{\rm SL}_* over local and adèle rings: *-euclideanity and Bruhat generators
aa r X i v : . [ m a t h . G R ] F e b SL ∗ OVER LOCAL AND AD`ELE RINGS: ∗ -EUCLIDEANITY ANDBRUHAT GENERATORS LUIS GUTI´ERREZ FREZ, LUIS LOMEL´I, AND JOS´E PANTOJA ∗ Abstract.
Let ( R, ∗ ) be a ring with involution and let A = M( n, R ) be thematrix ring endowed with the ∗ -transpose involution. We study SL ∗ (2 , A ) and thequestion of Bruhat generation over commutative and non-commutative local andad`elic rings R . An important tool is the property of a ring being ∗ -Euclidean. Inthis regard, we introduce the notion of a ∗ -local ring R , prove that A is ∗ -Euclideanand explore reduction modulo the Jacobson radical for such rings. Globally, weprovide an affirmative answer to the question wether a commutative ad`elic ring R leads towards the ring A being ∗ -Euclidean; while the non-commutative ad`elicquaternions are such that A is ∗ -Euclidean and SL ∗ is generated by its Bruhatelements if and only if the characteristic is 2. Introduction
We study SL ∗ groups, introduced by Pantoja and Soto-Andrade [5], over local andad`ele rings with involution, where we work over commutative and non-commutativerings with identity. At the base of the algebraic properties of SL ∗ groups are itsBruhat elements and the question of Bruhat generation. In [6], the notion of a ∗ -Euclidean ring is introduced, which provides a powerfult tool that allows us toinfer that SL ∗ (2 , A ) is generated by its Bruhat elements when A is such a ring withinvolution.The non-commutative ∗ -analogue of a special linear group, poses many interestingquestions that are in accordance with the classical theory of Weil representations,Bruhat presentations and the Langlands program. In this article, we study for thefirst time the questions of ∗ -Euclideanity and Bruhat generation for ad`ele rings,including the quaternions in characteristic zero and charactersitic p . We developgeneral machinery along the way, where we are in many places guided by globalquestions posed in the ad`elic setting. For example, we make a careful study of thelocal question of reduction modulo p , in fact, modulo the Jacobson radical. It iscurious that we quickly arrive at the notion of a ∗ -local ring , for which we introducea proper set of hypothesis in order to characterize a ring with a unique ideal that isstable under the involution and is maximal with this property. Date : February 2021.
Key words and phrases. SL ∗ groups; ∗ -Euclidean rings; Bruhat generators.2020 Mathematics Subject Classification . Primary 20G35, 20F05, 16L30.(*) The authors were partially supported by Fondecyt Grant 1171583.
Many examples arise by considering the ring of matrices A = M( n, R ), when R is a ∗ -local ring or an ad`ele ring with involution. The ring A is itself endowed withthe ∗ -transpose involution; the involution of A induced from that of R . We provethat A is ∗ -Euclidean for a list of examples of involutive rings ( R, ∗ ) provided inTheorem 4.3.1:(i) R a ∗ -local ring.(ii) R the ring of ad`eles A F of a global field F , with trivial involution.(iii) R = A E , where E/F is a separable quadratic field extension with the invo-lution dictated by the non-trivial Galois automorphism.(iv) R = A H , where H is a quaternion algebra over a global field F , char( F ) = 2.In all of these cases, SL ∗ (2 , A ) is generated by its Bruhat elements. It is noteworthyto observe in case (iv), of a quaternion algebra over a global field F , that the ringM( n, A H ) is ∗ -Euclidean and is generated by its Bruhat elements if and only ifchar( F ) = 2.We now give a more detailed account of the contents of this article. The SL ∗ functor of Pantoja and Soto-Andrade for not necessarily commutative rings withinvolution and basic properties are reviewed in §
1; the main references being [5, 6].In particular, in § ∗ -Euclidean ring.We arrive at the notion of a ∗ -local ring R in §
2, where we quickly characterizethese rings in the basic Lemma 2.1.1. They are 1- or 2-local rings with involution.We refer to the former simply as a local ring with involution, where the Jacobsonradical is the unique maximal prime ideal J = p . While the latter case has twodistinct maximal prime ideals p and p ∗ , its Jacobson radical being J = p ∩ p ∗ .We then proceed to study symmetric and invertible elements for the matrix ring A = M( n, R ), endowed with the involution induced from the ∗ -local ring R , andtheir behavior under reduction modulo the Jacobson radical J of A . More precisely,we treat both cases of 1- and 2-local rings with involution in a single swoop, studythe projection map π : A π −−→ A = A/ J and produce a section map σ that iscompatible with symmetry and invertibility in Lemma 2.2.1.We prove that the matrix ring A = M( n, R ) over a ∗ -local ring is ∗ -Euclidean inTheorem 3.1.1; a result that Soto-Andrade proved when R is a field [8], and, for adivision ring R , it is part of the “Co-prime Lemma” of Pantoja and Soto-Andrade[5]. For this, we show in Lemma 3.1.2 that if A = A/ J is ∗ -Euclidean, then so is A .And, in Lemma 3.1.3 we establish ∗ -Euclideanity for the new case of R = D × D ,with D and D division rings related by an anti-automorphism ϕ : D → D andthe ϕ -flip transpose involution on A = M( n, R ).The ad`eles over a commutative global field are studied in § A =M( n, A F ) is proved to be ∗ -Euclidean in Theorem 3.2.1. A quadratic global fieldextension E/F , which we address in § A = M( n, A E ) being ∗ -Euclideanfor the involution induced from the non-trivial Galois element of Gal( E/F ), The-orem 3.3.1. Locally, we require Lemmas 3.1.2 and 3.1.3 for the semi-local ring R = O F × O F and a finite residue ring R/ J = k F × k F , respectively, each of themwith the flip involution. L ∗ OVER LOCAL AND AD`ELE RINGS 3
The ∗ -Euclidean property for the quaternions H over a local or a global field F ,is studied in § H . We here encounter fundamental differences depending on the character-istic being 2 or not, see Theorem 3.4.3. Locally, we may have non-split or splitquaternions at a place v of F . The finitely many non-split cases H v are divisionalgebras over a local field F v , while at the remaining infinitely many split placeswe have that H v are matrix quaternions. The case of a finite place v of F , leadsto a non-Archimedean local field F v with ring of integers O v . And we also havequaternioninc rings Q v at the non-Archimedean places, which can also be split ornon-split.Before continuing to inspect the global quaternions, we need a couple of locallemmas that we prove in the slightly more general settings of quaternion matrix ringsand quaternion division algebras that exhibit the main difference of the characteristicbeing different than 2 or not, Lemmas 3.4.1 and 3.4.2, respectively. The main ∗ -Euclideanity result for the ad`elic quaternions is Theorem 3.4.3, whose proof alsorequires two general local lemmas that are of independent interest for local splitmatrix rings of quaternions and for division algebras, namely Lemmas 3.4.4 and3.4.5. In short, the ring of matrices A = M( n, A H ) over the quaternion ad`ele ring is ∗ -Euclidean if and only if char( H ) = 2.In the final section, we begin with Dieudonn´e and his non-conmutative deter-minant, and we extend a basic criterion for invertibility to general linear groupsover ∗ -local rings. The case of a commutative 2-local ring with involution naturallyarises while studying unitary groups at a split place of a quadratic extension ofglobal fields, where there is a known connection to general linear groups. We reviewthe connection between SL ∗ groups and the even unitary groups in § § ∗ -Euclidean property,that is, SL ∗ groups and Bruhat generation for a list of examples over local andad`ele rings, which includes unitary groups and SL ∗ over quaternion rings. We recallthe definition of the Bruhat elements, provide basic properties and state the mainapplication in the form of Theorem 4.3.1. Having done much of the work for localand ad`ele rings, a large part of the theorem follows from the results of §
3, oncewe incorporate the result of Pantoja and Soto-Andrade that SL ∗ (2 , A ) has a set ofBruhat generators when A is ∗ -Euclidean [6]. However, special care must be takenin the case of the quaternions.The quaternions over local and ad`ele rings, their connection to SL ∗ groups andthe question of Bruhat generation are scrutinized in § H over a non-Archimedean local field F leads towards special, Dieudonn´e special and ∗ -analogue special linear groups:SL(2 , F ), SL(2 , H ) and SL ∗ (2 , H ). All three groups are distinct, and the first andthe third are related by SL ∗ (2 , H ) = D H · SL(2 , F ) , L. GUTI´ERREZ, L. LOMEL´I, AND J. PANTOJA where D H is the subgroup of SL ∗ (2 , H ) consisting of diagonal elements. Furthermore,it is curious to observe that each of SL(2 , F ) and SL(2 , H ), by Ihara’s theoerem, hasa further decomposition as an amalgamated product involving the correspondingcongruence subgroup over the ring of integers and Iwahori subgroup; see [7], forexample.We conclude by refining our theorem on the algebra A = M( n, A H ), obtaining inthis case that SL ∗ (2 , A ) has Bruhat generation precisely when char( H ) = 2. Thisis done by incorporating a result of [3] on Bruhat generation for finite fields thatgeneralizes to the setting of a local non-Archimedean split quaternionic algebra ring,together with the results of § Preliminaries on SL ∗ Let R be a ring with 1, endowed with an anti-automorphism α : R → R , r r ∗ ,with α either trivial or of order 2. Notice that α = id is only possible when R iscommutative. In general, R is a not necessarily commutative ring with involution.1.1. SL ∗ groups. Let A R be the category of involutive rings with identity ( A, ∗ ),such that R ⊂ A and the involution of A is compatible with that of the involutivering R . Our main examples are obtained by taking the ring of n × n matrices A = M( n, R ) with entries in R . In this case, the involution is given by a ∗ = ( a ∗ ji ) , for a = ( a ij ) ∈ M( n, R ) . (1.1)Let ( A, ∗ ) ∈ A R . We denote the center of A by Z A and its group of invertibleelements by A × . We let A sym = { a ∈ A | a ∗ = a } , called the set of symmetric elements, and let Z × sym A = Z A ∩ A × ∩ A sym , be the set of central invertible symmetric elements of A .If we let G denote the category of groups, then we have a functor SL : A R → G ( A, ∗ ) SL ∗ (2 , A ) , originally defined by Pantoja and Soto-Andrade. We next recall several basic prop-erties of SL ∗ -groups, proved in [5].1.2. General setting.
Let ( A, ∗ ) ∈ A R . The involution on A induces the involutionon the ring of matrices M(2 , A ) given by (1.1). If we write J = (cid:18) − (cid:19) , then GL ∗ (2 , A ) denotes the set of invertible matrices in g ∈ M(2 , A ) such that g ∗ J g = δ ( g ) J, for some δ ( g ) ∈ Z × sym A . L ∗ OVER LOCAL AND AD`ELE RINGS 5
The set GL ∗ (2 , A ) forms a group under matrix multiplication, in fact, g ∈ GL ∗ (2 , A )implies δ ( g − ) = δ ( g ) − , g ∗ ∈ GL ∗ (2 , A ) and δ ( g ) = δ ( g ∗ ). Furthermore, we havean epimorphism det ∗ : GL ∗ (2 , A ) → Z × sym A , given by det ∗ ( g ) = ad ∗ − bc ∗ , g = (cid:18) a bc d (cid:19) ∈ GL ∗ (2 , A ) . Definition 1.2.1.
The group SL ∗ (2 , A ) is the kernel of the epimorphism det ∗ . Making explicit the conditions on the matrices, we see that SL ∗ (2 , A ) is the groupof matrices g = (cid:18) a bc d (cid:19) , (1.2)with a, b, c, d ∈ A , satisfying the following: ad ∗ − bc ∗ = a ∗ d − c ∗ b = 1 and ab ∗ , cd ∗ , a ∗ c, b ∗ d ∈ A sym . (1.3)We observe that SL ∗ (2 , A ) may often be viewed as the isometry group, (andGL ∗ (2 , A ) as the similitude group with multiplier δ ) of the hermitian form h ( x, y ) = x ∗ J y, x, y ∈ A . Here, x ∈ A is identified with a column vector, and the involution ∗ of A is extendedto a map from column vectors to row vectors, by taking the involution ∗ of A entry-wise and the transpose.Another observation is that one retrieves the groups GL(2 , A ) and SL(2 , A ) when A is commutative with trivial ∗ .1.3. ∗ -Euclidean rings. Let ( A, ∗ ) ∈ A R . We say that A is ∗ -Euclidean if given a, c ∈ A such that Aa + Ac = A, a ∗ c ∈ A sym , and setting a = r − , c = r , then there exist elements s , . . . , s n − ∈ A sym , r , . . . , r n − ∈ A and r n ∈ A × such that r i − = s i r i + r i +1 , for i = 0 , . . . , n −
1. When this is so, we say n is the decomposition length of thepair ( a, c ). The minimum length valid for all possible pairs, if it exists, is the lengthof the ∗ -Euclidean ring A .As examples of ∗ -Euclidean rings A we have: the ring of matrices M( n, F ) withentries in a field [8]; and, M( n, D ) with entries in a division ring [5]. In § n, R ) is ∗ -Euclidean, where R is a 1-local, 2-local or a commutativead`ele ring. L. GUTI´ERREZ, L. LOMEL´I, AND J. PANTOJA Local rings with involution
We say that a ring R with identity is l-maximal if every left maximal ideal is anideal. There is the similar notion of r-maximal , involving right maximal ideals.Assume that ( R, ∗ ) is a ring with involution, then R is l -maximal if and only if itis r -maximal. A stable ideal a of R is one such that a ∗ = a .An l -maximal ring with involution ( R, ∗ ) is Dedekind finite, i.e., a ring where everyelement with a right inverse also has a left inverse; equivalently, a left invertibleelement is invertible. To see this, suppose x ∈ R is right invertible, but not leftinvertible. Then there exists a left maximal ideal m of R containing x . But x beingright invertible implies m R = R , contradicting the assumption on R that m is alsoa right ideal.We observe that a finite direct product of local rings is l -maximal. However,the ring of matrices A = M( n, R ) over a local ring has left maximal ideals but nomaximal ideals for n >
1; its Jacobson radical is a stable ideal, the unique maximalstable ideal. In this case, we know that A is semilocal [4], hence Dedekind finite.2.1. On the notion of a ∗ -local ring. Define a ∗ -local ring to be an l -maximalring with involution ( R, ∗ ) having a unique stable maximal ideal. Lemma 2.1.1. A ∗ -local ring R has maximal spectrum MSpec( R ) = { p , p ∗ } ⊂ Spec( R ) , and unique stable maximal ideal given by the Jacobson radical J = p ∩ p ∗ . Furthermore, D = R/ p is a division ring, and we have x ∈ R × ⇐⇒ ¯ x ∈ ( R/ J ) × ⇐⇒ x / ∈ p , x ∗ / ∈ p . Proof.
Let s be the unique stable maximal ideal of R . Observe that s is containedin a maximal ideal m of R , and we have that m ∩ m ∗ is a stable ideal. By hypothesis s ⊃ m ∩ m ∗ . If x were an element of s \ m ∩ m ∗ , then x / ∈ m or x ∗ / ∈ m . If x ∗ / ∈ m ,for example, we would then have R = Rx ∗ + m ⊂ s + m ⊂ m , a contradiction; and, similarly if x / ∈ m . Hence, we must have s = m ∩ m ∗ .Now, if a ∈ MSpec( R ) were distinct from m , m ∗ , then so would a ∗ ∈ MSpec( R ).But then s ⊃ ( m ∩ m ∗ ) + ( a ∩ a ∗ ) = R, (2.1)where the last equality can be seen by using the Chinese Remainder Theorem.However, equation (2.1) gives a contradiction. Therefore, we must haveMSpec( R ) = { m , m ∗ } and s = J , the Jacobson radical.An application of the Chinese Remainder Theorem, tells us that x + J ∈ ( R/ J ) × ⇐⇒ x + m ∈ ( R/ m ) × , x + m ∗ ∈ ( R/ m ∗ ) × . L ∗ OVER LOCAL AND AD`ELE RINGS 7
Now, the Jacobson radical has the property that 1 + y is invertible for every y ∈ J .From here, we can infer that x ∈ R × ⇐⇒ x + J ∈ ( R/ J ) × ⇐⇒ x / ∈ m , x / ∈ m ∗ . Finally, R is semilocal, hence Dedekind finite [4]. Then a maximal ideal m ∈ MSpec( R ) is prime, i.e. m = p ∈ Spec( R ), and the quotient D = R/ p is a di-vision ring. (cid:3) We thus have two possibilities for a ∗ -local ring R , depending if it has one or twomaximal prime ideals. We refer to the former case as a 1- local ring with involution ,or simply a local ring with involution , since R has a unique maximal ideal p = p ∗ .And call the latter a 2 -local ring with involution , where p = p ∗ .To give an example of a 2-local ring with involution, take a local ring R withmaximal ideal p , then we form the semilocal ring S = R ⊕ R and provide it withthe flip involution ( r, r ′ ) ∗ = ( r ′ , r ). Then S is a ∗ -local ring that is not a local ring.2.2. Reduction mod p for ∗ -local rings. Let R be a ∗ -local ring withMSpec = { p , p ∗ } . We consider the ring S = R ⊕ R with flip ∗ involution( x, y ) ∗ = ( y ∗ , x ∗ ) . Then R is isomorphic to the diagonal subring R ∆ = { ( z, z ) | z ∈ R } ⊂ S. In this setting, the involution on R is compatible with the flip ∗ involution R RR ∆ R ∆ ∗≀ ≀ flip ∗ We fix a maximal prime ideal p of R , and reduce mod J . We write R = R/ J , S = R/ p ⊕ R/ p ∗ , where we have two projection maps x ∈ R ¯ x = x + p ∈ R/ p and y ∈ R ˜ y = y + p ∗ ∈ R/ p ∗ , giving rise to a projection from S to S , π : ( x, y ) (¯ x, ˜ y ). Let R p = { (¯ z, ˜ z ) | z ∈ R } ⊂ S, so that we obtain a non-canonical projection map π . R ∆ R R/ J R p ∼ π proj ≀ L. GUTI´ERREZ, L. LOMEL´I, AND J. PANTOJA
Here, R p is isomorphic to R/ p when R is a local ring, and is the degree 2 separablealgebra R/ p ⊕ R/ p ∗ ≃ R/ J when R is a 2-local ring, equipped with the involution(¯ z, ˜ z ) (¯ z ∗ , ˜ z ∗ ) . In the latter case, notice that the isomorphism R/ J ≃ R p is obtained via the ChineseRemainder Theorem, where every (¯ x, ˜ y ) corresponds to a (¯ z, ˜ z ) ∈ R p , z ∈ R .We wish to construct a non-canonical section map σ for π , in such a way that itis compatible with symmetry and invertibility. From Lemma 2.1.1, we know that z ∈ R × ≃ R × ∆ ⇐⇒ π ( z ) = (¯ z, ˜ z ) ∈ R × p . We build a set consisting of pairs of representatives R × σ = { ( z σ , z σ ) ∈ R × ∆ | (¯ z σ , ˜ z σ ) ∈ R × p } , which satisfy z = z ∗ ⇐⇒ z σ = z ∗ σ . We enlarge this set to obtain a section from R p to R ∆ , in such a way that it respectssymmetry, R σ = R × σ ∪ { ( z σ , z σ ) | z σ ∈ p ∪ p ∗ } . Hence, by construction, we have z ∈ R × ≃ R × ∆ ⇐⇒ z σ ∈ R × . and z ∈ R sym ≃ R sym∆ ⇐⇒ z σ ∈ R sym . We next extend this further to A = M( n, R ), where ( A, ∗ ) ∈ A R for the involutiongiven by (1.1), where in this context J denotes the Jacobson radical of A . We let A p = M( n, R p ) = { (¯ a, ˜ a ) ∈ M( n, R/ p ) ⊕ M( n, R/ p ∗ ) | a ∈ M( n, R ) } , which has involution (¯ a, ˜ a ) (¯ a ∗ , ˜ a ∗ ) . Hence, we also have a non-canonical projection π in this setting. A ≃ M( n, R ∆ ) A p ≃ A/ J . proj π (2.2)We continue by extending the section map on R , entry-wise for the elements of A . Setting A σ = { a σ = ( a ij ) ∈ M( n, R ) | a ij ∈ R σ } , allows for writing A as a sum of two sets A = A σ + J , with A σ ∩ J = { } . Given a ∈ A , we obtain a unique decomposition a = a σ + a , a σ ∈ A σ , a ∈ J . (2.3) L ∗ OVER LOCAL AND AD`ELE RINGS 9
Furthermore, symmetric elements are such that a = a ∗ ⇐⇒ a σ = a ∗ σ and a = a ∗ . (2.4)We summarize the basic properties of the above construction in the following. Lemma 2.2.1.
Let R be a ∗ -local ring, and form the matrix ring A = M( n, R ) . Forthe projection map A π −−→ A = A/ J , a a + J , there exists a non-canonical section map A σ −−→ A, a + J 7→ a σ . The maps preserve the involutions on A and A , and the following properties hold: (i) The projection π ( a ) is symmetric if and only if the section a σ is symmetric. (ii) The units satify a ∈ A × ⇐⇒ π ( a ) ∈ A × ⇐⇒ a σ , a ∗ σ ∈ A × . Proof.
By definition, π preserves the involutions. The existence of σ is due to thedecomposition (2.3), where we have a ∗ = a ∗ σ + a ∗ . Hence, the section preserves the involutions, namely,( a σ ) ∗ = ( a ∗ ) σ . Property (i) follows from (2.4).For invertibility, first suppose a σ , a ∗ σ ∈ A × . The elements of the form1 + x, with x ∈ J , are known to be invertible by Bass’ Lemma 6.4 of [2]. Hence a − σ a = 1 + a − σ a ∈ A × , and we conclude that a ∈ A × . Clearly a ∈ A × = ⇒ π ( a ) ∈ A × . Now, suppose that π ( a ) ∈ A × ; and, for brevity write ¯ a = π ( a ), ¯ b = π ( b ). Then ¯ a · ¯ b = ¯ b ¯ a = ¯1 for some¯ b ∈ A × . By writing a = a σ + a , b = b σ + b , we obtain a σ b σ = b σ a σ = 1. Thus a σ ∈ A × , and a ∗ σ ∈ A × . (cid:3) ∗ -Euclideanity over local and ad`ele rings We study involutive matrix rings over ad`elic rings and their ∗ -Euclidean property.This global situation, naturally poses questions for rings with involution in the localsetting. Hence, we begin by establishing for ∗ -local rings, a result that Soto-Andradeproved when R is a field in Chapitre III, § R , it is part of the “Co-prime Lemma” of Pantoja and Soto-Andrade [5]. Wethen extend to ad`elic rings over a global field F , we consider separable quadraticextensions of global fields E/F , in addition to quaternion algebras over F . Local setting.
Throughout this subsection, we let R be a ∗ -local ring, andform the ring A = M( n, R ) , ( A, ∗ ) ∈ A R , with the involution induced from that of R . We let J denote the Jacobson radicalof A , or sometimes the Jacobson radical of R , and it should be clear from contextwhich one is being used. Theorem 3.1.1.
The ring of n × n matrices A over a ∗ -local ring is ∗ -Euclidean. We need a couple of results in order to prove this theorem, where the key pointis reduction mod J . Lemma 3.1.2.
If the ring A = A/ J is ∗ -Euclidean, then so is A .Proof. We have elements a, c ∈ A , which we can reduce mod J , namely, we look at¯ a, ¯ c ∈ A = M( n, R ). We have the hypothesis a ∗ c ∈ A sym = ⇒ ¯ a ∗ ¯ c ∈ A sym Aa + Ac = A = ⇒ A ¯ a + A ¯ c = A. Identify A with M( n, R ∆ ) and A = A/ J with A p , as in (2.2), where we have aprojection map π . Setting ¯ a = ¯ r − and ¯ c = ¯ r , then, by assumption, there existelements ¯ s , . . . , ¯ s n − ∈ A sym , ¯ r , . . . , ¯ r n − ∈ A and ¯ r n ∈ A × such that¯ r i − = ¯ s i ¯ r i + ¯ r i +1 , for i = 0 , . . . , n − a σ , c σ , s i,σ , r i,σ ∈ A σ ⊂ A, where s i,σ ∈ A sym , r i,σ ∈ A sym and r n,σ ∈ A × . We then have r i − ,σ = s i,σ r i,σ + r i +1 ,σ in A = A/ J . Hence, there exists in every step an x i ∈ J such that r i − ,σ + r i − , = s i,σ r i,σ + ( r i +1 ,σ + x i ) . The last term parenthesis, when i + 1 = n , is the sum of a unit of A plus an elementof the radical of A , so it is a unit by Property (ii) of Lemma 2.2.1. Hence, the resultfollows. (cid:3) After reducing mod J , there are two possibilities for R : it is either a 1-local or a2-local ring with involution. The former gives A = M( n, D ), where D is a divisionring, a case proved by Pantoja and Soto-Andrade in Proposition 3.3 of [5]. Thelatter leads to a sum of two division rings after reduction mod J , and we now provethe ∗ -Euclidean property in this case. Lemma 3.1.3.
Let D and D be division rings, together with an anti-automorphism ϕ : D → D . Let A i = M( n, D i ) , i = 1 , , and extend ϕ : A → A , component-wise. Then, consider the ring A = A ⊕ A , with ϕ -flip transpose involution ( a , a ) ∗ = ( ϕ − ( a ) t , ϕ ( a ) t ) . L ∗ OVER LOCAL AND AD`ELE RINGS 11
Let a, c ∈ A be such that Aa + Ac = A, a ∗ c = c ∗ a. Then, there exist s ∈ A sym = { ( x, ϕ ( x ) t ) | x ∈ A } , r ∈ A × , satisfying a = sc + r. Proof.
Let us observe that any s ∈ A sym and r ∈ A × satisfying a = sc + r , mustalso be such that Ac + Ar = A. (3.1)And, the symmetry relation a ∗ c = c ∗ a leads to c ∗ r = r ∗ c. (3.2)We write a = ( a , a ) , c = ( c , c ) , s = ( s , s ) , r = ( r , r ) . Now, we have by hypothesis A a + A c = A . Since A consists of n × n matrices with entries in a division ring D , this equationtells us that a and c must satisfyrank( a ) + rank( c ) ≥ n. Because of this, we can multiply a and c by products of elementary matrices, e and f , in such a way that ea + f c = u, where u is a unipotent matrix, in particular, u ∈ A × . We can now go back andchoose s , where we note that we only need to define the first component, since thesymmetry requirement, s ∈ A sym , fixes the second component s = − e − f = ⇒ r = a − sc. With such a choice, r = u ∈ A × . Next, we need the second component of r to bea unit, and for this we look at equation (3.2), which gives ϕ ( r ) t c = ϕ ( c ) t r . And, incorporating (3.1) leads to A c + A r = A = ⇒ A ϕ ( r ) t c + A r = A = ⇒ A ϕ ( c ) t r + A r = A = ⇒ A r = A = ⇒ r ∈ A × . (cid:3) With the two lemmas at hand, the proof of Theorem 3.1.1 is complete. We observethat, in both cases, the ∗ -Euclidean length is 1. The ad`eles.
Let F be a global field, i.e., either a number field or a functionfield, and let O F be its ring of integers and A F its ring of ad`eles [10]. Given a placeor valuation v of F , we let F v denote its completion. If v is non-Archimedean, welet O v be the corresponding ring of integers.We wish to study SL ∗ groups over the ad`eles, in fact, over A = M( n, A F ). For this,we fix some notation. At every place v of F , we write A v for M( n, F v ). There arefinitely many infinite places of F , where we write v | ∞ , and have two possibilities: A v = M( n, R ) or A v = M( n, C ). The condition v | ∞ being empty in the case offunction fields. On the other hand, finite places of F are in correspondence withnonzero prime ideals of the ring of integers O F : p ←→ v. At every finite place v of F , we write O v for the maximal compact open subgroupM( n, O v ). We recall that the matrix ring of ad`eles is a restricted direct product A = M( n, A F ) = Y ′ ( A v : O v ) . Let S be a finite set of places containing all v | ∞ , and let A S = Y v ∈ S A v × Y v / ∈ S O v ⊂ A. Given an element a ∈ A , there is an S as above such that a ∈ A S , and we write a = ( a v ) = a S · a S , where a S has coordinates a v ∈ A v at every place v ∈ S and is 1 for v / ∈ S ; and, a S has 1 for coordinate at every v ∈ S and a v ∈ O v at every v / ∈ S . Theorem 3.2.1.
The ring of matrices A = M( n, A F ) is ∗ -Euclidean.Proof. With notation as above, let S be a finite set of places of k such that a, c ∈ A S . For a S , c S ∈ Q v ∈ S A v ֒ → A , we can go place by place where the local result, includedin Theorem 3.1.1, is known for each of the fields F v by Soto-Andrade [8], and thereare only finitely many places v ∈ S . We thus have the ∗ -Euclidean property with s S ∈ Q v ∈ S A sym v and r S ∈ Q v ∈ S A × v .Now, at places v / ∈ S , Theorem 3.1.1 gives the ∗ -Euclidean property, where thedecomposition length is 1. Thus we can solve for the equation a v = s v c v + r v , with s v ∈ O sym v and r v ∈ O × v . In this way, we obtain s S ∈ Q v / ∈ S O v ֒ → A and r S ∈ Q v / ∈ S O × v ֒ → A .Finally, by setting s = s S · s S ∈ A and r = r S · r S ∈ A × , we obtain the desired ∗ -Euclidean property for the ring of ad`eles with decompositionlength 1. (cid:3) L ∗ OVER LOCAL AND AD`ELE RINGS 13
Quadratic extensions.
We now let
E/F be a separable quadratic field ex-tension of the global field F , where we take the involution given by the non trivialGalois element α ∈ Gal(
E/F ). For every finite absolute value v of F , there are twopossibilities, either v remains inert with respect to E or v is split. pP P P p vw w w v (3.3)In one case, P = p O E is a prime ideal of O E and we have corresponding prime ideals p v of O v = O F v and P w of O w = O E w , where we say there is one place w above v ,written w | v . In the other case P P = p O E , and we obtain two places w , w | v .Now, every infinite place v of F , written v | ∞ , leads towards two possibilities: oneplace w | v , when we must have E w /F v = C / R ; or two places w , w | v , when E ⊗ F F v ≃ R × R or E ⊗ F F v ≃ C × C .For every valuation v of F , finite or infinite, we let E v = E ⊗ F F v ≃ Y w | v E w . We thus have that E v /F v is a separable quadratic F v -algebra with involution. When E v /F v is a field extension, the involution for E v is given by the non-trivial Galoisautomorphism. When E v ≃ E w × E w , we have E w ≃ E w ≃ F v , hence we fix E v to be the F v -algebra E v = F v × F v with the flip involution. By the ˇCebotarevdensity theorem, each case of one or two places of E above one for F happens withdensity 1 /
2. In the former case v is inert, while in the latter v is split.We thus endow A E with the involution ∗ obtained from the involution of the F v -algebra E v at every place v of F . The involution extends to the ad`elic ring ofmatrices A = M( n, A E ), as in (1.1), giving ( A, ∗ ) ∈ A A F .Using w to denote places of E and v for places of F , and writing O w = M( n, O w )for finite w , the ring of matrices over the ad`eles of E is the restricted direct productas before A = M( n, A E ) = Y w ′ ( A w : O w ) . However, in order to incorporate the involution, we group the places of E accordingto the places of F , by setting for finite places R v = Y w | v O w , K v = M( n, R v ) ⊂ G v = M( n, E v ) . Then, we can rearrange the restricted direct product to obtain A = Y v ′ ( G v : K v ) . Theorem 3.3.1.
Let
E/F be a quadratic extension of global fields. The ring ofmatrices A = M( n, A E ) is ∗ -Euclidean of decomposition length 1. Proof.
Let a , c ∈ A be such that Aa + Ac = A and a ∗ c = c ∗ a . There exists a finiteset of places S of F , which includes all v | ∞ , such that a, c ∈ G S = Y v ∈ S G v × Y v / ∈ S K v ⊂ A. We write a = a S · a S , c = c S · c S , similar to what we did before, however, we are now grouping the places w of E that lie above each place v of F . With these observations, we then follow the sameargument used to prove Theorem 3.2.1 in order to prove the result. (cid:3) Quaternions.
We work over a local or global field, where the basic theory isexpounded in [10] for central simple algebras in a manner that is independent ofthe characteristic, and is detailed in [9] for quaternion algebras. We do, however,encounter differences with regards to the ∗ -Euclidean property depending on thecharacteristic being 2 or not, see Theorem 3.4.3.Given a local or a global field F , we let ( H , ∗ ) ∈ A F be a quaternion algebraover F . Up to isomorphism, there are two options for H : either it is a division ringover F or it is the ring of matrices M(2 , F ). Both options are possible except when F = C , where the only quaternion algebra is H = M(2 , C ). The case of a matrixalgebra over F in general is called the split quaternion algebra, where we take theinvolution to be h ∗ = J h t J − , h ∈ H = M(2 , F ) . (3.4)The division ring case is called the non-split quaternion algebra over F .If F is a non-Archimedean local field, we denote its ring of integers by O , andwe denote by ( Q , ∗ ) ∈ A O the quaternionic ring of H . The ring Q is M(2 , O ) withthe involution given by (3.4) when H is split, and it is a non-commutative local ringwith involution when H is a division ring. In fact, in the non-split case the ring Q is locally profinite much like the p -adic integers O .If F is a global field, at every place v of F we let H v = H ⊗ F F v , where indeed, each H v is an F v -quaternion algebra. If v is a finite place of the globalfield F , we then write F v for the resulting non-Archimedean local field with ring ofintegers O v ; furthermore, we denote by Q v the quaternionic ring of H v . The ring H v is split at almost every place, Chapter XI of [10].Writing A F for the ring of ad`eles of a global field F , we have the ad`elic quaternions A H = H ⊗ F A F . They can equivalently be seen as a restricted direct product A H = Y v ′ ( H v : Q v ) , We have that ( A H , ∗ ) ∈ A A F with the involution obtained from that of H v at everyplace, i.e., a ∗ = ( a ∗ v ) , a = ( a v ) ∈ A H . L ∗ OVER LOCAL AND AD`ELE RINGS 15
Before continuing to inspect ∗ -Euclideanity for these rings, we record two lemmasthat arise in the local setting and already mark a difference between working incharacteristic 2 or not. For this, we extend the involution given by (3.4) to M(2 , R )over any ring with identity R . However, a ∗ a does not define a quaternion norm forgeneral R as in the commutative case. Lemma 3.4.1.
Let R be such that is a regular element and let Q = M(2 , R ) , sothat ( Q, ∗ ) ∈ A R with the involution obtained from (3.4) . Then Q is not ∗ -Euclidean.Proof. Because of the hypothesis on R , we have Q sym = { αI | α ∈ R } . The elements a = (cid:18) (cid:19) , c = (cid:18) (cid:19) ∈ Q, are such that Qa + Qc = Q and a ∗ c = c ∗ a = 0 . However, a and c cannot satisfy the ∗ -Euclidean property. (cid:3) When the underlying ring is a division algebra D , we recall that the Dieudonn´edeterminant on M(2 , D ) is given bydet (cid:18) α βγ δ (cid:19) = (cid:26) αδ if γ = 0 αγδγ − − γβ if γ = 0 . Lemma 3.4.2.
Let D be a division ring of characteristic and let H = M(2 , D ) ,so that ( H, ∗ ) ∈ A D with the involution obtained from (3.4) . If a , c ∈ H are suchthat Ha + Hc = H, then there exists an s ∈ H sym such that r = a + sc ∈ H × . Proof. If a is invertible or zero, the Lemma is immediate. Hence, we assume thatrank( a ) = 1. For any unit u ∈ H × and s ∈ H sym , we observe that r = a + sc ∈ H × ⇐⇒ ru = au + scu ∈ Q × . Thus, after taking u to be a suitable product of elementary matrices, we can assume a is of the form a = (cid:18) x y (cid:19) . (3.5)Also, we must have rank( c ) = 1 or 2.Assume c is invertible. When D = F is a field, this case is easy because c ∗ c ∈ F isthe quaternion norm. For D in general, when c is invertible, one can take s ∈ H sym to be of the form (cid:18) α β α (cid:19) , (cid:18) α γ α (cid:19) or (cid:18) αα (cid:19) , (3.6) to get sc in triangular or anti-triangular form with non-zero diagonal or anti-diagonalentries, respectively. Depending on the form of a , one can choose α , β and γ so that r = a + sc ∈ H × . Note that a careful consideration of the three cases for a of the form (3.5) is neededwhen D = F .Now, suppose rank( c ) = 1, then one can choose an appropriate s ∈ A × of one ofthe forms in (3.6), so that sc = (cid:18) x y (cid:19) or (cid:18) x y (cid:19) . We can take one or the other, depending on the form of a , to obtain a = sc + r, s ∈ H sym , r ∈ H × . (cid:3) We now study the ring A of n × n matrices with entries in A H . The proof of thenext theorem gives another example of how a global question requires us to inspectwhat is happening locally in detail, where we prove a pair of lemmas along the way.Note that the involution on A , globally or locally, is obtained by combining theinvolution on the quaternion ring with the involution given by (1.1). Theorem 3.4.3.
Let H be a quaternion algebra over a global field F , and let A =M( n, A H ) so that ( A, ∗ ) ∈ A A H with the involution induced from that of A H . Then (i) A is ∗ -Euclidean of decomposition length when char( F ) = 2 . (ii) A is not ∗ -Euclidean when char( F ) = 2 . Let a = ( a v ), c = ( c v ) ∈ A be such that Aa + Ac = A and a ∗ c = c ∗ a. At finite non-split places, Q v is a local ring with involution, where we know theresult locally holds. Now, the proof of Theorem 3.4.3 follows the general outlinethat is present in the proofs of Theorems 3.2.1 and 3.3.1. However, care needs tobe taken when inspecting M( n, Q v ) at finite split places.Before we continue, we have two lemmas that we write in a slightly more generalsetting. Let R be a ring with identity where 2 = 0. In this setting, we have that J = (cid:18) (cid:19) = J − , and we take the block diagonal matrix B = J · · · ... ... · · · J = B − . Consider the matrix ring M(2 n, R ), with involution given by a ∗ = Ba t B. (3.7)We first address the case when R = F is a field. L ∗ OVER LOCAL AND AD`ELE RINGS 17
Lemma 3.4.4.
Let F be a field of characteristic and let H = M(2 , F ) be the splitquaternions. Form the matrix ring A = M( n, H ) with the involution induced fromthat of H . Then A is ∗ -Euclidean of length .Proof. Consider A as the ring of 2 n × n matrices with entries in F , whose involutionis given by (3.7). We then have a non-degenerate bilinear pairing h· , ·i : F n × F n → F, h v, w i = v t Bw, where v , w are seen as column vectors of F n . The involution a a ∗ , is precisely theadjoint for the pairing h· , ·i , to which we can apply Transversality and the coprimeLemma of Pantoja and Soto-Andrade, in particular, Proposition 3.3 of [5]. In thisway, given a , c ∈ A , such that Aa + Ac = A, a ∗ c = c ∗ a, we obtain an s ∈ A sym such that r = a + sc ∈ A × . (cid:3) We now require to extend the above to the case of a local ring.
Lemma 3.4.5.
Let R be a local ring of characteristic with maximal prime ideal p .Let Q = M(2 , R ) , so that ( Q, ∗ ) ∈ A R with the involution obtained from (3.4) . Formthe matrix ring A = M( n, Q ) with the involution induced from that of Q . Then A is ∗ -Euclidean of length .Proof. We can identify A with M(2 n, R ), where the involution is given by (3.7).We next follow our general construction of § J , A π −−→ A = A/ J , which produces a section map A σ −−→ A . Morespecifically, equations (2.3) and (2.4) lead towards Lemma 2.2.1 where π and σ preserve the involutions on A and A , and the units satisfy a ∈ A × ⇐⇒ π ( a ) ∈ A × ⇐⇒ a σ ∈ A × . Lemma 3.1.2, extended to this setting, shows that if A is ∗ -Euclidean the so is A .After reducing mod p , we apply Lemma 3.4.4. (cid:3) We now continue the proof of Theorem 3.4.3. At every split place of F , we have anon-Archimedean local field F v and split quaternions H v = M(2 , F v ). We also havea ring of integers O v and ring of quaternions Q v = M(2 , O v ).When char( F ) = 2, thanks to Lemma 3.4.5, we now have ∗ -Euclideanity for thering M( n, Q v ). However, when char( F ) = 2, we use Lemma 3.4.1, which gives animmediate negative answer to the case n = 1 at every split non-Archimedean place;a result that directly extends to the case of n >
1. This concludes the proof ofTheorem 3.4.3. Examples and Bruhat generators
We begin with the Dieudonn´e determinant and ∗ -local rings, observing a con-nection between SL ∗ groups and the general linear group. We also provide twonon-trivial examples for the theory, namely, unitary groups arising as SL ∗ groupsand SL ∗ over local and ad`elic quaternions.The ∗ -Euclidean property is a strong one for a ring A , and leads to the importantresult of Pantoja and Soto-Andrade that SL ∗ (2 , A ) has a set of Bruhat generators[6]. We thus establish Bruhat generation over local and ad`elic rings, to concludewith Theorem 4.3.1.4.1. GL( n ) and ∗ -local rings. Given a division ring D , the Dieudonn´e determinant[1], gives a criterion for the invertibility of n × n matrices with entries in D : x ∈ GL( n, D ) ⇐⇒ x ∈ M( n, D ) and det( x ) = 0 . Now, let R be a ∗ -local ring, with MSpec( R ) = { p , p ∗ } . From § R to R/ p and, depending on p = p ∗ or p = p ∗ , also a projectionfrom R to R/ p ∗ . This leads to the following map of matrix ringsM( n, R ) −→ M( n, R/ p ) ⊕ M( n, R/ p ∗ ) , where we write a (¯ a, ˜ a ) . Over the ∗ -local ring R , the general linear group consists of invertible n × n matrices GL( n, R ), with its principal congruence subgroup K = { a ∈ M ( n, R ) | π ( a ) = I n + J } , where π : R → R/ J is the canonical projection map. We now observe that invert-ibility is compatible with the Dieudonn´e determinant and reduction mod J . Lemma 4.1.1.
Let R be a ∗ -local ring. The following are equivalent for a ∈ M( n, R ) : (i) a ∈ GL( n, R ) . (ii) There exists b ∈ M( n, R ) such that ba ∈ K . (iii) det(¯ a ) = 0 and det(˜ a ) = 0 .Proof. This follows from Lemma 2.2.1; notice that M( n, R ) is Dedekind finite. (cid:3)
A basic and important example of a 2-local ring with involution is S = R × R ,obtained from local rings R i that are linked by an anti-automorphism ϕ : R → R ,equipped with the ϕ -flip involution r ∗ = ( ϕ − ( r ) , ϕ ( r )) for r = ( r , r ) ∈ S. Form the matrix ring A = M( n, S ) = A ⊕ A , A i = M( n, R i ) , with the ϕ -flip transpose involution a ∗ = ( ϕ − ( a ) t , ϕ ( a ) t ) for a = ( a , a ) ∈ A, L ∗ OVER LOCAL AND AD`ELE RINGS 19 so that ( A, ∗ ) ∈ A R . We recall that Proposition 5.1 of [5], gives an isomorphismSL ∗ (2 , A ) = { a = ( a , J ( ϕ ( a ) − ) t J − ) ∈ A | a ∈ GL(2 , A ) } ≃ GL(2 , A ) . (4.1)4.2. Unitary groups.
Classically, one works over a base field F , where we havethe unitary group U n ( F ) associated to a separable quadratic algebra E over F withinvolution α : E → E , x ¯ x , given by the non-trivial Galois element α ∈ Gal(
E/F )if
E/F is a field extension, and is obtained from the flip involution ( x, y ) ( y, x )if E ≃ F × F . Let Φ n = (cid:18) J n − J n (cid:19) , where J n is the n × n matrix J n = ( δ i,n − j +1 ), where δ i,j denotes Kronecker’s deltafunction, and define the hermitian form h n on the 2 n -dimensional vector space V ofcolumn vectors with entries in E , defined by h n ( x, y ) = ¯ x t Φ n y, x, y ∈ V. Then U n ( F ) is the group of isometries of h n .There is a more general setting for defining unitary groups by considering ( A, ∗ ) ∈A R and let ε = ±
1. Then, a ε -hermitian form over a free left A -module V of finiterank is a bi-additive map h : V × V → A , linear in the second variable and satisfies h ( v, u ) ∗ = εh ( u, v ) , u, v ∈ V. The unitary group U( h ) associated to h consists then of all g ∈ GL( V ) such that h ( gu, gv ) = h ( u, v ) , u, v ∈ V. One of our examples arises by taking A = M( n, A E ), where we have the connectionbetween unitary groups and SL ∗ groups,U n ( A F ) = SL ∗ (2 , A E ) . We thus may observe that, at every split place v of F , we further have a connectionbetween SL ∗ groups and general linear groups. Indeed, in this case we can fix E v = F v × F v and take the map ϕ of (4.1) to be the identity. More precisely,SL ∗ (2 , E v ) is isomorphic to GL( n, F v ) via the projection map( g , J ( g − ) t J − ) g . Bruhat Generators.
We have the Bruhat elements of SL ∗ (2 , A ), which arethe natural extension of those for SL ( F ) when F is a field. Namely, the matrices h a = (cid:18) a ∗ a − (cid:19) , ( a ∈ A × ) , u b = (cid:18) b (cid:19) , ( b ∈ A sym ) , and ω = (cid:18) − (cid:19) are the Bruhat elements for SL ∗ (2 , A ).We observe a formal Bruhat relation, valid when one of the entries is a unit. Inparticular, if g = (cid:18) a bc d (cid:19) ∈ SL ∗ (2 , A ) , and a ∈ A × , then g = w − h − a u − a ∗ c w u a − b . (4.2)Note that if b , c or d is in A × , then multipliying g on the left or right by w , leads toa matrix in the previous situation. Notice that each one of the elements appearingin (4.2) is indeed inside SL ∗ (2 , A ), follows from the defining relations given by (1.3).For example, ab ∗ = ba ∗ = ⇒ a − b = b ∗ ( a ∗ ) − ∈ A sym . Hence u a − b ∈ SL ∗ (2 , A ).However, an element of SL ∗ (2 , A ) does not in general satisfy the property thatone of its entries is a unit. When A = M( n, R ), with R a ∗ -local ring, this onlyhappens when n = 1, because one can reduce the ∗ -determinant relation mod J :¯ a ¯ d ∗ − ¯ b ¯ c ∗ = ¯1 , forcing ¯ a ¯ d ∗ or ¯ b ¯ c ∗ to be a unit. Hence ad ∗ or bc ∗ is a unit, and two of the entriesare thus units in this case. For n >
1, we can arrange for a product u b w u b w − u b to have all of its entries non-invertible, after suitable choices for b , b , b ∈ A sym .Now, a very interesting problem is to determine when the matrices h t , u s , w generate the group SL ∗ (2 , A ). In this sense, there exists an important relationbetween ∗ -Euclidianity and Bruhat elements. More precisely, Pantoja and Soto-Andrade proved that if A is ∗ -Euclidean, then h a , u b and w , with a ∈ A × and b ∈ A sym , generate SL ∗ (2 , A ) [6], § ∗ groups over the quaternions in the following subsection. Theorem 4.3.1.
Consider involutive rings ( R, ∗ ) in the following cases: (i) R is a ∗ -local ring; or (ii) R is the ring of ad`eles A F of a global field F , with trivial involution; or (iii) R = A E , where E/F is a separable quadratic field extension with the invo-lution dictated by the non-trivial Galois automorphism; or (iv) R = A H , where H is a quaternion algebra over a global field F , char( F ) = 2 .Let A = M( n, R ) , so that ( A, ∗ ) ∈ A R with the involution induced from that of R .Then SL ∗ (2 , A ) is generated by the Bruhat elements h a , u b and w . When the underlying ring R is local, it was proved in [3], where many interestingproperties are explored like the Weil representations of these groups. Part (i) ofthe previous theorem is an extension to the case of ∗ -local rings and provides andalternate proof when R is local.The unitary groups, as in § §
3. And finally, the quaternions provide aninteresting non-conmutative example of Theorem 4.3.1 for case (i) in general, andcase (iv) when char( F ) = 2. We now inspect SL ∗ groups over the quaternions moreclosely, explore some interesting properties and complete the proof of the theorem. L ∗ OVER LOCAL AND AD`ELE RINGS 21
Quaternions.
Let H be a quaternion division algebra over a local or a globalfield F . We have three related, yet distinct groups in this setting, namelySL(2 , F ) , SL(2 , H ) and SL ∗ (2 , H ) . The first two are the special linear groups of F and H , respectively, where SL(2 , H )is given by the kernel of the Dieudonn´e determinant. The third is the SL ∗ groupof Pantoja and Soto-Andrade over the division algebra H endowed with the quater-nionic involution. Case of a p -adic field . Let F be a non-Archimedean local field F , with ring ofintegers O and maximal ideal p . We have a finite residue field k F = O / p .In this setting, H is the unique (up to isomorphism) 4 dimensional central divisionalgebra over F . As in § Q denote the ring of integers of H , and q itsmaximal ideal. We obtain a finite quotient field k H = Q / q .Consider the ring of matrices A = M( n, Q ) with the involution induced from( Q , ∗ ) ∈ A O . Then A is ∗ -Euclidean by Theorem 3.1.1. Hence, by [6] we knowthat SL ∗ (2 , A ) is generated by the Bruhat elements of § Q . In particular, SL ∗ (2 , H ) is generated by theBruhat elements of § , F ) and SL(2 , H ) are generated by its Bruhat ele-ments, see for example [1]. We further observe that we have group homomorphismsobtained from reduction modulo the corresponding maximal ideal, namely ϕ F : SL(2 , O ) → SL(2 , k F )and ϕ H : SL(2 , Q ) → SL(2 , k H ) . The Borel subgroups, B ( k F ) of SL(2 , k F ) and B ( k H ) of SL(2 , k H ), consist of therespective upper triangular matrices. These lead towards the Iwahori subgroups,defined as the pre-images of the above maps: I F = ϕ − F ( B ( k F )) and I H = ϕ − H ( B ( k H )) . Now, consider the subgroups of diagonal elements, D F = (cid:8) h a | a ∈ F × (cid:9) and D H = (cid:8) h a | a ∈ H × (cid:9) . It then follows from the definitions that we have the following relationSL ∗ (2 , H ) = D H · SL(2 , F ) . (4.3)It is interesting to note a result of Ihara, which gives an expression of the speciallinear group as an amalgamated product, valid over F ,SL(2 , F ) = SL(2 , O ) ∗ I F SL(2 , O ) , (4.4)and over H , SL(2 , H ) = SL(2 , Q ) ∗ I H SL(2 , Q ) . (4.5) Where a proof of equations (4.4) and (4.5) can be found in [7]. Also note thatequation (4.3) can also be written as an amalgamaded product,SL ∗ (2 , H ) = D H ∗ D F SL(2 , F ) . (4.6) Case of a global field.
Let F be either a number field or a global function fieldof characteristic p . At each place v of F , we obtain a quaternion algebra H v overthe local field F v ; H v may be split or non-split. At a non-Archimedean place v of F , we have the ring of integers O v of F v , in addition to the quaternionic ring Q v of H v . At these places, we let Q v = M( n, Q v ) ∈ A Q v .Let A = M( n, A H ), and at every place v of F let A v = M( n, H v ). The matrixring A (resp. A v at each v ) is equipped with the involution induced from that of A H (resp. H v ). A straightforward inspection tells us that we have Bruhat generationfor the group SL ∗ (2 , A ) if and only if we have Bruhat generation for SL ∗ (2 , A v ) atevery Archimedean place and for SL ∗ (2 , Q v ) at every non-Archimedean place.The quaternionic ring H over the global field F forms a central division algebraover F . There is at least one place v of F where H v is a division algebra, hencenon-split; and, H v can be non-split at only finitely many v [10].Fix a split place v of F , where we have that H v and Q v are matrix algebras over F v and O v , respectively. The involutions on H v and Q v come from (3.4). These,in turn, induce the involutions on A v and Q v , respectively. When char( F ) = 2, inLemma 3.4.5 we showed that Q v is ∗ -Euclidean of length 1. And, Lemma 3.4.4 saysthat A v is also ∗ -Euclidean of length 1. By [6], § ∗ (2 , Q v ) is inthis case generated by its Bruhat elements at every non-Archimedean place and sois SL ∗ (2 , A v ).However, continuing with the case of a split place v , if char( F ) = 2, then itfollows directly from Lemma 3.4.1 that Q v is not ∗ -Euclidean. Furthermore, from[3] Example 13.3 for a finite field, we deduce that SL ∗ (2 , Q v ) is not generated by itsBruhat elements.At a non-split place, we have either a division algebra H v or a local ring withinvolution Q v . Hence, we know that SL ∗ (2 , A v ) and SL ∗ (2 , Q v ) are generated by itsBruhat elements at every non-split place.Finally, we observe that only in the case of a global function field F with char( F ) =2, do we have that A v is generated by its Bruhat elements at every place and sois Q v at all non-Archimedean places. This concludes the proof of Theorem 4.3.1 inthe remaining case. In particular, we have the following: Let H be a quaternion algebra over a global field F and let A H beits ring of ad`eles. Let A = M( n, A H ) be endowed with the involutioninduced from that of A H . Then SL ∗ (2 , A ) is generated by its Bruhatelements if and only if char( F ) = 2 . L ∗ OVER LOCAL AND AD`ELE RINGS 23
References [1] E. Artin,
Geometric Algebra , Interscience Tracts in Pure and Applied Mathematics , NewYork, 1957.[2] H. Bass, K -theory and stable algebra , Pub. Math. I.H.´E.S. (1964), 5-60.[3] J. Cruickshank, L. Guti´errez Frez and F. Szechtman, Weil representations via abstract dataand Heisenberg groups: A comparison , J. Algebra (2020), 129-161.[4] T. Y. Lam,
A first course in noncommutative rings , 2 nd ed., Graduate Texts in Mathematics , Springer-Verlag, New York, 2001.[5] J. Pantoja and J. Soto-Andrade, A Bruhat decomposition of the group SL ∗ (2 , A ), J. Algebra (2003), 401-412.[6] J. Pantoja and J. Soto-Andrade, Bruhat presentations for ∗ -classical groups , Comm. Algebra (2009), 4170-4191.[7] J.-P. Serre, Trees , Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1980.[8] J. Soto-Andrade,
Repr´esentations de certains groupes symplectiques finis , M´emoires S.M.F. (1978), 334 p.[9] M.-F. Vign´eras,
Arithm´etique des alg`ebres de quaternions , Lecture Notes in Mathematics ,Springer-Verlag, Berlin, 1980.[10] A. Weil
Basic Number Theory , Classics in Math., Springer-Verlag, Berlin, 1995.
Luis Guti´errez Frez, Instituto de Ciencias F´ısicas y Matem´aticas, Universidad Australde Chile, Campus Isla Teja SN, Valdivia, Chile.
E-mail address: [email protected]
Luis Lomel´ı, Instituto de Matem´aticas, Pontificia Universidad Cat´olica de Valpara´ıso,Blanco Viel 596, Cerro Bar´on, Valpara´ıso, Chile.
E-mail address: [email protected]
Jos´e Pantoja, Instituto de Matem´aticas, Pontificia Universidad Cat´olica de Val-para´ıso, Blanco Viel 596, Cerro Bar´on, Valpara´ıso, Chile.