Discrete Subgroups Generated by Lattices in Opposite Horospherical Subgroups

JOURNAL OF ALGEBRA 203, 621᎐676Ž. 1998 ARTICLE NO. JA977188

Hee Oh

Department of Mathematics, Yale Uni¨ersity, New Ha¨en, Connecticut 06520

View metadata, citation and similar papersCommunicated at core.ac.uk by Walter Feit brought to you by CORE Received June 19, 1996 provided by Elsevier - Publisher Connector

Contents.

0. Introduction. 1. Preliminaries. 1.1. Notation and terminology. 1.2. Some known algebraic lemmas. 1.3. Adjoint representation and maximal subgroups. 1.4. ޑ- forms of algebraic groups and ޑ-rational representations. 1.5. Exten- sion of ޑ-forms. 2. The subgroups of the form ⌫ and ޑ-forms. 2.1. Discrete subgroups in F12, F algebraic groups. 2.2. Generators of arithmetic groups. 2.3. Reflexive horospherical subgroups. 2.4. Margulis’ theorem on representation the-

ory and extension of ޑ-forms. 2.5. The subgroup generated by ZUŽ.1 and ZUŽ.2 . 3. Adjoint action on the space of lattices. 3.1. The space of lattices in algebraic unipotent groups. 3.2. Adjoint action. 3.3. Ratner’s theorem on orbit closures. 3.4. Closedness of some orbits and ޑ-forms. 4. The proof of the main theorem. 4.1. Commutative horospherical subgroup cases. 4.2. Heisenberg horospherical subgroup cases. 4.3. Non-ޒ- Heisenberg horospherical subgroup cases. 4.4. Arithmetic subgroups of the form ⌫ . F12, F

0. INTRODUCTION

Let G be a center-free connected semisimple real algebraic group with no compact factors. The unipotent radical of a proper parabolic subgroup of G is called horospherical. Two horospherical subgroups are called opposite if they are the unipotent radicals of two opposite parabolic subgroups.

621

0021-8693r98 $25.00 Copyright ᮊ 1998 by Academic Press All rights of reproduction in any form reserved. 622 HEE OH

We recall the following theorem:

THEOREM 0.1Žwxwx 12, Theorem 7.1.1 , also see 18, Theorem 4.2. . Suppose that the real rank of G is at least 2. Then for any irreducible non-uniform lattice ⌫ in GŽ.ޒ , there exists a pair of opposite horospherical subgroups U1 and U2 defined o¨er ޒ such that ⌫ l UiiŽ.ޒ is a lattice in U Ž.ޒ for i s 1, 2. This theorem was one of the main steps in proving the arithmeticity of a non-uniform lattice in such groups, without the use of the superrigidity theoremwx 13 which had settled the arithmeticity of both uniform and non-uniform lattices at once. In this paper we study the converse problem, which may be stated as follows: suppose that one is given opposite horospherical real subgroups U1 and U21and lattices F and F212inside U Ž.ޒ and U Ž.ޒ , respectively. Then under what conditions is the group generated by F12and F discrete? What discrete subgroups of G can arise in this way? Our main result is that if G is absolutely simple, thenŽ under some additional assumptions on U12and U . any discrete group generated by F12and F is a non-uniform lattice in GŽ.ޒ . In particular we prove the following:

THEOREM 0.2. Let G be an adjoint absolutely simple ޒ-algebraic group with real rank at least 2, U12, U a pair of opposite horospherical ޒ-subgroups of G. Suppose that G is split oër ޒ and that U1 is not the unipotent radical of a Borel subgroup in a group of type A2 . Let F12 and F be lattices in U1Ž.ޒ and U 2 Ž.ޒ , respectiëly. If F12 and F generate a discrete subgroup, then there exists a ޑ-form of G with respect to which U12 and U are defined oër ޑ and Fii is commensurable to U Ž.ޚ for each i 1, 2. Furthermore the discrete subgroup ⌫ generated by F and s F12, F 1 F2 is commensurable to GŽ.ޚ . Let us remark that Theorem 0.2 is not true in the group of real rank one; in fact there exist discrete subgroups of the form ⌫ which are not F12, F Ž. Ž.1n lattices. For example, the subgroup ⌫n of SL2 ޒ generated by01 and Ž.10 n 1 for some nonzero n g ޚ is not a lattice if n ) 2. To see this, note that the subgroup ⌫n is contained in the subgroup generated by the ele- Ž.1 n Ž0 y1 . ments01 and 1 0 , and the fundamental domain in the upper half plane for the latter subgroup is the set Äz g ވqN <

THEOREM 0.3. Let Zii denote the center of U for each i s 1, 2. In Theorem 0.2 the assumption that G is split o¨er ޒ may be replaced by one of DISCRETE SUBGROUPS 623 the following:

Ž.1 U1 is commutati¨e, and either rank ޒ ŽG .G 3 or G is not of type E6.

Ž.2 U1 is Heisenberg, and either rank ޒŽ.G G 3 or G is not of type A2 , Bnn, or D .

Ž.3wxU11,UsZ 1,Z 1 is not the root group of a highest real root, and either rank ޒŽ.G00G 3, or G is not of type E 60, where G is the subgroup generated by Z12 and Z .

Ž.4Ifwx U11, U / Z 1, Z 1 is the root group of a highest real root, and X XX either rank ޒŽG00.G 3 or G is not of type E 60, where G is the subgroup X X X generated by the center of U12and of U , and Uiis the centralizer of y1y1 U˜iisÄggU< gug u g Z i for all u g UinU ii4. Since Theorem 0.2 follows from Theorem 0.3, we will refer to Theorem 0.3 as the main theorem hereafter. As a corollary, we obtain a complete classification of discrete subgroups generated by lattices in opposite horospherical subgroups U12and U considered in the main theorem. In particular, we note the following:

COROLLARY 0.4. Let G, U12, and U be as in the main theorem. Then any discrete subgroup generated by two lattices in U12Ž.ޒ and U Ž.ޒ is an arithmetic subgroup. Therefore for the ޒ-split groups, combining Theorem 0.2 with Theorem 0.1, we obtain the following criterion for a discrete subgroup to be a non-uniform lattice:

COROLLARY 0.5. Let G be an adjoint absolutely simple ޒ-split group with rank at least 2 and ⌫ a discrete subgroup. In addition, assume that G is not of type A2 . Then ⌫ is a non-uniform lattice if and only if there exists a pair of opposite horospherical subgroups U12 and U of G such that ⌫ l Ui is Zariski dense in Ui for each i s 1, 2. ŽŽNote that any arithmetic subgroup of G ޒ.which has a non-trivial unipotent element is a non-uniform lattice in GŽ.Žޒ e.g.,w 16, Theorem 10.18x.. . Remark. We note that the main theorem presents a strong necessary condition for discreteness of a subgroup generated by lattices in U1Ž.ޒ and U2Ž.ޒ. We refer the readers towx 9 for discreteness criteria of this kind in SL2Ž.ޒ . We call a horospherical subgroup U ޒ-Heisenberg if wxU, U is equal to the center ZUŽ.of U, i.e., 2-step nilpotent and ZU Ž.is the root group of a highest real root of G.IfUis ޒ-Heisenberg and dim ZUŽ.s1, then U is Heisenberg. It should be noted that the main theorem would not cover the 624 HEE OH

cases when U1 is either ޒ-Heisenberg with dimŽŽ..ZU )1 or Heisenberg in an ޒ-split group of type A2 , even if we were to drop the assumption on ޒ-anisotropic factors. On the other hand we can see that the main theoremŽ. Theorem 0.3 implies Theorem 0.2 as follows: if G is split over X ޒ, none of the subgroups G00, G , and H has ޒ-anisotropic factors. Therefore the case when U1 is either commutative or Heisenberg follows fromŽ. 1 . When U11is neither of those, U satisfies assumptionŽ. 2 or Ž. 3 according to whether U1 is 2-step nilpotent or not, respectively. The proof of the main theorem is given in three parts according to whether the horospherical subgroups involved are commutativeŽ Theorem 4.1.1.Ž , Heisenberg Theorem 4.2.11 . , or non-ޒ-Heisenberg Ž Theorem 4.3.4 . . One of the main ideas for the first two cases is to use Raghunathan’s conjecture proved by RatnerŽ. Theorem 3.3.1 for the action of the group of real points of the commutator subgroup of NUŽ.12lNU Ž.on the space of lattices in UiŽ.ޒ for each i s 1, 2. A theorem of Margulis on the construc- tion of a representationŽ. Theorem 2.4.2 enables us to reduce the non-ޒ- Heisenberg horospherical subgroup cases to the commutative cases. In fact the assumptions on ޒ-anisotropic factors inŽ.Ž. 1 ᎐ 3 arise because of the dependence of our proof on Ratner’s theorem. The main theorem of this paper was announced inwx 14 together with a detailed sketch of the proof, in the case when GŽ.ޒ s SLn Ž.ޒ , n G 3, and the horospherical subgroups involved are commutative.

1. PRELIMINARIES

1.1. Notation and Terminology 1.1.1. As usual, ރ, ޒ, ޑ, ޚ, and ގ denote the complex numbers, the reals, the rationals, the integers, and the positive integers. 1.1.2. For a group H, ZHŽ.denotes the center of H. For any subset F;H,NFŽ.and CF Ž.denote the normalizer and the centralizer of F in H, respectively. For a Lie algebra ᒅ, the commutator of two elements X, Y of ᒅ is denoted by wxX, Y and for any two subsets A and B of ᒅ, wxA, B denotes the linear subspace generated by all the commuators wxX, Y , X g A, YgB.ZŽ.ᒅdenotes the center of ᒅ, that is, ÄX g ᒅ N wxX, Y s 0 for all Y g ᒅ4. 1.1.3. Two subgroups H12and H are called commensurable if H12l H has a finite index in both of H1212and H . H = H denotes the direct product of H12and H and L h N the semi-direct product of L and a normal subgroup N. DISCRETE SUBGROUPS 625

1.1.4. For an algebraic group H, RHŽ.and RHu Ž.denote the radical and the unipotent radical of H, respectively. For a Lie group H, we denote by H 0 the connected component of the identity of H.

1.1.5. For a subfield k of ޒ, a linear algebraic ޒ-group G ; GLnŽ.ރ is defined over k if it consists of all matrices whose entries annihilate some set of polynomials on MnŽ.ރ with coefficients in k. In this case, we denote y1 by GJŽ.the subgroup Äg s Ž.gij g G N g, g g GLnŽ. J , g ijs ␦ ij mod J4 for any subring J of k and GŽ.ޒ s G l GLn Ž.ޒ . 1.1.6. A k-form of an algebraic ޒ-group G is a pair Ž.G˜˜, f where G is an algebraic group defined over k and f an isomorphism G ª G˜ defined over ޒ. For a given k-form Ž.G˜, f of G, we denote fGJy1 ŽŽ..Ž.˜ by GJ for a subring J of k. If an algebraic ޒ-subgroup H of G is such that fHŽ.is a k-subgroup of G˜, we say that H is defined over k and denote H l GJŽ. by HJŽ.. A subgroup commensurable to G Ž.ޚ is called an arithmetic subgroup of GŽ.ޒ . 1.1.7. We extend the definition of an arithmetic group to a semisimple Lie group with finite center as follows. Let G be a connected semisimple Lie group with finite center. A discrete subgroup ⌫ of G is called an arithmetic subgroup if there exist a connected adjoint semisimple algebraic 0 ޑ-group G and an isogeny p : G ª GŽ.ޒ such that pŽ.⌫ is commensu- 0 rable to GŽ.ޚ l G Ž.ޒ . 1.1.8. A connected algebraic k-group G is called absolutely almost simple if it has no connected normal subgroup, and almost k-simple if it has no connected normal k-subgroup. A connected semisimple algebraic group G is called simply connectedŽ. resp. adjoint if every central isogeny ␾:GЈªGŽ.resp. ␾ : G ª GЈ , for GЈ connected, is an algebraic group isomorphism. In characteristic 0 case, a connected semisimple algebraic group is adjoint if and only if its center is trivial. 1.1.9. For a locally compact group G, a discrete subgroup ⌫ of G is called a lattice if the quotient Gr⌫ has a finite invariant measure. A lattice ⌫ in G is called uniform if Gr⌫ is compact, and non-uniform, otherwise. A lattice ⌫ in a connected semisimple Lie group G with finite center is called irreducible if for every normal subgroup N of positive dimension, ⌫ is dense when projected onto GrN. 1.1.10. A discrete subgroup F in a real vector space V is called a quasi-lattice and in particular, if F spans V, then we call it a lattice. The determinant of a lattice F in V is the volume of the quotient VrF. 1.1.11. A connected semisimple k-group G is isotropic oër k if it contains a non-trivial k-split torus and is anisotropic oër k otherwise. For k s ޒ, G is anisotropic over ޒ if and only if GŽ.ޒ is compact. 1.1.12. Let G be a connected semisimple algebraic k-group. We call a subgroup U a horospherical k-subgroup if it is the unipotent radical of a parabolic k-subgroup P such that P l GЈ is a proper parabolic subgroup 626 HEE OH of GЈ for each semisimple normal k-subgroup of G. It is known that the normalizer of a horospherical k-subgroup is a parabolic k-subgroup. We note that, in our definition, the existence of a horospherical ޒ-subgroup of a semisimple ޒ-group G implies that G does not have any ޒ-anisotropic factors. 1.1.13. In a connected simple algebraic group G, a horospherical subgroup U is called Heisenberg if wxU, U s ZUŽ.and ZU Ž.has dimension one. We call a horospherical subgroup U ޒ-Heisenberg if wxU, U s ZUŽ. and ZUŽ.is the root group of a highest real root of G Žalso see Subsection 1.3.1. . A parabolicŽ. resp. horospherical subgroup of a semisimple algebraic group is called reflexië if its conjugacy class contains an opposite parabolic Ž.resp. horospherical subgroup to it. 1.1.14. Let G be a connected semisimple algebraic k-group and P a parabolic k-subgroup. A Levi decomposition P s LRuŽ. P of P is called a k-Levi decomposition if L is defined over k. We denote by Ad : L ª GLŽ.U the representation of L in U s Lie ŽRPu Ž..which is the restriction of the adjoint representation of P.

1.1.15. Let G be a connected semisimple algebraic ޒ-group and U1 and U2 a pair of opposite horospherical ޒ-subgroups. For lattices F12and F in UŽ.ޒand U Ž.ޒ , respectively, we denote by ⌫ the subgroup of GŽ.ޒ 12 F12,F generated by F12and F . 1.1.16. Let Ž.G, U12, U be a triple where G is a connected semisimple adjoint algebraic ޒ-group with no ޒ-anisotropic factors and U12and U a pair of opposite horospherical ޒ-subgroups. In order to avoid the lengthy repetition, we say the triple Ž.G, U12, U has property Ž. A if every discrete subgroup of the form ⌫ is an arithmetic subgroup of GŽ.ޒ 0where F F12, F 1 and F212are lattices in U Ž.ޒ and U Ž.ޒ , respectively. We use this terminology only in Section 4 where we prove the main theorem. 1.1.17. All algebraic groups are assumed to be connected and all fields, usually denoted by k or K, have characteristic 0. We freely use terminology from the theory of algebraic groups inwx 1, 5 and of Lie algebras in wx 8 .

1.2. Some Known Algebraic Lemmas 1.2.1Ž cf.wx 5, Chaps. 3᎐4. . Let G be a connected semisimple algebraic k-group, S a maximal k-split torus of G, and T a maximal k-torus containing S. Denote by k⌽ s ⌽Ž.ŽS, G resp. ⌽ s ⌽ Ž..T, G the set of roots of G with respect to S Ž.resp. T . An element in k⌽ is called a k-root. We choose compatible orderings on ⌽ and kk⌽, and let ⌬ and ⌬ be the simple roots for these orderings. Let j be the map ⌽ ªk⌽ j Ä40 induced by the restriction onto S. DISCRETE SUBGROUPS 627

For each b g ⌽Ž.T, G , we denote by Ub the unique one parameter root subgroup associated with b. We observe that LieŽ.Ubbs U [ Ä¨ g LieŽ.G < Ad tŽ.bt Ž.,t T4, dimŽ.U 1, and LieŽ.G U ¨ s ¨ g bbs s [ g⌽ŽT,G.b[ LieŽ.T .

A subset ⌿ ;kk⌽ is called closed if a, b g ⌿ and a q b g ⌽ imply aqbg⌿.If ⌿is closed, then we denote by G⌿ the subgroup generated y1 U by T and the subgroups Ua, a g j Ž⌿ j Ä40. , and by G⌿ the subgroup y1 generated by all the subgroups Ua, a g j Ž.⌿ . The subgroups G⌿ and U G⌿ are algebraic and do not depend on the choice of a maximal torus T U containing S. The groups G⌿⌿and G are defined over k if and only if ⌿ is invariant under GalŽ.Krk , K the separable closure of k w5, Proposition U 3.14x . If G⌿⌿is unipotent, then it will be also denoted by U and the set ⌿ in this case will be called unipotent. For a closed unipotent subset ⌿ ;k⌿, the unipotent subgroup U⌿ is defined and split over k. 1.2.2. For ⌰ ;k ⌬, wx⌰ denotes the ޚ-linear combinations of ⌰ which are k-roots.

We define the following closed subsets of k⌽:

qy y ␲␪swx⌰jk⌽, ␲␪ swx⌰jk⌽, q yy ␤␪sk⌽ywx⌰, ␤␪ sk⌽ywx⌰.

y y For the sake of simplicity, we shall denote by k PP⌰, k ⌰ ,kVV⌰, k ⌰ , the y y subgroups G , G y, U , U y . The subgroups PP,V, and V are ␲ ⌰⌰⌰⌰␲␤␤ k⌰,k⌰k⌰ k⌰ connected and defined over k.

The subgroups k P⌰ Ž.resp. kV⌰ , ⌰ ;k ⌬ are called standard parabolic q Ž.resp. horospherical k-subgroups of G associated with S and k⌽ . Every parabolicŽ. resp. horospherical k-subgroup of G is conjugate by an element of GkŽ.to a unique standard parabolic Ž resp. horospherical . subgroup.

y y LEMMA.1Ž.RPuk Ž⌰ .skV⌰,RPuk Ž ⌰.skV⌰ for any ⌰ ;k⌬. Ž.2 Any pair of opposite parabolicŽ resp. horospherical. k-subgroups is y y conjugate by an element of GŽ. k to the pairk P⌰, k P⌰ Ž resp. kV⌰, k V⌰ . for some ⌰ ;k ⌬. Ž.3 Any two parabolic k-subgroups opposite to P are conjugate by a unique element of RuŽ.Ž. P k . 1.2.3. The root system ⌽Ž.ŽŽ..S, G resp. ⌽ T, G is irreducible if and only if G is almost k-simpleŽ. resp. absolutely almost simple . The type of an irreducible root system ⌽ is by definition the type of its Dynkin diagram. For a k-simple k-group G, we refer the type of ⌽Ž.T, G by the absolute type, or simply the type of G, and the type of ⌽Ž.S, G by the k-type of G. 628 HEE OH

1.2.4. PROPOSITION wx21, 3.1.2 .Ž. 1 A connected simply connected Ž resp. adjoint. semisimple k-group decomposes uniquely into a direct product of simply connectedŽ. resp. adjoint almost k-simple normal k-groups. Ž.2 A connected semisimple k-group decomposes into an almost direct product of almost k-simple normal k-groups.

1.2.5. PROPOSITION. For a connected semisimple k-group G, there exists a ˜˜ sequence G ªpp˜G ª G, where G is a simply connected k-group, Gisan adjoint k-group, and˜ p and p are central k-isogenies. The groups G˜ and G and the isogenies˜ p and p are determined uniquely up to k-isomorphism. 1.2.6. Let kЈ be a finite separable field extension of k. We denote by Ј Ž RkЈr k the restriction of scalar functor from k to k. For definition, see wx5, 6.17᎐6.21 ..

ROPOSITION Ž. Ј P .1If H is a k -group, then the functor RkЈr k defines a bijection of the set of parabolicŽ. resp. horospherical kЈ-subgroups of H onto Ž. Ž. the set of parabolic resp. horospherical k-subgroups of the k-group RkЈr k H Ž. Ž Ž.. and if H is reducti¨e, then rank kЈH s rank kkRHЈrk . Ž.2 If G is a simply connectedŽ. resp. adjoint almost k-simple k-group, then there exists a finite separable field extension kЈ of k and a connected simply connectedŽ. resp. adjoint absolutely almost simple kЈ-group GЈ such Ј that G s RGkЈrk . 1.2.7. The next proposition is useful when we want to determine whether some algebraic groups and algebraic maps are defined over a subfield.

PROPOSITION Žseewx 27, 3.1.8, 3.1.10.Ž. . 1 Suppose that G ; GLn Ž.ރ is an algebraic group and that G l GLnŽ. k is Zariski dense in G for some subfield k of ރ. Then G is defined oër k. Ž.2 Suppose that V, W are k-ärieties and that f : V ª W is a regular map. Suppose also that there is a set A ; Vk which is Zariski dense in V such that fŽ. A ; Wk . Then f is defined oër k.

1.3. Adjoint Representation and Maximal Subgroups 1.3.1. Let G be a connected almost k-simple k-group, S a maximal k-split torus, and ⌽Ž.S, G a corresponding root system. We fix a basis k ⌬ of ⌽Ž.S, G . Since G is k-simple, ⌽Ž.S, G is an irreducibleŽ but not necessarily reduced. root system. We note that a horospherical k-subgroup

U is k-Heisenberg if and only if U is conjugate to kV⌰ such that VV ZVŽ.and ZV Ž.U where ␣ is the highest root in wxk ⌰, k ⌰ sk ⌰ k ⌰ s␣ h h ⌽Ž.S,G. Note that the center of a k-Heisenberg subgroup is the root group of a highest k-root. DISCRETE SUBGROUPS 629

We note that if U has dimension one, a k-Heisenberg subgroup is in ␣h fact a Heisenberg subgroup. If kV⌰ is k-HeisenbergŽ. resp. commutative , we call k ⌬ y ⌰ the set of k-HeisenbergŽ. resp. commutative roots. In an irreducible root system ⌽Ž.S, G , there exist a unique highest k-root ␣h and a unique set of simple roots ⌬ ⌬ such that V is the kH; k kkkH⌬y⌬ unique k-Heisenberg standard horospherical k-subgroup of G. It is not difficult to prove the following lemma which characterizes the commutative and k-Heisenberg roots in each irreducible root system.

LEMMA.1Ž. A simple root ␣ is a commutati¨e root if and only if its coefficient in ␣h is 1.

Ž.2 The set kH⌬ is determined uniquely by the condition that if ␣ gkH⌬ , then ␣hky ␣ is a root and the sum of the coefficients of ⌬ Hin ␣his 2.

Commutative k-type ␣hk⌬ Hroots

An ␣12q␣qиии q␣␣n Ä4Ä12,␣␣ 1,...,␣n 4 Bn ␣12q2␣qиии q2␣␣␣n 21 ␣␣иии ␣ ␣␣␣ Cn 212q2qq2 ny1q n 1 n ␣␣иии ␣ ␣ ␣␣␣Ä4␣␣ Dn 12q2qq2 ny2q ny1q n 21,ny1,n E6 ␣123456q2␣q2␣q3␣q2␣q␣␣␣2Ä4 16,␣ E7123456712␣q2␣q3␣q4␣q3␣q2␣q␣␣ ␣ 7 E81234567882␣q3␣q4␣q6␣q5␣q4␣q3␣q2␣␣ л F412␣q3␣2q4␣3q2␣␣4 1л G213␣q2␣␣2 2л ␣ ␣ иии ␣ ␣␣ BCn 2 12q 2 q q2 ny1q 2 n 1 л

1.3.2Ž cf.wx 7, 5.5.1. . Among maximal parabolic subgroups, the adjoint representation of a Levi component on the Lie algebra of the unipotent radical is absolutely irreducible if and only if the unipotent radical is commutative. These cases are quite restricted, and the list is well known Ž.Table I . 1.3.3Ž cf.wx 7, 5.5.2. . The commutator quotient of the Lie algebra of a Heisenberg horospherical subgroup carries a sympletic form defined in terms of the Lie bracket. This form is preserved by the action of the Levi component. Except for the groups of type An, the representation of the Levi component on the commutator quotient is absolutely irreducible Ž.Table II . 1.3.4. The following two theorems of Dynkin classify the maximal connected subgroups of SLnŽ.ރ into the three categoriesᎏreducible Ž as linear groups. , irreducible non-simple, and irreducible simple. It is known that any irreducible connected Lie subgroup of SLnŽ.ރ is semisimplew 15, Theorem 1.1.1x . 630 HEE OH

TABLE I List of Parabolic Subgroups with Commutative Unipotent Radical

Group Levi component Unipotent radical

ރ knރ yk GLnknGL = GL yk m ރ 2ny1 SO2 nq12O ny11= GL 2n Sp2 nn GL S Ž.ރ ރ 2Žny1. SO2 n O2 ny11= GL n SO2 nn GL ⌳Ž.ރ E61Spin 0= GL1 spin 10q EE76=GL1 V 27

THEOREM Žcf.wx 15, 3.3.1᎐3.3.2.Ž . Let M be a maximal connected complex.Ž. Lie subgroup of SLn ރ .

Ž.1 If M is reducible, then it is a maximal parabolic subgroup of

SLnŽ.ރ . Ž.2 If M is non-simple irreducible, then it is conjugate to the subgroup

SLstŽ.ރ m SL Ž.ރ where n s st, s, t P 2.

1.3.5. THEOREM Žcf.wx 15, 3.3.3.Ž. . Let R : M ª GL V be a non-triïal irreducible linear representation of a simply connected simple complex linear group M. If there are no nondegenerate bilinear forms in V inäriant under R, then RŽ. M is a maximal connected subgroup of SL Ž. V , and if R is orthogonal or symplectic, then R is a maximal connected subgroup of SOŽ. V or Sp Ž. V , respectiëly. The only exceptions are the representations listed in Table 7 in wx15 .

TABLE II List of Parabolic Subgroups with Heisenberg Unipotent Radical

Group Levi component Commutator quotient

ރ ny2 Ž.ރ ny2 GLn GL1 = GLny21= GL [ * ރ 2Žny4. OOnny42=GL ރ 2Žny1. Sp2 n Sp2Ž ny1. = GL1 36 EGL66 ⌳Ž.ރ E71Spin 2= GL1 spin 12q EE87=GL1 V 56 36 FSp46=GL1 ⌳ primŽ.ރ 32 GGLS22 Ž.ރ DISCRETE SUBGROUPS 631

1.3.6. As a corollary of the above theorem and Subsections 1.3.2᎐1.3.3, we obtain the following two propositions. We denote by P a parabolic ޒ-subgroup of an adjoint absolutely simple algebraic group G,byLa Levi 0 ޒ-subgroup of P and H s wxL, L , and H s HŽ.ޒ . The notation Ad is as in Subsection 1.1.14.

PROPOSITION. If RuiŽ. P is commutati¨e, the representation Ad : H ª SLŽ.Uiiis absolutely irreducible and Ad Ž.H is a maximal connected closed subgroup of SLŽŽ..Ui ޒ . Proof. The first statement is due to the well known fact that all the representations listed in Subsection 1.3.2 are absolutely irreducible. By Proposition 1.2.5, there exist a simply connected simple ޒ-group H˜ and an ˜˜ ޒ-isogeny p : H ª H. Consider the representation Adi ( p of H. By the previous theorem, AdiiŽŽ..p H˜ , which is the same as Ad Ž.H , is a maximal subgroup of SLŽUii .. Since Ad ŽH Žޒ ..; SL ŽU i Žޒ .., it is clear that AdiŽ.H is a maximal connected closed subgroup of SLŽŽ..Ui ޒ .

1.3.7. Consider the case when RPuŽ.is Heisenberg. Except for groups of type Ani, U has two Adi -invariant subspaces one of which is ZŽ.Ui. Denote by V i the other invariant subspace. For groups of type Ani,Ad decomposes into the direct sum ␳ [ ␳* [ id where ␳ is the Ž.n y 1- ␳ ␳ dimensional standard representation of Any2 , * is the dual of , and id U is the one-dimensional trivial representation. Let Wiiand W be the invariant subspaces of ␳ and ␳*, respectively, and in this case, let V i U X denote Wii[ W . We shall denote by Adithe restriction of Adi on V i. Set SpŽ.V iis Äg g SLŽ.V N wxwxg¨, gw s ¨, w for all ¨, w g V i4. X PROPOSITION.1Ž. If G is not of type Ani,Ad is absolutely irreducible X and AdiŽ.H is a maximal connected closed subgroup of SpŽŽ..V i ޒ or SpŽŽ..V i ޒ itself. X X Ž.2 For Ani,Ad:HªSpŽ.V iis equi¨alent to ␳ [ ␳* and AdiŽ.His the maximal connected semisimple closed subgroup of SpŽŽ..V i ޒ . Proof. PartŽ. 1 can be shown by the same argument as the proof of the X previous proposition. To seeŽ. 2 , it is enough to note that AdiŽ.H is the semisimple part of a Levi subgroup of the maximal parabolic subgroup

Ä ggSpŽ.V iiiN gW s W 4which stabilize Wi.

1.3.8. LEMMA. Let H be a connected simple Lie group and for each is1, 2, ␾ii: H ª G an isomorphism for some Lie group Gi. Then ␦Ž.H s ÄŽŽ.␾12h,␾ Ž..hNhgH4 is a maximal closed connected subgroup of G12= G . Proof. If L is a closed connected subgroup which contains ␦Ž.H properly, then there exists an element Ž.g, e g L. It follows that L y1 contains the subgroup ÄŽŽ.␾11hg␾ Žh ..,eNhgH4. Since H is simple 632 HEE OH

y1 hence so is G11, ÄŽŽ.␾ hg␾ 1 Žh ..,eNhgH4Ä4sG1=e. In the same way, we can show that L contains Ä4e = G21; hence L s G = G2. ÄŽ.A0 Ž.4 1.3.9. PROPOSITION. Set H sN0 A AgSLn ރ and n P 4. Then a connected semisimple complex proper Lie subgroup of SL2 nŽ.ރ containing H ÄŽ.A 0 Ž.4 Ž. Ž. properly is either 0 B N A, B g SLn ރ or SL2 ރ m SLn ރ .

Proof. Let M be a maximal closed semisimple subgroup of SL2 nŽ.ރ containing H.If Mis reducible, then M is a semisimple part of a Levi subgroup, say L, of some maximal parabolic subgroup. Then L is the centralizer of a one-dimensional torus S. Since S should centralize H,we have that S ÄŽ.t 0 tރ*4Ä . Therefore L Ž.A 0 SL Ž.ރ A, B sN01rt g sg0 B 2 n N gGLnŽ.ރ 4and hence M s wxL, L . Since H is a maximal subgroup of wxL,L, in this case wxL, L is the only semisimple subgroup containing H. Suppose that M is irreducible and non-simple. Then M s gSLmŽ.ރ m y1 SLkŽ.ރ g for m ) 1 and k ) 1 such that mk s 2n. Since the Lie algebra of M is isomorphic to the direct sum of the Lie algebras of SLmŽ.ރ and SLkŽ.ރ , we have that n O m or n O k. It follows that k s n and hence ms2. Now we will show that g g SLnŽ.ރ m SL20 Ž.ރ . Denote by T the diagonal subgroup of SL2nŽ.ރ and set S s T0l SLnŽ.ރ m SL2 Ž.ރ .Bya direct calculation, we can see that CSŽ.sT0. Let TM be a maximal torus of M which contains S. Then TM ; CSŽ.sT0. Since any two maximal tori are conjugate, there exists h g SLnŽ.ރ m SL2 Ž.ރ such that TM s ghŽ T0 l y1 y1 SLnŽ.ރ m SL2 Ž.Ž.ރ gh . That ghŽ Tonl SL Žރ .m SL20 Žރ .Žgh . ; T im- y1 plies that gh g NTŽ0 .. Since Žgh .Ž SLn Žރ .m SL2 Žރ ..Žgh . contains H,it follows that gh and hence g belongs to SLnŽ.ރ m SL2 Ž.ރ . We can now exclude the case when M is simple irreducible. We recall the well-known fact that the smallest possible dimensions of linear representations of the groups of types G24, F , E 6, E 7, and E 8are 7, 26, 27, 133, and 248 respectively. If M is of type G2 , it implies that 2n is at least 8 and hence the rank of H is at least 4, yielding the contradiction. A similar argument can be applied for other types of exceptional groups to show that M is not of exceptional type. Now assume that M is of classical type. If it is of type Bkk, C ,orD k, then k is at least n y 1 since the rank of H is ny1. This excludes the type of Bk since the smallest possible dimension of linear representations of the groups of type Bny1 is 2n q 1. If M is of type Ckkor D , then we have k s n since the groups of type Cny1or Dny1 cannot contain a subgroup of type Any1 and k cannot be bigger than n since the dimension of the representation should be at most 2n. It follows that M is either a symplectic or an orthogonal group. But by a simple calculation we can see that there is no bilinear form which is invariant under H. Therefore M can be neither Cnnnor D . Now let M be of type Ak . Since H is reducible, there exists a maximal parabolic subgroup PM of DISCRETE SUBGROUPS 633

M which contains H. It is not difficult to see that a maximal parabolic subgroup of SL2 nŽ.ރ containing H is either

AC P SL Ž.ރ A, B GL Ž.ރ or 12sg½5ž/0B nnN g A0 P SL Ž.ރ A, B GL Ž.ރ . 22sg½5ž/CB nnN g

Denote by L the common Levi subgroup of P12and P . We can also easily see that L is the only Levi subgroup of Pi which contains H. Therefore a Levi subgroup, say LMM,of P which contains H is contained in L.Itis known that there is no semisimple subgroup of maximal rank in the groups ÄŽ.A 0 Ž. of type AkM. It follows that L is either L02sg0ASL nރ N A g GLnŽ.ރ 4or L since H is a maximal semisimple subgroup of wxL, L . Hence the type of M is either An or A2 ny1, respectively. But we can check from the character formula that thre is no irreducible representation of dimension 2n for the groups of type An unless n s 3. ÄŽ.A 0 Ž.4 1.3.10. PROPOSITION. Set H sN0 A AgSOn ރ and n P 8. Then a connected semisimple complex Lie subgroup of Sp2 nŽ.ރ containing H is either A 0 ÄŽt . Ž. Ž.4 Ž. Ž. 0Ay1 gSp2 nnރ N A g SL ރ or SL2ރ m SOnރ . The proof of the above proposition goes exactly the same as the proof of Proposition 1.3.9.

1.4. ޑ-Forms of Algebraic Groups and ޑ-Rational Representations We prove some basic propositions on the ޑ-forms of semisimple groups and ޑ-rational representations. The main references for this section are wx21, 22 . 1.4.1Ž seewx 21, Chap. 2. . Let G be a semisimple algebraic k-group, ⌫sGalŽ.Krk where K is the separable closure of k, and S, T, ⌬, and k ⌬ be as in Subsection 1.2.1. Denote by ⌬ 0 the subset of ⌬ which vanishes on S. We define the so-called )-action of ⌫ on ⌬. For ␥ g ⌫, there exists a ␥ unique element w in the Weyl group such that wŽ ⌬. s ⌬ and we set ␥ ␥␣Ž.sw Ž␣.. If the )-action is trivial, the k-form is called inner and otherwise, outer. The orbits of ⌫, whose elements do not belong to ⌬ 0 , are called distinguished orbits. The Tits index of a group G is the data consisting of ⌬, together with Dynkin diagram, ⌬ 0 , and the )-action of ⌫. The group G is determined, up to k-isomorphism, by its K-isomorphism class, the Tits index, the commutator subgroup of CSŽ., called the semisimple anisotropic kernel of G, given up to k-isogeny. 634 HEE OH

PROPOSITION wx21, 2.5.4 . Let P⌰ denote the parabolic subgroup generated Ž.␣ ⌬ Ž.␣ ⌰ by T and U␣ g and Uy␣ g . Then P⌰ is defined o¨er k if and only if ⌰ contains ⌬ 0 and is in¨ariant under the )-action of ⌫. 1.4.2. For an adjoint semisimple k-group G, there exist an adjoint semisimple k-split group Gd and an adjoint semisimple k-quasi-split group q G whose )-action of ⌫ on the Dynkin diagram is the one given by the index of G, both of which are K-isomorphic to G. The k-form of G is obtained by twisting Gd Žresp. G q . by a cocycle c of ⌫ with values in d q Aut KKŽG .Žresp. Int ŽG .. Žseewx 21, 3.4.2. . The proof of the following proposition is due to G. Prasad.

PROPOSITION. Let G be an adjoint semisimple algebraic ޒ-group. Then for a giën minimal parabolic ޒ-subgroup P, there exists a ޑ-form on G with respect to which eëry parabolic ޒ-subgroup of G which contains P is defined oër ޑ. Proof. The group G decomposes into a direct product of adjoint ޒ-simple normal ޒ-groups by Proposition 1.2.4 and a parabolic ޒ-subgroup of G is a product of parabolic ޒ-subgroups of each ޒ-simple factor of G. Therefore it is enough to prove the proposition for an ޒ-simple group. Case Ž.1. Gis absolutely simple. Let G q be the adjoint ޑ-split ޑ-group if the ޒ-form of G is of inner type, and the quasi-split ޑ-form of G, splitting over k s ޑŽ.i , if the ޒ-form of G is of outer type. Let P s MU be a minimal parabolic q q q ޒ-subgroup of G and P s M U be the corresponding parabolic ޑ-subgroup of Gq, where M q is a Levi ޑ-subgroup of P q and U q is the qЈ qЈqЈ unipotent radical. Set MЈ s wxM, M and M s wM , M x. Since the two ޒ-forms of G and Gq differ only by their semisimple anisotropic kernels MЈ and M qЈ and the indices of MЈ and M qЈ coincide, the ޒ-form of G is q the twist of the ޒ-form of G by a cocycle c of GalŽ.ރrޒ with values in qqЈ Int ރ M s M Žseewx 21, 3.4.2.Ž . But it is known seewx 3, Theorem 1.7. that the natural homomorphism of H 1Žޑ, M qЈ. to H 1Žޒ, M qЈ. is surjective. qЈ Thus there is an M -valued cocycle d on GalŽ.ރrޑ cohomologous to c over ޑ. The twist of G q by the cocycle d is a ޑ-form of G with the following properties:Ž. i over ޒ, it coincides with the ޒ-form of G; Ž ii . its distinguished orbits of GalŽ.ރrޑ contain all the distinguished orbits of GalŽ.ރrޒ of the ޒ-form of G. Therefore by Proposition 1.4.1, this ޑ-form has the required property. Case Ž.2. Gis ޒ-simple but not absolutely simple. There exists an adjoint absolutely simple group GЈ such that G s Ј Јd ޑŽ. Ј RGރ r ޒ by Proposition 1.2.6. Let G be the adjoint i -split form of G DISCRETE SUBGROUPS 635

d Јd d ޑ and G s RGޑŽi.r ޑ . It is clear that G is a -form of G with the desired property. 1.4.3. Let D be a central simple division algebra over k of degree d. For

any r, there exists a k-form GLr, Drdrof GL such that GL,DrŽ. k s GL Ž D .. Let V be a finite dimensionalŽ. left vector space over D.A D-representation or Ž.D, k -representation of G in V is a k-homomorphism G ª GLV, D. A representation of G is called D-rational if it is equivalent to a D-representation.

1.4.4. PROPOSITION wx22, Theorem 3.3 . For any dominant weight ␭ fixed by ⌫, there exists a di¨ision algebra D, unique up to k-isomorphism, such that an absolutely irreducible representation of G with the highest weight ␭ is D-rational.

1.4.5. PROPOSITION. Let G be an absolutely simple algebraic k-group of

type Cn. If the standard 2n-dimensional representation of G is k-rational, then G is split o¨er k. Proof. By the classification of k-forms given inwx 21 , GkŽ. is Ž. SU2 n r d D, h where D is a division algebra of degree d and h is a nondegenerate antihermitian sesquilinear form of index r relative to a ␴ first kind involution ␴ such that D has dimension 1r2 ddŽ.q1 . The Ž. canonical representation SU2 n r d D, h ª SL2 n, D is an absolutely irreducible representation with the same highest weight as the standard 2n-dimensional representation. Since the standard representation is rational over k, by the uniqueness of such a D, D s k. It implies that G is split over k.

1.4.6. Let ␣ be a dominant weight and k␣ the invariant field by the stabilizer of ␣ in ⌫. Then Proposition 1.4.4 gives a central division algebra D over k such that ␳ : G GL is an absolutely irreducible ␣ ␣␣ª n,D␣ Ž.D␣ ,k␣ -representation of dominant weight ␣. Define restD kn: GL ,D ␣r ␣␣k GL2Ž. K GL Ž D K .and similarly rest . We put ␳ ª nd s n ␣ m kk␣ ␣rk␣s restŽ. rest ␳ . k␣r kD␣rk( ␣ THEOREM wx22, Theorem 7.2, Lemma 7.4 .Ž. 1 For each dominant weight k of G, ␳␣ is irreducible oër k. Ž.2 Each k-irreducible representation of G is k-equiälent to the repre- k sentation of the form ␳␣ . kk Ž.3Let ␣ and ␣ Ј be dominant weights. Then ␳␣ and ␳␣ Ј are k-equi¨- alent iff there exists ␥ g ⌫ such that ␥␣Ž.s␣Ј. k Ž.4If ⌫ Ž␣ .s Ä4␣1,...,␣n , then ␳␣ is equiälent oër K to the direct sum of d irreducible representations of dominant weight ␣1,...,d irreducible 2 representations of dominant weight ␣n where d s wxD␣␣: k . 636 HEE OH

1.4.7. Let n P 3, and G be either SLnŽ.ރ or SO2 n Ž.ރ , and ␺ be the representation of SL2Ž.ރ = G defined by ␺ Ž.A, B s A m B for A g SL2Ž.ރ and B g G.

PROPOSITION. Let H s SL2Ž.ރ = G. Assume that the representation ␺ is rational o¨er ޑ.

Ž.1 If G s SLnŽ.ރ , the group H is equipped with either a split ޑ-form t or an inner ޑ-form such that HŽ.ޑ is conjugate to SL1Ž.Ž Dޑ = SLs Dޑ . where D is a diïsion algebra defined oër ޑ such that Dޑ is a central simple t t quarternion diïsion algebra oër ޑ,2ssn,and Dޑ [ Ä X g M2Ž.ރ N X g Dޑ4.

Ž.2 If G s SOnŽ.ރ , the group H is equipped with either a split ޑ-form t or a ޑ-form such that HŽ.ޑ is conjugate to SL1Ž.Ž Dޑ = SUs Dޑ , h. where D is a diïsion algebra defined oër ޑ such that Dޑ is a central simple quaternion diïsion algebra oër ޑ,2ssn,and h is a non-degenerate ␴ hermitian form relatiëtoaninölution ␴ of the first kind such that D has dimension 3. In particular, with respect to those ޑ-forms, ÄŽ.I, A N A g Gisa4 ޑ-subgroup of H. Proof. Ž.1 Let T be a maximal torus consisting of the diagonal ele-

ments of SL2Ž.ރ = SLn Ž.ރ . Then the highest weight ␭1 of ␺ sends tsdiagŽ.Ž.t1,...,tm =diag s1,...,sk to t11q s . It is known that if an absolutely irreducible representation ␺ is rational over ޑ, then its highest weight is invariant under the )-action of GalŽ.ޑrޑ and the element ␤␺ in the Brauer group BrŽ.ޑ associated to ␺ is trivialŽ seewx B-T, 12 for the definition of ␤␺ .. Since n / 2, H cannot be not almost simple over ޑ and hence there exist two subgroups H12and H which are ޑ-forms of SL2Ž.ރ and SLnŽ.ރ respectively such that H s H12= H . Since the highest weight ␭1 is invariant under the )-action of GalŽ.ޑrޑ , the )-action must be trivial on the Dynkin diagrams of both H12and H , from which it follows that the ޑ-forms of H12and H are inner. Therefore HŽ.ޑ is conjugate to SLrŽ. D1ޑ = SLs Ž. D2ޑ where D12and D are as described inŽ. 1 . If ␺ is rational having this ޑ-structure of H, then the triviality of ␤␺ implies that t the two algebras Ž.D1 ޑ and Ž.ŽD2 ޑ resp. Ž..D2 ޑ represent the same element in the Brauer group BrŽ.ޑ . Since they areŽ. central division

algebras, this implies that they are isomorphic. In the case when Ž.D1 ޑ t t and Ž.D2 ޑ are isomorphic, an isomorphism of Ž.D2 ޑ onto Ž.D1 ޑ extends t to an isomorphism of the algebra MDsŽ.2ޑonto the algebra MDsŽ.1ޑ and hence to an automorphism of the algebra MkŽ.ރ through taking the tensor product with ރ over ޑ. It follows from the Skolem᎐Noether

theorem that this automorphism is inner by an element g g GLkŽ.ރ . Thus y1 t we have gSLsŽ. D2 ޑ g s SLs Ž. D1 ޑ. DISCRETE SUBGROUPS 637

The proof ofŽ. 2 goes exactly as above, using the classification of

ޑ-forms of groups of Dn-type.

1.4.8. PROPOSITION. Let G s SLnnŽ.ރ and let ␺ : SL Ž.ރ ª SL2n Ž.ރ be the direct sum ␳ [ ␳* where ␳ is the standard representation of SLnŽ.ރ and t 1 ␳*the dual of ␳, i.e., ␳*Ž.A s Ay for A g G. Then the ޑ-forms of G with respect to which ␺ is ޑ-rational and irreducible o¨er ޑ are such that GŽ.ޑsSUn Ž k, h . where k is a quadratic extension of ޑ and h g GLn Ž. k t ␴ such that h s h where ␴ is the non-tri¨ial element in GalŽ.krޑ . Proof. Since the highest weights of ␳ and ␳* are obviously different, the )-action of ⌫ of the ޑ-form is non-trivial, proving the form of the outer. That the division algebra attached to this outer ޑ-form is in fact a field is a direct consequence ofŽ. 4 in Theorem 1.4.6. 1.4.9. The following propositions are easy consequences of Theorem 1.4.6; their derivation from the theorem is similar to the proof of Proposi- tion 1.4.8.

PROPOSITION. Let H s SLnnŽ.ރ = SL Ž.ރ and let ␺ be the representation Ž.Ž.A0 such that ␺ A, B s 0 B . If H is equipped with a ޑ-form with respect to which ␺ is rational o¨er ޑ, then HŽ.ޑ is up to conjugation either SLnŽ.ޑ = Ž.ޑ Ž. SLnkor Rrޑ SLn k .

1.4.10. PROPOSITION. LetGbeSLnŽ.Žރ resp. SO2 n Ž..ރ and ␺ be the Ž. ŽA 0 . representation of G defined by ␺ A s 0 A . If G is equipped with a ޑ-form with respect to which ␺ is rational oër ޑ, then GŽ.ޑ is, up to conjugation, either SLnŽޑ . Žresp. SO2 ns Žޑ ..or SL Ž Dޑ . Ž resp. SUs Ž Dޑ, h .. where D is a diïsion algebra described in Ž.Ž1 resp. where D and h are described in Ž..2 in the aboë proposition.

1.5. Extension of ޑ-Forms We show that for an adjoint absolutely simple group G and its parabolic subgroup P such that RPuŽ.is commutative or Heisenberg, a ޑ-form of wxP,Pextends to a ޑ-form of G. The following theorem of Raghunathan plays a key role. 1.5.1. THEOREM ŽRaghunathanwx 18, 3.31. . Let G be a complex semisimple Lie algebra and P the Lie algebra corresponding to a parabolic subgroup P of G Ž.s a semisimple Lie group with ᒄ its Lie algebra . Let Pޑ be a Lie algebra oër ޑ and i : Pޑ ª P an injectië homomorphism such that i m id : Pޑޑm ރ ª P is an isomorphism. Then there exists a ޑ-Lie algebra ᒄޑޑand injectië homomorphisms j : ᒄ ª G and ␣ : Pޑª ᒄ ޑsuch that

Ž.1 jmid : ᒄ ޑޑm ރ ª G is an isomorphism and Ž.2 j(␣s␤(i where ␤ : P ª G is the natural inclusion. 638 HEE OH

1.5.2. Considering the adjoint reprsentation of G, the following is a direct consequence of the above theorem.

COROLLARY. Let G be an adjoint semisimple algebraic ޒ-group and P its parabolic ޒ-subgroup. If P is equipped with a ޑ-form, then there exists a ޑ-form of G which extends the ޑ-form of P. 1.5.3. It is well known that for every unipotent subgroup U, the logarith- mic mapping ln U ª LieŽ.U is equivariant in the following sense: if f is a biregular automorphism of U, then ln( f s df (lnŽ seewx 11, 1.1.3. . In particular, we have the following:

LEMMA. Let G be a semisimple algebraic group and P a parabolic subgroup with a Le¨i subgroup L. Then AdŽ.h (ln s ln(Int Ž.h where Ad : LªGLŽ.U is the representation as in Subsection 1.1.14. 1.5.4. LEMMA. Let G be an adjoint semisimple algebraic group and P, L, Ube as in the abo¨e lemma. Then the representation Ad : L ª GLŽ.U is faithful. Proof. Applying the previous lemma to the conjugation map of U by each element of L, we can see that the kernel of Ad : L ª GLŽ.U is equal to CUŽ.lL. But it is known Ž e.g.,wxw 10, Lemma 11.16 or 16, Proposi- tion 11.19x.Ž. that CU ;U. Therefore CU Ž.lL is trivial, proving the lemma.

1.5.5. PROPOSITION. Let G be an adjoint absolutely simple algebraic

ޒ-group, P a parabolic ޒ-subgroup with an ޒ-Leï decomposition LRuŽ. P , and H s wxL, L . Assume that H and RuŽ. P haë ޑ-forms such that H Ž.ޑ normalizes RuuŽ.Ž. P ޑ . If R Ž. P is commutatië or Heisenberg and H is non-triïal, there exists a ޑ-form of G which extends the ޑ-forms of H and RPuŽ.. Proof. By Corollary 1.5.2, it is enough to extend a ޑ-form of H and RPuŽ.to a ޑ-form of P. Consider the representation Ad : L ª GLŽ.U , UsLieŽŽ..RPu defined in Subsection 1.1.14. The ޑ-form on RPuŽ. defines a ޑ-structure on the vector space U, through the logarithm map, and hence a ޑ-form on GLŽ.U . Since H Ž.ޑ normalizes RPu Ž.Ž.ޑand the logarithm map is ޑ-rational, AdŽŽ..H ޑ preserves UŽ.ޑ . Since H is defined over ޑ, and hence HŽ.ޑ is Zariski dense in H, it implies that AdŽ.H is a ޑ-subgroup of GLŽ.U .

Case Ž.1 . Ad is absolutely irreducible, or equivalently, RPuŽ.is commutative. Since Ad is absolutely irreducible, it follows from Schur’s lemma that the centralizer CŽAd ŽH ..of Ad ŽH .coincides with ZGL Ž ŽU ... Since AdŽZL Ž ..;C ŽAd ŽH ..and Ad ŽZL Ž ..has dimension 1, Ad ŽZL Ž .. s ZGLŽ ŽU ... It follows that Ad ŽZL Ž ..is a ޑ-subgroup of GL ŽU .. Since DISCRETE SUBGROUPS 639

AdŽH .l GL ŽŽ..U ޑ and ZL Ž.lGL ŽŽ..U ޑ are Zariski dense in Ad ŽH . and ZLŽ ., respectively, Ad ŽH .Ad ŽZL Ž ..lGL ŽU Žޑ .. is Zariski dense in AdŽ.H Ad ŽŽ..ZL . Since L s HZ Ž. L , we obtain that Ad Ž.L is a ޑ-subgroup of GLŽ.U . Since Ad is faithful, Ž Ad, Ad Ž..L now provides a ޑ-form of L, which extends the given ޑ-form of H. Since LŽ.ޑ normalizes Ru Ž.ޑ with respect to this ޑ-form, we have a ޑ-form of P.

Case Ž.2. RPun Ž .is Heisenberg and G is not of type A . Then U decomposes into V [ ZŽ.U where V is the AdŽ.H -invariant subspace other than ZŽ.U . Since Z Ž.U has dimension one and H has no rational characters, AdŽ.H acts trivially on ZŽ.U . Denote by AdЈ the restriction of Ad on V so that Ad s AdЈ [ Id. We note that V is a ޑ-subspace of U; in fact, let Ž.¨, z g U Ž.ޑ where ¨ g V and z g ZŽ.U , and h g HŽ.ޑ . Then Ad Ž.Ž.h ¨, z s ŽAdЈ Ž.h ¨, z .g U Ž.ޑ and hence ŽŽ.AdЈ h ¨ y ¨,0 .gU Ž.ޑ , giving a non-trivial element in V l U Ž.ޑ .It follows from the fact that AdЈ acts absolutely irreducibly on V that AdŽ.Ž.ŽH ޑ V l U Ž..ޑ generates a Zariski dense subspace of V , proving the claim. Since the restriction of AdŽ.H on V is absolutely irreducible, by the same argument as the previous case, we can extend the ޑ-form of H to L with respect to which AdŽL Žޑ .. preserves V Žޑ .. Since Z ŽU Žޑ .. s wV Ž.ޑ , V Ž.ޑ x,AdŽL Žޑ ..also preserves Z ŽU Žޑ ... It follows that L Žޑ . normalizes RPuŽ.Ž.ޑ, yielding the desired ޑ-form of P.

Case Ž.3. RPun Ž .is Heisenberg in a group of type A , n G 3. In this case, H is of type Any2 and Ad is equivalent to the direct sum ␳[␳*[id where ␳ Ž.resp. ␳* denotes the Ž resp. dual of . standard ޑ representation of Any2 . By Proposition 1.4.7, the -forms of H with respect to which Ad is ޑ-rational are, up to conjugation and isogeny, such Ž.ޑ Ž.ޑ Ž.Ž.ޑ Ä Ž. t ␴ 4 that H s SLny1 , SU h s g g SLny1 k N ghgsh or tŽ Ž .Žޑ ..y1 Ä Ž. ty1 Ž.Ž.ޑ 4 SU h s g g SLny1 k N g g SU h where k is a real quadratic extension field of ޑ, ␴ is a non-trivial element in GalŽ.krޑ , Ž. t ␴ and h g GLny1 k such that h s h. We observe that each of those ޑ-forms of H extends to a ޑ-form of G with respect to which GŽ.ޑ Ž.ޑ ŽЈ .Ž.ޑ Ä Ž. ty1 Ј Ј4 is isogenous to SLnq1 , SU h s g g SLnq1 k N ghgshor tŽSU Ž hЈ .Žޑ ..y1 for 001 hЈs0h0, ž/100 respectively. 1.5.6. Remark. If the rank of G is one, H is trivial. For the groups with rank at least 2, H is trivial only if G is of type A2 and P is a Borel subgroup, or equivalently, RPuŽ.is Heisenberg. 640 HEE OH

2. THE SUBGROUPS OF THE FORM ⌫ F12, F AND ޑ-FORMS

2.1. Discrete Subgroups in Algebraic Groups 2.1.1. The following two theorems are the well known Borel density theorem and a theorem of Borel and Harish-Chandra, respectively. THEOREM ŽBorel Density Theoremwx 2. . Any lattice in a connected semisimple Lie group without compact factors is Zariski dense. 2.1.2. THEOREM wx4. Let G be a semisimple linear algebraic group defined o¨er ޑ. Then GŽ.ޚ is a lattice in GŽ.ޒ . 2.1.3. LEMMA. Let G be a connected semisimple algebraic ޒ-group, 0 GsGŽ.ޒ,and ⌫ a discrete and Zariski dense subgroup in G. Then the normalizer NŽ.⌫ of ⌫ in G is discrete. Proof. Let N 0 be the connected component of the identity of NŽ.⌫ . Since N 0 normalizes ⌫ and ⌫ is discrete, N 0 centralizes ⌫. That is, 0 0 0 N;CŽ.⌫and therefore ⌫ ; CNŽ .Ž. Since CN.is algebraic and ⌫ is 0 0 Zariski dense, G ; CNŽ . and hence N is contained in the center of G, 0 which is finite. Thus N s Ä4e , implying that NŽ.⌫ is discrete. 2.1.4. LEMMA.1Ž. If G is a locally compact group and ⌫ is a discrete subgroup of G containing a lattice in G, then ⌫ is a lattice in G. 0 Ž.2 If G is a semisimple algebraic ޑ-group, G s G Žޒ ., and ⌫ is a discrete subgroup containing an arithmetic subgroup GŽ.ޚ , then ⌫ is also an arithmetic subgroup of G. Ž.3 LetGbeasin Ž.2 and in addition, suppose that G has no compact factors. Then the normalizer of an arithmetic subgroup of G is again an arithmetic subgroup of G. Proof. PartŽ. 1 is an immediate corollary of Lemma 1.6 ofwx 16 . By Theorem 2.1.2, GŽ.ޚ is a lattice in G. Again by Lemma 1.6 ofwx 16 , the subgroup GŽޚ .is a subgroup of a finite index in ⌫, proving Ž. 2 . Part Ž. 3 is a direct consequence of Borel density theorem and Lemma 2.1.3. 2.1.5. We state a well known arithmeticity theorem of Margulis for the real field case, followed by his finiteness theorem. THEOREM ŽMargulis’ Arithmeticity Theorem, seewx 13 or w 27, Theorem 6.1.2x. . Let G be a connected semisimple Lie group with tri¨ial center and no compact factors. Suppose that the rank of G is at least 2 and that ⌫ is an irreducible lattice in G. Then there exists a semisimple algebraic ޑ-group H 0 and an epimorphism p : HŽ.ޒ ª G with compact kernel such that pŽŽ. H ޚ 0 lHŽ.ޒ.is commensurable to ⌫. Furthermore if ⌫ is a non-uniform lattice, p can be taken as an isomorphism. DISCRETE SUBGROUPS 641

2.1.6. THEOREM ŽMargulis’ Finiteness Theorem, seewxw 11, Chap. 8 or 27, Theorem 8.1.2x. . Let G and ⌫ be as in Theorem 2.1.5. Then any non-central normal subgroup of ⌫ has a finite index in ⌫.

2.1.7. Let G be a locally compact group and Sn a sequence of subsets of G. We say that Sn converges S if for every compact subset K ; G and a neighborhood U of e in G, there exists an integer r s rKŽ.,U such that for all n P r and x g Sn l K, xU l S / л and for all y g S l K, yU l Sn / л.

THEOREM ŽChabautywx 16, Theorem 1.20. . Let G be a Lie group and ⌫n a sequence of lattices in G such that for some open set W of G with e g W, W ⌫ Ä4e for all n. Then a subsequence ⌫ of ⌫ con erges to ⌫ and ⌫ l nis nn¨ is a discrete subgroup. Furthermore, if ␮ is a right Haar measure on G, ␮Ž.G⌫Lim inf ␮ ŽG ⌫ .. rO r i n 2.1.8. A subgroup H of a Lie group G is said to have propertyŽ. P if every AdŽ.H -stable subspace of ᒄ ރ is AdŽ.G -stable where Ad denotes the adjoint representation of G in the complexification ᒄ ރ of the Lie algebra ᒄ. THEOREM ŽWangwx 16, Lemma 9.5. . Let G be a semisimple Lie group and K a compact set of G. Then there is a neighborhood U of e in G such that the following holds: if ⌫ is any discrete subgroup of G such that ⌫ l U generates a subgroup with propertyŽ. P , then ⌫ l U s Ä4e . For example, a Zariski dense subgroup in an algebraic group has propertyŽ. P since the adjoint representation is algebraic. 2.1.9. In the following, we list well known lemmas on lattices in unipotent algebraic groupsŽ seewx 12, Chap. 3. . LEMMA. Let U be a unipotent algebraic ޒ-group and F a discrete subgroup of UŽ.ޒ . Then F is Zariski dense in U if and only if the factor space UŽ.ޒrF is compact. 2.1.10. LEMMA. Assume further that F is a lattice in UŽ.ޒ . Then Ž.1 U Žޒ .rF is compact. Ž.2 F is an arithmetic subgroup of UŽ.ޒ . Ž.3 FlZU ŽŽ..ޒ is a lattice in ZŽŽ.. U ޒ . 2.1.11. LEMMA. Let U be a unipotent ޑ-group. Ž.1 For any n g ގ, UnŽ.ޚ is a lattice in UŽ.ޒ ; Ž.2 if F ; U Žޑ .and F is a lattice in UŽ.ޒ , then F is commensurable to UŽ.ޚ ; Ž.3 e¨ery subgroup of finite index in UŽ.Ž.ޚ contains U nޚ for some n g ގ. 642 HEE OH

2.2. Generators of Arithmetic Subgroups 2.2.1. Let k be a number field and G a simply connected and absolutely almost simple algebraic group defined over k with respect to which U1 and U2 are opposite horospherical k-subgroups. Let S be a finite set of valuations of k containing all archimedean valuations and ⌳ be the ring of

S-integers in k and the S-rank of G is at least 2. If U12and U are maximal horospherical k-subgroups, then for any ideal A of ⌳, the subgroup generated by UA12Ž.and UA Ž.is of fine index in G Ž.Ž⌳ for k-rank G G 2 by Raghunathanwx 17 and for k-rank G s 1 by Venkataramanawx 25. . This result was known for Chevalley groups for k-rank G P 2 by Titswx 23 and for classical groups of k-rank G P 2 by Vasersteinwx 24 . In fact, this result holds for an arbitrary pair of opposite horospherical k-subgroups. The argument for this was explained to the author by T. N. Venkataramana.

2.2.2. COROLLARY. Let G be as abo¨e and U12, U a pair of opposite horospherical k-subgroups. Then for any ideal A of ⌳, the subgroup generated by U12Ž. A and U Ž. A is of finite index in G Ž.⌳ .

X X Proof. Let P11; NUŽ.and P 22; NUŽ.be a pair of opposite minimal parabolic k-subgroups. This can be done as follows: if the pair NUŽ.1 , y NUŽ.2 is conjugate to k P⌰ , k P⌰ by g g GkŽ.as in Lemma 1.2.2, set X y1 X X Pi s gPл g for each i s 1, 2. Set Uiuis RPŽ . for each i s 1, 2 and X X LsNUŽ.12lNU Ž.. Then Uiiis the semi-direct product of ŽL l U .and X UiA. Denote by ⌫ the subgroup generated by UA12Ž.and UA Ž., and by ⌫B X X the subgroup generated by UB12Ž.and UBŽ.. By the result mentioned X above for maximal horospherical k-subgroups, ⌫B is an arithmetic sub- X X group. We note that Ž L l UAUAii.Ž . Ž . has a finite index in UAiŽ.. X Therefore there exists an ideal B ; A such that UBiŽ.; ŽLl X XX UAUAii.Ž . Ž .. We claim that ⌫Bnormalizes ⌫A. Let x g UB1Ž.and write X y1 it as ly where l g ŽL l UA111.Ž .,ygUA Ž .. For u g UA11Ž.,yu y g y1y1 UA11Ž.and hence lyu l y g UA1Ž.since LA Ž.normalizes UA1Ž.. y1 y1 So, xU11Ž. A x s UAŽ..For u 2gUA2Ž.,since xu 2 x s y1y1 y1 y1 y1 y1 Žlyl.Ž lu21 l.Ž lyl . , lyl g UAŽ., and lu2 l g UA2Ž., we have y1X xu2 x g ⌫A. It shows that UB1 Ž.normalizes ⌫A and similarly we can show X X that UB2Ž.normalizes ⌫AAB. Therefore ⌫ ; NŽ⌫ .. By the finiteness theorem of MargulisŽ. Theorem 2.1.6 , ⌫A has a finite X index in NŽ⌫B . and hence an arithmetic subgroup. It is now clear that ⌫A has a finite index in GAŽ..

2.2.3. COROLLARY. Let G be a connected semisimple ޑ-subgroup such that each ޑ-simple factor has ޑ-rank at least 1 and ޒ-rank at least 2. Let U1, U2 be a pair of opposite horospherical ޑ-subgroups and F12, F lattices in DISCRETE SUBGROUPS 643

U12Ž.ޒand U Ž.ޒ which are commensurable to U12 Ž.ޚ and U Ž.ޚ , respecti¨ely. If ⌫ is discrete, then it is an arithmetic subgroup of GŽ.ޒ . F12, F

Proof. Suppose that G is simply connected. Using Proposition 1.2.4, we may assume that G is almost ޑ-simple. By Proposition 1.2.6, there exist a finite separable extension k of ޑ, a connected simply connected absolutely almost simple k-group GЈ, and a pair of opposite horospherical k-sub- X X Ј Ј X groups U12, U of G such that G s RGkrޑ and Uiks RUrޑifor each is1, 2. We also note that ޒ-rank of G is equal to the S-rank of GЈ where S is the set of archimedean valuations of k. In fact, if k has r real embeddings and 2 s imaginary embeddings, then S has r q s elements. Both the ޒ-rank of G and the S-rank of GЈ are equal to Ž.Žr q s k-rank of Ž..GЈ. Since there exists n g ގ such that Unii Ž.ޚ;Fby Lemma 2.1.11, we may assume that Fiis UnŽ.ޚfor each i s 1, 2. On the other hand, we can ⌳ ŽXŽ.. find an ideal A of , the ring of the integers of k, such that RUAkrޑi ;UniŽ.ޚ. By the previous corollary, the subgroup of GЈ generated by UAXŽ.and UAXŽ.has a finite index in GЈ⌳Ž.. Since ⌫ contains a 12 F12,F ŽXŽ.. ŽXŽ.. Ž.ޚ subgroup generated by RUAk r ޑ 1 and RUAk r ޑ 2 and G s RGŽŽ..Ј⌳ ,⌫ contains a subgroup of finite index in GŽ.ޚ . k r ޑ F12, F In general, there exists a simply connected semisimple ޑ-group G˜and a ˜˜y1 central ޑ-isogeny p : G ª G by Proposition 1.2.5. Then each Uiis pUŽ. is a horospherical ޑ-subgroup of G˜and there exists m g ގ such that pUŽŽ˜ mޚ ..Un Žޚ .for each i 1, 2. Since pŽ⌫ . ii; s U˜˜12Žmޚ.,UŽmޚ.; ⌫and the image of an arithmetic subgroup under an isogeny U12Žnޚ., U Žnޚ. map is an arithmetic subgroup, it only remains to apply the previous case to ⌫ to obtain that ⌫ is an arithmetic subgroup. U˜˜12Žmޚ.,UŽmޚ. U12Žnޚ.,UŽnޚ.

2.2.4. EXAMPLE. We present an example which shows that the assumption on the discreteness of ⌫ in the above corollary is essential. F12, F ޒ Let U and U be the subgroups of the formŽ.Ž.IMmm=kŽ. and Im 0 12 0 Ik Mk=mkŽ.ޒ I for some m, k g ގ, respectively. By Mm=kŽ.ޒ , we mean the set of all m = k matrices whose coefficients lie in ޒ. In fact any pair of minimal opposite horospherical subgroups in SLnŽ.ޒ , n s m q k, is conjugate to such a pair U12, U . Let F 1s Mm=kޚ and F2s Ž.1rpMk=mޚfor some pgގ. Obviously F2 is commensurable to Mk=mޚ. We claim that unless p 1, then ⌫ is not discrete. The subgroup s F12, F ⌫ contains a subgroup, which is isomorphic to the subgroup ⌫ of F12, F p Ž.ޒ Ž.Ž.11 10 SL2 generated by01 and1rp 1 . Applying Jorgenson’s inequality given below to ⌫pp, we obtain that if ⌫ is discrete, then p O 1, yielding ps1 since p g ގ. 644 HEE OH

Jorgenson’s Inequality Žcf.wx 7.Ž. . Let A and B be matrices in SL2 ރ which generate an infinite subgroup. If the subgroup generated by A and 2 1 1 Bis discrete, then< tr Ž.A y 4<

2.2.5. EXAMPLE. We close this section by giving some examples which demonstrate how the discreteness assumption on ⌫ restricts the choice F12, F of lattices F121and F in U Ž.ޒ and U 2 Ž.ޒ , respectively.

Ž. Ž1x . Ž10 . Ž. 1 Let g s 01 , h s y 1 , and ⌫x, y be the subgroup of SL2 ޒ generated by g and h. Jorgenson’s inequality says that if ⌫x, y is discrete, 2 2 then x y P 1.

Ž.2 Let U12and U be as in Remark 2.2.4. Let F1s Mm=kޚ and F2 s␣Mk=mޚfor some ␣ g ޒ. We claim that if ␣ is an irrational number, then ⌫ is not discrete. F12, F Let Eij denote the elementary matrix whose entries are all 0 but 1 in Ž. ␣ i,j-entry. Note that the commutator wxI q Emq1, 1, I q E 1, n of two ␣ ␣ elements I q Emq1, 1and I q E 1, nmis the element I q E q1, n. Let’s denote this element by g. Then

IMmm=kޚ IMmmŽ.Ž.=kޚIq␣E1, k ggy1 . 0Is0I ž/k ž/k

Therefore ⌫ l U1 is not discrete if ␣ is an irrational number.

Ž.3 Let G s SL2 nnŽ.ރ with an inner ޒ-form given by GŽ.ޒ s SL Ž Dޒ . where Dޒ is a Hamiltonian quaternion division algebra over ޒ and n P 3. Let U Ž.Ž.IMmm=kŽ. Dޒ and U Im 0 where m k n. Consider 12s 0IMkks =mŽ.DޒIkq s two quaternion division algebras D12and D defined over ޑ such that DiŽ.ޑis a central simple division algebra over ޑ and DiŽ.ޒ s Dޒ for each is1, 2. The notation DiiŽ.ޚ denotes a ޚ-order of the algebra D Ž.ޑ . Let ޚ FŽ.Ž.IMmm=kŽ. D1Ž. and F Im 0. We claim that if ⌫ is 12s0 Ikks ␣MD=mŽ.2Ž.ޚIk F12,F discrete, then D12Ž.ޚ s ␣D Ž.ޚ up to commensurability. Since n is at least 3, we have either that m G 2 or that k G 2. Set j s m if m P 2; j s k otherwise. We denote by Ets the elementary matrix in MDnŽ.ޒ whose Ž.t,s-entry is the identity matrix and 0 elsewhere. Then for each x, ygDޒ , the commutator Cxy [ wxI q xE1, nn, I q yE ,jis the element I q xyE since j / 1. For any x D Ž.ޚ and y ␣D Ž.ޚ , C ⌫ . Since 1, j g 12g xyg F12,F y1 CFCxy 1 xy ;F11, we obtain that xy g D Ž.ޚ . Therefore we obtain that ␣D12Ž.ޚD Ž.ޚ;D 1 Ž.ޚ. It follows that ␣D2Ž.ޚ s D 1 Ž.ޚ up to commensurability. DISCRETE SUBGROUPS 645

2.3. Reflexi¨e Horospherical Subgroups 2.3.1. Let G be a connected semisimple algebraic ޒ-group without any 0 ޒ-anisotropic factors and G s GŽ.ޒ . Let U12, U be a pair of opposite horospherical ޒ-subgroups and F12and F lattices in U 1Ž.ޒ and U 2 Ž.ޒ , respectively.

LEMMA.1Ž. G is a normal subgroup of finite index in GŽ.ޒ .

Ž.2 U12 Ž.ޒ and U Ž.ޒ generate G. In particular G is generated by one-parameter unipotent subgroups. Ž.3 The subgroup ⌫ is Zariski dense in GŽ.ޒ . F12, F Proof. PartŽ. 1 is well known. Since G has no ޒ-anisotropic factors,Ž. 2 follows fromwx 5, 6.14-15Ž or seewx 11, Chap. 1, Theorem 2.3.1. . Since the Zariski closure of ⌫ contains U and U ,Ž. 3 is a direct corollary F12, F 12 ofŽ. 2 .

2.3.2. LEMMA. Suppose that P1 is a reflexi¨e parabolic k-subgroup of a connected semisimple k-group G and that P2 is a parabolic k-subgroup y1 opposite to P11. Then the set MŽ. P , P2s Äh g RPuŽ.21NhP h and P 1 are opposite4 is Zariski dense and open in RuŽ. P212. In particular, MPŽ.,P l RuŽ.Ž. P2 k is non-empty. Proof. It follows from Proposition 4.10 and Lemma 4.12 inwx 5 that the y1 set M s Äh g G N hP11 h and P are opposite4 is Zariski dense and open in G. Since RPPuŽ.21is open and M is right P 1-invariant, we have that RPuŽ.2lMis open in RPuŽ.2.

y1 2.3.3. LEMMA. Suppose that U11 is reflexië, i.e., wU w s U2 for some wgGŽ.ޒ.If U1 is commutatiëŽ resp. Heisenberg., there exist u g U1Ž.ޒ , y1 x g ŽNU Ž12 .lNU Ž ..Žޒ . Žindependent of F12 and F. such that u⌫F,Fu XX 12 contains ⌫ y1y1Žresp. ⌫ XXy1where F and F are lattices whose F11, xŽwF w .xF12,xF x 12 y1 quotients by commutators are the same as those of F11 and wF w , respec- tiëly.. Proof. Since ⌫ is Zariski dense, it follows from Lemma 2.3.2 that F12, F there exists ␥ ⌫ such that ␥ NUŽ.␥y1is opposite to NUŽ.. Then g F12, F 11 y1 y1 there exists a unique element u g U11Ž.ޒ such that u␥ NUŽ.␥ u s y1 y1 NUŽ.212. Since wN Ž. U w s NUŽ., it follows that u␥ w is in NUŽ.2.By y1 Levi decomposition we can write u␥ w s xy for x g NUŽ.Ž.1 ޒ l y1 y1 y1 NUŽ.Ž.22ޒ and y g U Ž.ޒ . Now Ž.Ž.u␥ Fu 11␥ sŽ.xy wF wŽ. xy . Set Xy1 X y1 y1 y1 y1 y1 F11suF u and F 2s ywF 1 w y . Then u⌫u s u␥ ⌫␥ u contains y1X y1 Xy1 the subgroup generated by uF11 u s F and Ž.Ž.u␥ Fu 1␥ sxF 2 x .If X X y1 y1 U11is commutative, then F s F1and xF2x s xF2 x .IfU1is Heisen- XX berg, it is clear that F12and F satisfy the desired properties. 646 HEE OH

2.3.4. Remark. For an adjoint absolutely simple group G and its root system ⌽ with a choice of basis ⌬, there exists a unique involutory permutation i, so-called opposition in¨olution of ⌬ such that ␣ ª yiŽ.␣ y1 y extends to an operation of a Weyl element, say n. Then nP⌰ n s PiŽ⌰. for each subset ⌰ ; ⌬. Therefore if iŽ.⌰ s ⌰, then P⌰⌰Ž.resp. V is yy conjugate to P⌰⌰Žresp. V .. It is known that i induces a non-trivial automorphism on ⌬ if and only if Ž . Gis type of An, D2 nq16,orE cf.wx 21, 1.5.1 . In such cases, we have non-reflexive commutative horospherical subgroups. On the other hand, it follows from the classification given in Subsection 1.3.1 that iŽ.⌬ HHs ⌬ . Therefore every Heisenberg horospherical subgroup is reflexive.

2.4. Margulis’ Theorem on Representation Theory and Extension of ޑ-Structures 2.4.1. Let K be an algebraically closed field, H a connected algebraic K-group, ⌳ ; H a Zariski dense subgroup, T a faithful linear representation of H into a finite dimensional linear space L over K, M and N linear subspaces of L, and W a non-empty Zariski-open subset of H. We put ŽŽ.. Ž Ž.. dsminhgH dim N l ThM and assume that dim M l TwN s 1 dimŽŽN l Twy.. M sdfor all w g W. In addition, we assume that M, Ž N, and L are generated as linear subspaces by the unions D w g W M l Ž. .1 Ž Ž. . Ž. TwN,Dwyg WhNlTwM and D gHThM, respectively. For a subfield k of K, any k-structure on M Ž.resp. N induces in a natural way a k-structure on TŽ.␭ M Žresp. T Ž.␭ N .for any ␭ g ⌳. We say that two k-structures on M and N are compatible if M l TŽ.␭ N for any ␭g⌳lWis a k-subspace of both M and TŽ.␭ N.

THEOREM ŽMarguliswx 12, Lemma 8.6.2. . If there exist compatible k-structures on M and N, then H can be gi¨enak-structure such that ⌳ ; Hk .

2.4.2. PROPOSITION. Let G be an adjoint semisimple algebraic ޒ-group,

U12, U a pair of opposite horospherical ޒ-subgroups and F12, F lattices in UŽ.ޒand U Ž.ޒ , respecti ely. Suppose that ⌫ is discrete and that there 12 ¨ F12,F exists a non-tri¨ial connected semisimple normal ޒ-algebraic subgroup S of NUŽ.N Ž. U with a ޑ-form with respect to which SŽ.ޒ N Ž⌫ .is a 12l l F12,F Zariski dense arithmetic subgroup of SŽ.ޒ and the projection of S onto each simple factor of G is infinite. Then there exists a ޑ-form of G such that ⌫ GŽ.ޑ. F12,F; Proof. Set ⌫ NŽ.⌫ , B Äg G gNŽ. U gy1and NUŽ.are op- s F12, F s g N 21 posite4 , Aiiiis S h U , and ᑛ s LieŽ.A for each i s 1, 2. Denote by M and N the subspaces of the Lie algebra ᒄ of G generated, respectively, by

ŽŽ..␥ y1 ŽŽ..␥ D␥ g B l ⌫ ᑛ12lAd ᑛ and D␥gBl⌫ᑛ 21l Ad ᑛ and by V the DISCRETE SUBGROUPS 647

ŽŽ.. Ž set of g00g G for which dim M l Ad gNsminggG dim M l AdŽ.gN .. We set W s B l V. We claim that the assumptions of Theorem 2.4.1 hold for G s H, ⌫ s ⌳,AdsT,Lsᒄ,M,N, and W. First of all, since G is adjoint, equivalently ZGŽ.is trivial, the adjoint representation Ad of G is faithful.

Since ᑛ12l ᑛ is the Lie algebra of S, ᑛ12l ᑛ ; M is non-trivial. Since ᒄis simple and the projection of S onto each simple factor of G is finite, ᒄis generated by AdŽ.gM, ggG. We observe that M l Ad Ž.␥ N s ᑛ12lAdŽ.␥ ᑛ and N l Ad Ž.␥ M s ᑛ 21l Ad Ž.␥ ᑛ for all ␥ g B l ⌫. Since the set B is of the form NUŽ.Ž.12 NU wx5, Lemma 4.1.2 and Aiis a y1 normal subgroup of Pi, we have that A11s ŽA l ␥ A 2␥ .h U 1for all ␥gB. This implies that dimŽŽ..ŽŽ..M l Ad ␥ N s dim ᑛ12l Ad ␥ ᑛ s y1 dimŽA12l ␥ A ␥ .Ž.Ž.s dim A1y dim U 1for all ␥ g B l ⌫. But W ; B, Wis open, and ⌫ is Zariski dense. Therefore d s dimŽ.A11y dim Ž.U and hence B l ⌫ s W l ⌫. It follows that M and N are generated ŽŽ..ŽŽ..␥ y1 ␥ by D␥ g W ᑛ12l Ad ᑛ and D␥gWᑛ 21l Ad ᑛ , respectively, proving the claim.

Since Fiiis a Zariski dense arithmetic subgroup of U Ž.ޒ for i s 1, 2 and Sl⌫is a Zariski dense arithmetic subgroup of SŽ.ޒ normalizing F1 and F21, we can give ޑ-forms on A and A21with respect to which A l ⌫ and A21l⌫are arithmetic subgroups of A Ž.ޒ and A2 Ž.Žޒ e.g., Lemma 3.4.9. . In a natural way, these ޑ-forms induce ޑ-structures on ᑛ12and ᑛ and also on M and N. Since Aiil ⌫ is a lattice in A Ž.ޒ for each i s 1, 2, and y1 ⌫is discrete, we have that for any ␥ g ⌫, ŽA12l ␥ A ␥ l ⌫.h F 1has a finite index in A11l ⌫. Since A l ⌫ is Zariski dense in A1, the Zariski y1 y1 y1 closure of A12l ␥ A ␥ l ⌫ is A 12l ␥ A ␥ and hence A 12l ␥ A ␥ is a ޑ-subgroup of A11. Since M l AdŽ.␥ N s ᑛ l AdŽ.␥ ᑛ 2for all ␥gBl⌫, it follows that the ޑ-structures on M and N are compatible. Now by Theorem 2.4.1, there exists a ޑ-form of G such that ⌫ GŽ.ޑ . F12, F ;

2.4.3. LEMMA. If ⌫ GŽ.ޑ , then U is defined o er ޑ and there F12, Fi; ¨ exists n g ގ such that UiiŽ. nޚ ; F for each i s 1, 2.

Proof. Since Fii; U l GŽ.ޑ and Fiis Zariski dense in Uiby Lemma 2.1.9, Ui is defined over ޑ by Proposition 1.2.7. The second claim follows from Lemma 2.1.11.

2.4.4. LEMMA.1Ž.If ⌫ G Žޑ .and for e ery infinite proper normal F12, F ; ¨ ޒ-subgroup GЈ of G, Fi l GЈŽ.ޒ is finite for each i s 1, 2 then G is ޑ-simple.

Ž.2 If G is ޑ-simple and Ui is defined o¨er ޑ, then for e¨ery infinite proper normal ޒ-subgroup GЈ of G, Ui l GЈŽ.ޒ is finite. 648 HEE OH

Proof. Ž.1IfGis not ޑ-simple, there exist non-trivial semisimple normal ޑ-subgroups GЈ and GЉ such that G is the direct product of GЈ and GЉ by Proposition 1.2.4. By Lemma 2.4.3, Fiicontains UnŽ.ޚ for some n g ގ. Since Uiil GЈ is a ޑ-subgroup, U l GЈŽ.nޚ is infinite. But UniiŽ.ŽޚsUlGЈ Ž..Žnޚ U ilGЉ Ž..nޚ, yielding that Fil GЈŽ.ޒ is infinite. This contradicts the assumption. Ž.2 There exist a finite extension field k of ޑ, an adjoint absolutely X simple k-group G0 , and a horospherical k-group Ui such that G s Ž. ŽX. RGk r ޑ 0 and Uiks RUrޑiby Proposition 1.2.6. Any infinite normal algebraic ޒ-subgroup GЈ is then isomorphic to the product of suitable Ž.ޚ Ž X.Ž . n-copies of G0. Since Uiks RUJrޑi,Jthe ring of the integers of k, the claim follows.

2.4.5. COROLLARY. Let G, U1212, U , F , F , and S be as in Proposition 2.4.2. Assume that the real rank of each ޒ-simple factor of G is at least 2 and

Fi lGЈŽ.ޒis finite for eëry infinite proper normal algebraic ޒ-subgroup GЈ of G and each i 1, 2. If ⌫ is discrete, then it is an arithmetic subgroup s F12, F of GŽ.ޒ . Proof. By Proposition 2.4.2 and Lemma 2.4.3, there exists a ޑ-form of Gsuch that ⌫ GŽ.ޑ and Un Žޚ . Ffor some n ގ. By Lemma F12, Fii; ; g 2.4.4, G is almost ޑ-simple. It follows from Corollary 2.2.3 that the subgroup ⌫ generated by Un12Ž.ޚand Un Ž.ޚis an arithmetic subgroup of GŽ.ޒ. By Lemma 2.1.4, ⌫ is an arithmetic subgroup of GŽ.ޒ . F12, F 2.4.6. PROPOSITION. LetGbeaޑ-simple algebraic ޑ-group with real rank at least 2 and U12, U be defined oër ޑ. Suppose that the semisimple part of NŽ U12 .Ž.ޒ l NU Ž .Ž.ޒ does not haë compact factors. If F12 and F y1 are commensurable to U12Ž.ޚ and wU Ž.ޚ w for some w g ZLŽ.Ž.ޒ,respecti ely, and ⌫ is discrete, ⌫ is an arithmetic subgroup of GŽ.ޒ . ¨ F12, FF12,F y1 Proof. We may assume that F11s U Ž.ޚ and F 2s wU 2Ž.ޚ w . Let N NŽ.⌫ and H the semisimple part of NUŽ .Ž.ޒ NU Ž .Ž.ޒ. Then s F12, F 12l Nis discrete by Lemma 2.1.3 and H is a semisimple subgroup defined over ޑ. Since w g ZLŽ.,H Ž.ޚnormalizes F12and F and hence H Ž.ޚ ; N. Since HŽ.ޚ is a Zariski dense arithmetic subgroup of HŽ.ޒ , it follows from Proposition 2.4.2 that there exists a ޑ-form of G such that N ; GŽ.ޑ . Since ⌫ GŽ.ޑ , we can show the rest of claim by the same argument F12, F ; as the proof of Corollary 2.4.5.

2.5. The Subgroup Generated by ZŽ. U12 and Z Ž. U We prove some algebraic lemmas which we will need later in applying Proposition 2.4.2. 2.5.1. Let G be an adjoint k-simple algebraic k-group. This section has content only when the k-rank of G is at least 2. We continue the notation DISCRETE SUBGROUPS 649 from Subsection 1.3.1 for S, T, ⌽ s ⌽Ž.T, G , kk⌽ s ⌽ Ž.S, G , j : ⌽ ª ⌽ j y Ä40,⌬,kk⌬, V⌰,KV⌰for ⌰ ;kh⌬ and so on. We recall that ␣ denotes the highest k-root in kkH⌽ and ⌬ the unique set of k-Heisenberg roots in k⌬. The following lemma can be checked case by case in each irreducible root system.

LEMMA.1Ž. The only positi¨ek-root the sum of whose coefficients with respect to which the roots in kH⌬ is 2 is ␣h. q Ž.2 If ␤ gk⌽ is such that the sum of the coefficients with respect to kH⌬ is 1 and there exists a simple root in k⌬ with respect to which the q coefficient of ␤ is 0, then ␣hky ␤ g ⌽ . 2.5.2. The commutator law over an algebraically closed field is well known. The following is a result of Vinberg which says that the commutator law holds for an arbitrary field. Hereafter, we shall also refer to this lemma as the commutator law.

EMMA Ž . ⌽ Le.g.,wx 12, 4.5.1 . If a, b, a q b gkk, then wxUa, kUbks Uaqb and moreo¨er, for any nonzero x g Ž.kaU , we ha¨exwx,kbU/0. 2.5.3. We recall that the notation U denotes the k-root group corre- ␣h sponding to the highest k-root ␣hHand U denotes the k-Heisenberg horospherical k-subgroup V . kk⌬ykH⌬ q We observe that ZVŽ.k ⌰ sU␺ where ␺ s Ä␤ gkk⌽ ywx⌰ N ␤ q ␣ f ⌽ q for any ␣ gkh⌽ ywx⌰ 4. In particular ␣ g ␺. LEMMA. ZVŽ.U if and only if ⌬ ⌬ ⌰. k ⌰ s␣ h kH; k y

Proof. If ␥ gkH⌬ l ⌰, ␣ hy ␥ is a k-root by Lemma 2.5.1; hence q qq ␣hky␥g⌽ywx⌰. We claim that Ž.␣hy ␥ q ␤ f k⌽ for any ␤ gk⌽ ywx⌰. Suppose not. Then by the height calculation, we have Ž.␣h y ␥ q ␤ s ␣h. It implies that ␥ s ␤, contradicting the assumption ␥ g ⌰. There- fore if ZUŽ. U , or equivalently ␺ Ä4␣ as the above notation, then s ␣h s h kH⌬ l ⌰ is empty. c To see the converse, suppose that kH⌬ ; ⌰ . Then UHis contained in kV⌰. Since the centralizer of UHkin V⌰is contained in RNUuHŽŽ ..sU H and hence coincides with ZUŽ.U , we have ZVŽ.U. H s ␣hhk ⌰ ;␣

2.5.4. For a gkk⌽, denote by Uathe root space corresponding to a and Ž. ␺ ⌽ U put kaG s exp kaU . Note that for any subset ;k, the subgroup Gw␺x Ž.as in the notation of Subsection 1.2.1 coincides with the subgroup generated by all the kaGaŽ.g␺. A closed subset ⌽0 of ⌽ is called an ideal if the following holds: for all a g ⌽0 and ␤ g ⌽ such that ␣ q ␤ g ⌽, ␣ q ␤ g ⌽0. It follows from the commutator law that if ⌽ is an ideal of ⌽, then the subgroup GU is 0 ⌽ 0 a normal subgroup of G. 650 HEE OH

In proving the following proposition, we use the factŽ e.g.,w 26, Proposi- tion 1.1.10x. that if ␾ is a closed subset of ⌽ such that if ␣ g ␾ then y␣ g ␾, then the subalgebra generated by Ub, b g ␾ is semisimple.

y PROPOSITION. Let U1 skV⌰ and U2 skV⌰ for some ⌰ ;k ⌬. Denote by G01 the subgroup of G generated by ZŽ. U and Z Ž. U2, H the commutator subgroup of NŽ. U12l N Ž. U and S s H l G0. Then

Ž.1 G0 is a connected almost k-simple algebraic k-subgroup, ZŽ. U1 and ZŽ. U2 are opposite commutatië horospherical k-subgroups of G0 , and S is a connected semisimple normal algebraic k-subgroup of H. Moreoër, if G is absolutely almost simple, so is G0. Ž.2 If Z Ž U . is strictly bigger than U , then S is isotropic o er k and the 1 ␣ h ¨ k-rank of G0 is at least 2. Ž.3 If H does not haë any k-anisotropic factors, neither does S.

q q Proof. Set ␺ s Ä ␤ gkkk⌽ ywx⌰ N ␤ q ␣ f ⌽ for any ␣ g ⌽ ywx⌰ 4. ␺ ⌽qŽ. Ž. Note that is a closed subset of k and ZU1 sU␺ and ZU2 sUy␺. Ž. ␺ ⌽ U 1 Note that wxis a closed set of k and the subgroup Gw␺ x y1 y1 generated by Ub, b g j Žwx␺ .Žor equivalently, b g wj Ž.␺ x.coincides with G0. This group is algebraic, connectedŽ any algebraic group generated by unipotent subgroups is connected.Ž and semisimple by the fact mentioned preceding the statement of this proposition. . On the other hand, ␺ is invariant under GalŽ.Krk since ZU Ž.1 is defined over k Žsee Subsection 1.2.1. . Therefore wx␺ is also invariant under GalŽ.Krk and hence G0 is defined over k.

To show that G0 is almost k-simple, assume that ⌿ s wx␺ is the disjoint union ⌿12j ⌿ j иии j ⌿siwhere each ⌿ is a non-empty irreducible ideal of ⌿. We will show that each ⌿ihcontains the unique highest k-root ␣ , which is an obvious contradiction if s P 2. Note that ⌿i l ␺ is non-empty y1 for each i. For any ␣ g ⌿i l j Ž.␺ , there exists simple k-roots ␣,␣,...,␣ Ž.not necessarily different such that ␣ ␣ иии ␣ ii12 ik q i1q q ijg ⌽qfor each j 1, 2, . . . , k and ␣ ␣ иии ␣ ␣ . kis s 1q q iks h Since ␣ ␣ ⌽q, ␣ ␣ ␺. It follows that ␣ ⌿. Since ⌿ is q ik11g q ig i1g i an ideal of ⌿, we have ␣ ␣ ⌿ . Therefore by induction ␣ q ii1g q ␣иии ␣ ⌿ for each j 1, 2, . . . , k; hence ␣ ⌿ . This proves iii1qq jg s hig that G0 is almost k-simple. If G is absolutely almost simple, then ⌽ has the unique highest root. Then using the same argument as above we can show that w jy1 Ž.␺ xcannot be a disjoint union of ideals. It follows that G0 is absolutely almost simple. We note that Lie G00is decomposed intoŽ.Ž Lie L l Lie G [ Lie ZU1. [Lie ZUŽ.212. Clearly ZU Ž.are ZU Ž.are opposite horospherical subgroups of G0. DISCRETE SUBGROUPS 651

By the same argument we made for G0 , we can show that S is a semisimple algebraic k-group. We now show that S is a normal subgroup of H. Let s g S and h g H. Then S s aa12иии akifor some a g ZUŽ.12jZU Ž.. Since H ; NU Ž. 1l y1 NUŽ.2 , hai h g U12j U for each i. Suppose that aig ZUŽ.1; hence y1 y1 y1 hai h g U1. To see hai h g ZUŽ.11, let u g U and consider uhahŽ i. y1y1 y1y1 shhŽ uhah.iii, which is equal to haŽ h uh. h since ␣ g ZUŽ.i. y1y1 So uhahŽii.Žs ha h.Ž u. It proves that H normalizes ZU1.and sim- y1 y1 ilarly ZUŽ.20. Therefore hSh g G ; hence hSh g S, proving H normalizes S. Ž.2 Suppose that U is a proper subgroup of ZUŽ.. By Lemma ␣h 1 q 2.5.3, we have an element ␣ g ⌰ lkH⌬ . Then ␣hy ␣ g k⌽ by Lemma 2.5.1. We claim that this root belongs to ␺. Otherwise, for some element q ␤gkh⌽ywx⌰,Ž.␣y␣q␤is a k-root. By comparing height, it follows that Ž.␣hhy ␣ q ␤ s ␣ hence ␣ s ␤, contradicting ␣ g ⌰. Therefore ␣sŽ.␣hhy Ž␣y␣ .gwx␺, showing that S is isotropic over k, since Lie S clearly contains U␣ for all ␣ g wx␺ l ⌰. Ž.3 Since S is a normal algebraic k-subgroup of H, S has no k-anisotropic factors unless H does.

2.5.5. EXAMPLE. The following example illustrates the power of Propo- sition 2.4.2, which will be also one of the most fundamental ideas of the proof of the main theorem.

Let U12, U be the pair of opposite horospherical subgroups in SLnŽ.ޒ , consisting of the elements of the form

IXl 11 Z Il00

0IYm 12and XIm0 , 00 IZYI 00k 22k

respectively, where l q m q k s n. Note that U12and U are two step nilpotent groups, i.e., wxUii, U s ZUŽ. ifor each i s 1, 2. If lk P 2 and F12and F are lattices in U12and U such that F 1l ZUŽ. 1 sMl=kŽ.ޚand F22l ZUŽ .sMk=ln Ž.ޚ, there exists a ޑ-form of SL Ž.ޒ such that ⌫ GŽ.ޑ . To see this, note that the subgroup G generated F12, F ; 0 by ZUŽ.12and ZU Ž.consists of the elements

X0XЈ

0Im0 0ZЈ0 Z 652 HEE OH in SLnŽ.ޒ and S s H l G0 consists of the elements X 00

0Im 0, 000Z where H is the commutator subgroup of NUŽ.12lNU Ž.. Observe that S⌫is lF12, F

Ml=lŽ.ޚ 00

0 Im 0 . 00Mޚ 0k=kŽ. It only remains to apply Proposition 2.4.2. Ž.ޑ ␣ ␣ Since S is split in this case and hence at least one of 1 and ny1 belongs to ⌬ y ⌬ 0 Ž.see the notation in Subsection 1.4.1 , it follows from the classification of ޑ-formswx 21 that GŽ.ޑ is conjugate to SLn Ž.ޑ .

3. ADJOINT ACTION ON THE SPACE OF LATTICES

3.1. The Space of Lattices in Algebraic Unipotent Groups We collect some well known lemmas.

3.1.1. PROPOSITION Že.g.,wx 10, Lemma 1.3.1. . The group of real points of a unipotent algebraic ޒ-group is a topologically connected, simply connected, nilpotent Lie group, and e¨ery unipotent subgroup of an algebraic group is nilpotent. 3.1.2. We state a well known criterion of compactness due to Mahler.

LEMMA ŽMahlerwx 6. . A set A of lattices of some Euclidean space is relatiëly compact if and only if there are positië constants d and ⑀ such that the ⑀-neighborhood in the Euclidean space does not contain any nonzero element from any lattice belonging to A and det x - difxgA. 3.1.3. LEMMA ŽMinkowskiwx 6.Ž. . For any n, there is a constant c n such that in any quasi-lattice ⌬ lying in an Euclidean space of dimension not exceeding n, there exists a basis e12, e ,...,eofk ⌬Ž.ksdim ⌬ such that for Ž.l55 Ž.Ž⌬ .1rk any l 1 F l F k , Ł is1 ei - cnd . 3.1.4. LEMMA. Let W be the space of lattices of some fixed Euclidean space. For any relatiëly compact subset M ; W, there are positië constants d12 and d such that in any lattice M, there is a basis consisting of elements of norm less than d12 and greater than d . DISCRETE SUBGROUPS 653

Proof. It is easily deduced from Mahler’s compactness criterion and the previous lemma.

3.1.5. PROPOSITION wx10, Lemma 5.2 . Let Z be a simply connected nilpotent Lie group and ᑴ its Lie algebra. Then there exists an integer b such that if ⌫ is a subgroup of Z and ln ⌫ is the subring in ᑴ generated by ln ⌫, then bŽ.ln ⌫ ; ln ⌫. Furthermore b depends only on the length of the lower central series.

0 3.1.6. Let G be an algebraic ޒ-group, G s GŽ.ޒ , and ᒄ the Lie algebra of G. We choose a Euclidean metric ␳ᒄ on ᒄ and a left invariant metric ␳G on GŽ.ޒ corresponding ␳ᒄ . Denote by X the set of all discrete unipotent subgroups of G. Since in an algebraic ޒ-group, any two maximal unipotent ޒ-groups are conjugate to each other, there exists an integer N g ގ such that for any subgroup F g X, the length of the lower central series of F is at most N. Therefore there exists an integer b g ގ such that if F g X, then bŽ.ln F ; ln F. We denote bŽ.ln F by ⌬ F . We introduce in the following topology on the space of quasi-lattices ⍀ in that a subset B ; ⍀ is open if, for any F g B, there exists some ⑀ ) 0 such that any quasi-lattice of the form TFŽ.belongs to B for each linear transformation T with 55T y Id O ⑀. Let f : X ª ⍀ be the mapping defined by fFŽ.s⌬F. LEMMA wx10, Lemma 5.7 . For any subset L ; X for which fŽ. L is relati¨ely compact in ⍀, there are constants ⑀ ) 0 and c such that if F g L, then

Ž.1 ␳G Že,x .)⑀for all non-tri¨ial x g F;

Ž.2 there exists x1,..., xkGigF such that ␳ Ž.e, x - c for any i and any algebraic subgroup of G containing x1,..., xk also contains the Zariski closure of F in G.

3.2. Adjoint Action 3.2.1. Let G be a connected adjoint semisimple algebraic ޒ-group with no ޒ-anisotropic factors and U12, U a pair of opposite horospherical 0 ޒ-subgroups. Set L s NUŽ.12lNU Ž.,HswxL,L, and H s HŽ.ޒ as well 0 as G s GŽ.ޒ . As before set Uiis LieŽ.U and denote by Adi : L ª GLŽ.U i the restriction of the adjoint representation of NUŽ.ii. We recall that Ad is faithful. We mention that this section has no content if H is trivial.

3.2.2. Denote by ⍀ iithe space of lattices in the Euclidean space U Ž.ޒ . The group H acts through Adii on U Ž.ޒ and hence on ⍀ i. The notation H.Jiidenotes the orbitÄ Ad Ž.hJig⍀iNhgH4of Jiunder H for a lattice Jiiin U Ž.ޒ . We also diagonalize the action of H on ⍀ idescribed above to 654 HEE OH the space ⍀12= ⍀ of pairs of lattices in U1Ž.ޒ and U 2 Ž.ޒ by hJ Ž 12,J .s ŽŽ.Ad112hJ,Ad Ž..hJ 2. the image of H under this action is the subgroup ÄŽAd12 Žh .,Ad Žh ..gSL ŽU 1 Žޒ ..= SL ŽU 2 Žޒ .. N h g H4 and will be denoted by ␦Ž.H . The notation ␦Ž.ŽH . J12, J .denotes the H-orbit of the pair Ž.J12,Jvia this action. 3.2.3. Let F be a lattice in U Ž.ޒ and ⌬ be as in Subsection 3.1.6 for iiFi i 1, 2. Denote by ⌳ the stabilizer Äg H Ad Ž.g ⌬ ⌬ 4of ⌬ and s Fiiig N Fs FiFi by ⌳ the stabilizer Äg H Ad Ž.g ⌬ ⌬ ,AdŽ.g⌬ ⌬ 4of F12, F g N 1 FF1s 1 2FF2s 2 Ž.⌬,⌬. FF12 The following lemma then immediately follows from the definition of ⌬ and Lemma 1.5.3. Fi

LEMMA.1Ž. ⌬ y1 Ad Ž.h ⌬ for any h H and any lattice F in hFii h s i F g i UiŽ.ޒ. Ž.2 ⌳ Äg H gF gy1 F 4. Fiiis g N s Ž.3 ⌳ Äg H gF gy1 F , gF gy1 F 4. F12, F s g N 1122s s

3.2.4. We choose a Euclidean metric ␳ᒄ on the Lie algebra ᒄ of G and a left invariant metric ␳G on G corresponding to this metric. We define a topology on ⍀ i and ⍀12= ⍀ in the following way: a sequence of lattices Ä4⌬kig⍀Nks1, 2, . . . converges to a lattice ⌬ if and only if there is ⑀)0 such that ␳ᒄŽ.0, x ) ⑀ for all nonzero x g ⌬ k and for all k and there exist bases in the lattices ⌬ k which converge to some basis in ⌬.Itis not difficult to see that this topology coincides with the topology defined in Subsection 3.1.6.

We note that not every lattice in the Euclidean space UiŽ.ޒ is of the form ⌬ for a lattice F in U Ž.ޒ unless U is commutative. In fact any Fii ii ޚ-linear span of n-linearly independent vectors forms a lattice in UiŽ.ޒ where n s dimŽ.Uii; while a lattice in U Ž.ޒ should satisfy certain relations from the group structure of Uiisince by definition a lattice in U Ž.ޒ is a subgroup. On the other hand, the following lemma says that any lattice in H.⌬ is of the form ⌬ for some lattice E in U Ž.ޒ . FEii ii For a subset M of the space of lattices, we denote by M the closure of M.

LEMMA. For any lattice J H.⌬ , there exists a lattice E in U Ž.ޒ such iFg i ii that ⌬ J and the determinant of J is equal to the determinant of ⌬ . Eiis i Fi Proof. Let x H be a sequence such that Ad Ž.x ⌬ converges to nig nFi J. SinceÄ Ad Ž.x ⌬ n ގ4is relatively compact, by Lemma 3.1.6, there iinFiN g y1 is a neighborhood W of e such that W l xFxnin sÄ4e for each n g ގ. By Chabauty’s theoremŽ. see Theorem 2.1.7 , there exists a discrete sub- y1 group Eiiin U Ž.ޒ to which a subsequence of xFxninconverges. Since H y1 is a semisimple group, the volume of UininŽ.ޒ rxFx coincides with the DISCRETE SUBGROUPS 655 volume of UiiŽ.ޒ rF for all n. It follows from Chabauty’s theorem Ž Theo- rem 2.1.7.Ž that E is a lattice in U ޒ.Ž. Since a subsequence of Ad x .⌬ ii inFi converges to ⌬ , ⌬ J , proving the lemma. EEiis i

3.2.5. PROPOSITION. Suppose that ⌫ is discrete. Then for any pair F12, F Ž.⌬,⌬lying in ␦ H . ⌬ , ⌬ , the subgroup ⌫ is discrete. EE12 Ž.Ž.FF12 E12,E Proof. Consider a sequence x H such that AdŽ.Žx ⌬ , ⌬ .con- iig F12F verges to Ž.⌬ , ⌬ . Since the setÄ Ad Ž.x ⌬ ,Ad Ž.x ⌬ ,⌬ ,⌬ i 14 EE12 1iF1 2iF2 E 12 EN G is relatively compact in the space of lattices, there exist d12) 0 and d ) 0 such that for any lattice in this set, there exists a basis consisting of X elements of norm less than d12and greater than d by Lemma 3.1.4. Let d 1 XXX and d22be such that d F ␳GŽ.e, x F d12implies that d F ␳ᒄŽ.0, ln x F d1 X X for all x g U12j U . Set K s Äg g G N d2F ␳GŽ.e, g F d14. For each xig H, the set x ⌫ xy1 K generates the subgroup x ⌫ xy1. Since iF12,Fil iF12,Fi ⌫ and hence x ⌫ xy1 are Zariski dense and hence have property F12, FiF12,Fi Ž.P , and K is compact, there exists a neighborhood V of e such that V x⌫ xy1 Ä4efor each x by Theorem 2.1.8. Theorem 2.1.7 implies l iF12,Fis i that the limiting subgroup ⌫ of the sequence x ⌫ xy1 is discrete. E12, EiF12,Fi

3.2.6. We will need the following lemma for the proof of Proposition 3.2.7.

LEMMA Že.g.,wx 10, Lemma 1.6.1. . There is an ⑀-neighborhood V of the y1 y1 identity e in G such that for any x, y g V, we ha¨e ␳GŽe, xyx y . - 1 2minŽŽ␳GGe, x ., ␳ Že, y ...

3.2.7. PROPOSITION. Suppose that ⌫ is discrete. Then for a sequence F12, F x H,the set M ÄAd Ž.x ⌬ H.⌬ i 14is relati ely compact if i g 11s iF11g FN G ¨ and only if the set M ÄAd Ž.x ⌬ H.⌬ i 14is relati ely compact. 22s iF22g FN G ¨

Proof. Suppose that M12is relatively compact and M is not. Choose a positive ⑀ and V as in Lemma 3.2.6. Since the set M1 is relatively compact, by Lemma 3.1.3, there exists a d ) 0 such that for any lattice Ad1Ž.xFi 1 in M1, there exists a basis ᑜ iGsuch that ␳ Ž.e, exp x - d for all x g ᑜ i. Let ᑜbe the union of all ᑜ i’s. It is not difficult to see that we can find y1 ggCHŽ.such that ␳G Že, gug .Ž- ⑀ for all u g U1 ޒ.Žsuch that ␳G e, u. - d. By Mahler’s criterion of compactness, there exists ⑀0 ) 0 such that ⑀011-⑀and Ad Ž.gM does not contain any non-trivial element x such that ␳GŽ.e, exp x - ⑀022. Since Ad Ž.gM is not relatively compact and the determinants of the lattices in Ad22Ž.gM are equal, there exists some j ގsuch that Ad Ž.g Ad Žx .⌬ contains a non-trivial element y with g 22jF2 ␳G Ž.e,exp y F⑀0.Set nsxj.SetU˜sÄxgU1 Nln x g ᑜ l Ad Ž.g Ad Ž.n ⌬ 4Ä. With U gUg˜ y1, U exp y4and U Ästsy1 ty1 22F1 y10s s is N 656 HEE OH

sU,tU4, sup ␳ Ž.e, u d tends to 0 as i goes to ϱ,asU giigy1ugUGi s i y1 and U00lie in V. On the other hand, both U and Uy1belong to the discrete subgroup gn⌫ ny1 gy1 and hence U gn⌫ ny1 gy1 for all F12, FiF; 12,F iP0. Thus there exists k ) 0 such that Ukkis not trivial and U q1is trivial. Let z be a nontrivial element of UkG. Then ␳ Ž.e, z F ⑀0and z belongs to Ž. CUy1 . Since the centralizer of any subset is algebraic, the elements of y1 y1 gnU11 n g commute with z since U is the Zariski closure of expŽ.ᑜ . Therefore, ny1 gy1 zgn CUŽ. ⌫ . Since CUŽ.U Žsee the proof g 1 l F12, F 11; of Lemma 1.5.4.Ž. , CU ⌫ F. But ␳ Ž.e, z - ⑀ , contradicting 1 l F12, F s 1 G 0 that the ⑀01-neighborhood intersects AdŽ.gM trivially. This proves the proposition. In fact, we have proved that if the sequence Ad Ž.x ⌬ converges, then 1 iF1 we can find a convergent subsequence of Ad Ž.x ⌬ . The following is an 2 iF2 immediate corollary of Propositions 3.2.5 and 3.2.7.

3.2.8. PROPOSITION. If ⌫ is discrete, then for any lattice ⌬ in F12, FE1 H.⌬,there exists a lattice E in U Ž.ޒ such that Ž⌬ , ⌬ .lies in F1 22 EE12 ␦H.⌬,⌬and the subgroup ⌫ is discrete. Ž.Ž.FF12 E12,E 3.3. Ratner’s Theorem and Orbit Closures 3.3.1. Let G be a connected Lie group and ⌫ a lattice in G. Consider the natural action of G by left translation on the homogeneous space Gr⌫. For x g Gr⌫, Gx s Ä4g g G N g. x s x is the stabilizer of x. M. Ratner proved the following theorem which was known as Raghunathan’s conjecture. THEOREM ŽRatnerwx 19. . Let H be a connected closed subgroup of G generated by unipotent one-parameter subgroups in it. Then for x g Gr⌫, there exists a connected and closed subgroup L of G containing H such that H. x coincides with L. x and L l Gx is a lattice in L. 3.3.2. THEOREM Žseewx 16, Theorem 1.13. . Let G and ⌫ be as abo¨e and H be a closed subgroup of G. Then if H l Gx is a lattice for some x g Gr⌫, then H. x is closed.

3.3.3. PROPOSITION wx20, Proposition 3.2 . Let G ; SLnŽ.ރ be an alge- 0 braic ޑ-group, G s G ޒ , ⌫ s G l SLnŽ.ޚ , and H be a subgroup generated by algebraic unipotent one-parameter subgroups of G contained in H. Suppose that H.⌫s M.⌫ for a connected Lie subgroup M of G. Let M be the smallest algebraic ޑ-subgroup containing M. Then the radical of M is a unipotent 0 algebraic ޑ-subgroup and M s Mޒ . Furthermore, if M s SU, U s RuŽ.M is 0 0 an ޒ-Le¨i-decomposition of M, then M s SŽ.ޒ UŽ.ޒ . In particular, we note that the Levi component of M is semisimple in the above proposition since the radical of M is unipotent. DISCRETE SUBGROUPS 657

3.3.4. PROPOSITION. With the same notation as the abo¨e proposition, 0 there exists an element u g UŽ.ޒ such that the orbit S Ž.ޒ . Ž.u⌫ is closed. Proof. Since M is defined over ޑ, there exists u g UŽ.ޒ such that uSuy1 is defined over ޑ. It follows that uy1Ž.S ޒ 0u.⌫ is closed. Therefore its left-translation by u, SŽ.Žޒ 0. u⌫ ., is closed.

3.4. Closedness of Some Orbits and ޑ-Structures 3.4.1. We keep the notation for G, L, G, and H from Section 3.2.1. In addition, we assume that the ޒ-rank of G is at least 2 and H is non-compact. We fix a non-trivial semisimple normal ޒ-subgroup HЈ of H without 0 any ޒ-anisotropic factors and set HЈ s HЈŽ.ޒ . Then HЈ is generated by unipotent one-parameter subgroups by Lemma 2.3.1.

In this section, we fix lattices F121and F in U Ž.ޒ and U 2 Ž.ޒ , respectively. Let ⍀ iidenote the space of lattices in U Ž.ޒ of the same determinant as ⌬ for each i 1, 2. Since SLŽŽ..U ޒ acts transitively on ⍀ , ⍀ Fiiis i can be identified with the homogeneous space SLllŽ.ޒ rSL Ž.ޚ , through the l isomorphism of Uiiwith the Euclidean space ޒ where l s dimŽŽ..U ޒ . XX 3.4.2. PROPOSITION. The orbit H .⌬ ŽŽresp. ␦ H .Ž. ⌬ , ⌬ .. is closed if FFi 12F and only if ⌳ HXŽ. resp. ⌳ HЈ is a Zariski dense arithmetic FFil 12,Fl subgroup of H. Proof. Suppose that H X.⌬ is closed. Choose a representative of ⌬ in FFi i SL Ž.ޒ SL Ž.ޚ , say gSL Ž.ޚ so that Ad Ž⌳ H X.Ž.gSL ޚ gy1.By llr il iFiliil ; Ratner’s theorem, ⌳ is a lattice in H X. Since H X has no compact factors Fi by the assumption, it follows from Borel density theorem that gy1Ad Ž⌳ iiFi X y1 lHg.Ž.ii;SL ޑ is Zariski dense in g AdiŽ.HЈ gi. Therefore by Proposi- y1 tion 1.2.7, giiAd Ž.HЈ g iis a ޑ-subgroup of SLnŽ.ރ , providing a ޑ-form of HXXsuch that H Ž.ޚ is commensurable to ⌳ H X.If␦ŽHX.Ž.⌬ ,⌬ . is FFil 12F closed and ⌬ gSLŽ.ޚ for i 1, 2, by the same argument as above, Filis s y1y1 X Žg12,g.ÄŽŽ.Ž..Ad 1h ,Ad 2h NhgHg4Ž.12,gis a ޑ-subgroup of SLlŽ.ރ = X XX SLlŽ.ރ , providing a ޑ-form of ␦ŽH .and hence of H such that H Ž.ޚ is commensurable to Ž.⌳ HЈ . The converse is a direct consequence of F12, F l Theorem 2.1.2 and Theorem 3.3.2.

3.4.3. COROLLARY. Let Fii and E be commensurable lattices in UiŽ.ޒ . Then H X.⌬ is closed if and only if H X.⌬ is closed. FEii Proof. If F and E are commensurable, then ⌳ and ⌳ are com- ii FEii mensurable. Therefore ⌳ H X is a lattice in H X if and only if ⌳ HЈ FEil il is a lattice in H.XThe corollary now follows from Theorem 3.4.2.

3.4.4. PROPOSITION. Assume that for any infinite proper normal ޒ-subgroup GЈ of G, F GЈŽ.ޒ is finite and that ⌫ is discrete. Then iFl 12,F ␦ŽHX.Ž. ⌬ ,⌬ .is closed for some non-tri ial semisimple normal ޒ-subgroup FF12 ¨ 658 HEE OH

H X of H without any compact factors if and only if ⌫ is an arithmetic F12, F subgroupŽ. also, irreducible lattice of G. X Proof. Suppose that ␦Ž H .Ž. F12, F .is closed. Then the stabilizer ⌳ F,F X X 12 lHis a Zariski dense arithmetic subgroup in H by Proposition 3.4.2. Let N be the normalizer of ⌫ H X. Then N contains ⌳ HЈ by F12, FFl 12,Fl Lemma 3.2.3. By Corollary 2.4.5, there exists a ޑ-form of G such that ⌫ is commensurable to GŽ.ޚ . Since G is ޑ-simple by the assumption F12, F on F Ž.Lemma 2.4.4 , it follows that ⌫ is an irreducible lattice. iF12,F Suppose that ⌫ is an arithmetic subgroup of G, that is, there exists a F12, F ޑ-form of G such that ⌫ is commensurable to GŽ.ޚ . It follows that U F12, F 1 and U2 are defined over ޑ and Fiiis commensurable to U Ž.ޚ for each X X is1, 2. Therefore H is also defined over ޑ and ⌳ F , F l H is commen- XX12 surable to H Ž.ޚ . By Theorem 3.4.2, ␦ŽH .Ž. F12, F .is closed. 3.4.5. COROLLARY. Suppose that G is a ޑ-simple group with respect to y1 which U12 and U are ޑ-subgroups and that E11s U Ž.ޚ and E 22s zU Ž.ޚ z for some z CHŽ.Ј. If ⌫ is discrete, the orbit ␦ŽH X.Ž. ⌬ , ⌬ . is g E12, EE1E 2 closed. X XX Proof. Since z CHŽ .,⌳ H ⌳ y1H.On g U12Žޚ.,UŽޚ.l s U1Žޚ.,zU 2Žޚ.z l the other hand, ⌳ HЈ is commensurable to H XŽ.ޚ . By Proposi- U12Žޚ., U Žޚ. l tion 3.4.2, the orbit ␦Ž.ŽHЈ . ⌬ , ⌬ .is closed. EE12 3.4.6. Since H Xis a normal subgroup of finite index in HЈŽ.ޒ , the following is another corollary of Theorem 3.3.2 and Proposition 3.4.2. X X LEMMA.1Ž. HŽ.ޒ.⌬is closed if and only if H .⌬ is closed. FFii Ž.2 ␦ ŽHЈ .Ž.Žޒ. ⌬ ,⌬ .is closed if and only if ␦ŽH X.Ž. ⌬ , ⌬ . is FF12 FF12 closed. X 3.4.7. PROPOSITION. Suppose that ⌫ is discrete and that H .⌬ and F12, FF1 HXX.⌬are closed. If ␦ H ޒ . ⌬ , ⌬ contains a closed orbit FF21Ž.Ž.Ž.F2 ␦ŽHX.Ž. ⌬ ,⌬ . and ⌫ is an irreducible lattice in G, then EE12 E12,E ␦ŽHX.Ž. ⌬ ,⌬ . is closed. FF12 Proof. Let HXŽ.ޒ gHX and ␦ ŽH X.Ž. ⌬ , ⌬ . s D is 1,...,ni E12 E ␦HXޒ.⌬,⌬ be a closed orbit. Then ␦Ž.ŽH X. ⌬ , ⌬ . ;Ž.Ž.Ž.FF12 EE12; ␦ŽHX.ŽŽ..Ad g ⌬ ,Ad Ž.g ⌬ .for some 1 k n.Let⌬X 1 kF122kFO O Eis Ad Žgy1.⌬ for each i 1, 2. Since H XXis a normal subgroup of H Ž.ޒ , ik Ei s we have that ␦Ž.ŽH XX. ⌬ , ⌬ X .␦ H X. ⌬ , ⌬ . Since gy1Ž.⌫ g EE12; Ž.Ž.FF12 k E12,Eks ⌫XX, the subgroup ⌫ XXis an irreducible lattice in G. E12, EE12,E Therefore we may replace HXŽ.ޒ by H Xin the statement of the proposition. By Ratner’s theorem, there exists a connected closed subgroup MX of SLŽU Žޒ ..= SL ŽU Žޒ .. containing ␦ŽAd ŽH XX .. such that ␦ H . ⌬ , ⌬ 12 Ž.Ž.FF12 is MX.Ž.⌬ , ⌬ . Since H XX.⌬ and H .⌬ are closed by the assumption, FF12 F1 F2 DISCRETE SUBGROUPS 659

MXXXH =H. Assume that ␦ŽH X.Ž. ⌬ , ⌬ .is not closed. We claim that ; FF12 ␦ŽHX.Ž.⌬,⌬ .is open in MX.Ž.⌬ , ⌬ . It is enough to show that for a EE12 EE12 sequenceŽŽ. Ad x ⌬ ,Ad Ž..y ⌬ , i 1in MX.Ž.⌬ ,⌬ converging to 1 iE122iEG E1 E2 Ž.⌬,⌬, there exists n ގ such thatŽŽ.Ž.. Ad x ⌬ ,Ad y ⌬ EE12 g iE12 iEg ␦ŽHX.Ž. ⌬ ,⌬ . for all i ) n. Denote by ⌫ the subgroup generated by EE12 i xExy1and yE yy1. Since MX.Ž.⌬ , ⌬ lies inside MX.Ž.⌬ , ⌬ , each i 1 ii2iEEFF12 12 ⌫i is discrete by Proposition 3.2.5 and the sequence of ⌫i’s converges to ⌫ . Therefore there exists a neighborhood V of the identity e such that E12, E Vl⌫i sÄ4eŽ.see the proof of Proposition 3.2.5 . It is well known that an irreducible lattice in a semisimple Lie group with real rank at least 2 is finitely presentable and locally rigidŽ for example, seewx 11. . Let h,h,...,h be generators of ⌫ with relations whŽ.,h,...,h e, 12 sE12,Ej12 ss js1, 2, . . . , t. We may assume the generators lie in the set E12j E . y1 y1 Denote by hikg xEx i 1 iijyE2 yiithe element in ⌫ such that y1 y1 xhiiki x ªh kif h kg E1, and yhiiki y ªh kif h kg E2as i ª ϱ, for each i P 1 and k s 1, 2, . . . , s. Since the number of generators is finite, we can find a sufficiently large n such that whjiŽ.1,hi2,...,hisgVl⌫ i sÄ4e for each j s 1, 2, . . . , t, and i ) n. Thus we can define a homomorphism r : ⌫ ⌫ which carries h to h for each k 1, 2, . . . , s and for all iE12,Eiª kik s i)n. Note that the sequence Ä4ri N i ) n tends to the identity map id of ⌫ in the limit. It follows from the local rigidity of ⌫ that there is E12, E E12, E n0 )nsuch that ri is an inner automorphism for each i ) n0. Now if y1 riiiisintŽ.g for g g G, g g NUŽ.Ž.12lNU sL, since gEgi1i;U1Ž.ޒ y1 and gEi 2 gi ;U2Ž.ޒ. Therefore AdŽ.gi s Ad1 Ž.xiiand Ad Ž.g s Ad2 Ž.yiii, and hence Žx , y .g XX X X ŽH =H .ŽŽ..Žl␦ Lޒ s␦ H .Žfor all i ) n0. It follows that ␦ H .Ž. ⌬ E , X 1 ⌬EE.Žis open. On the other hand ␦ H .Ž. ⌬ , ⌬ E . is also closed by the 2X X 12 X X assumption. Since both ␦ŽH .and M are connected, ␦ŽH .s M . This contradiction establishes the closedness of the orbit ␦Ž.ŽH X . ⌬ , ⌬ .. FF12

3.4.8. PROPOSITION. There exists a lattice E in U Ž.ޒ such that ⌬ ii Eig SLŽŽ..U ޒ .⌬ and the orbit H X.⌬ is closed. iFiiE Proof. By Proposition 1.4.2, there exists a ޑ-form of G such that every parabolic ޒ-subgroup is defined over ޑ. Therefore NUŽ.12and NU Ž.are defined over ޑ. We may assume that the determinant of ⌬ is the same UiŽޚ. as ⌬ by modifying the ޑ-form by the inner automorphism by an element Fi of L. It is now enough to set Eiis U Ž.ޚ to conclude the proof since XX HlHŽ.ޚis a lattice in H . 3.4.9. LEMMA. Let H be an algebraic group, H s L h N, ⌳ a Zariski dense arithmetic subgroup of L, and F a Zariski dense arithmetic subgroup of N which normalizes ⌳. Then ⌳ h F is a Zariski dense arithmetic subgroup of H. 660 HEE OH

Proof. Consider the ޑ-forms of L and N such that ⌳ and F are commensurable to LŽ.ޚ and N Ž.ޚ , respectively. Since ⌳ normalizes F, LŽ.ޑnormalizes N Ž.ޑ and hence they define a ޑ-form of H. The lemma now follows from the well-known fact that LŽ.ޚ N Ž.ޚ is of finite index in HŽ.ޚ .

3.4.10. PROPOSITION. Suppose that G is absolutely simple, that U1 is either X commutati¨e or Heisenberg and that H has no compact factors, i.e., H s H . Then the orbit H.⌬ is closed if and only if there exists a ޑ-form of G with Fi respect to which Ui and NŽ. U12l N Ž. U are defined o¨er ޑ and Fi is commensurable to UiŽ.ޚ . Proof. Suppose that the orbit H.⌬ is closed. By Proposition 3.4.2, Fi there exists a ޑ-form of H such that ⌳ is commensurable to HŽ.ޚ . Since Fi F is a Zariski dense arithmetic subgroup of U and ⌳ normalizes F ,we iiFii obtain a ޑ-form on HUiiii. Since HU s wNUŽ.,NU Ž.x, it follows from Proposition 1.5.3 that there exists a ޑ-form of G which we are looking for, proving the claim since in the proof we see NUŽ.12lNU Ž.is defined over ޑ.

Let G have a ޑ-form with respect to which Ui and H are defined over ޑ.If F is commensurable to U Ž.ޚ , then ⌳ is commensurable to HŽ.ޚ . iiFi It follows from Proposition 3.4.2 that H.⌬ is closed. Fi

3.4.11. LEMMA. Let G be a semisimple ޑ-group such that U1 and NUŽ.12lN Ž. U are defined o¨er ޑ. Then U2 is also defined o¨er ޑ.

Proof. The map PЈ ª PЈ l P is a bijection between the set of opposite parabolic subgroups to a parabolic subgroup P and the Levi subgroups of P wx5, 4.8 . It follows that an opposite parabolic subgroup PЈ to P is defined over ޑ if P and P l PЈ are defined over ޑ. Therefore NUŽ.2 is defined over ޑ; hence so is U2 .

3.4.12. We close this section by presenting specific lattices Fi such that the orbit H X.⌬ is closed, which will be also used in the proof of the main Fi theorem. Ž.ރ ޒ Ž . Let G s SL2Ž kq1. with the -form defined by G s SLkq1 Dޒ , k P 2, Dޒ is a Hamiltonian quaternion algebra over ޒ,

1 MD1=kޒ U1Ž.ޒs , ž/0Ik 10 ޒ U2Ž.sMD I ž/k=1ޒk DISCRETE SUBGROUPS 661 and

SL1Ž. Dޒ 0 H . s 0SL D ž/kŽ.ޒ Then

X 10 Hs . ž/0SLkŽ. Dޒ

Let F be a lattice in U Ž.ޒ . We will show that if the orbit H X.⌬ is 11 F1 closed, then there exists a central simple division algebra D1 over ޑ and ޚ Ž.I ␣ MD1=k 1Ž. D1 is isomorphic to Dޒ over ޒ such that F1 s 0 I for some ␣ g ޒ and up to conjugation by an element in HЈ. X X That the orbit H .⌬ F is closed implies that there exists a ޑ-form of H 1 X such that the representation Ad14 restricted to H to SL kŽ.ރ given by AªŽ.A,Ais ޑ-rational. By Proposition 1.4.10, the ޑ-form of HЈ is up to X conjugation such that H Ž.ޑ s SLk Ž D11 .Ž.ޑ for some D described above. X Without loss of generality, we may assume that H Ž.ޑ s SLk Ž D1 .Ž.ޑ . This ޑ-form of HЈ obviously extends to a ޑ-form of H such that HŽ.ޑ s ޚ Ž.Ž. Ž.Ž. ŽIM1=k D1Ž. . Ž. SL11 D ޑ = SLk D11ޑ . Now set E s 0I. Then H ޚ preserves E11by conjugation. Now the claim that E s F1up to a constant multiple and commensurability follows from the fact that any two irreducible ޑ-rational representations which are isomorphic over ރ are isomorphic over ޑ.

4. THE PROOF OF THE MAIN THEOREM

For the entire Section 4, we assume that G is an adjoint absolutely simple algebraic ޒ-group with real rank at least 2 and U12, U is a pair of 0 opposite horospherical ޒ-subgroups. We recall the notation G s GŽ.ޒ , 0 LsNUŽ.12lNU Ž., HswxL,L, and H s HŽ.ޒ from Subsection 3.2.1 and the terminology PropertyŽ. A from Subsection 1.1.16.

4.1. Commutati¨e Horospherical Subgroup Cases We are now ready to give the proof of the main theorem for commutative horospherical subgroup cases. A main feature in this case is the fact that AdiŽ.H is a maximal connected closed subgroup of SL ŽŽ..Ui ޒ ; hence by Ratner’s theorem H.⌬ is either closed or dense in ⍀ . Moreover, Fii unless G is of type An, ␦Ž.H is a maximal connected closed subgroup of Ad Ž.H = Ad Ž.H ; hence ␦ Ž.ŽH . ⌬ , ⌬ .is either closed or dense in 12 FF12 662 HEE OH

H.⌬=H.⌬. This feature facilitates finding a closed ␦Ž.H -orbit lying in FF12 ␦H.⌬,⌬, which is enough to prove the main theorem by Proposi- Ž.Ž.FF12 tions 3.4.4 and 3.4.7. 4.1.1. Remark. The case when H does have compact factors occurs 1 only when the ޒ-type of G is An, n s 2k q 1, and U1 is conjugate to the minimal parabolic subgroup V⌰ where ⌰ is either ⌬ y ␣1 or ⌬ y ␣n.

4.1.2. PROPOSITION. Let U12, U be a pair of commutati¨e horospherical ޒ-subgroups of G and F12 and F be lattices in U1Ž.ޒ and U 2 Ž.ޒ respecti¨ely. Let HЈ be the maximal semisimple normal ޒ-subgroup of H without any XX0 ޒ-anisotropic factors and H s H Ž.ޒ . Then there exists a lattice Eii in U Ž.ޒ such that H XX.⌬ H .⌬ and H X.⌬ is closed for each i 1, 2. EFiig E i s X Proof. Suppose that H s H. Since AdiŽ.H is a maximal connected closed subgroup of SLŽŽ..ŽUi ޒ see Proposition 1.3.6. it follows from Ratner’s theorem that H.⌬ is either H.⌬ or SLŽŽ..U ޒ .⌬ . It then FFi11F 1 follows from Proposition 3.4.8 that H.⌬ contains a closed orbit H.⌬ for FEii some lattice Eiiin U Ž.ޒ . According to the previous remark, it remains to consider when the 1 1 1 ޒ-form of G is of type A2 k 112, and H and HЈ are of types A = A k1 1 q y and A2 ky1 respectively. Without loss of generality, we may realize G as follows: G s SLk 1Ž. Dޒޒ, D is a Hamiltonian quarternion division alge- q X bra over ޒ, H s SL1Ž. Dޒ = SLk Ž. Dޒ , and H s SLkŽ. Dޒ . By Theorem 3.3.1, there exists a connected closed subgroup MЈ of SLŽŽ..U1 ޒ contain- XX X ing Ad1Ž.HЈ such that H .⌬ FFs M .⌬ . Denote by S a Levi subgroup of X X11 X Mwhich contains Ad1ŽH .. By Proposition 3.3.3, S is a semisimple subgroup, and for some u RMŽ X .,SX.Ž.u⌬ is closed. We claim that g uF1 HX.Ž.u⌬is closed. Note that Ad ŽHX.Ž.I SL ރ and since Ad is F1 122s m k 1 X y1 rational over ޒ,Ad14ŽH.Ž.;xSL kޒ x where x is a basis transformation k 4k y1 of U1Ž.ޒ s Dޒ to ޒ . Let g g xSL4 kŽ.ޒ x be a representative of the lattice u⌬ in the space xSL Ž.ޒ xy1 xSL Ž.ޚ xy1. We consider the F1 4 k r 4 k ޑ-form of SLŽ.U SL Ž.ރ given by this choice of this lattice u⌬ in 14s kF1 U14. Denote by MЈ the smallest ޑ-subgroup of SL kŽ.ރ containing MЈ and XXXX0 by S the smallest ޑ-subgroup containing S . Then M s M Ž.ޒ and XX0 SsSŽ.ޒ. Denote by ␳11the block diagonal embedding of S s SL2kŽ.ރ =SL2 kŽ.ރ into SL4 k Ž.ރ . Also denote by ␳2 the tensor product representa- X tion of S22s SL Ž.ރ = SL 2k Ž.ރ . By Proposition 1.3.9, we have that S is X either ␳11Ž.S or ␳ 2 Ž.S 2. That S s ␳iiŽ.S would imply that there exists a 0 X ޑ-form of Si , compatible with the ޒ-form of SiiŽŽ.i.e., S ޒ s S .such that y1 ␳iibecomes ޑ-rational in such a way that ␳ ŽŽ..Siޑ ; gSL4k Ž.ޑ g .We first observe that this condition on compatibility with the ޒ-form cannot be satisfied by any ޑ-form of S11. By Proposition 1.4.9, the ޑ-forms of S with respect to which ␳11is rational over ޑ are such that S Ž.ޑ is DISCRETE SUBGROUPS 663

conjugate to the subgroup SL2 kŽ.ޑ = SL2 k Ž.ޑ or to the subgroup Ž. ޑ Ž.Ј RSLkk r ޑ n where k is a quadractic extension of . Since Ad1 H must X X be an ޒ-subgroup of S , it follows that Ad1Ž H . s ÄŽ.A, AA

X X 4.1.3. LEMMA. Let U1212, U , F , F , H , and H be the same as in Proposition 4.1.2. Assume that ⌫ is discrete. Then there exists Ž.⌬ , ⌬ F12, FEE1 2 inside ␦ H XX. ⌬ , ⌬ such that H .⌬ and H X.⌬ are closed. Ž.Ž.FF12 EE1 2

Proof. By the previous proposition, H X.⌬ contains a closed orbit F1 HX.⌬ for some lattice E in U Ž.ޒ . By Proposition 3.2.8, there exists a E1 11 lattice E in U Ž.ޒ such that Ž⌬ , ⌬ . ␦ H . ⌬ , ⌬ . By the re- 22 EE12g Ž.Ž. FF12 mark preceding Proposition 3.2.8, in fact Ž.⌬ EE, ⌬ g ␦ Ž.HЈ .Ž.⌬ FF, ⌬ . XX12 12 If H .⌬ is not closed, then let E be a lattice in U Ž.ޒ such that ⌬ X E2 22E2 XX X H.⌬ and H .⌬ X is closed. Again we get a lattice E in U Ž.ޒ such g EE22 11 XXX that ⌬ EEXXg H .⌬ such that Ž.⌬ EE, ⌬ Xg ␦ Ž.H .Ž.⌬ EEE, ⌬ . Since H .⌬ 11X 12 121 is closed, so is H .⌬ X, proving the lemma. E1

4.1.4. THEOREM. Let U12, U be a pair of commutati¨e horospherical ޒ-subgroups of G. Assume that if rank ޒŽ.G s 2, then G is not of type E6. Then the triple Ž.G, U12, U has property Ž. A .

Proof. Let F12, F be arbitrary lattices in U1Ž.ޒ and U 2 Ž.ޒ respectively. To prove that ⌫ is arithmetic subgroup, it is enough to find a closed F12, F orbit ␦Ž.ŽHЈ . ⌬ , ⌬ .lying in ␦ HЈ . ⌬ , ⌬ by Propositions 3.4.4 and EE12 Ž.Ž.FF12 3.4.7. By the previous lemma, there exists Ž.⌬ , ⌬ ␦ HЈ . ⌬ , ⌬ EE12g Ž.Ž.FF12 such that H X.⌬ and H X.⌬ are closed. EE12

Case 1: U1 Is Reflexive. In this case, we always have that HЈ s H. Since H.⌬ is closed, by Proposition 3.4.10 there exists a ޑ-form of G E1 with respect to which U11and H are defined over ޑ and E is commensurable to U12Ž.ޚ . Since U is also defined over ޑ by Lemma 3.4.11, there y1 exists w g GŽ.ޑ such that wU12Ž.ޑ w s U Ž.ޑ by Lemma 1.2.2. By y1y1 Lemma 2.3.3, we may assume that E21s xwEwŽ. x for some x g Ž.Ž.Lޒ. Write x s yz for y g HŽ.ޒ and z g ZLŽ.Ž.ޒ. Suppose that ␦Ž.ŽH.⌬,⌬ .is not closed, and denote by M a connected closed EE12 subgroup of Ad12Ž.H = Ad Ž.H such that ␦ Ž.H ; M and ␦H.⌬,⌬M.Ž.Ž.⌬,⌬Theorem 3.3.1 . Ž.Ž.EE12s EE12 664 HEE OH

Subcase Ž.i: MsAd12 Ž.H = Ad Ž.H . In that case, it is clear that ␦Hޒ .⌬ ,⌬ is HŽ.ޒ .⌬ = H Ž.ޒ .⌬ . Therefore we have found a Ž.Ž.Ž.EE12 E1 E2 closed orbit ␦Ž.ŽH . ⌬ , ⌬ y1.lying inside ␦ H ޒ . ⌬ , ⌬ up U12Žޚ.zU Žޚ.zEEŽ.Ž.Ž.12 to commensurability. It follows from Proposition 3.4.5 that ␦Ž.ŽH . ⌬ , ⌬ . EE12 is closed

Subcase Ž.ii : M Is a Proper Subgroup of Ad12Ž.Ž.H = Ad H . If the absolute type of G is not An, then H is a simple group and hence ␦Ž.H is a maximal closed connected subgroup of Ad12Ž.H = Ad Ž.H . Therefore the case when M is a proper subgroup of Ad12Ž.H = Ad Ž.H can happen only when G has type An. Let us now suppose that G has type An. From Fig. 1.3.2., H is the product of two absolutely simple ޒ-groups, say, N12= N . Also Ad 1Ž.H s N12mN and Ad 2Ž.H s N 21m N . There are only two proper connected closed subgroups of Ad12Ž.H = Ad Ž.H which properly contain ␦Ž.H : 0 0 M11sÄAd Ž.Ž..A, B ,Ad 2C,BA<,CgN1Ž.ޒ,BgN 2Ž.ޒ4and M 2s 0 0 ÄŽŽAd12A, B .,Ad ŽA,DA ..

Subcase Ž.i: G and G˜ Are Isomorphic over ޑ. Since U12and U are defined over ޑ with respect to both of the ޑ-forms, we may assume that y1 G˜Ž.Ž.ޑsxG ޑ x for some x g NUŽ12 .Ž.ޒ lNU Ž .Ž.ޒ. It follows that E2 y1 is commensurable to xUŽ21lG Žޚ .. x . Let x s yz for z g ZNUŽŽ. l NUŽ ..Žޒ .. and y HŽ.Žޒ . Then ⌬ , ⌬ y1 . H.⌬ = H.⌬ while E 2 g EyEy12g E1 E 2 1 y1 y1 and yE21 y are commensurable to U l GŽ.ޚ and zU Ž2lG Ž..ޚ z re- DISCRETE SUBGROUPS 665 spectively. By the same argument as in the previous case, we can show that ␦Ž.ŽH.⌬,⌬ .is closed. EE12

Subcase Ž.ii : G and G˜Are Not Isomorphic over ޑ. By the uniqueness of ޑ-split forms of G, G is not split over ޑ; hence the ޑ-form of G has 1 1 28 type either An or E6, 2. There exist a maximal torus T containing a maximal ޑ-split torus S and ⌬ ⌽Ž. y Gand a basis of T, G such that U1 s V⌬y␣ and U2 s V⌬y␣ for some commutative root ␣ g ⌬. Since U1 is a ޑ-subgroup of G, by Proposition 1.4.1, ␣ is fixed by the )-actionŽ. in the Tits index of G of GalŽރrޑ.and X ␣ f ⌬ 00, ⌬ the subset of ⌬ whose elements vanish on S. Set U1s X y ⌬ V⌬yÄ␣ , iŽ ␣ .4 and U2 s V⌬yÄ␣ , iŽ ␣ .4 where i is the opposition involution on Ž.see 2.3.4 . Since i commutes with the )-action of the Galois group, iŽ.␣ is XX also fixed by the )-action and iŽ.␣ f ⌬ 01. It follows that U and U2are X ޑ-subgroups of G˜. In particular, U1 is a maximal horospherical ޑ-sub- 1 28 group of G when G is of type E6, 2 . Since U1 is a ޑ-subgroup of G˜, it is enough to note the fact that i is invariant under an isomorphism between two Dynkin diagrams to see that XX XX U12and U are also ޑ-subgroups of G˜. Therefore E1s GŽ.ޚ l U 1 Ž.ޒ and X XXX E22sG˜Ž.ޚlU Ž.ޒare lattices in U 12Ž.ޒ and U Ž.ޒ respectively. Note that X for each k s 1, 2, Ekkl U Ž.ޒ s E kup to commensurability. Ž. Ž. y Ž. ŽX. Set iU1 sV⌬yiŽ␣ . and iU2 sV⌬yiŽ ␣ .. Then Ukkl iU sZU kfor each k s 1, 2Ž. see the following Lemma 4.1.5 . In particular, we have X X X ZUŽ kk.Ž;U. Then ZU kŽ..ޒ lE kis a lattice in ZUŽkŽ..ޒ for each k s 1, 2 X since ZUŽ.k is a ޑ-subgroup with respect to both ޑ-forms G and G˜. X Denote by G0 the algebraic ޒ-subgroup of G generated by ZUŽ 1 . and X 0 X ZUŽ 20.Ž.,G sG0ޒ , and by ⌫011the subgroup generated by ZUŽŽ..ޒ lE and ZUŽXŽ..ޒ E. Note that ⌫ is discrete since ⌫ ⌫ . 22l 0 0; E12,E By Proposition 2.5.4, G 0 is an absolutely simple algebraic ޒ-group and X X ZUŽ 12.Žand ZU.are opposite commutative horospherical ޒ-subgroups of X G0. Since i obviously preserves the set Ä␣, iŽ.␣ 4, U1 is reflexive and it X follows that ZUŽ 1 . is also reflexive. X We now claim that the commutator subgroup, say SЈ,of NZUŽŽ1.. l X NZUŽŽ20.. lG has no ޒ-anisotropic factors. Recall from Proposition 2.5.4 X X that SЈ is a semisimple normal algebraic ޒ-subgroup of NUŽ12.ŽlNU..If the absolute type of G is E6 , the ޒ-form of G is split by the assumption 1 28 that the ޒ-form of G is not E6, 2 , and hence so are the ޒ-forms of G 0 and SЈ, proving the claim in that case. For reference the types of G 0 and SЈ are in this case D54and D respectively. Now if the absolute type of G is An and hence H is H12= H , where H 1and H 2are respectively of types Ak Ž.ޑ and Anyky1, k ) 1 from the assumption that G is not -split , then in that case G0 is of type Amjjfor some m - n and the type of SЈ is H = H 666 HEE OH where j s 1if k-nyky1; otherwise, j s 2. The claim follows from the assumption that H has no ޒ-anisotropic factors.

It also follows that the ޒ-rank of G 0 is at least 2 since the ޒ-rank of SЈ is at least 2.

Therefore there exists a ޑ-form of G 00with respect to which ⌫ is commensurable to G 0Ž.ޚ by the previous resultŽ. Case 1 on the case of reflexive horospherical subgroup cases. Note that ⌫ ⌫ and ⌫ 0 ; E12, EE12,El SЈŽ.ޒsSЈlG0 Ž.ޚis a Zariski dense arithmetic subgroup of SЈ since SЈŽ.ޒhas no compact factors. Observe that GŽ.ޑ l SЈ s G˜ Ž.ޑ l SЈ since SЈ GŽ.ޚ ⌫ , up to commensurability. In a group G of type A ,an l 0 ; E12,E n inner ޑ-form of G is completely determined by a Q-form of SЈ up to

ޑ-isomorphism. In a group G of type E6 , it implies that both G and G˜ are 1 28 of type E6, 2 with the same semisimple anisotropic kernel determined by ˜ SЈlG0Ž.ޚ. In conclusion, there exists a ޑ-isomorphism F: G ª G.By the same argument as that in SubcaseŽ. i , we can prove that the orbit ␦Ž.ŽH.⌬,⌬ .is closed. EE12

Case 3: U1 Is Not Reflexive and H / HЈ. Without loss of generality, Ž.Žރ. we may realize G as follows see Section 3.4.12 : G s SL2Ž kq1. , k P 2, Ž. ޒ GsSLkq1 Dޒޒ, D is a Hamiltonian quaternion division algebra over , HsSL1Ž. Dޒ = SLk Ž. Dޒ , and HЈ s SLkŽ. Dޒ . The closedness of the orbit H X.⌬ implies that there exists a ޑ-form of HЈ such that the representa- Ei tion of HЈ to SL4 kŽ.ރ defined by A ª Ž.A, A is ޑ-rational. It follows that the ޑ-form of HЈ is up to conjugation such that HЈŽ.ޑ s SLki Ž D .Ž.ޑ , where Di is a central simple division algebra over ޑ and Di is isomorphic to D over ޒ. Assume that ␦Ž.ŽHЈ . ⌬ , ⌬ .is not closed. Since HЈ is ޒ EE12 simple, then it follows that the orbit is dense in H X.⌬ = H X.⌬ . There- EE12 fore the pair Ž.⌬ , ⌬ ␦ HЈ . ⌬ , ⌬ where FF12g Ž.Ž.EE1 2 1MDŽ.ޚ 10 E 1=k1and E . 12s0I s ␭MDŽ.ޚ I ž/k ž/k=12 k Since the subgroup ⌫ is discrete, by Example 2.2.5, we have that up to E12, E commensurability D Ž.ޚ ␭D Ž.ޚ . Denote DЈ D . Therefore ⌫ is 12s s 1 E12,E Ž.Ž.ޚ an arithmetic subgroup commensurable to SLkq1 D . By Proposition 3.4.4, this proves that ␦Ž.Ž.HЈ . ⌬ , ⌬ as well as that ␦ Ž.Ž.H . ⌬ , ⌬ is EE12 EE12 closed. 4.1.5. We now give a proof of the lemma used in the proof of the above theorem. X LEMMA. Ukkl iUŽ.sZ Ž U k. for each k s 1, 2. Proof. It suffices to prove that Ž⌽qywx⌬ y ␣ .Žl ⌽qyw⌬ y iŽ.␣ x.s Ä ␤g⌽qyw⌬yÄ␣,iŽ.␣4xN␤q␥f⌽qqfor any ␥ g ⌽ yw⌬ y Ä␣,iŽ.␣4x4. That the coefficient of ␤ is nonzero with respect to each ␣ and DISCRETE SUBGROUPS 667 iŽ.␣implies that the coefficient is 1 in each case by Lemma 1.3.1 and by the assumption that ␣ is a commutative root hence so is iŽ.␣ . Since the highest root has the coefficient 1 with respect to each ␣ and iŽ.␣ , ␤ q ␥ q cannot be a root for any ␥ gޒ ⌽ yw⌬ y Ä␣, iŽ.␣ 4x, proving the inclusion X ; . Suppose that U␤ ; ZUŽ 1 . and the coefficient of ␤ with respect to ␣ is zero. There exist simple roots ␣ ,...,␣ such that ␤ ␣ иии ␣ ii1 m q i1q q ijg ⌽qfor each j 1, 2, . . . , m and ␤ ␣ иии ␣ is the highest ޒ s q ii1q q m q root. Then for some j, ␣i s ␣. Since ␣ g ⌽ yw⌬ y Ä␣, iŽ.␣ 4xand X j U␤ ␣ иии ␣ ZUŽ 1., we obtain a contradiction to the assumption q ii1q q j1; Xy U␤ ;ZUŽ1 .. Similarly we can show that the coefficient of ␤ with respect to iŽ.␣is nonzero. This proves the other inclusion relation.

4.1.6. Remark. We remark that inwx 14 , we proved the closedness of the orbits H.⌬ and H.⌬ in a direct way, in the case of GŽ.ޒ SL Ž.ޒ and FF12 s n this method works for the ޒ-split groups of classical types

4.2. Heisenberg Horospherical Subgroup Cases

4.2.1. In this section, let U12and U be a pair of opposite Heisenberg horospherical ޒ-subgroups. We recall the representations Adi : L ª X GLŽ.Uiifrom Subsection 1.1.14 and Ad : H ª SLŽ.V ifrom Subsection 1.3.7. Denote by ␲ the projection of Uiionto V . The following lemma easily follows from Lemma 2.1.10.

X LEMMA. Let F be a lattice in U Ž.ޒ and ⌬ ␲ Ž.⌬ . Then ⌬ Z Ž.U iiFFFiis il i and ⌬Xare lattices in ZŽ.U and V , respecti ely. Fii i¨

X The group H acts through Adii and Ad on the space of lattices in Ui X and V ii, respectively. The notation H. J and H. Jidenotes the orbits X X AdiiŽ.Ž.HJand Ad iiŽ.ŽHJ., respectively, under this action. 4.2.2. To prove the main theorem in this case, we investigate the closures of the orbits H X.⌬ and ␦Ž H X .Ž. ⌬ , ⌬ .as we did in Section 4.1. FFi 12F Unlike the case when U12and U are commutative, we have various closed subgroups between AdiiŽ.H and SL ŽŽ..U ޒ . The key point of narrowing the possibilities for the closures of those orbits will be thatŽ. 1 we may assume that the closures are the homogeneous spaces under semisimple subgroups of SLŽŽ..UiFޒ , after replacing ⌬ and ⌬ Fby suitable lattices inside the X 12 closures andŽ. 2 AdiŽ.H is a maximal connected closedŽ resp. semisimple. subgroup of SpŽŽ..V innޒ if G is not of type A Žresp. if G is of type A ..

4.2.3. Since wxV ii, V s ZŽ.U i, we have a skew-symmetric bilinear product²: , : V ii Ž.ޒ = V Ž.ޒ ª Z ŽU i Ž..ޒ defined by ²¨, w :s wx¨, w . We set SpŽ.V iis Äg g SLŽ.²V N g¨, gw :s ²¨, w :for all ¨, w g V i4. 668 HEE OH

LEMMA. Let Fiiii be a lattice in U Ž.ޒ and ⌽ ŽF .s Äg g SLŽ.V N ²Ž.g ,gw Ž .:⌬ Z ŽU .for all , w ⌬X 4. Then the connected compo- ¨ g Fiiil ¨ g F nent of identity in ⌽Ž.FisSpii ŽV . Proof. In a Lie group, the connected components coincide with the path-wise connected components. Denote by ⌽ 0 the path-wise connected 0 component of the identity in ⌽Ž.Fi and let g g ⌽ . Then there exists a 0 continuous family gt g ⌽ , t g wx0, 1 , such that g01s e and g s g. For any ,w ⌬X,²Ž.Ž.:g ,gw ⌬ Z Ž.U. Since ⌬ ZŽ.U is discrete, it ¨ gFtiii¨ tg Fl i Fl i follows that ²Ž.gtt¨ , gw Ž.:²s¨,w :for all t g wx0, 1 . In particular, it implies that ²Ž.g , gw Ž .: ²,w :for all , w ZŽ.U since ⌬X is a ¨ s ¨ ¨ g iFi lattice in V ii. Hence g g SPŽ.V .

4.2.4. Denote by Piithe parabolic subgroup of SLŽ.U which stabilizes the line ZŽ.Uii. Then wxP , Piis isomorphic to SLŽ.V iiiuW where W s RP Ž.i. XX X LEMMA.1Ž. H.⌬ SpŽV Žޒ ...⌬ . FiFii; Ž.2 HX.⌬ SpŽV Žޒ ..ŽW ޒ ..⌬ . FiiFii; Proof. By Ratner’s theorem, there exists a connected closed subgroup XXXXXXX Lii;SLŽŽ..V ޒ containing AdiŽ.H such that H .⌬ FiFs L .⌬ . Since XX Xii AdiiŽH .ŽŽ..; Sp V ޒ , it is not difficult to see that Lii; ⌽Ž.F and hence Ž.1 follows from Lemma 4.2.3. Since ⌬ ZŽ.U is a lattice in ZŽ.U ,it Fiil i follows that Pi is defined over ޑ with respect to the ޑ-form of SLŽ.Ui given by ⌬ . ThenŽ. 2 follows fromŽ. 1 and Proposition 3.3.3. Fi X 4.2.5. PROPOSITION. For any lattice E in U Ž.ޒ , the orbit Sp ŽV Ž..ޒ .⌬ ii i Ei is closed. Proof. Similarly to proof of Lemma 4.2.4, we can show that if Sp V ޒ .⌬XXMЈ.⌬ for some connected closed subgroup MЈ of Ž.iEŽ. 11s E SLŽŽ..V iiޒ , MЈ ; Sp ŽŽ..V ޒ . Therefore MЈ s SpŽŽ..V iޒ , proving that the orbit is closed.

4.2.6. PROPOSITION. If SpŽŽ..V ޒ .⌬ is closed, it contains a closed orbit iFi HX.⌬. Ei

Proof. By Proposition 1.4.2, there exists a ޑ-form of G such that U1 and U are defined over ޑ. Set E U Ž.ޚ . Since ⌳ HŽ.ޚ and ⌳ H X 2 iis Eiis El are lattices in H and H X, respectively, H.⌬ and H X.⌬ are closed. We EEii will show that ⌬ is contained in SpŽŽ..V ޒ .⌬ . It is enough to show that EiiiF ⌬Xis contained in SpŽŽ..V ޒ .⌬X. Fixing an isomorphism of V Ž.ޒ to ޒ n, EiiiF i let J be the standard lattice ޚ n. Since SpŽŽ..V ޒ .⌬X is closed, it follows iiFi that ⌬Xis commensurable to gSp Ž.ޚ gy1 for some g SpŽŽ..V ޒ .It Fnii g implies that ⌬X is commensurable to gޚ n. For the same reason, ⌬X is Fi Ei n commensurable to hޚ for some h g SpŽŽ..V i ޒ , proving the claim. DISCRETE SUBGROUPS 669

4.2.7. If we assume that the real rank of G is at least 3, then the cases when H has compact factors occur only when G is of type Dnq2 , n P 4. Furthermore, the nonsplit real forms of G admitting Heisenberg horo- ޒ Ž. spherical -subgroups are such that G s SUnq2 Dޒ , h where D is a ޒ Ž. Hamiltonian quaternion division algebra over and h g GLnq2 Dޒ is a nondegenerate hermitian form relative to an involution ␴ of the first kind such that the space of ␴-symmetric elements in Dޒ has dimension 3. Since the real rank of G is at least 2, we may assume that 001 hs0h0 0 0100 where h0g GLnŽ. Dޒ is a non-degenerate hermitian form relative to ␴ .In this case, ¡ A 00 ¦ Hs~¥0 B 0 AgSL1Ž. Dޒ, B g SUn Ž Dޒ, h0 . ¢§000␴A and 100

HЈs0SUnŽ. Dޒ, h00 . 0001 4.2.8. PROPOSITION. Suppose that H s HЈ or that the real rank of G is at least 3. Then HЈ.⌬ contains a closed orbit HЈ.⌬ . FEii Proof. By Theorem 3.3.1, there exists a connected closed subgroup M containing Ad Ž.HЈ such that H X .⌬ MЈ.⌬ . By Proposition 3.3.4, we iFiis F may assume that MЈ is a semisimple subgroup of SpŽŽ..V i ޒ . Suppose that HsHЈ. Then M is either AdiiŽ.H or Sp ŽŽ..V ޒ . Therefore the claim follows from Proposition 4.2.6. Now if H / HЈ, we consider the realization X of the ޒ-form of G given in the previous section. Note that Ad1Ž.HЈ s ÄŽ.A 0 Ž.4 0 A

4.2.9. LEMMA. If H.⌬ is closed, then ⌬ V Ž.ޒ is a lattice in V Ž.ޒ . FFiil i i Furthermore ⌬ is, up to commensurability, determined by ⌬ V Ž.ޒ . FFi il i

Proof. It follows from the ޑ-rationality of Adii that V is a ޑ-subspace with respect to the ޑ-structure of U given by ⌬ Žsee the proof of iFi 670 HEE OH

Proposition 1.5.5.Ž. . It implies that ⌬ V ޒ is Zariski dense and hence Fiil is a lattice in V Ž.ޒ . The second claim follows from the fact that ⌬ i w Fi l VŽ.ޒ,⌬ V Ž.ޒis commensurable to ZŽ.U ⌬ . iFiil x iFl i 4.2.10. Remark.Ž. 1 The subgroup H is trivial only when G is ޒ-split of type A2 . Ž.2 The subgroup H has compact factors only when G is of the

Ž.absolute type Bnnwith the real rank 2 and D . The following theorem covers the cases not only when H has no compact factors but also when G is of type Dn as long as the real rank of G is greater than 2. For example, it covers all Heisenberg horospherical subgroup cases for the groups with real rank at least 3.

4.2.11. THEOREM. Let U12, U be a pair of opposite Heisenberg horospherical ޒ-subgroups. Assume that if rank ޒ G s 2, theŽ. absolute type of G is none of A2 , Bnn and D for some n. Then Ž.G, U12, U has property Ž. A . Proof. By the same argument as in Theorem 4.1.4, it is enough to find a closed orbit ␦Ž.ŽH XX. ⌬ , ⌬ .inside ␦ H . ⌬ , ⌬ . By Proposition EE12 Ž.Ž.FF12 4.2.8, H XX.⌬ contains a closed orbit H .⌬ . By the same argument as the FE11 proof of Theorem 4.1.3, we may assume that there exists a lattice E2 such XX that H .⌬ EEis closed and Ž.⌬ , ⌬ EFg ␦ Ž.H .Ž.⌬ , ⌬ F. Therefore there 21212X exist a ޑ-form of G with respect to which U12, U , and H are defined over ޑand E11is commensurable to U Ž.ޚ . If H s HЈ, this follows from Proposition 3.4.10. It only remains to consider the case described in 4.2.7. That the orbit HЈ.⌬ is closed E1 implies that there exists a ޑ-form of HЈ such that the representation Ž.A 0 Ž. ŽŽ..X Aª0 A is rational over ޑ. It follows that HЈ ޑ s SUn DЈ ޑ , h0 X where DЈ and h0 are described as in Proposition 1.4.10. Therefore it is enough to observe that this ޑ-form of HЈ naturally extends to a ޑ-form of Ž.ޑ ŽЈ Ž.ޑ Ј . Gby declaring that G s SUnq2 D , h where 001 X hЈs0h0 0. 0100 X X Assume that the orbit ␦ŽH .Ž. ⌬ EE, ⌬ . is not closed. Denote by M a 12X X X connected closed subgroup of Ad12ŽH .Ž= Ad H .Žsuch that ␦ H .; MЈ and ␦ H XX. ⌬ , ⌬ M .Ž.⌬ , ⌬ . Ž.Ž.EE12s EE12 X X Case Ž.1. MsAd12 ŽH .Ž= Ad H .. Since a Heisenberg horospherical subgroup is reflexiveŽ. see Remark 2.3.4 , there exists w g G Žޑ.such that y1 wU12Ž.ޑ w is conjugate to U Ž.ޑ . By Lemma 2.3.3, we may assume that for some x g NUŽ.Ž.12ޒ lNU Ž.Ž.ޒ,⌬ElV2is commensurable to X 2 X ⌬y1V. By Lemma 2.3.3, ⌬ is commensurable to ⌬ y1 . xU22Žޚ.x l2 ExU2Žޚ.x DISCRETE SUBGROUPS 671

Write x as yw where y g H and w g ZŽŽ..L ޒ . Again y s yy12where y12gHЈand y g CHŽ.Ј since HЈ commutes with compact factors of H. XX Set z yw. Then Ž␦ , ⌬ y1␦ H ޒ . ⌬ , ⌬ . Since the H - s 2 U12Žޚ.zEzg Ž.Ž.Ž. E12 E orbits of ⌬ y1 and ⌬ y1 are closed and their projections onto yEy22 zUŽޚ.z VŽ.ޒ are commensurable, by Lemma 4.2.9, Ž⌬ , ⌬ y1. i U12Žޚ. zU Žޚ.z ␦HXޒ.⌬,⌬up to commensurability, contradicting that gŽ.Ž.Ž.EE12 ␦ŽHX.Ž. ⌬ ,⌬ .is not closed by Proposition 3.4.5. EE12

X X X Case Ž.2. M is a proper subgroup of Ad12ŽH .Ž= Ad H .. This case happens only when the type of G is Bnnor D ; otherwise H is a simple Lie group, and hence ␦ŽH X.is a maximal connected closed X X subgroup of Ad12Ž H .Ž= Ad H .. Since in this case H is a product of two simple groups, by the same argument as CaseŽ.Ž. 1 ii in Theorem 4.1.4, we can show that ␦ H XX. ⌬ , ⌬ contains a closed ␦Ž.H -orbit. This con- Ž.Ž.EE12 cludes the proof.

4.3. Non-ޒ-Heisenberg Horospherical Subgroup Cases We deal with non-ޒ-Heisenberg subgroup cases by reducing into commutative horospherical subgroup cases. If ZUŽ.1 is not the root group of a highest real root, this can be done by considering ZUŽ.12and ZU Ž.. X X Otherwise we show that we can replace U12, U by another pair U12, U of X opposite horospherical subgroups such that ZUŽ1 .is not a root group of a highest real root. 4.3.1. In Subsections 4.3.1᎐4.3.2, we assume that G is almost ޒ-simple. X X We define the pair of horospherical ޒ-subgroups U12, U mentioned in the beginning of this section and prove their needed properties.

For each i s 1, 2, set Uiiiiiis LieŽ.U , U˜ s ÄX g U N wxU , X ; ZŽ.U 4and X X UiiisÄXgUNwxX,U˜˜s04 . Denote by Uiiand U the connected sub- XX groups of Uiisuch that LieŽ.U˜˜s Uiiand LieŽ.U s Ui. Equivalently, U˜ is y1y1 X ÄugUiiiNgug u g ZUŽ.for all g g U 4and Uithe centralizer of U˜iiin U .

X PROPOSITION.1Ž. The subgroups U˜ii and U are ޒ-subgroups of Ui. X Ž.2 If Uii is defined o¨er ޑ, then U˜ and Ui are ޑ-subgroups. X X Ž.3 If Fii is a lattice in U Ž.ޒ , Fil UiŽ.ޒ is a lattice in U iŽ.ޒ .

Proof. It is well known that the normal subgroups in the ascending central series of a unipotent ޒ-group are defined over ޑ. Hence U˜i is X defined over ޒ. Since Uiiis the centralizer of U˜ in Ui, it is defined over ޒ. This argument works in the same way for the field ޑ. PartŽ. 3 directly follows fromŽ. 2 and Lemma 2.1.11. 672 HEE OH

4.3.2. The notation in this section is the same as in Subsection 2.5.1 with q ksޒ. Set ⌿1 s Ä␣ gޒ ⌽ ywx⌰ N the coefficient of ␣ with respect to q ޒޒ⌬yŽ.⌰j⌬His 04 and ⌿2s Žޒ⌽ ywx⌰ .y ⌿112. Then ⌿ and ⌿ are qq closed sets of ޒ ⌽ . It is easy to see that ⌿2 sޒޒ⌽ ywx⌰ j ⌬ H. To simplify the notation, let ޒޒ⌬ s ⌬ and ⌬ HHs ⌬ in the following proposition.

PROPOSITION. For U V and U Vy , suppose that ZŽ. U U 1 sޒ ⌰ 2 sޒ ⌰ 1 s ␣ h andwx U11, U / ZUŽ. 1.Then

q Ž.1 U˜1 sU␺ where ␺ s Ä␣ gޒ ⌽ ywx⌰ N ␣h y ␣ g ⌿14. Ž.2UXXV,UVy.In particular, UXX and U are opposite 1 sޒ ⌰ j ⌬ HH2 sޒ ⌰ j ⌬ 12 horospherical ޒ-subgroups of G and ZŽ UX. is strictly bigger than U . 1 ␣ h

Proof. The last claim directly follows from Lemma 2.5.3. Since ZUŽ.1 s U, we have that ⌬ ⌰c and that ⌬ Ž.⌰ ⌬ is non-empty by the ␣hHH; y j q hypothesis that U111/ wxU , U . We observe that Ä␣ gޒ⌽ ywx⌰ N U␣; U˜14 q qq sÄ␣gޒޒ⌽ywx⌰Nif ␣ q ␤ g ⌽ for ␤ gޒ⌽ ywx⌰ , then ␣ q ␤ s ␣h4, qXq which we will denote by ⌽˜. Also Ä␣ gޒ ⌽ ywx⌰ N U␣ ; U14Äs ␣ gޒ ⌽ q ˜ ywx⌰N␣q␤fޒ⌽ for all ␤ with U␤ ; U14, which we will denote by ⌽Ј. It is enough to show that ⌽˜ s ␺ forŽ. 1 and ⌽Ј s ⌿2 forŽ. 2 .

Ž.1 To show that ␺ ; ⌽˜ , let ␤ s ␣hy ␣ with ␣ g ⌿1.If ␣q␥g qq ޒޒ⌽ for some ␥ g ⌽ ywx⌰ , then the coefficient of ␥ with respect to q each simple ޒ-root in ⌬ y Ž.⌰ j ⌬ H must be 0. Since ␥ gޒ ⌽ ywx⌰ , the sum of the coefficients of ␥ with respect to ⌬ H should be non-zero and hence 1. Therefore the sum of all coefficients of ␤ q ␥ with respect to ⌬ H is 2, yielding ␤ q ␥ s ␣h by Proposition 1.3.1. To show the converse, assume that ␣ g ⌽˜.If ␣f␺, then htŽ.␣ O htŽ.␣ 2. We can find ޒ-simple roots ␣ , ␣ ,...,␣ Žnot necessarily hiy 12iik different. such that ␣ ␣ иии ␣ ⌽qfor each j 1, 2, . . . , k q ii1q q jgޒ s and ␣ ␣ иии ␣ ␣ . Let j be the smallest number such that q iih1q q ks ␣⌬⌰. Then j - k since ␣ ␺. Set ␣ Ј ␣ ␣ иии ␣ . i jgy f s q ii1q q jy1 It follows from the property of ascending series of U that if U␣ ; U˜and ␣ ␣ ⌽q for ␣ ⌬, then U U˜. Therefore by induction we q m gޒ m g ␣q␣ m ; qq have U␣ Ј ; U˜. But ␣ Ј q ␣i gޒ ⌽ , ␣i gޒ ⌽ ywx⌰ while ␣ Ј q ␣ih/ ␣ . ˜˜jj j Therefore ␣ Ј f ⌽, contradicting U␣ Ј ; U.

Ž.2 To show that ⌿22; ⌽Ј, assume ␣ g ⌿ y ⌽Ј. Then ␣ q ␤ gޒ q ⌽for some ␤ g ⌽˜. Therefore ␣ q ␤ s ␣hh. Since ␤ s ␣ y ␣ g ␺ by q Ž.1,␣g⌿1 sޒ ⌽ y⌿2 , contradicting the assumption. To show the converse, suppose that ␣ g ⌽Ј y ⌿2 . Since ⌬ y Ž.⌰ j ⌬ H is non-empty and ␣ g ⌿1, the sum of the coefficients of ␣ with respect to ⌬ H is 1 and there exists a simple root in ⌬ with respect to which the q coefficient of ␣ is 0. It follows from Lemma 2.5.1 that ␣h y ␣ gޒ ⌽ . DISCRETE SUBGROUPS 673

Hence ␣ ␣ ␺ ; U U˜. But ␣ ␣ Ž.␣ ␣ , contradicting the hhy g ␣hy␣ ; s q y assumption that ␣ ; ⌽˜byŽ. 1 .

4.3.3. PROPOSITION. Let G be a connected adjoint ޒ-simple algebraic

ޒ-group with real rank at least 2 and U12, U a pair of opposite horospherical ޒ-subgroups. Let the notation G0 , H, and S be the same as Proposition 2.5.4 and Ziis ZŽ. U for each i s 1, 2. Suppose that S is non-tri¨ial and has no ޒ-anisotropic factors. If the tripleŽ. G012, Z , Z has property Ž.A,then so does Ž.G,U12,U.

Proof. By Proposition 2.5.4, G0 is a connected almost ޒ-simple ޒ-group with real rank at least 2 and S is a semisimple normal algebraic ޒ-group of

H. Now let F12and F be lattices in U1Ž.ޒ and U 2 Ž.ޒ , respectively, such that the subgroup ⌫ generated by them is discrete. Note that Z Ž.ޒ F F12, Fiil is a lattice in ZiŽ.ޒ for i s 1, 2, by Lemma 2.1.10. By the assumption that Ž.G012,Z,Zhas property Ž. A , there exists a ޑ-form of G0with respect to which Z12and Z are defined over ޑ and the subgroup ⌫0generated by Z11lF and Z 22l F is commensurable to the subgroup G0Ž.ޚ . Since SŽ.ޒhas no compact factors, S ⌫ Ž.ޚ ⌫ is a Zariski dense arith- l 0 ; F12, F metic subgroup of SŽ.ޒ . Therefore by Corollary 2.4.5, ⌫ is an F12, F arithmetic subgroup of G.

4.3.4. THEOREM. Let G be an adjoint absolutely simple algebraic ޒ-group with real rank at least 2 and U12, U a pair of opposite horospherical ޒ-subgroups which satisfies either of the following condition:

Ž.1wxU11,UsZ 1,Z 1 is not the root group of a highest real root, and if rank ޒŽ.G00s 2, G is not of type E 6, where G 0 is the subgroup generated by Z12 and Z .

Ž.2wxU11,U/Z 1,Z 1 is the root group of a highest real root, and if XX X rank ޒŽ.G00s 2, G is not of type E 6, where Uii is the centralizer of U˜s Äg g y1y1 X Uiii< gug u g Z for all u g UinU4i,and G0 is the subgroup generated by X X the center of U12and of U .

Then the triple Ž.G, U12, U has Property A.

Proof. Ž.1 Let U12, U satisfy the first condition. By Theorem 4.1.4 and Proposition 2.5.4, the triple ŽŽ.Ž..G01, ZU ,ZU 2has property Ž. A . This proves the theorem by the previous proposition. y Ž.2 Since U12, U is conjugate to the pair ޒV⌰and ޒV⌰for some X X ⌰;ޒ ⌬, by Proposition 4.3.2, U12and U are opposite horospherical ޒ-sub- X groups and ZUŽ1 .Žis not the root group of a highest real root. By part 1. , X X X we have that the triple ŽG01, ZUŽ .Ž,ZU 2.. has propertyŽ. A . Now let F12and F be lattices in U1Ž.ޒ and U 2 Ž.ޒ , respectively, and ⌫ be discrete. Then FXXF U Ž.ޒ is a lattice in UXŽ.ޒ by Proposi- F12, Fiiiis l 674 HEE OH tion 4.3.1 and ⌫ XXis discrete since ⌫ XX ⌫. Therefore the sub- F12, FF12,FF; 12,F group ⌫ XXis an arithmetic subgroup; hence so is ⌫ . F12, FF12,F 4.4. Arithmetic Subgroups of the Form ⌫ F12, F

We discuss, for a given pair U12and U of opposite horospherical subgroups, how specifically we can determine the discrete subgroups of the form ⌫ from the main theorem. F12, F 4.4.1. COROLLARY. Let G be an absolutely simple algebraic ޑ-group with y the Tits index ŽŽ⌬, ⌬ 01, )-action of Gal ރrޑ..and U s V⌰, U2s V⌰ for some ⌰ ⌬. If ⌫ is a subgroup commensurable to GŽ.ޚ for some ; F12, F lattices F121 and F in U Ž.ޒ and U 2 Ž.ޒ , respecti¨ely, then ⌰ contains ⌬ 0and is in¨ariant under the )-action of GalŽ.ރrޑ . Proof. It follows from the assumption that ⌫ is commensurable to F12, F GŽ.ޚthat F11is commensurable to U Ž.ޚ s U11l G Ž.ޚ . Since F is a lattice in U11Ž.ޒ , it follows that U Ž.ޚ is Zariski dense in U 1. By Proposition 1.2.7, U1 is defined over ޑ. Therefore the normalizer NUŽ.1 is defined over ޑ. Since NUŽ.1 sP⌰ , the corollary follows from Proposition 1.4.1.

4.4.2. Let U12, U be as in the Example 2.2.4 and F1and F 2lattices in Mm=kkŽ.ޒand M =m Ž.ޒ , respectively. COROLLARY. Let n 3 and G SL Ž.ޒ . If the subgroup ⌫ is G s nF12,F y1 discrete, there exist elements g g GLmkŽ.ޒ and h g GL Ž.ޒ such that gF1 h y1 and hF2 g are, up to commensurability, one of the following pairs:

Ž.1 the pair consisting of Mr=sŽ.Ž. Dޚ and Ms=r Dޚ where D is an ޒ-algebra defined oër ޑ with Dޒ s MdŽ.ޒ such that Dޑ is a central diïsion algebra oër ޑ, d s DegޑޑD , rd s m, sd s k, Disaޚ ޚ-order of the algebra Dޑ , and Mr=sŽ. Dޚ denotes the set of r = s matrices oër the ring Dޚ;

Ž.2the pair consisting ofÄŽ. Xijg MD r Ž. JNX ij q␴ Ž.Xji s04re- peated twice, where K is a real quadratic extension field of ޑ, J is the ring of integers of K, and D is an ޒ-algebra defined oër K with Dޒ s MdŽ.ޒ such that DK is a central diïsion algebra with an inölution of the second kind ␴ , d s Deg KKD , rd s m s k, and DJ is a J-order of the algebra DK compatible with ␴ . Moreo er by conjugation by the element Ž.g 0 , the subgroup ⌫ is ¨ 0 h F12, F Ž. Ž. commensurable to either the subgroup SLrqs Dޚ or the subgroup SU h0 ޚ t ␴ ÄYŽ.SL D YhY h4 where h Ž.0 Ir , respecti ely. sg 2 rJN 00s 0s Ir0 ¨ Proof. By Theorem 4.1.4, there exists a ޑ-form of G such that ⌫ is F12, F Ž.ޚ ⌬ Ä4␣ ␣ ␣ commensurable to G . Let s 12, ,..., ny1be the set of simple ␣ ŽŽ .. roots such that i diag t12, t ,...,tniist ytq11for each i. Then U s DISCRETE SUBGROUPS 675

V . By the previous corollary, ␣ ⌬ ⌬ . If the ޑ-form G is inner, ⌬y␣ m m g y 0 GŽ.ޑsSLj Ž Dޑ ., up to conjugation, where D is described as above and jd s n. Since ⌬ y ⌬ 0 s Ä4ajd g ⌬ , we have m s rd for some r and hence ksŽ.jyrd. If the ޑ-form G is outer and has a minimal horospherical subgroup defined over ޑ, then n should be even and the standard minimal horospherical ޑ-subgroup is V⌬ ␣ . Therefore m n 2, followed by y n r2 s r k s nr2.

ACKNOWLEDGMENT

I express my deep gratitude to my advisor, G. Margulis, for suggesting this problem and for his constant encouragement and guidance. I very much thank G. Prasad for many valuable comments and for providing the proof of Proposition 1.4.2. I am also very thankful to T. N. Venkataramana for useful discussions and especially for pointing out that Corollary 0.5 follows from the main theorem.

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