P of G , we denote the simplex corresponding to P in Δk by σP .Throughout, we identify Δk with its geometric realization, which is a bouquet of spheres [13].An apartment in Δk consists of the simplices σP corresponding to all k-defined parabolic subgroups P of G that containa fixed maximal k-split torus of G; it is a subcomplex whose geometric realization is a sphere. Any two points of Δk liein a common apartment. We say that x, y ∈ Δk are opposite if they are opposite in some apartment that contains themboth. It can be shown that if x and y are opposite in some apartment that contains them both, then they are opposite inany apartment that contains them both. If x, y ∈ Δk are not opposite, then there is a unique geodesic joining them, [13,§2.1.4]. Two simplices σP and σQ are said to be opposite if every point of σP is opposite a point of σQ , [13, §2.1.4]. Interms of parabolic subgroups of G , the simplices σP and σQ corresponding to k-defined parabolic subgroups P and Q of Gare opposite in Δk if and only if P ∩ Q is a common Levi subgroup of P and Q (this Levi subgroup is then automaticallyk-defined).Let Σ be a subcomplex of Δk . We say that Σ is convex if whenever x, y ∈ Σ are not opposite, then Σ contains thegeodesic between x and y, [13, §2.1].Suppose Σ is a convex subcomplex of Δk . Serre has shown that Σ is contractible — that is, Σ has the homotopy type ofa point — if and only if there exists a point of Σ which has no opposite in Σ ; see [13, §2.2]. The following terminology isdue to Serre [13, Def. 2.2.1]:Definition 2.1. Let Σ be a convex subcomplex of Δk . We say that Σ is Δk-completely reducible (or Δk-cr) if every simplexin Σ has an opposite in Σ .Serre has shown that the group-theoretic definition of G-complete reducibility over k has the following building-theoreticinterpretation, [13]: Given a subgroup H of G , letΔHk = {σP | P is a k-defined parabolic subgroup containing H}.Then ΔHk is a convex subcomplex of Δk [13, Prop. 3.1], the fixed point subcomplex of Δk under the action of H , and H isG-completely reducible over k if and only if ΔHk is Δk-cr, [13, 2.3.1, 3.2]. Equivalently, H is not G-completely reducible overk if and only if ΔHk is contractible.Definition 2.2. Let Σ be a subcomplex of Δk and let x ∈ Σ . Let Γ be a group which acts on Δk by means of building auto-morphisms, [15], i.e., suppose there is a homomorphism Γ → Aut Δk , where Aut Δk is the group of building automorphismsof Δk . We say that x is a Γ -centre of Σ if x is fixed by any element of Γ that stabilizes Σ setwise.The following theorem is known as the “Centre Conjecture” of J. Tits, cf. [14, Lem. 1.2], [12, §4], [13, §2.4], [16], [9, Ch. 2,§3], [11, Conj. 3.3]. It has recently been proved in a series of intricate case-by-case arguments by B. Mühlherr and J. Tits[8] (G of classical type or type G2), B. Leeb and C. Ramos-Cuevas [7] (G of type F4 or E6) and C. Ramos-Cuevas [10] (G oftype E7 or E8).Theorem 2.3 (Tits’ Centre Conjecture). Let Σ be a convex contractible subcomplex of Δk. Then Σ has an Aut Δk-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 alsodefined over k. In [4, Thm. 5.31], we give a uniform proof of the following special case of the Centre Conjecture: Let H be asubgroup of G . If ΔHk is contractible, i.e., if ΔHk is not Δk-cr, then ΔHk admits an (Aut G)(k)-centre. The proof of this resultin [4] utilizes methods from geometric invariant theory and the concept of optimal destabilizing parabolic subgroups.Let k be a field, let ks denote its separable closure, and let k denote its algebraic closure. Note that ks = k if k is perfect.Thanks to [15, 5.7.2], Γ := Gal(ks/k) acts on Δk via building automorphisms. In [4, Thm. 5.33], we show that if H is ak-defined subgroup of G such that ΔHkis contractible, then ΔHkadmits a Γ -centre. The proof of [4, Thm. 5.33] rests on arationality result concerning G-cr subgroups of G , [4, Prop. 5.14(iii)].Both [4, Thm. 5.31] and [4, Thm. 5.33] improve on [2, Thm. 3.1].M. Bate et al. / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 495–497 4973. Proof of Theorem 1.1As noted above the reverse implication of Theorem 1.1 is proved in [4, Thm. 5.11]. We deduce the other direction withthe aid of Theorem 2.3.Suppose k1/k is an algebraic separable extension of fields and let Δk and Δk1 denote the buildings of G over k andk1, respectively. By the reverse implication, one may suppose that k1/k is Galois. Then the Galois group Γ := Gal(k1/k)acts simplicially on Δk1 , i.e., Γ permutes the set of k1-defined parabolic subgroups of G . Moreover, the subcomplex of Δk1consisting of Γ -stable simplices is just Δk .It is convenient to reduce to the case when H is not contained in any k-defined Levi subgroup of any proper k-definedparabolic subgroup of G . To do this, we let L be minimal such that L is a k-defined Levi subgroup of some k-definedparabolic subgroup P of G and H ⊆ L. Then L is also k1-defined, and by a result of Serre, [13, Prop. 3.2], H is G-completelyreducible over k (resp. k1) if and only if H is L-completely reducible over k (resp. k1). Now if L′ is a k-defined Levi subgroupof some proper k-defined parabolic subgroup Q of L, then Q Ru(P ) is a k-defined parabolic subgroup of G , and L′ is aLevi subgroup of Q Ru(P ), [6, Prop. 4.4]. Since L is minimal among those k-defined Levi subgroups of k-defined parabolicsubgroups of G that contain H , H cannot be contained in L′ . By replacing G with L, we can now assume that H is notcontained in any k-defined Levi subgroup of any proper k-defined parabolic subgroup of G .Suppose that H is not G-completely reducible over k1. Then ΔHk1 is contractible, and since H is k-defined, ΔHk1is Γ -stable. Since ΔHk1 is a convex contractible subcomplex of Δk1 , it follows from Theorem 2.3 that Γ fixes a point of ΔHk1, andthis point lies in some (minimal) simplex σP , where P is a proper k1-defined parabolic subgroup of G . Since the action ofΓ on Δk1 is simplicial, P is stabilized by Γ , which is equivalent to saying that P is k-defined. Now, by assumption, H isnot contained in any k-defined Levi subgroup of P , so H is not G-completely reducible over k. This completes the proof ofTheorem 1.1.Remark 3.1. In [5, Thm. 4.13], we prove a generalization of the reverse implication of Theorem 1.1 in the setting of “relativecomplete reducibility”. The arguments above used to derive the forward direction of Theorem 1.1 do not apply to this moregeneral situation, as the relevant subset in Δk1 is only a convex subset but not a subcomplex of Δk1 . Thus Theorem 2.3does not apply.AcknowledgementsThe paper was written during a stay of the first and third authors at the Max Planck Institute for Mathematics in Bonn.The authors also acknowledge financial support from the DFG-priority program SPP1388 “Representation Theory”, MarsdenGrant UOC0501, and The Royal Society.References[1] M. Bate, B. Martin, G. Röhrle, A geometric approach to complete reducibility, Invent. Math. 161 (1) (2005) 177–218.[2] M. Bate, B. Martin, G. Röhrle, On Tits’ centre conjecture for fixed point subcomplexes, C. R. Math. Acad. Sci. Paris, Ser. I 347 (2009) 353–356.[3] M. Bate, B. Martin, G. Röhrle, R. Tange, Complete reducibility and separability, Trans. AMS 362 (8) (2010) 4283–4311.[4] M. Bate, B. Martin, G. Röhrle, R. Tange, Closed orbits and uniform S-instability in geometric invariant theory, preprint, arXiv:0904.4853v3 [math.AG],2009.[5] M. Bate, B. Martin, G. Röhrle, R. Tange, Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras, preprint,arXiv:0905.0065v2 [math.GR], 2009.[6] A. Borel, J. Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. 27 (1965) 55–150.[7] B. Leeb, C. Ramos-Cuevas, The center conjecture for spherical buildings of types F4 and E6, preprint, arXiv:0905.0839v1.[8] B. Mühlherr, J. Tits, The Centre Conjecture for non-exceptional buildings, J. Algebra 300 (2) (2006) 687–706.[9] D. Mumford, J. Fogarty, F. Kirwan, Geometric Invariant Theory, third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 34, Springer-Verlag, Berlin, 1994.[10] C. Ramos-Cuevas, The center conjecture for thick spherical buildings, preprint, arXiv:0909.2761v1.[11] G. Rousseau, Immeubles sphériques et théorie des invariants, C.R.A.S. 286 (1978) 247–250.[12] J-P. Serre, La notion de complète réductibilité dans les immeubles sphériques et les groupes réductifs, Séminaire au Collège de France, résumé dans[17, pp. 93–98], 1997.[13] J-P. Serre, Complète réductibilité, Séminaire Bourbaki, 56ème année (2003–2004) no. 932.[14] J. Tits, Groupes semi-simples isotropes, Colloq. Théorie des Groupes Algébriques, Bruxelles, 1962.[15] J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Math., vol. 386, Springer-Verlag, 1974.[16] J. Tits, Quelques cas d’existence d’un centre pour des ensembles de chambres qui sont convexes, non vides et ne contiennent pas de paires de chambresopposées, Séminaire au Collège de France, résumé dans [17, pp. 98–101], 1997.[17] J. Tits, Théorie des groupes, Résumé des Cours et Travaux, Annuaire du Collège de France, 97e année (1996–1997) 89–102.

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arXiv:1002.4319v3  [math.GR]  14 Apr 2010COMPLETE REDUCIBILITY ANDSEPARABLE FIELD EXTENSIONSMICHAEL BATE, BENJAMIN MARTIN, AND GERHARD RO¨HRLEAbstract. Let G be a connected reductive linear algebraic group.The aim of this note is to settle a question of J-P. Serre concern-ing the behaviour of his notion of G-complete reducibility underseparable field extensions. Part of our proof relies on the recentlyestablished Tits Centre Conjecture for the spherical building of thereductive group G.1. IntroductionThroughout, G denotes a connected reductive linear algebraic groupdefined over a field k. Following Serre, [13], a subgroup H of G is calledG-completely reducible over k (G-cr over k) if whenever H is containedin a k-defined parabolic subgroup P of G, there exists a k-defined Levisubgroup of P containing H . In case V is a finite dimensional k-vector space and G = GL(V ), a subgroup H of G is G-completelyreducible over k precisely when V is a semisimple H-module, [13, 1.3,3.2.2]. In this sense, Serre’s notion generalizes the usual concept ofcomplete reducibility in representation theory. For more details andfurther results on this notion, see [12], [13], [1], [3], and [4].The following theorem answers a question of Serre.Theorem 1.1. Suppose k1/k is a separable extension of fields. Let Gbe a reductive group defined over k, and let H be a k-defined subgroupof G. Then H is G-completely reducible over k if and only if H isG-completely reducible over k1.The reverse implication in Theorem 1.1 is proved in [4, Thm. 5.11].The proof of [4, Thm. 5.11] rests on a general rationality result, [4,Thm. 3.1], concerning G-orbits in an affine variety. We present a proofof the forward direction of the statement in Section 3 based on therecently established Tits Centre Conjecture, Theorem 2.3.2010 Mathematics Subject Classification. 51E24, 20E42, 20G15.Key words and phrases. G-complete reducibility, spherical buildings, Tits CentreConjecture.12 M. BATE, B. MARTIN, AND G. RO¨HRLERemarks 1.2. (i). In [4, Ex. 5.12], we showed that Theorem 1.1 holdswhen G = GL(V ).(ii). Theorem 1.1 was proved in [1, Thm. 5.8] for k perfect, by passingback and forth between k and its algebraic closure k and between k1and k. In general this approach fails, because the extension k/k neednot be separable.(iii). There are examples showing that each implication in Theorem1.1 fails without the separability assumption on the extension k1/k; see[1, Ex. 5.11] and [3, Ex. 7.22].2. The Centre Conjecture for spherical buildingsLet ∆k denote the spherical building ofG over k, [15, Sec. 5]: the sim-plices of ∆k correspond to k-defined parabolic subgroups of G. Givena k-defined parabolic subgroup P of G, we denote the simplex cor-responding to P in ∆k by σP . Throughout, we identify ∆k with itsgeometric realization, which is a bouquet of spheres [13].An apartment in ∆k consists of the simplices σP corresponding to allk-defined parabolic subgroups P of G which contain a fixed maximalk-split torus of G; it is a subcomplex whose geometric realization isa sphere. Any two points of ∆k lie in a common apartment. We saythat x, y ∈ ∆k are opposite if they are opposite in some apartment thatcontains them both. It can be shown that if x and y are opposite insome apartment that contains them both, then they are opposite in anyapartment that contains them both. If x, y ∈ ∆k are not opposite, thenthere is a unique geodesic joining them, [13, §2.1.4]. Two simplices σPand σQ are said to be opposite if every point of σP is opposite a pointof σQ, [13, §2.1.4]. In terms of parabolic subgroups of G, the simplicesσP and σQ corresponding to k-defined parabolic subgroups P and Q ofG are opposite in ∆k if and only if P ∩Q is a common Levi subgroupof P and Q (this Levi subgroup is then automatically k-defined).Let Σ be a subcomplex of ∆k. We say that Σ is convex if wheneverx, y ∈ Σ are not opposite, then Σ contains the geodesic between x andy, [13, §2.1].Suppose Σ is a convex subcomplex of ∆k. Serre has shown that Σ iscontractible — that is, Σ has the homotopy type of a point — if andonly if there exists a point of Σ which has no opposite in Σ; see [13,§2.2]. The following terminology is due to Serre [13, Def. 2.2.1]:Definition 2.1. Let Σ be a convex subcomplex of ∆k. We say thatΣ is ∆k-completely reducible (or ∆k-cr) if every simplex in Σ has anopposite in Σ.COMPLETE REDUCIBILITY AND SEPARABLE FIELD EXTENSIONS 3Serre has shown that the group-theoretic definition of G-completereducibility over k has the following building-theoretic interpretation,[13]: Given a subgroup H of G, let∆Hk = {σP | P is a k-defined parabolic subgroup containing H}.Then ∆Hk is a convex subcomplex of ∆k ([13, Prop. 3.1]), the fixedpoint subcomplex of ∆k under the action of H , and H is G-completelyreducible over k if and only if ∆Hk is ∆k-cr, [13, 2.3.1, 3.2]. Equivalently,H is not G-completely reducible over k if and only if ∆Hk is contractible.Definition 2.2. Let Σ be a subcomplex of ∆k and let x ∈ Σ. Let Γ bea group which acts on ∆k by means of building automorphisms, [15],i.e., suppose there is a homomorphism Γ → Aut∆k, where Aut∆k isthe group of building automorphisms of ∆k. We say that x is a Γ-centreof Σ if x is fixed by any element of Γ that stabilizes Σ setwise.The following theorem is known as the “Centre Conjecture” of J.Tits, cf. [14, Lem. 1.2], [12, §4], [13, §2.4], [16], [9, Ch. 2, §3], [11, Conj.3.3]. It has recently been proved in a series of intricate case-by-casearguments by B. Mu¨hlherr and J. Tits [8] (G of classical type or typeG2), B. Leeb and C. Ramos-Cuevas [7] (G of type F4 or E6) and C.Ramos-Cuevas [10] (G of type E7 or E8).Theorem 2.3 (Tits’ Centre Conjecture). Let Σ be a convex contractiblesubcomplex of ∆k. Then Σ has an Aut∆k-centre.Remark 2.4. Suppose G is semisimple and k is a perfect field. It followsfrom [15, 5.7.2] that AutG is an algebraic group also defined over k. In[4, Thm. 5.31], we give a uniform proof of the following special case ofthe Centre Conjecture: LetH be a subgroup ofG. If ∆Hk is contractible,i.e., if ∆Hk is not ∆k-cr, then ∆Hk admits an (AutG)(k)-centre. Theproof of this result in [4] utilizes methods from geometric invarianttheory and the concept of optimal destabilizing parabolic subgroups.Let k be a field, let ks denote its separable closure, and let k denoteits algebraic closure. Note that ks = k if k is perfect. Thanks to [15,5.7.2], Γ := Gal(ks/k) acts on ∆k via building automorphisms. In [4,Thm. 5.33], we show that if H is a k-defined subgroup of G such that∆Hkis contractible, then ∆Hkadmits a Γ-centre. The proof of [4, Thm.5.33] rests on a rationality result concerning G-cr subgroups of G, [4,Prop. 5.14(iii)].Both [4, Thm. 5.31] and [4, Thm. 5.33] improve on [2, Thm. 3.1].4 M. BATE, B. MARTIN, AND G. RO¨HRLE3. Proof of Theorem 1.1As noted above the reverse implication of Theorem 1.1 is proved in[4, Thm. 5.11]. We deduce the other direction with the aid of Theorem2.3.Suppose k1/k is an algebraic separable extension of fields and let∆k and ∆k1 denote the buildings of G over k and k1, respectively. Bythe reverse implication, one may suppose that k1/k is Galois. Thenthe Galois group Γ := Gal(k1/k) acts simplicially on ∆k1 , i.e., Γ per-mutes the set of k1-defined parabolic subgroups of G. Moreover, thesubcomplex of ∆k1 consisting of Γ-stable simplices is just ∆k.It is convenient to reduce to the case when H is not contained in anyk-defined Levi subgroup of any proper k-defined parabolic subgroup ofG. To do this, we let L be minimal such that L is a k-defined Levisubgroup of some k-defined parabolic subgroup P of G and H ⊆ L.Then L is also k1-defined, and by a result of Serre, [13, Prop. 3.2], H isG-completely reducible over k (resp. k1) if and only ifH is L-completelyreducible over k (resp. k1). Now if L′ is a k-defined Levi subgroup ofsome proper k-defined parabolic subgroup Q of L, then QRu(P ) isa k-defined parabolic subgroup of G, and L′ is a Levi subgroup ofQRu(P ), [6, Prop. 4.4]. Since L is minimal among those k-defined Levisubgroups of k-defined parabolic subgroups of G that contain H , Hcannot be contained in L′. By replacing G with L, we can now assumethat H is not contained in any k-defined Levi subgroup of any properk-defined parabolic subgroup of G.Suppose that H is not G-completely reducible over k1. Then ∆Hk1is contractible, and since H is k-defined, ∆Hk1 is Γ-stable. Since ∆Hk1is a convex contractible subcomplex of ∆k1 , it follows from Theorem2.3 that Γ fixes a point of ∆Hk1 , and this point lies in some (minimal)simplex σP , where P is a proper k1-defined parabolic subgroup of G.Since the action of Γ on ∆k1 is simplicial, P is stabilized by Γ, whichis equivalent to saying that P is k-defined. Now, by assumption, His not contained in any k-defined Levi subgroup of P , so H is notG-completely reducible over k. This completes the proof of Theorem1.1.Remark 3.1. In [5, Thm. 4.13], we prove a generalization of the re-verse implication of Theorem 1.1 in the setting of “relative completereducibility”. The arguments above used to derive the forward direc-tion of Theorem 1.1 do not apply to this more general situation, as therelevant subset in ∆k1 is only a convex subset but not a subcomplex of∆k1. Thus Theorem 2.3 does not apply.COMPLETE REDUCIBILITY AND SEPARABLE FIELD EXTENSIONS 5Acknowledgements: The paper was written during a stay of thefirst and third authors at the Max Planck Institute for Mathematics inBonn. The authors also acknowledge financial support from the DFG-priority program SPP1388 “Representation Theory”, Marsden GrantUOC0501, and The Royal Society.References[1] M. Bate, B. Martin, G. Ro¨hrle, A geometric approach to complete reducibility,Invent. Math. 161, no. 1 (2005), 177–218.[2] , On Tits’ Centre Conjecture for fixed point subcomplexes, C. R. Math.Acad. Sci. Paris 347 (2009), 353–356.[3] M. Bate, B. Martin, G. Ro¨hrle, R. Tange, Complete reducibility and separabil-ity, Trans. AMS, (2010), in press.[4] , Closed orbits and uniform S-instability in geometric invariant theory,preprint (2009) arXiv:0904.4853v3 [math.AG].[5] , Complete reducibility and conjugacy classes of tuples in algebraicgroups and Lie algebras, preprint (2009) arXiv:0905.0065v2 [math.GR].[6] A. Borel, J. Tits, Groupes Re´ductifs, Inst. Hautes E´tudes Sci. Publ. Math. 27(1965), 55–150.[7] B. Leeb, C. Ramos-Cuevas, The center conjecture for spherical buildings oftypes F4 and E6, Preprint. arXiv:0905.0839v1.[8] B. Mu¨hlherr, J. Tits, The Centre Conjecture for non-exceptional buildings, J.Algebra 300 (2006), no. 2, 687–706.[9] D. Mumford, J. Fogarty, F. Kirwan, Geometric invariant theory. Third edi-tion. Ergebnisse der Mathematik und ihrer Grenzgebiete, 34. Springer-Verlag,Berlin, 1994.[10] C. Ramos-Cuevas, The center conjecture for thick spherical buildings, Preprint.arXiv:0909.2761v1.[11] G. Rousseau, Immeubles sphe´riques et the´orie des invariants, C.R.A.S. 286(1978), 247–250.[12] J-P. Serre, La notion de comple`te re´ductibilite´ dans les immeubles sphe´riqueset les groupes re´ductifs, Se´minaire au Colle`ge de France, re´sume´ dans [17, pp.93–98], (1997).[13] , Comple`te Re´ductibilite´, Se´minaire Bourbaki, 56e`me anne´e, 2003-2004,no 932.[14] J. Tits, Groupes semi-simples isotropes, Colloq. The´orie des GroupesAlge´briques, Bruxelles, 1962.[15] , Buildings of spherical type and finite BN -pairs, Lecture Notes inMath. 386, Springer-Verlag (1974).[16] , Quelques cas d’existence d’un centre pour des ensembles de chambresqui sont convexes, non vides et ne contiennent pas de paires de chambres op-pose´es, Se´minaire au Colle`ge de France, re´sume´ dans [17, pp. 98–101], (1997).[17] , The´orie des groupes, Re´sume´ des Cours et Travaux, Annuaire duColle`ge de France, 97e anne´e, (1996–1997), 89–102.6 M. BATE, B. MARTIN, AND G. RO¨HRLEDepartment of Mathematics, University of York, York YO10 5DD,United KingdomE-mail address : meb505@york.ac.ukMathematics and Statistics Department, University of Canterbury,Private Bag 4800, Christchurch 8140, New ZealandE-mail address : B.Martin@math.canterbury.ac.nzFakulta¨t fu¨r Mathematik, Ruhr-Universita¨t Bochum, 44780 Bochum,GermanyE-mail address : gerhard.roehrle@rub.de

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