Some reductive anisotropic groups that admit no non-trivial split spherical BN-pairs
SSOME REDUCTIVE ANISOTROPIC GROUPS THAT ADMIT NONON-TRIVIAL SPLIT SPHERICAL BN-PAIRS
PETER ABRAMENKO AND MATTHEW C. B. ZAREMSKY
Abstract.
We prove, for any infinite field k , that any virtually trivial splitspherical BN -pair in the group G ( k ) of k -rational points of a reductive k -group G is already trivial. We then inspect the case when G is k -anisotropic and showthat in many situations G ( k ) admits no non-trivial split spherical BN -pairs.This improves results and contributes to a conjecture of Caprace and Marquis,which can be viewed as a converse to a well-known result of Borel and Tits. Introduction
We prove a variety of results inspired by a conjecture of Caprace and Marquis,formulated in [CM].
Conjecture . Let G be a reductive algebraic k -group that is anisotropic over k .Then every split spherical BN -pair of G ( k ) is trivial [CM].Here and in the following, a reductive group is always connected by definition.A BN -pair ( B, N ) of a group G is called saturated if T := B ∩ N is equal to (cid:92) w ∈ W wBw − , and split if it is saturated and B decomposes as B = U (cid:111) T , where U is nilpotent. We say ( B, N ) is trivial if B = G , or equivalently if the building ∆ =∆( G, B ) is trivial. See [AB] for the relevant background on buildings. Conjecture 1can be phrased as a converse to a well-known result of Borel and Tits [BT]. Namely,for any reductive algebraic k -group G that is isotropic over k , the group of k -rationalpoints G ( k ) admits a canonical (non-trivial) split spherical BN -pair. In their paperCaprace and Marquis focus on the following weaker conjecture. Conjecture . Let G be a reductive algebraic k -group that is anisotropic over k .Then every split spherical BN -pair of G ( k ) is virtually trivial [CM2].We say a BN -pair ( B, N ) of G is virtually trivial if [ G : B ] < ∞ , or equivalentlyif the building ∆ = ∆( G, B ) is finite. In [CM2] Conjecture 2 is shown to hold if k iseither local or perfect. In the present context we prove that for any infinite field k ,if G ( k ) consists of semisimple elements then Conjecture 1 holds. We also discoverthat, contrary to expectations, Conjecture 1 actually follows from Conjecture 2, a r X i v : . [ m a t h . G R ] A ug PETER ABRAMENKO AND MATTHEW C. B. ZAREMSKY and so the two conjectures are equivalent. As explained in [CM2], Conjecture 1 isrelated to a conjecture of Prasad and Rapinchuk that has so far only been provedin the D × case, namely that any finite quotient of an anisotropic reductive groupmust be solvable [RSS].We phrase our main results without explicit reference to anisotropic groups, forthe sake of full generality. By [B, Corollary 18.3], for any infinite field k , G ( k ) isZariski-dense in G . Also, if G is k -anisotropic then in many cases G ( k ) containsno non-trivial unipotent elements, or even consists solely of semisimple elements.These are the only properties we will need to prove our main results, though oneshould keep in mind the example of H = G ( k ), and we will explicitly state ourresults with respect to the conjectures at the end.The main results here are the following: Theorem 1.1.
Let G be a reductive group over an algebraically closed field. Let H be a subgroup of G that is Zariski-dense in G . Then any virtually trivial split BN -pair of H is trivial. In particular if we take H = G ( k ) for some infinite field k , Conjecture 2 impliesConjecture 1. Note however that Theorem 1.1 still applies even if G ( k ) is isotropic. Theorem 1.2.
Let G be a reductive group over an algebraically closed field. Let H be a subgroup that is Zariski-dense in G and contains no non-trivial unipotentelements. Then any split spherical BN -pair of H having irreducible rank ≥ istrivial. Theorem 1.3.
Let G be a reductive group over an algebraically closed field. Let H be a subgroup that is Zariski-dense in G and consists only of semisimple elements.Then any split spherical BN -pair of H is trivial. In Section 2 we prove Theorem 1.1. In particular if k is local or perfect thenby [CM2] Conjecture 2 holds, and so we conclude that even Conjecture 1 holds.We next present a proof of Theorem 1.2 in Section 3, and prove Theorem 1.3 inSection 4. Note that applying Theorem 1.3 with H = G ( k ), the conclusion ofTheorem 1.2 can be sharpened in some cases, e.g. if k is perfect. In this wayConjecture 1 is shown to hold for the case when k is perfect using a differentline of attack than in [CM2]. We also establish Conjecture 1 for certain G withno restriction on k , for example if G ( k ) is the multiplicative group of a divisionalgebra; see Section 5.We mention that a much more general program is carried out by Prasad in [P],which in particular establishes all the results proved here, and more. Our methods NISOTROPIC GROUPS ADMITTING NO SPLIT SPHERICAL BN-PAIRS 3 are very different, however, and could possibly be useful in other contexts, so thepresent work is still of interest.2.
Virtually trivial implies trivial
In this section we prove Theorem 1.1. The following key lemma plays an impor-tant role here as well as in Proposition 4.2 below. The first part of the proof belowmimics parts of the proof of Theorem 4 in [CM2].
Lemma 2.1.
Let G be a reductive group over an algebraically closed field K , and let H be any subgroup of G . Suppose that H possesses a split BN -pair ( B = U (cid:111) T, N ) such that G = BZ ( G ) , where B is the Zariski closure of B in G . Then U is trivial.Proof. We first show that U is finite. Since U is nilpotent and normal in B , we knowthat U is nilpotent and normal in B [B, Section 2.1]. By hypothesis G = BZ ( G ), soin fact U is normal in G . This implies that the identity component U is containedin the radical of G , which coincides with Z ( G ) [B, Proposition 11.21]. Now as inthe proof of [CM2, Theorem 4] we have[ U : U ∩ Z ( G )] ≤ [ U : U ∩ U ] = [ U U : U ] ≤ [ U : U ] < ∞ . Also, if u ∈ U ∩ Z ( G ), then for any g ∈ H we have ugC = guC = gC where C isthe fundamental chamber in ∆ = ∆( H, B ). Of course every chamber of ∆ is of theform gC for some g ∈ H , so u acts trivially on ∆. But T contains the kernel of theaction and U ∩ T = { } , so in fact u = 1. We conclude that U is finite.We now want to show that U , or equivalently ∆, is even trivial. Since U isfinite, U = U , which we know is normal in G . Thus U is normal in H . Nowsuppose S (cid:54) = ∅ . Let s ∈ S , so by the BN -axioms sBs (cid:54)≤ B . But s normalizes T ,and since U is normal in H we know that s also normalizes U , so this is impossible.We conclude that S = ∅ , so N = T ≤ B and in fact B = H . Since the chambers of∆ are in one-to-one correspondence with H/B , we conclude that ∆, and thus U , istrivial. (cid:3) Lemma 2.2.
Let G be a connected linear algebraic group, H a Zariski-dense sub-group of G and B any subgroup of H . Then either B has infinite index in H or B is Zariski dense in G .Proof. Assume that B has finite index in H . Let h , . . . , h n be a set of cosetrepresentatives for H/B , and let B be the Zariski closure of B in G . Then G = H is the union of the cosets h B, . . . , h n B . Since G is connected, G = h i B for some i , and hence also G = B . (cid:3) We are now in a position to prove Theorem 1.1.
PETER ABRAMENKO AND MATTHEW C. B. ZAREMSKY
Proof of Theorem 1.1.
Let G be a reductive group and H a Zariski-dense subgroup.Let ( B = U (cid:111) T, N ) be a split BN -pair of H such that [ H : B ] < ∞ . By Lemma 2.2 B is Zariski dense in G , and by Lemma 2.1, U is trivial. (cid:3) In particular if H = G ( k ) this shows that Conjectures 1 and 2 are equivalent.We immediately get the following: Corollary 2.3 (Extension of Theorems 3 and 4 from [CM2]) . Let k be either alocal field or a perfect field. Let G be a reductive k -group that is anisotropic over k . Then G ( k ) does not admit any non-trivial split spherical BN -pairs. (cid:3) Unipotent-free groups and split spherical BN-pairs without rank1 factors
Let G be a group with a BN -pair ( B, N ) of spherical type (
W, S ). In this section,our standing assumption is that the Coxeter system (
W, S ) has no direct rank 1factors, which means that the corresponding Coxeter diagram has no isolated nodes.As usual, we set T = (cid:92) w ∈ W wBw − . By the main result of [DMHTVM], the existenceof a splitting B = U (cid:111) T with nilpotent U implies that the building ∆ = ∆( G, B ) isMoufang and U = U + , the group generated by all the root groups U α for α ∈ Φ + .Here Φ is the root system of a fixed fundamental apartment corresponding to N ,and Φ + is the set of all α ∈ Φ that contain the fundamental chamber correspondingto B . In this section we prove Theorem 1.2 by inspecting the root structure.Recall that the concepts of spherical Moufang buildings and spherical RGD sys-tems are equivalent [AB, Example 7.83 and Theorem 7.116]. The following is ageneral lemma about spherical (or just 2-spherical) RGD systems. Lemma 3.1.
Let ( G, ( U α ) α ∈ Φ ) , T ) be an RGD system of spherical type ( W, S ) ,such that the Coxeter diagram of ( W, S ) has no isolated nodes. Let Φ + be a choiceof positive roots and Π ⊆ Φ + a choice of simple roots. Then for any simple root α ∈ Π , there exists a simple root β such that no non-trivial elements of U β commutewith any non-trivial elements of U α .Proof. Let U + and B + = U + T be the usual subgroups. We consider the Moufangbuilding ∆ = ∆( G, B + ) with fundamental chamber C and fundamental apartmentΣ . Let α = α s be the simple root corresponding to s ∈ S , so C ∈ α but sC (cid:54)∈ α .Hence there exists a panel P of C in the boundary of α , and P has cotype s . Choose t ∈ S \{ s } such that s and t are connected in the Coxeter diagram, and let Q bethe panel of C of cotype t . Denote by β the root containing C and having Q in itsboundary, i.e. β is the simple root α t . We set A := P ∩ Q , so A has cotype { s, t } . NISOTROPIC GROUPS ADMITTING NO SPLIT SPHERICAL BN-PAIRS 5
Let ∆ (cid:48) := lk ∆ ( A ) be the link of A in ∆, and set α (cid:48) := α ∩ ∆ (cid:48) , β (cid:48) := β ∩ ∆ (cid:48) , C (cid:48) := C ∩ ∆ (cid:48) , and Σ (cid:48) := Σ ∩ ∆ (cid:48) . By [AB, Proposition 7.32], ∆ (cid:48) is strictly Moufangand we can identify U α (cid:48) and U β (cid:48) with U α and U β respectively, via the naturalrestriction map. In fact, since A has codimension 2, ∆ (cid:48) is a generalized Moufang n -gon, and since s and t are connected in the Coxeter diagram we know that n > (cid:54) = a ∈ U α (cid:48) and b ∈ U β (cid:48) and suppose ab = ba . Then baβ (cid:48) = abβ (cid:48) = aβ (cid:48) and C (cid:48) ∈ aβ (cid:48) is fixed by b , so b fixes aβ (cid:48) pointwise. Since α (cid:48) ∩ β (cid:48) = { C (cid:48) } , n > a (cid:54) = 1, we know that β (cid:48) ∪ aβ (cid:48) contains a pair of opposite chambers. If Σ (cid:48) is theapartment containing these, clearly b fixes Σ (cid:48) pointwise. But Σ (cid:48) also contains somechamber D (cid:48) of β (cid:48) that has no panels in ∂β (cid:48) . Since b fixes Σ (cid:48) and all chambers of∆ (cid:48) adjacent to D (cid:48) , by the rigidity theorem [AB, Corollary 5.206] we conclude that b is the identity on ∆ (cid:48) , and hence b = 1. See Figure 1 for an idea of the n = 3situation.Now we return to the original building ∆. If 1 (cid:54) = a ∈ U α and b ∈ U β commute,then by the above argument b acts trivially on lk ∆ ( A ). In particular b fixes anysimplex joinable to Q . Since Q ∈ ∂β this implies that b = 1, and the resultfollows. (cid:3) Figure 1.
Example for n =3 Lemma 3.2.
Let H be any nilpotent linear group having no non-trivial unipotentelements. Then H is virtually abelian.Proof. Assume H ≤ GL n ( K ) for some algebraically closed field K . Since H isnilpotent, so is the Zariski closure H . By [B, Theorem 10.6(3)] then, the connectedcomponent H decomposes as a direct product of its semisimple and unipotentparts, H = ( H ) s × ( H ) u . Since ( H ) s is abelian and the product is direct,we know that [ H , H ] = [( H ) u , ( H ) u ]. Now set H (cid:48) := H ∩ H . Of course[ H (cid:48) , H (cid:48) ] ≤ H , and also we see that [ H (cid:48) , H (cid:48) ] ≤ [( H ) u , ( H ) u ], which consists of PETER ABRAMENKO AND MATTHEW C. B. ZAREMSKY unipotent elements. Since H has no non-trivial unipotent elements, in fact H (cid:48) isabelian. Also, [ H : H (cid:48) ] < ∞ so indeed H is virtually abelian. (cid:3) Proof of Theorem 1.2.
Let G be a reductive group and H a Zariski-dense subgroupsuch that H ∩ G u = { } . Let ( B = U (cid:111) T, N ) be a split spherical BN -pair of H of type ( W, S ) without rank 1 factors. We claim that B = H . Since the Coxeterdiagram of ( W, S ) has no isolated nodes, we can use Lemma 3.1. Let ∆ = ∆(
H, B )with fundamental apartment Σ and fundamental chamber C . Denote the rootsystem by Φ and the root groups by U α , so U = U + = (cid:104) U α | α ∈ Φ + (cid:105) . We knowthat U is nilpotent, and so by Lemma 3.2 we can choose V ≤ U abelian such that[ U : V ] < ∞ .Given any system of positive roots Φ + and any simple root α ∈ Φ + , Lemma 3.1shows that we can choose a simple root β such that no non-trivial elements of U α commute with any non-trivial elements of U β . Since V is abelian, this shows thateither U α ∩ V = { } or U β ∩ V = { } . In either case at least one of the root groupsis finite. However, by the classification of Moufang polygons [TW], if one of theroot groups is finite then they must all be finite. Since ∆ is Moufang, this impliesthat ∆ is locally finite , i.e. each panel in ∆ is contained in finitely many chambers.Thus for any w ∈ W there are finitely many chambers of ∆ at Weyl distance w from C . Since W itself is finite, we conclude that ∆ is finite. By Theorem 1.1 it iseven trivial, and so the BN -pair is trivial. (cid:3) We now take a moment to discuss some examples of unipotent-free anisotropicgroups. Let D be a finite dimensional central k -division algebra and suppose x ∈ D is unipotent. Then x − D , which since D is a divisionalgebra implies that x − x = 1. In particular the anisotropic groups D × and SL ( D ) are unipotent-free. The theorem tells us they admit no split spherical BN -pairs of rank greater than 1, though as we will see in the last section we caneliminate the rank-1 case as well.Also, if k is perfect then any anisotropic G ( k ) is unipotent-free, as explained in[CM2, Proposition 4.1]. Thus the proof of Theorem 1.2 constitutes an alternateproof of Conjecture 1 in case k is perfect, at least for BN -pairs of rank ≥
2. Again,in the next section the rank-1 case will also be eliminated.Note that the only time we used our precise setup in the proof of Theorem 1.2was to see that if U is nilpotent then it is already virtually abelian. In fact, thesame proof yields the following Proposition 3.3.
Let G be a group with a split spherical BN -pair ( B = U (cid:111) T, N ) of type ( W, S ) without rank 1 factors. If U is virtually abelian then U is finite. (cid:3) NISOTROPIC GROUPS ADMITTING NO SPLIT SPHERICAL BN-PAIRS 7
It seems likely to us that this last proposition can be applied to other reductiveanisotropic groups as well, even perhaps ones with unipotent elements, in whichcase we would conclude, applying Theorem 1.1 again, that these BN -pairs are eventrivial.4. Groups consisting of semisimple elements and split BN-pairs ofany rank
In this section, we prove Theorem 1.3. It is based on a more general criterion(see Proposition 4.2) that might even be applicable in more generality.
Proposition 4.1.
Let G be a nonabelian reductive group over an algebraicallyclosed field K , H a Zariski-dense subgroup of G and A ≤ H such that | A \ H/A | < ∞ . Then A is not virtually nilpotent. We will not individually cite every result quoted in the proof, but all the detailscan be found in Chapters 13 and 14 of [B], unless otherwise cited.
Proof. If | A \ H/A | < ∞ and H = G is connected, then there exists a double coset AhA ( h ∈ H ) that is Zariski-dense in G . Now if A were virtually nilpotent, alsoits Zariski closure A would be, and there would exist a closed connected nilpotentsubgroup M of A of finite index. So AhA would be a finite union of double cosetsmodulo M , and one of these would have to be dense in G . Hence our claim willfollow if we can show that M gM is not dense in G , for any closed connectednilpotent subgroup M of G and any g ∈ G . For this it suffices to show dim M gM < dim G (where we set dim M gM = dim
M gM ), and this is what we are going to donow.First consider the case when G is semisimple . Choose a Borel subgroup B of G that contains M , and denote by U the unipotent radical of B . As a closedconnected nilpotent group, M is the direct product of a torus T (cid:48) and a unipotentsubgroup U (cid:48) , see [B, Theorem 10.6]. Necessarily, U (cid:48) is a subgroup of U . Choose amaximal torus T of B that contains T (cid:48) . Let Φ be the root system associated to G and T . Then B determines a positive system Φ + in Φ. Recall that dim U = | Φ + | and dim G = dim T + | Φ | .We want to show that dim M ≤ dim U , which is clear if T (cid:48) = { } . The simpleidea of the following argument is to deduce dim U (cid:48) ≤ dim U − dim T (cid:48) from the factthat the product T (cid:48) × U (cid:48) is direct. Order the positive roots Φ + = { α , . . . , α m } ,and for each 1 ≤ i ≤ m denote the corresponding root group by U i := U α i . Thenthe map U × · · · × U m → U given by multiplication is a bijection [B, Proposi-tion 14.4]. Next choose isomorphisms x i : ( K, +) → U i for all i . It follows (see PETER ABRAMENKO AND MATTHEW C. B. ZAREMSKY [B, Section 10.10]) that tx i ( λ ) t − = x i ( α i ( t ) λ ) for t ∈ T , λ ∈ K . Then for any u ∈ U (cid:48) , there exist uniquely determined λ i ∈ K such that u = m (cid:89) i =1 x i ( λ i ), and tut − = (cid:81) mi =1 x i ( α i ( t ) λ i ) for all t ∈ T . Since tut − = u for all t ∈ T (cid:48) , we obtain m (cid:89) i =1 x i ( α i ( t ) λ i ) = m (cid:89) i =1 x i ( λ i ) for t ∈ T (cid:48) . By uniqueness, for each i such that λ i (cid:54) = 0 we must have T (cid:48) ≤ ker α i . In particularwe get that U (cid:48) ≤ (cid:89) i : T (cid:48) ≤ ker α i U i . (4.1)Now consider the character groups X ( T ) and X ( T (cid:48) ). Let π : X ( T ) (cid:16) X ( T (cid:48) ) bethe restriction map, so ker π = { χ ∈ X ( T ) | T (cid:48) ≤ ker χ } . Also,dim T = rk( X ( T )) = rk(ker π ) + rk( X ( T (cid:48) )) = rk(ker π ) + dim T (cid:48) so rk(ker π ) = dim T − dim T (cid:48) .Now let (cid:96) := dim T = rk( X ( T )). By [B, Theorem 13.18(3)] (cid:104) Φ (cid:105) Z has rank (cid:96) ,so without loss of generality the first (cid:96) positive roots α , . . . , α (cid:96) can be chosen tobe linearly independent. Since rk(ker π ) = dim T − dim T (cid:48) , only (cid:96) − dim T (cid:48) of theroots α , . . . , α (cid:96) can be contained in ker π . In particular at least dim T (cid:48) of them are not contained in ker π . By 4.1 we conclude that the dimension of U (cid:48) is less thanor equal to dim U − dim T (cid:48) , so indeed dim M = dim T (cid:48) + dim U (cid:48) ≤ dim U . Since T is non-trivial (if B = U , then also G = B = U by [B, Corollary 11.5], whichcontradicts our assumptions), it follows as claimed thatdim M gM ≤ M ≤ U < dim T + 2 dim U = dim G. We now consider the case when G is reductive . By [B, Proposition 14.2], G decomposes as G = SG with S := Z ( G ) and semisimple G := [ G, G ]. Note that G has the same root system Φ as G . Since G is not abelian, G is non-trivialand hence not nilpotent, since semisimple. Choose a maximal torus T of G ; it hasto contain S since ST is a torus. If we had S = T , then this were the uniquemaximal torus of G , implying (again by [B, Corollary 11.5]) that G is nilpotent, acontradiction. Now let M be a closed connected nilpotent subgroup of G , whichwithout loss of generality contains S . Obviously M = S ( M ∩ G ), and since G issemisimple, the previous case implies dim( M ∩ G ) ≤ | Φ + | . Thusdim( M gM ) = dim(( M ∩ G ) gM ) ≤ dim( M ∩ G ) + dim M ≤ | Φ + | + dim S. Also, G has dimension 2 | Φ + | + dim T > | Φ + | + dim S , so we conclude that M gM is not dense in G . (cid:3) NISOTROPIC GROUPS ADMITTING NO SPLIT SPHERICAL BN-PAIRS 9
Proposition 4.2.
Let G and H be as in Proposition 4.1. Let ( B = U (cid:111) T, N ) bea split spherical BN -pair of H , and suppose that the kernel of the action of H onthe building ∆ = ∆( H, B ) has finite index in T . Then U and ∆ are trivial.Proof. Denote by Q the kernel of the action of H on ∆, so [ T : Q ] < ∞ . Since Q isnormal in H , if we pass to the Zariski closure we get that Q is normal in H = G .Consider the canonical projection π : G (cid:16) G/Q . It follows from [B, Corollary 14.11]that
G/Q = π ( G ) is reductive. Also π ( H ) is dense in G/Q since
G/Q = π ( H ) ⊆ π ( H ). Since ( B, N ) is a spherical BN -pair in H , | B \ H/B | < ∞ , which implies | π ( B ) \ π ( H ) /π ( B ) | < ∞ . Next we note that B/Q is virtually nilpotent since it isa semidirect product of U and T /Q , and | T /Q | < ∞ by assumption. But thenalso π ( B ), which is isomorphic to B/ ( B ∩ Q ) and hence to a quotient of B/Q , isvirtually nilpotent. Combining all these facts, Proposition 4.1 now implies that
G/Q is abelian . As above we have a decomposition G = [ G, G ] Z ( G ) where [ G, G ] is thecommutator subgroup. Since
G/Q is abelian, clearly [
G, G ] ≤ Q , so G = QZ ( G ).Since Q ≤ B , we can now apply Lemma 2.1 to conclude that U and ∆ are trivial. (cid:3) Lemma 4.3.
Let G be any linear algebraic group. Let U be a nilpotent subgroupof G consisting of semisimple elements, and T any subgroup of N G ( U ) . Then [ T : C T ( U )] < ∞ .Proof. We first claim that [ U : Z ( U )] < ∞ . Of course by Lemma 3.2 we alreadyknow that U is virtually abelian, but in the present context we can do even better.Indeed, the Zariski closure U is nilpotent, and so by [B, Proposition 12.5] thesemisimple part of the connected component ( U ) s is central in U . In particular U ∩ ( U ) s is central in U . But since U consists of semisimple elements, U ∩ ( U ) s = U ∩ U . This has finite index in U and so indeed [ U : Z ( U )] < ∞ .Let { u , . . . , u n } be a set of coset representatives of U/Z ( U ). For any 1 ≤ i ≤ n ,the group V i := (cid:104) u i , Z ( U ) (cid:105) is abelian and consists of semisimple elements, i.e. isdiagonalizable. In particular by rigidity [B, Corollary 8.10(2)], we see that [ T : C T ( V i )] < ∞ . This implies that the group n (cid:92) i =1 C T ( V i ) has finite index in T . But if t ∈ T centralizes every V i then of course t centralizes U , so we conclude that indeed[ T : C T ( U )] < ∞ . (cid:3) We are now in a position to prove Theorem 1.3. From now on G is a reductivegroup over an algebraically closed field, and H be a subgroup that is Zariski-densein G and consists only of semisimple elements. Proof of Theorem 1.3.
Let ( B = U (cid:111) T, N ) be a split spherical BN -pair of H , andlet ∆ = ∆( H, B ). We want to show that ∆ is trivial. Let Σ be the fundamental apartment in ∆, with fundamental chamber C . Let D be the chamber opposite C in Σ , so any other chamber opposite C in ∆ is of the form uD for some u ∈ U (since H acts strongly transitively on ∆ and B = U T ). Now, if t ∈ T commuteswith every element of U then clearly t ( uD ) = utD = uD . Thus the centralizer C T ( U ) of U in T is contained in the kernel of the action. Since U is nilpotent and,by virtue of being contained in H , consists of semisimple elements, by Lemma 4.3in fact [ T : C T ( U )] < ∞ . In particular T acts virtually trivially on ∆, and so byProposition 4.2 ∆ is trivial. (cid:3) Corollary 4.4.
Let k be an infinite field. Let G be reductive k -group such G ( k ) consists of semisimple elements. Then G ( k ) admits no non-trivial split spherical BN -pairs. (cid:3) In particular Conjecture 1 holds in case G ( k ) consists of semisimple elements. If G is k -anisotropic and k is perfect then this is the case, by [B, Proposition 4.2(5)]and the fact discussed earlier that G ( k ) has no non-trivial unipotent elements. Wethus have a second, different proof that Conjecture 1 holds when k is perfect.5. Division algebras, and some conclusions
As promised, in case H = G ( k ) is the multiplicative or norm-1 group of a finitedimensional k -division algebra, we can also eliminate the rank-1 case. By the proofof Theorem 1.3 it suffices to show that if ( B = U (cid:111) T, N ) is any split spherical BN -pair of H , then [ T : C T ( U )] < ∞ . We prove this in the following Lemma 5.1.
Let H be the multiplicative or norm-1 group of a finite dimensional k -division algebra D . Let U ≤ H be nilpotent and T ≤ N H ( U ) . Then [ T : C T ( U )] < ∞ .Proof. Clearly k [ U ] is a division subalgebra of D . Then the canonical faithfulrepresentation k [ U ] (cid:44) → GL( k [ U ]) ∼ = GL r ( k ) for r = dim k k [ U ] is irreducible. Hencealso the restricted representation U (cid:44) → GL r ( k ) is faithful and irreducible. By[S, Theorem 27] we conclude that [ U : Z ( U )] < ∞ . Now consider the action of T on U by conjugation. Clearly Z ( U ) is normalized by this action, so we get ahomomorphism T → Aut(
U/Z ( U )). Let T denote the kernel of this map, so T has finite index in T . Choose a transversal { u , . . . , u r } of U/Z ( U ). For any i , T normalizes the abelian group V i = (cid:104) u i , Z ( U ) (cid:105) . If k [ V i ] denotes the subfield of D spanned by V i , we get a homomorphism φ i : T → Aut( k [ V i ] | k ), the kernel of whichhas finite index in T (since k [ V i ] | k is a finite extension). In particular r (cid:92) i =1 ker φ i NISOTROPIC GROUPS ADMITTING NO SPLIT SPHERICAL BN-PAIRS 11 has finite index in T , and thus in T . Of course any element of this subgroupcentralizes each V i , and so centralizes U . We conclude that indeed [ T : C T ( U )] < ∞ . (cid:3) Corollary 5.2. If H is the multiplicative or norm-1 group of a division ring, then H admits no non-trivial split spherical BN -pairs. (cid:3) To summarize, at this point Conjecture 1 stands completely proved in case k is local or perfect, or if G ( k ) consists of semisimple elements, or if G ( k ) is themultiplicative or norm-1 group of a finite dimensional k -division algebra. Also theconjecture stands proved modulo rank-1 BN -pairs if G ( k ) contains no non-trivialunipotent elements.Lastly we note that we have two main tools now to demonstrate that a splitspherical BN -pair ( B = U (cid:111) T, N ) of a reductive anisotropic group is trivial.Namely, if the rank is greater than 1 it suffices to show U is virtually abelian, byProposition 3.3, and if the rank is arbitrary it suffices to show that [ T : C T ( U )] < ∞ ,by Proposition 4.2. It is our hope that these criteria could prove useful in eventuallyestablishing the full conjecture. References [AB] Peter Abramenko and Kenneth S. Brown,
Buildings: Theory and Applications , Grad-uate Texts in Mathematics, vol. 248, Springer-Verlag, New York, 2008.[B] Armand Borel,
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Groupes r´eductifs , Inst. Hautes ´Etudes Sci. Publ.Math. (1965), 55-150.[CM1] Pierre-Emmanuel Caprace and Timoth´ee Marquis, Can an anisotropic reductivegroup admit a Tits system? (2009), available at arxiv.org/PS_cache/arxiv/pdf/0908/0908.2577v1.pdf .[CM2] ,
Can an anisotropic reductive group admit a Tits system? , Pure and Appl.Math. Quart. (2011), 539-558.[DMHTVM] Tom De Medts, Fabienne Haot, Katrin Tent, and Hendrik Van Maldeghem,
Split BN -pairs of rank at least 2 and the uniqueness of splittings , J. Group Theory (2005), 1-10.[P] Gopal Prasad, Weakly-split spherical Tits systems in pseudo-reductive groups (2011), available at arxiv.org/PS_cache/arxiv/pdf/1103/1103.5970v2.pdf .[RSS] Andrei S. Rapinchuk, Yoav Segev, and Gary M. Seitz,
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Department of Mathematics, University of Virginia, Charlottesville, VA 22904
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