The subnormal structure of classical-like groups over commutative rings
aa r X i v : . [ m a t h . K T ] S e p The subnormal structure of classical-like groupsover commutative rings
Raimund Preusser
Abstract
Let n be an integer greater than or equal to 3 and p R, ∆ q a Hermitian formring where R is commutative. We prove that if H is a subgroup of the odd-dimensional unitary group U n ` p R, ∆ q normalised by a relative elementarysubgroup EU n ` pp R, ∆ q , p I, Ω qq , then there is an odd form ideal p J, Σ q suchthat EU n ` pp R, ∆ q , p J I k , Ω JI k min . ` Σ ˝ I k qq ď H ď CU n ` pp R, ∆ q , p J, Σ qq where k “
12 if n “ k “
10 if n ě
4. As a conseqence of this resultwe obtain a sandwich theorem for subnormal subgroups of odd-dimensionalunitary groups.
Recall that if H is a subgroup of a group G and d is a nonnegative integer, then onewrites H ⊳ d G if H “ H ⊳ H ⊳ ¨ ¨ ¨ ⊳ H d ´ ⊳ H d “ G for some subgroups H , . . . , H d ´ of G . If H ⊳ d G for some d , then H is called a subnormal subgroup of G . In 1972, J.Wilson [10] proved that if H ⊳ d GL n p R q where n ě R is a commutative ring,then there is an ideal I such thatE n p R, I k q ď H ď C n p R, I q (1)where k “ p d ´ q´ if n “ k “ d ´ if n ě
4. In (1), E n p R, I k q denotes therelative elementary subgroup of level I k and C n p R, I q the full congruence subgroupof level I . Wilson’s result was subsequently improved by L. Vaserstein [6, 7] and N.Vavilov [8]. For more information on the general linear case we refer the reader tothe introduction in [9].A natural question is if similar results hold for other classical and classical-likegroups. The (even-dimensional) hyperbolic unitary groups U n p R, Λ q were definedby A. Bak in 1969 [1]. They embrace the classical Chevalley groups of type C m and D m , namely the even-dimensional symplectic and orthogonal groups Sp n p R q and O n p R q . In 2006, Z. Zhang [12] obtained a “sandwich” result similar to (1) for Keywords and phrases: classical-like groups, unitary groups, subnormal subgroups.The work is supported by the Russian Science Foundation grant 19-71-30002. p R, Λ q “ lim ÝÑ n ě U n p R, Λ q .In 2012, H. You [11] obtained a sandwich result for subnormal subgroups of nonstablehyperbolic unitary groups.In 2018, A. Bak and the author [2] defined odd-dimensional unitary groups U n ` p R, ∆ q . These groups generalise the even-dimensional unitary groups U n p R, Λ q and em-brace all classical Chevalley groups. In this article we prove that if H ⊳ d U n ` p R, ∆ q where p R, ∆ q is a commutative Hermitian form ring and n ě
3, then there is an oddform ideal p I, Ω q such thatEU n ` pp R, ∆ q , p I k , Ω I k min . ` Ω ˝ I k ´ qq ď H ď CU n ` pp R, ∆ q , p I, Ω qq (2)where k “ d ´ if n “ k “ d ´ if n ě
4, see Theorems 43 and 49. Forthe even-dimensional unitary groups this improves the result obtained by You in [11].Moreover, we obtain our result (2) only by straightforward computations, withoutusing localisation.The rest of the paper is organised as follows. In Section 2, we recall some standardnotation that is used throughout the paper. In Section 3, we recall the definitions ofthe groups U n ` p R, ∆ q and some important subgroups. In Section 4, we define thelower and upper level of a subgroup H ď U n ` p R, ∆ q . In Section 5, we prove ourmain results for the groups U p R, ∆ q , namely Theorems 41, 42 and 43. In Section 6,we prove our main results for the groups U n ` p R, ∆ q , n ě
4, namely Theorems 47,48 and 49. N denotes the set of all positive integers. If G is a group and g, h P G , we let g h : “ h ´ gh , h g : “ hgh ´ and r g, h s : “ ghg ´ h ´ . By a ring we mean an associativering with 1 such that 1 ‰
0. By an ideal we mean a two-sided ideal. If n P N and R is a ring, then the set of all n ˆ n matrices with entries in R is denoted by M n p R q .If a P M n p R q , we denote the entry of a at position p i, j q by a ij , the i -th row of a by a i ˚ and the j -th column of a by a ˚ j . The group of all invertible matrices in M n p R q is denoted by GL n p R q and the identity element of GL n p R q by e . If a P GL n p R q , thenthe entry of a ´ at position p i, j q is denoted by a ij , the i -th row of a ´ by a i ˚ and the j -th column of a ´ by a j . Furthermore, we denote by n R the set of all row vectors oflength n with entries in R and by R n the set of all column vectors of length n withentries in R . We consider n R as left R -module and R n as right R -module. We describe Hermitian form rings p R, ∆ q and odd form ideals p I, Ω q first, then theodd-dimensional unitary group U n ` p R, ∆ q and its elementary subgroup EU n ` p R, ∆ q over a Hermitian form ring p R, ∆ q . For an odd form ideal p I, Ω q , we recall thedefinitions of the following subgroups of U n ` p R, ∆ q ; the preelementary subgroupEU n ` p I, Ω q of level p I, Ω q , the elementary subgroup EU n ` pp R, Λ q , p I, Ω qq of level2 I, Ω q , the principal congruence subgroup U n ` pp R, Λ q , p I, Ω qq of level p I, Ω q , thenormalised principal congruence subgroup NU n ` pp R, Λ q , p I, Ω qq of level p I, Ω q , andthe full congruence subgroup CU n ` pp R, Λ q , p I, Ω qq of level p I, Ω q . First we recall the definitions of a ring with involution with symmetry and a Hermitianring.
Definition 1.
Let R be a ring and ¯: R Ñ Rx ÞÑ ¯ x an anti-isomorphism of R (i.e. ¯ is bijective, x ` y “ ¯ x ` ¯ y , xy “ ¯ y ¯ x for any x, y P R and ¯1 “ λ P R such that ¯¯ x “ λx ¯ λ for any x P R . Then λ iscalled a symmetry for ¯ , the pair p ¯ , λ q an involution with symmetry and the triple p R, ¯ , λ q a ring with involution with symmetry . A subset A Ď R is called involutioninvariant iff ¯ x P A for any x P A . A Hermitian ring is a quadruple p R, ¯ , λ, µ q where p R, ¯ , λ q is a ring with involution with symmetry and µ P R is a ring element suchthat µ “ ¯ µλ . Remark 2.
Let p R, ¯ , λ, µ q be a Hermitian ring.(a) It is easy to show that ¯ λ “ λ ´ .(b) The map ¯: R Ñ Rx ÞÑ x¯ : “ ¯ λ ¯ xλ is the inverse map of ¯. One checks easily that p R, ¯ , λ ¯ , µ ¯ q is a Hermitian ring.Next we recall the definition of an R ‚ -module. Definition 3. If R is a ring, let R ‚ denote the underlying set of the ring equipped withthe multiplication of the ring, but not the addition of the ring. A (right) R ‚ -module is a not necessarily abelian group p G, . `q equipped with a map ˝ : G ˆ R ‚ Ñ G p a, x q ÞÑ a ˝ x such that the following holds:(i) a ˝ “ a P G ,(ii) a ˝ “ a for any a P G ,(iii) p a ˝ x q ˝ y “ a ˝ p xy q for any a P G and x, y P R and3iv) p a . ` b q ˝ x “ p a ˝ x q . ` p b ˝ x q for any a, b P G and x P R .Let G and G be R ‚ -modules. A group homomorphism f : G Ñ G satisfying f p a ˝ x q “ f p a q˝ x for any a P G and x P R is called a homomorphism of R ‚ -modules . A subgroup H of G which is ˝ -stable (i.e. a ˝ x P H for any a P H and x P R ) is called an R ‚ -submodule . Moreover, if A Ď G and B Ď R , we denote by A ˝ B the subgroup of G generated by t a ˝ b | a P A, b P B u . We treat ˝ as an operator with higher prioritythan . ` .An important example of an R ‚ -module is the Heisenberg group, which we definenext. Definition 4.
Let p R, ¯ , λ, µ q be a Hermitian ring. Define the map. . ` : p R ˆ R q ˆ p R ˆ R q Ñ R ˆ R pp x , y q , p x , y qq ÞÑ p x , y q . ` p x , y q : “ p x ` x , y ` y ´ ¯ x µx q . Then p R ˆ R, . `q is a group, which we call the Heisenberg group and denote by H .Equipped with the map ˝ : p R ˆ R q ˆ R ‚ Ñ R ˆ R pp x, y q , a q ÞÑ p x, y q ˝ a : “ p xa, ¯ aya q H becomes an R ‚ -module. Remark 5.
We denote the inverse of an element p x, y q P H by . ´p x, y q . One checkseasily that . ´p x, y q “ p´ x, ´ y ´ ¯ xµx q for any p x, y q P H .In order to define the odd-dimensional unitary groups we need the notion of aHermitian form ring. Definition 6.
Let p R, ¯ , λ, µ q be a Hermitian ring. Let p R, `q have the R ‚ -modulestructure defined by x ˝ a “ ¯ axa . Define the trace map tr : H Ñ R p x, y q ÞÑ ¯ xµx ` y ` ¯ yλ. One checks easily that tr is a homomorphism of R ‚ -modules. Set∆ min : “ tp , x ´ xλ q | x P R u and ∆ max : “ ker p tr q . An R ‚ -submodule ∆ of H lying between ∆ min and ∆ max is called an odd form pa-rameter for p R, ¯ , λ, µ q . Since ∆ min and ∆ max are R ‚ -submodules of H , they arerespectively the smallest and largest odd form parameters. A pair pp R, ¯ , λ, µ q , ∆ q iscalled a Hermitian form ring . We shall usually abbreviate it by p R, ∆ q .4ext we define an odd form ideal of a Hermitian form ring. Definition 7.
Let p R, ∆ q be a Hermitian form ring and I an involution invariantideal of R . Set J p ∆ q : “ t y P R | D z P R : p y, z q P ∆ u and ˜ I : “ t x P R | J p ∆ q µx Ď I u .Obviously ˜ I and J p ∆ q are right ideals of R and I Ď ˜ I . Moreover, setΩ I min : “ tp , x ´ ¯ xλ q | x P I u . ` ∆ ˝ I and Ω I max : “ ∆ X p ˜ I ˆ I q . An R ‚ -submodule Ω of H lying between Ω I min and Ω I max is called a relative odd formparameter of level I . Since Ω I min and Ω I max are R ‚ -submodules of H , they are respec-tively the smallest and the largest relative odd form parameters of level I . If Ω isa relative odd form parameter of level I , then p I, Ω q is called an odd form ideal of p R, ∆ q . Definition 8.
Let p R, ∆ q be a Hermitian form ring where R is commutative. Let p I, Ω q be an odd form ideal of p R, ∆ q and J an involution invariant ideal of R . Wedefine p I, Ω q ˚ J as the odd form ideal p IJ, Ω IJ min . ` Ω ˝ J q . Furthermore, we define p I, Ω q : J as the odd form ideal p I : J, Ω I : J min . ` t α P Ω I : J max | α ˝ J Ď Ω uq where I : J “ t x P R | xJ Ď I u is the usual quotient of ideals. Let p R, ∆ q be a Hermitian form ring and n P N . Set M : “ R n ` . We use the followingindexing for the elements of the standard basis of M : p e , . . . , e n , e , e ´ n , . . . , e ´ q .That means that e i is the column whose i -th coordinate is one and all the othercoordinates are zero if 1 ď i ď n , the column whose p n ` q -th coordinate is oneand all the other coordinates are zero if i “
0, and the column whose p n ` ` i q -thcoordinate is one and all the other coordinates are zero if ´ n ď i ď ´
1. If u P M ,then we call p u , . . . , u n , u ´ n , . . . , u ´ q t P R n the hyperbolic part of u and denote itby u hb . We set u ˚ : “ ¯ u t and u ˚ hb : “ ¯ u t hb . Moreover, we define the maps B : M ˆ M Ñ R p u, v q ÞÑ u ˚ ¨˝ π µ πλ ˛‚ v “ n ÿ i “ ¯ u i v ´ i ` ¯ u µv ` ´ ÿ i “´ n ¯ u i λv ´ i and Q : M Ñ H u ÞÑ p Q p u q , Q p u qq : “ p u , u ˚ hb ˆ π ˙ u hb q “ p u , n ÿ i “ ¯ u i u ´ i q where π P M n p R q denotes the matrix with ones on the skew diagonal and zeroselsewhere. 5 emma 9. (i) B is a λ -Hermitian form , i.e. B is biadditive, B p ux, vy q “ ¯ xB p u, v q y @ u, v P M, x, y P R and B p u, v q “ B p v, u q λ @ u, v P M .(ii) Q p ux q “ Q p u q ˝ x @ u P M, x P R , Q p u ` v q ” Q p u q . ` Q p v q . ` p , B p u, v qq mod∆ min @ u, v P M and tr p Q p u qq “ B p u, u q @ u P M .Proof. Straightforward computation.
Definition 10.
The group U n ` p R, ∆ q : “t σ P GL n ` p R q | B p σu, σv q “ B p u, v q ^ Q p σu q ” Q p u q mod ∆ @ u, v P M u is called the odd-dimensional unitary group . Remark 11.
The groups U n ` p R, ∆ q include as special cases the even-dimensionalunitary groups U n p R, Λ q and all classical Chevalley groups. On the other hand,the groups U n ` p R, ∆ q are embraced by Petrov’s odd unitary groups U l p R, L q . Fordetails see [2, Remark 14(c) and Example 15]. Definition 12.
We define the sets Θ ` : “ t , . . . , n u , Θ ´ : “ t´ n, . . . , ´ u , Θ : “ Θ ` Y Θ ´ Y t u and Θ hb : “ Θ zt u . Moreover, we define the map ǫ : Θ hb Ñ t˘ u i ÞÑ , if i P Θ ` , ´ , if i P Θ ´ . Remark 13.
We will sometimes use expressions of the form ř i P A f p i q (resp. ś i P A f p i q )where A Ď Θ is a subset and f : A Ñ X is a map where X is a set with a fixedaddition ` (resp. multiplication ¨ ). In such expressions we assume that the order ofthe summands (resp. factors) corresponds to the strict total order 1 ă ¨ ¨ ¨ ă n ă ă ´ n ¨ ¨ ¨ ă ´ Lemma 14 ([2, Lemma 17]) . Let σ P GL n ` p R q . Then σ P U n ` p R, ∆ q iff Condi-tions (i) and (ii) below are satisfied.(i) σ ij “ λ ´p ǫ p i q` q{ ¯ σ ´ j, ´ i λ p ǫ p j q` q{ @ i, j P Θ hb ,µσ j “ ¯ σ ´ j, λ p ǫ p j q` q{ @ j P Θ hb ,σ i “ λ ´p ǫ p i q` q{ ¯ σ , ´ i µ @ i P Θ hb and µσ “ ¯ σ µ. (ii) Q p σ ˚ j q ” p δ j , q mod ∆ @ j P Θ . .3 The polarity map Definition 15.
The map r : M ÝÑ M ˚ u ÞÝÑ ` ¯ u ´ λ . . . ¯ u ´ n λ ¯ u µ ¯ u n . . . ¯ u ˘ where M ˚ “ n ` R is called the polarity map . Clearly r is involutary linear , i.e. Ć u ` v “ ˜ u ` ˜ v and Ă ux “ ¯ x ˜ u for any u, v P M and x P R . Lemma 16. If σ P U n ` p R, ∆ q and u P M , then Ă σu “ ˜ uσ ´ .Proof. Follows from Lemma 14.
We introduce the following notation. Let p R, ¯ , λ ¯ , µ ¯ q be the Hermitian ring definedin Remark 2(b) and H ´ the corresponding Heisenberg group. Note that the un-derlying set of both H and H ´ is R ˆ R . We denote the group operation (resp.scalar multiplication) of H by . ` (resp. ˝ ) and the group operation (resp. scalarmultiplication) of H ´ by . ` ´ (resp. ˝ ´ ). Furthermore, we set ∆ : “ ∆ and∆ ´ : “ tp x, y q P R ˆ R | p x, ¯ y q P ∆ u . One checks easily that pp R, ¯ , λ ¯ , µ ¯ q , ∆ ´ q is a Hermitian form ring. Analogously, if p I, Ω q is an odd form ideal of p R, ∆ q , we setΩ : “ Ω and Ω ´ : “ tp x, y q P R ˆ R | p x, ¯ y q P Ω u . One checks easily that p I, Ω ´ q isan odd form ideal of p R, ∆ ´ q .If i, j P Θ, let e ij denote the matrix in M n ` p R q with 1 in the p i, j q -th positionand 0 in all other positions. Definition 17. If i, j P Θ hb , i ‰ ˘ j and x P R , the element T ij p x q : “ e ` xe ij ´ λ p ǫ p j q´ q{ ¯ xλ p ´ ǫ p i qq{ e ´ j, ´ i of U n ` p R, ∆ q is called an (elementary) short root transvection . If i P Θ hb and p x, y q P ∆ ´ ǫ p i q , the element T i p x, y q : “ e ` xe , ´ i ´ λ ´p ` ǫ p i qq{ ¯ xµe i ` ye i, ´ i of U n ` p R, ∆ q is called an (elementary) extra short root transvection . The extrashort root transvections of the kind T i p , y q “ e ` ye i, ´ i are called (elementary) long root transvections . If an element of U n ` p R, ∆ q is ashort or extra short root transvection, then it is called an elementary transvection .The subgroup of U n ` p R, ∆ q generated by all elementary transvections is called the elementary subgroup and is denoted by EU n ` p R, ∆ q .7 emma 18 ([2, Lemma 20]) . The following relations hold for elementary transvec-tions. T ij p x q “ T ´ j, ´ i p´ λ p ǫ p j q´ q{ ¯ xλ p ´ ǫ p i qq{ q , (S1) T ij p x q T ij p y q “ T ij p x ` y q , (S2) r T ij p x q , T kl p y qs “ e if k ‰ j, ´ i and l ‰ i, ´ j, (S3) r T ij p x q , T jk p y qs “ T ik p xy q if i ‰ ˘ k, (S4) r T ij p x q , T j, ´ i p y qs “ T i p , xy ´ λ p´ ´ ǫ p i qq{ ¯ y ¯ xλ p ´ ǫ p i qq{ q , (S5) T i p x , y q T i p x , y q “ T i pp x , y q . ` ´ ǫ p i q p x , y qq , (E1) r T i p x , y q , T j p x , y qs “ T i, ´ j p´ λ ´p ` ǫ p i qq{ ¯ x µx q if i ‰ ˘ j, (E2) r T i p x , y q , T i p x , y qs “ T i p , ´ λ ´p ` ǫ p i qq{ p ¯ x µx ´ ¯ x µx qq , (E3) r T ij p x q , T k p y, z qs “ e if k ‰ j, ´ i and (SE1) r T ij p x q , T j p y, z qs “ T j, ´ i p zλ p ǫ p j q´ q{ ¯ xλ p ´ ǫ p i qq{ q¨¨ T i p yλ p ǫ p j q´ q{ ¯ xλ p ´ ǫ p i qq{ , xzλ p ǫ p j q´ q{ ¯ xλ p ´ ǫ p i qq{ q . (SE2) In this subsection p I, Ω q denotes an odd form ideal of p R, ∆ q . Definition 19.
A short root transvection T ij p x q is called p I, Ω q -elementary if x P I .An extra short root transvection T i p x, y q is called p I, Ω q -elementary if p x, y q P Ω ´ ǫ p i q .The subgroup EU n ` p I, Ω q of EU n ` p R, ∆ q generated by the p I, Ω q -elementary trans-vections is called the preelementary subgroup of level p I, Ω q . Its normal closureEU n ` pp R, ∆ q , p I, Ω qq in EU n ` p R, ∆ q is called the elementary subgroup of level p I, Ω q .If σ P M n ` p R q , we call the matrix p σ ij q i,j P Θ hb P M n p R q the hyperbolic part of σ and denote it by σ hb . Furthermore, we define the submodule M p R, ∆ q : “ t u P M | u P J p ∆ qu of M . Definition 20.
The subgroup U n ` pp R, ∆ q , p I, Ω qq : “t σ P U n ` p R, ∆ q | σ hb ” e hb mod I and Q p σu q ” Q p u q mod Ω @ u P M p R, ∆ qu of U n ` p R, ∆ q is called the principal congruence subgroup of level p I, Ω q . Lemma 21 ([2, Lemma 28]) . Let σ P U n ` p R, ∆ q . Then σ P U n ` pp R, ∆ q , p I, Ω qq iff Conditions (i) and (ii) below are satisfied.(i) σ hb ” e hb mod I .(ii) Q p σ ˚ j q P Ω @ j P Θ hb and p Q p σ ˚ q . ´ p , qq ˝ a P Ω @ a P J p ∆ q . efinition 22. The subgroup NU n ` pp R, ∆ q , p I, Ω qq : “ Normaliser U n ` p R, ∆ q p U n ` pp R, ∆ q , p I, Ω qqq of U n ` p R, ∆ q is called the normalised principal congruence subgroup of level p I, Ω q . Definition 23.
The subgroup CU n ` pp R, ∆ q , p I, Ω qq : “t σ P NU n ` pp R, ∆ q , p I, Ω qq | r σ, EU n ` p R, ∆ qs ď U n ` pp R, ∆ q , p I, Ω qqu of U n ` p R, ∆ q is called the full congruence subgroup of level p I, Ω q . In this section n denotes an integer greater than or equal to 3 and p R, ∆ q a Hermitianform ring where R is commutative. We will define the lower and the upper level of asubgroup of U n ` p R, ∆ q . Definition 24.
Let H be a subgroup of U n ` p R, ∆ q . Set I : “ t x P R | T ij p xr q τ P H @ i, j P Θ hb , i ‰ ˘ j, r P R, τ P EU n ` p R, ∆ q^ T i p , xr ´ xrλ ´ ǫ p i q q τ P H @ i P Θ hb , r P R, τ P EU n ` p R, ∆ q^ T i p α ˝ xr q τ P H @ i P Θ hb , α P ∆ ´ ǫ p i q , r P R, τ P EU n ` p R, ∆ q^ T i p α ˝ ¯ xr q τ P H @ i P Θ hb , α P ∆ ´ ǫ p i q , r P R, τ P EU n ` p R, ∆ qu andΩ : “ Ω I min . ` tp y, z q P Ω I max | T i pp y, z q ˝ r q τ P H @ i P Θ ´ , r P R, τ P EU n ` p R, ∆ q^ T i pp y, ¯ z q ˝ r q τ P H @ i P Θ ` , r P R, τ P EU n ` p R, ∆ qu . The odd form ideal L p H q : “ p I, Ω q is called the lower level of H .If H is a subgroup of U n ` p R, ∆ q , then clearly EU n ` pp R, ∆ q , L p H qq ď H and L p H q is the greatest odd form ideal with this property. Definition 25.
Let H be a subgroup of U n ` p R, ∆ q . Set Y : “ t σ ij , σ ii ´ σ jj , σ i J p ∆ q , J p ∆ q µσ j , J p ∆ q µ p σ ´ σ jj q J p ∆ q | σ P H, i, j P Θ hb , i ‰ j u and Z : “ t Q p σ ˚ j q , p Q p σ ˚ q . ´ p , qq ˝ y . ` p y, z q . ´ p y, z q ˝ σ ii | σ P H, i, j P Θ hb , p y, z q P ∆ u . Let I be the involution invariant ideal generated by Y (i.e. the ideal generated by Y Y ¯ Y ) and set Ω : “ Ω I min . ` Z ˝ R . The odd form ideal U p H q : “ p I, Ω q is called the upper level of H . 9e will show that if H is a subgroup of U n ` p R, ∆ q , then H ď CU n ` pp R, ∆ q ,U p H qq and U p H q is the smallest odd form ideal with this property. Lemma 26.
Let σ P U n ` p R, ∆ q and p I, Ω q be an odd form ideal of p R, ∆ q . Then σ P NU n ` pp R, ∆ q , p I, Ω qq iff p Q p σ ˚ q . ´ p , qq ˝ x P Ω and p Q p σ q . ´ p , qq ˝ x P Ω for any x P J p Ω q .Proof. By [2, Corollary 35], we have σ P NU n ` pp R, ∆ q , p I, Ω qq iff Ω “ σ Ω where σ Ω “ tp Q p σ ˚ q . ´ p , qq ˝ x . ` p x, y q | p x, y q P Ω u . ` Ω I min (see [2, Definition 30]).By the definition of equality of sets, we have Ω “ σ Ω iff Ω Ď σ Ω and σ Ω Ď Ω.By [2, Lemma 33], the map U n ` p R, ∆ q ˆ FP p I q Ñ FP p I q , p τ, Σ q ÞÑ τ Σ, whereFP p I q denotes the set of all relative odd form parameters for I , is a left group action.Clearly this action preserves inclusions. It follows that Ω Ď σ Ω iff σ ´ Ω Ď Ω. Hence σ P NU n ` pp R, ∆ q , p I, Ω qq iff σ ´ Ω Ď Ω and σ Ω Ď Ω. The assertion of the lemmafollows.
Corollary 27.
Let H be a subgroup of U n ` p R, ∆ q . Then H ď NU n ` pp R, ∆ q , U p H qq .Proof. Write U p H q “ p I, Ω q and let σ P H . By the previous lemma, it suffices toshow that p Q p σ ˚ q . ´ p , qq ˝ x P Ω for any x P J p Ω q . Let x P J p Ω q . Then p x, y q P Ωfor some y P R . Hence p Q p σ ˚ q . ´ p , qq ˝ x ` p x, y q . ´ p x, y q ˝ σ P Z Ď Ω where Z isdefined as in Definition 25. It clearly follows that p Q p σ ˚ q . ´ p , qq ˝ x P Ω. Lemma 28.
Let p a, b q P ∆ k and x , . . . , x m P R where k P t˘ u and m P N . Then p a, b q ˝ m ÿ i “ x i “ p a, b q ˝ x . ` k . . . . ` k p a, b q ˝ x m . ` k p , m ÿ i,j “ ,i ą j ¯ x i bx j ´ ¯ x i bx j λ k q . Proof.
Straightforward computation.
Lemma 29.
Let p I, Ω q be an odd form ideal. Suppose σ P U n ` p R, ∆ q satisfiesConditions (i)-(v) in Lemma 31. Then p a, b q ” p a, b q ˝ σ ii σ ii mod Ω for any p a, b q P ∆ and i P Θ hb .Proof. By the previous lemma we have p a, b q“p a, b q ˝ p ´ ÿ s “ σ is σ si q“p ´ . ă s “ p a, b q ˝ σ is σ si q . ` p , ´ ÿ s,t “ ,s ą t σ is σ si bσ it σ ti ´ σ is σ si bσ it σ ti λ qq (3)Since σ satisfies Conditions (i) and (iv) in Lemma 31, all the summands in (3) except p a, b q ˝ σ ii σ ii are contained in Ω I min . 10 emma 30. Let p I, Ω q be an odd form ideal. Suppose σ P U n ` p R, ∆ q satisfiesConditions (i)-(v) in Lemma 31. Then p , ´ ¯ σ i, ´ i ¯ y ¯ σ µy ` ¯ σ i, ´ i ¯ y ¯ σ µyλ q P Ω forany i P Θ hb and y P J p ∆ q .Proof. First we note that by [4, Lemma 6.30], σ ´ also satisfies Conditions (i)-(v) inLemma 31. Clearly p , ´ ¯ σ i, ´ i ¯ y ¯ σ µy ` ¯ σ i, ´ i ¯ y ¯ σ µyλ q“p , ´ σ ii ¯ yµσ y ` σ ii ¯ yµσ yλ q“ p , ´ σ ii ¯ yµ p σ ´ σ ii q y ` σ ii ¯ yµ p σ ´ σ ii q yλ q loooooooooooooooooooooooooooomoooooooooooooooooooooooooooon X : “ . ` p , ´ σ ii ¯ yµσ ii y ` σ ii ¯ yµσ ii yλ q looooooooooooooooomooooooooooooooooon Y : “ , the first equality by Lemma 14. Since σ satisfies Condition (v), X lies in Ω I min . Onthe other hand, σ ii σ ii “ ´ ř j ‰ i σ ij σ ji ” I since σ satisfies satisfies Conditions(i) and (iii). Hence Y ” p , ´ ¯ yµy ` ¯ yµyλ q “ I min since µ “ ¯ µλ . Lemma 31.
Let σ P U n ` p R, ∆ q and p I, Ω q an odd form ideal. Then r σ, EU n ` p R, ∆ qs ď U n ` pp R, ∆ q , p I, Ω qq iff(i) σ ij P I @ i ‰ j P Θ hb ,(ii) σ ii ´ σ jj P I @ i, j P Θ hb ,(iii) σ i J p ∆ q P I @ i P Θ hb ,(iv) J p ∆ q µσ j P I @ j P Θ hb ,(v) J p ∆ q µ p σ ´ σ jj q J p ∆ q P I @ j P Θ hb ,(vi) Q p σ ˚ j q P Ω @ j P Θ hb and(vii) p Q p σ ˚ q . ´ p , qq ˝ y . ` p y, z q . ´ p y, z q ˝ σ ii P Ω @ i P Θ hb , p y, z q P ∆ .Proof. pñq Suppose that r σ, EU n ` p R, ∆ qs ď U n ` pp R, ∆ q , p I, Ω qq . Then r σ, EU n ` p R, ∆ qs ď U n ` pp R, ∆ q , p I, Ω I max qq and hence σ P CU n ` pp R, ∆ q , p I, Ω I max qq (notethat NU n ` pp R, ∆ q , p I, Ω I max qq “ U n ` p R, ∆ q , see [2, Remark 26]). It follows from[4, Lemma 6.30] that Conditions (i)-(v) above hold. By analysing [2, Lemma 63]we obtain Q p σ ˚ i q ˝ σ jj P Ω and p Q p σ ˚ q . ´ p , qq ˝ yσ ii . ` p y, z q ˝ σ ii . ´ p y, z q P Ωfor any i ‰ ˘ j P Θ hb and p y, z q P ∆. There are the following misprints in [2,Lemma 63], all on page 2866: a ´ k should be replaced by a ´ i (1 occurence), σ , ´ should be replaced by σ i, ´ i (4 occurences) and p , ¯ σ i, ´ i ¯ y ¯ σ µy ´ ¯ σ i, ´ i ¯ y ¯ σ µyλ q should be replaced by p , ´ ¯ σ i, ´ i ¯ y ¯ σ µy ` ¯ σ i, ´ i ¯ y ¯ σ µyλ q (1 occurence; note that p , ´ ¯ σ i, ´ i ¯ y ¯ σ µy ` ¯ σ i, ´ i ¯ y ¯ σ µyλ q P Ω by Lemma 30). It follows from Lemma 29that σ satisfies Conditions (vi) an (vii). pðq Now suppose that σ satisfies Conditions (i)-(vii). We have to show that r σ, EU n ` R, ∆ qs ď U n ` pp R, ∆ q , p I, Ω qq . Since U n ` pp R, ∆ q , p I, Ω qq is normalised by EU n ` p R, ∆ q (follows from [2, Corollary 36]), it suffices to show that r σ, τ s P U n ` pp R, ∆ q , p I, Ω qq for any elementary transvection τ . Since σ P CU n ` pp R, ∆ q , p I, Ω I max qq by [4,Lemma 6.30], we obtain that r σ, τ s satisfies Condition (i) in Lemma 21. It remainsto show that r σ, τ s satisfies Condition (ii) in Lemma 21. But that follows from [2,Lemma 63].If p I, Ω q and p J, Σ q are odd form parameters, then we write p I, Ω q Ď p J, Σ q if I Ď J and Ω Ď Σ. Proposition 32.
Let H be a subgroup of U n ` p R, ∆ q . Then H ď CU n ` pp R, ∆ q ,U p H qq and U p H q is the smallest form odd ideal with this property.Proof. It follows from Corollary 27 and Lemma 31 that H ď CU n ` pp R, ∆ q , U p H qq .Let Y and Z be defined as in Definition 25. If p I, Ω q is an odd form ideal such that H ď CU n ` pp R, ∆ q , p I, Ω qq , then Y Ď I and Z Ď Ω by Lemma 31. It follows that U p H q Ď p I, Ω q . Corollary 33.
Let H be a subgroup of U n ` p R, ∆ q . Then L p H q Ď U p H q .Proof. By Proposition 32 we haveEU n ` pp R, ∆ q , L p H qq ď H ď CU n ` pp R, ∆ q , U p H qq . It follows from Lemma 31 that L p H q Ď U p H q . Corollary 34.
Let H be a subgroup of U n ` p R, ∆ q and τ P EU n ` p R, ∆ q . Then U p H q “ U p H τ q .Proof. By Proposition 32 we have H ď CU n ` pp R, ∆ q , U p H qq . It follows that H τ ď CU n ` pp R, ∆ q , U p H qq since EU n ` p R, ∆ q ď NU n ` pp R, ∆ q , U p H qq by [2, Corollary36]. This implies that U p H τ q Ď U p H q , again by Proposition 32. Similarly one canshow that U p H q Ď U p H τ q . U p R, ∆ q In this section p R, ∆ q denotes a Hermitian form ring where R is commutative. Lemma 35.
Let p I, Ω q be an odd form ideal and H a subgroup of U p R, ∆ q normalisedby EU p I, Ω q . Suppose there is an x P R , r, s P Θ hb , r ‰ ˘ s and an m P N such that T r, ˘ s p xa q P H for all a P I m . Then T ij p xa q P H for all i, j P Θ hb , i ‰ ˘ j and a P I m ` .Proof. Choose a t P Θ hb such that t ‰ ˘ r, ˘ s . It follows from Relation (S4) in Lemma18 that • T r, ˘ t p xa q , T ˘ t, ˘ s p xa q P H for any a P I m ` , • T ˘ s, ˘ t p xa q , T ˘ t, ˘ r p xa q , T ´ r, ˘ s p xa q P H for any a P I m ` and12 T ˘ s, ˘ r p xa q , T ´ r, ˘ t p xa q P H for any a P I m ` .The assertion of the lemma follows.We recall some notation introduced in [5]. Let G be a group and p a , b q , p a , b q P G ˆ G . If there is a g P G such that a “ r a ´ , g s and b “ r g, b s , then we write p a , b q g ÝÑ p a , b q . If p a , b q , . . . , p a n ` , b n ` q P G ˆ G and g , . . . , g n P G such that p a , b q g ÝÑ p a , b q g ÝÑ . . . g n ÝÑ p a n ` , b n ` q , then we write p a , b q g ,...,g n ÝÝÝÝÑ p a n ` , b n ` q .If H ď G , g P G and h P H , then we call g h an H -conjugate of g . Lemma 36 ([5, Lemma 7]) . Let G be a group and p a , b q , p a , b q P G ˆ G . If p a , b q g ,...,g n ÝÝÝÝÑ p a , b q for some g , . . . , g n P G , then a b is a product of n H -conjugates of a b and p a b q ´ where H is the subgroup of G generated by t a , g , . . . , g n u . Lemma 37.
Let p I, Ω q be an odd form ideal and H a subgroup of U p R, ∆ q normalisedby EU p I, Ω q . Let σ P H and r, s, t P Θ hb such that r ‰ ˘ s and t ‰ ˘ r, ˘ s . Then(i) T ij p σ s, ´ t ¯ σ sr a q P H for any i, j P Θ hb , i ‰ ˘ j, a P I and(ii) T ij p σ rs ¯ σ rr a q P H for any i, j P Θ hb , i ‰ ˘ j, a P I .Proof. Let a , . . . , a P I . Set τ : “ T rt p σ ss ¯ σ sr ¯ a q T st p´ σ sr ¯ σ sr ¯ a q T r, ´ s p´ λ p ǫ p t q´ ǫ p s qq{ σ s, ´ t ¯ σ sr a q¨¨ T r p , λ p ǫ p t q´ ǫ p r qq{ σ s, ´ t ¯ σ ss a ´ λ p´ ǫ p t q´ ǫ p r qq{ σ ss ¯ σ s, ´ t ¯ a q and τ : “ T rt p σ rs ¯ σ rr a q T st p´ σ rr ¯ σ rr a q T r, ´ s p´ λ p ǫ p t q´ ǫ p s qq{ σ r, ´ t ¯ σ rr ¯ a q¨¨ T r p , λ p ǫ p t q´ ǫ p r qq{ σ r, ´ t ¯ σ rs ¯ a ´ λ p´ ǫ p t q´ ǫ p r qq{ σ rs ¯ σ r, ´ t a q . Clearly τ , τ P EU p I, Ω q . Set ξ : “ στ ´ σ ´ and ξ : “ στ ´ σ ´ . One checks easilythat p στ ´ q s ˚ “ σ s ˚ and p τ ´ σ ´ q ˚ , ´ s “ σ , ´ s . Hence p ξ q s ˚ “ e s ˚ and p ξ q ˚ , ´ s “ e ˚ , ´ s . Similarly p στ ´ q r ˚ “ σ r ˚ and p τ ´ σ ´ q ˚ , ´ r “ σ , ´ r . Hence p ξ q r ˚ “ e r ˚ and p ξ q ˚ , ´ r “ e ˚ , ´ r . A straightforward computation shows that p τ , ξ q T ´ s,t p a q ,T ´ s,r p a q ,T t, ˘ r p´ λ p ǫ p s q´ ǫ p t qq{ a q ÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÑ p T ´ s, ˘ r p σ s, ´ t ¯ σ sr a a a a q , e q and p τ , ξ q T sr p a q ,T tr p a q ,T r, ´ t p a q ,T ´ r,s p a q ÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÑ p T ´ r, ´ t p σ rs ¯ σ rr a a a a a q , e q . It follows from Lemma 36 that H contains the matrices T ´ s, ˘ r p σ s, ´ t ¯ σ sr a a a a q and T ´ r, ´ t p σ rs ¯ σ rr a a a a a q . Clearly H also contains T ´ r,t p σ rs ¯ σ rr a a a a a q (just re-place t by ´ t in the argument above). The assertion of the lemma follows fromLemma 35. 13 emma 38. Let p I, Ω q be an odd form ideal and H a subgroup of U p R, ∆ q normalisedby EU p I, Ω q . Then T ij p σ rs a q P H for any i, j P Θ hb , i ‰ ˘ j , σ P H , r, s P Θ hb , r ‰ ˘ s and a P I .Proof. Choose a t P Θ hb such that t ‰ ˘ r, ˘ s . Let a P I and set τ : “ r σ ´ , T tr p´ ¯ σ rs ¯ a qs P H. Let J be the involution invariant ideal generated by the set t a σ rs ¯ σ rr , a σ rs ¯ σ r, ˘ t u .Clearly τ tt “ ´ σ tt ¯ σ rs ¯ a σ rt ` σ t, ´ r λ p ǫ p r q´ ǫ p t qq{ σ rs a σ ´ t,t “ ´ σ tt ¯ σ rs ¯ a σ rt looomooon P J ` λ ´ ǫ p t q ¯ σ r, ´ t σ rs a loooomoooon P J σ ´ t,t ” J and τ tr “ ´ σ tt ¯ σ rs ¯ a σ rr ` σ t, ´ r λ p ǫ p r q´ ǫ p t qq{ σ rs a σ ´ t,r ` τ tt ¯ σ rs ¯ a “ ´ σ tt ¯ σ rs ¯ a σ rr looomooon P J ` λ ´ ǫ p t q ¯ σ r, ´ t σ rs a loooomoooon P J σ ´ t,r ` τ tt ¯ σ rs ¯ a ” ¯ σ rs ¯ a mod J by Lemma 14. Hence τ tt ¯ τ tr ” σ rs a mod J . By Lemma 37(ii) we have T ij p τ tr ¯ τ tt a q P H for any i, j P Θ hb , i ‰ ˘ j, a P I whence T ij p τ tt ¯ τ tr a q P H for any i, j P Θ hb , i ‰ ˘ j, a P I (because of Relation (S1) in Lemma 18). Since τ tt ¯ τ tr a “ σ rs a a ` xa for some x P J , we obtain the assertion of the lemma in view of Lemma 37. Lemma 39.
Let p I, Ω q be an odd form ideal and H a subgroup of U p R, ∆ q normalisedby EU p I, Ω q . Let p J, Σ q “ U p H q . Then EU p J I , Ω JI min q ď H .Proof. Let σ P U n ` p R, ∆ q and i, j, r, s P Θ hb such that i ‰ ˘ j and r ‰ ˘ s . Fur-thermore, let x, y P J p ∆ q . Suppose we have shown(i) T ij p σ rs a q P H for any a P I ,(ii) T ij p σ r, ´ r a q P H for any a P I ,(iii) T ij p σ r xa q P H for any a P I ,(iv) T ij p ¯ xµσ s a q P H for any a P I ,(v) T ij pp σ rr ´ σ ss q a q P H for any a P I ,(vi) T ij pp σ rr ´ σ ´ r, ´ r q a q P H for any a P I and(vii) T ij p ¯ xµ p σ ´ σ ss q ya q P H for any a P I .14hen it would follow that EU p J I , Ω JI min q ď H in view of Relations (S5) and (SE2)in Lemma 18. Hence it suffices to show (i)-(vii) above.(i) Follows from the previous lemma.(ii) Let b P I and c P I . Clearly the entry of T sr p b q σ P H at position p s, ´ r q equals σ r, ´ r b ` σ s, ´ r . It follows from the previous lemma that T ij pp σ r, ´ r b ` σ s, ´ r q c q P H .But T ij p σ s, ´ r c q P H , again by the previous lemma. Hence T ij p σ r, ´ r bc q P H .(iii) Let b P I and c P I . Choose a z such that p x, z q P ∆ ǫ p s q (possible since x P J p ∆ q ). Clearly the entry of σ T ´ s p xb, ¯ bzb q P H at position p r, s q equals σ r xb ` σ rs ` σ r, ´ s ¯ bzb . It follows from the previous lemma that T ij pp σ r xb ` σ rs ` σ r, ´ s ¯ bzb q c q P H . But T ij p σ rs c q , T rs p σ r, ´ s ¯ bzbc q P H , again by the previous lemma. It followsthat T ij p σ r xbc q P H .(iv) Let b P I and c P I . Choose a z such that p x, z q P ∆ ǫ p´ r q . Clearly the entryof T r p x ¯ b,bz ¯ b q σ P H at position p r, s q equals ´ λ ´p ` ǫ p r qq{ b ¯ xµσ s ` σ rs ` bz ¯ bσ ´ r,s . Itfollows from the previous lemma that T ij pp´ λ ´p ` ǫ p r qq{ b ¯ xµσ s ` σ rs ` bz ¯ bσ ´ r,s q c q P H . But T ij p σ rs c q , T rs p bz ¯ bσ ´ r,s c q P H , again by the previous lemma. It followsthat T ij p´ λ ´p ` ǫ p r qq{ b ¯ xµσ s c q P H .(v) Let b P I and c P I . One checks easily that the entry of T sr p b q σ P H at position p s, r q equals p σ rr ´ σ ss q b ` σ sr ´ σ rs b . It follows from the previous lemma that T ij ppp σ rr ´ σ ss q b ` σ sr ´ σ rs b q c q P H . But T ij p σ sr c q , T rs p´ σ rs b c q P H , again bythe previous lemma. It follows that T ij pp σ rr ´ σ ss q bc q P H .(vi) Follows from (v) since T ij pp σ rr ´ σ ´ r, ´ r q a q “ T ij pp σ rr ´ σ ss q a q T ij pp σ ss ´ σ ´ r, ´ r q a q .(vii) Let b P I and c P I . Choose a z such that p y, z q P ∆ ǫ p s q . One checks easilythat the entry of σ T ´ s p yb, ¯ bzb q P H at position p , s q equals p σ ´ σ ss q yb ` σ s ´ σ s y b ` σ , ´ s ¯ bzb ´ σ s, ´ s yb ¯ bzb . It follows from (iv) that T ij p ¯ xµ pp σ ´ σ ss q yb ` σ s ´ σ s y b ` σ , ´ s ¯ bzb ´ σ s, ´ s yb ¯ bzb q c q P H. But T ij p ¯ xµσ s c q , T ij p´ ¯ xµσ s y b c q , T ij p ¯ xµσ , ´ s ¯ bzbc q , T ij p´ ¯ xµσ s, ´ s yb ¯ bzbc q P H by(ii),(iii) and (iv). It follows that T ij p ¯ xµ p σ ´ σ ss q ybc q P H .We introduce the following notation. If p x, y q P ∆, we set p x, y q : “ p x, y q and p x, y q ´ : “ p x, ¯ λy q . Note that ∆ contains the element . ´p x, y q ˝ p´ q “ p´ x, λ ¯ y q ˝p´ q “ p x, λ ¯ y q . Hence p x, y q ´ P ∆ ´ . On the other hand any element of ∆ ´ is equalto p x, y q ´ for some p x, y q P ∆. Hence ∆ ´ “ tp x, y q ´ | p x, y q P ∆ u . Similarly, if Ω isa relative odd form parameter, then Ω ´ “ tp x, y q ´ | p x, y q P Ω u . One checks easilythat p α . ` β q ´ “ α ´ . ` ´ β ´ and p α ˝ x q ´ “ α ´ ˝ x for any α, β P ∆ and x P R .In the proof of Theorem 41 we will use the matrices T ˚ j p u, x q defined below. Thesematrices are examples of ESD transvections, cf. [3].15 efinition 40. Let p I, Ω q be an odd form ideal and j P Θ hb . Moreover, let u P M and x P R such that u i P I for any i P Θ hb , u j “ Q p u q ǫ p j q . ` p , x q P Ω ǫ p j q . Wedefine the matrix T ˚ j p u, x q : “ e ` ue tj ´ e ´ j λ p ǫ p j q´ q{ ˜ u ` xe ´ j,j “p ź i P Θ hb zt˘ j u T ij p u i qq T ´ j p Q p u q ǫ p j q . ` p , x q . ` p , u ´ j ´ λ ǫ p j q ¯ u ´ j qqP EU p I, Ω q . Instead of T ˚ j p u, q we may write T ˚ j p u q . Clearly T ˚ j p u q ´ “ T ˚ j p´ u q (note that˜ uu “ tr p Q p u qq “ Q p u q P Ω Ď ∆ max ) and σ T ˚ j p u q “ e ` σue tj σ ´ ´ σe ´ j λ p ǫ p j q´ q{ ˜ uσ ´ “ e ` λ p´ ǫ p j q´ q{ σu Ć σ ˚ , ´ j ´ λ p ǫ p j q´ q{ σ ˚ , ´ j Ă σu (4)for any σ P U p R, ∆ q , the last equality by Lemma 16. Theorem 41.
Let p I, Ω q be an odd form ideal and H a subgroup of U p R, ∆ q nor-malised by EU p I, Ω q . Then EU p U p H q ˚ I q ď H ď CU pp R, ∆ q , U p H qq . Proof.
The inclusion H Ď CU pp R, ∆ q , U p H qq holds by Proposition 32. It remainsto show the inclusion EU p U p H q ˚ I q Ď H . Recall that if U p H q “ p J, Σ q , then U p H q ˚ I “ p J I , Ω JI min . ` Σ ˝ I q . Because of Lemma 39 it suffices to show that T i p α q P H for any i P Θ hb and α P p Σ ˝ I q ´ ǫ p i q , or equivalently T i p α ´ ǫ p i q q P H for any i P Θ hb and α P Σ ˝ I . (5)Recall that Σ “ Ω J min . ` Z ˝ R where Z “ t Q p σ ˚ s q , p Q p σ ˚ q . ´ p , qq ˝ y . ` p y, z q . ´ p y, z q ˝ σ ss | σ P H, s P Θ hb , p y, z q P ∆ u . In order to prove (5) it suffices to show (i) and (ii) below (note that Ω J min ˝ I Ď Ω JI min ).(i) T i pp Q p σ ˚ s q ˝ a q ´ ǫ p i q q P H for any σ P H , i, s P Θ hb and a P I .(ii) T i pppp Q p σ ˚ q . ´ p , qq ˝ y . ` p y, z q . ´ p y, z q ˝ σ ss q ˝ a q ´ ǫ p i q q P H for any σ P H , i, s P Θ hb , p y, z q P ∆ and a P I .We first show (i) and then (ii).(i) Let σ P H . In Step 1 below we show that T i pp Q p σ ˚ s q ˝ σ ss a q ´ ǫ p i q q P H forany i, s P Θ hb , i ‰ ˘ s and a P I . In Step 2 we use Step 1 to show that T i pp Q p σ ˚ s q ˝ a q ´ ǫ p i q q P H for any i, s P Θ hb , i ‰ ˘ s and a P I . In Step 3 we useStep 2 to prove (i). Step 1.
Let r, s P Θ ` , r ‰ s and b P I . Set u : “ e ´ r σ s, ´ s ´ e ´ s σ s, ´ r “ e ´ r ¯ σ ss ´ e ´ s ¯ σ rs P M u : “ σ ´ u b P M . Then clearly u ´ s “
0. Moreover, u i P I for any i P Θ hb and Q p u q “ Q p σ ´ u b q “ Q p σ ´ u q ˝ b P Ω I min since Q p u q “ σ ´ preserves Q modulo ∆. Hence the matrices T ˚ , ´ s p u q and T ˚ , ´ s p´ u q are defined and arecontained in EU p I, Ω q (see Definition 40). Set ξ : “ σ T ˚ , ´ s p´ u q p q “ e ´ σu Ă σ ˚ s ` σ ˚ s ¯ λ Ă σu “ e ´ u b Ă σ ˚ s ` σ ˚ s ¯ λ Ă u b. Choose a t P Θ hb such that t ‰ ˘ r, ˘ s . Set τ : “ T tr p´ σ ts σ ss ¯ b q T ts p σ ts σ rs ¯ b q and ζ : “ ξτ . Note that by Lemma 38 we have τ P H . With a little effort onecan check that ζ t ˚ “ e t ˚ , ζ ˚ , ´ t “ e ˚ , ´ t and ζ ˚ r “ e r ` p σ ˚ s ´ e t σ ts q σ ss ¯ b ` e ´ s p ¯ σ rs ¯ σ ´ r,s bλ ´ σ ts σ ss ¯ b ¯ σ rs ¯ σ ´ t,s bλ p ǫ p t q` q{ q` e ´ r p´ ¯ σ ss ¯ σ ´ r,s bλ ` σ ts σ ss ¯ b ¯ σ ss ¯ σ ´ t,s bλ p ǫ p t q` q{ q . Let c P I . Clearly p T ˚ , ´ s p u q , ζ q T rt p´ c q ÝÝÝÝÑ p φ, ψ q where φ “r T ˚ , ´ s p´ u q , T rt p´ c qs“ T s p , ´ u t ¯ u ´ r c ` u t ¯ u ´ r c ¯ λ q T s, ´ r p ¯ λ ¯ u t ¯ c q T st p´ ¯ u ´ r c q and ψ “r T rt p´ c q , ζ s“ T rt p´ c q T ˚ t p ζ ˚ r c q“ T ˚ t pp ζ ˚ r ´ e ˚ r q c, λ p ǫ p t q´ q{ ¯ cζ ´ r,r c q“p ź i P Θ hb zt˘ t u T it pp ζ ir ´ δ ir q c qq T ´ t p , ζ ´ t,r c ´ λ ǫ p t q ζ ´ t,r c q¨¨ T ´ t p Q pp ζ ˚ r ´ e ˚ r q c q ǫ p t q . ` p , λ p ǫ p t q´ q{ ¯ cζ ´ r,r c qqq . It follows from Lemma 36 that φψ P H (since T ˚ , ´ s p u q ζ “ r T ˚ , ´ s p u q , σ s τ P H ).Clearly φψ “ T s p , ´ u t ¯ u ´ r c ` u t ¯ u ´ r c ¯ λ q T s, ´ r p ¯ λ ¯ u t ¯ c q T st pp ζ sr ´ ¯ u ´ r q c q¨¨ p ź i P Θ hb zt s, ˘ t u T it pp ζ ir ´ δ ir q c qq T ´ t p , ζ ´ t,r c ´ λ ǫ p t q ζ ´ t,r c q¨¨ T ´ t p Q pp ζ ˚ r ´ e ˚ r q c q ǫ p t q . ` p , λ p ǫ p t q´ q{ ¯ cζ ´ r,r c qqq . (6)We leave it to the reader to deduce from Lemma 38 that all the factors on theright hand side of Equation (6) except the last one are contained in H (cf. theproof of Lemma 39). Since φψ P H , it follows that T ´ t p Q pp ζ ˚ r ´ e ˚ r q c q ǫ p t q . ` p , λ p ǫ p t q´ q{ ¯ cζ ´ r,r c qq P H. Q pp ζ ˚ r ´ e ˚ r q c q ǫ p t q . ` p , λ p ǫ p t q´ q{ ¯ cζ ´ r,r c q “ p Q p σ ˚ s q ˝ σ ss ¯ bc q ǫ p t q . ` p , y ´ ¯ yλ ǫ p t q q for some y that lies in the ideal generated by σ ´ r,s ¯ bc and σ ts ¯ bc . It follows that T ´ t pp Q p σ ˚ s q ˝ σ ss ¯ bc q ǫ p t q q P H . Thus we have shown that T ´ t pp Q p σ ˚ s q ˝ σ ss a q ǫ p t q q P H for any s P Θ ` , t P Θ hb , t ‰ ˘ s and a P I . Analogously one can show that T ´ t pp Q p σ ˚ s q ˝ σ ss a q ǫ p t q q P H for any s P Θ ´ , t P Θ hb , t ‰ ˘ s and a P I . Step 2.
Let i, s P Θ hb , i ‰ ˘ s and a P I . By Lemma 28 we have T i pp Q p σ ˚ s q ˝ a q ´ ǫ p i q q“ T i p Q p σ ˚ s q ´ ǫ p i q ˝ a q“ T i p Q p σ ˚ s q ´ ǫ p i q ˝ ´ ÿ p “ σ sp σ ps a q“ ´ ź p “ T i p Q p σ ˚ s q ´ ǫ p i q ˝ σ sp σ ps a q loooooooooooooooomoooooooooooooooon X : “ ¨¨ ´ ź p,q “ ,p ą q T i p , σ sp σ ps aQ p σ ˚ s q σ sq σ qs a ´ σ sp σ ps aQ p σ ˚ s q σ sq σ qs aλ ´ ǫ p i q q loooooooooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooooooooon Y : “ . By Step 1, T i p Q p σ ˚ s q ´ ǫ p i q ˝ σ ss σ ss a q P H . All the other factors of X are alsocontained in H (see the proof of Lemma 39). Similarly Y is contained in H . Itfollows that T i pp Q p σ ˚ s q ˝ a q ´ ǫ p i q q P H . Step 3
By Step 2 we know that T i pp Q p σ ˚ s q ˝ a q ´ ǫ p i q q P H for any i, s P Θ hb , i ‰ ˘ s and a P I . In order to show (i) it remains to show that T i pp Q p σ ˚ s q ˝ a q ´ ǫ p i q q P H for any s P Θ hb , i P t˘ s u and a P I . (7)Let s, r P Θ hb , s ‰ ˘ r , b P I and set τ : “ r σ, T sr p b qs P H . Applying Step 2 to τ we obtain T i pp Q p τ ˚ r q ˝ c q ´ ǫ p i q q P H for any i P t˘ s u and c P I . We leave it tothe reader to deduce from [2, Lemma 63] that T i pp Q p σ ˚ s q ˝ bcσ rr q ´ ǫ p i q q P H forany i P t˘ s u and c P I . It follows as in Step 2 that T i pp Q p σ ˚ s q ˝ bc q ´ ǫ p i q q P H for any i P t˘ s u and c P I . Thus (7) holds.(ii) Let σ P H , s P Θ hb , p y, z q P ∆ and b P I . Set ρ : “ r σ, T s ppp y, z q ˝ b q ´ ǫ p s q qs P H .By Step 2 in the proof of (i) above, we have T i pp Q p ρ ˚ , ´ s q ˝ c q ´ ǫ p i q q P H for any18 ‰ ˘ s and c P I . We leave it to the reader to deduce from [2, Lemma 63]that T i ppp Q p σ ˚ q . ´ p , qq ˝ yσ s, ´ s . ` p y, z q ˝ σ s, ´ s . ´ p y, z qq ˝ bc q ´ ǫ p i q q P H for any i ‰ ˘ s and c P I . It follows as in Step 2 that T i ppp Q p σ ˚ q . ´ p , qq ˝ y . ` p y, z q . ´ p y, z q ˝ σ ´ s, ´ s q ˝ bc q ´ ǫ p i q q P H (8)for any i ‰ ˘ s and c P I (cf. the proof of Lemma 29). Let r P Θ hb . Onechecks easily that p y, z q ˝ σ ´ s, ´ s “ p y, z q ˝ σ rr . ` p y, z q ˝ p σ ´ s, ´ s ´ σ rr q . ` α (9)where α “ p , σ ´ s, ´ s ´ σ rr zσ rr ´ σ ´ s, ´ s ´ σ rr zσ rr λ q . Clearly T i pppp y, z q ˝ p σ ´ s, ´ s ´ σ rr q . ` α q ˝ bc q ´ ǫ p i q q P H. (10)It follows from (8), (9) and (10) that T i ppp Q p σ ˚ q . ´ p , qq ˝ y . ` p y, z q . ´ p y, z q ˝ σ rr q ˝ bc q ´ ǫ p i q q P H (11)for any i ‰ ˘ s and c P I . Since (11) holds for any s, r P Θ hb , we have shown(ii). Theorem 42.
Let p I, Ω q be an odd form ideal and H a subgroup of U p R, ∆ q nor-malised by EU pp R, ∆ q , p I, Ω qq . Then EU pp R, ∆ q , U p H q ˚ I q ď H ď CU pp R, ∆ q , U p H qq . Proof.
By Proposition 32 we only have to show that EU pp R, ∆ q , U p H q˚ I q ď H . Let τ P EU p R, ∆ q . Then clearly H τ is normalised by EU pp R, ∆ q , p I, Ω qq . By Corollary34 we have U p H τ q “ U p H q . It follows from the previous theorem that EU p U p H q ˚ I q ď H τ whence EU p U p H q ˚ I q τ ´ ď H . Since this holds for any τ P EU p R, ∆ q ,we obtain EU pp R, ∆ q , p U p H q ˚ I qq ď H . Theorem 43.
Suppose that H ⊳ d G where d is a positive integer and G a subgroupof U p R, ∆ q containing EU p R, ∆ q . Suppose U p H q “ p I, Ω q . Then EU pp R, ∆ q , U p H q ˚ I k q ď H ď CU pp R, ∆ q , U p H qq where k “ d ´ ´ .Proof. By Proposition 32 we only have to show that EU pp R, ∆ q , U p H q ˚ I k q ď H .We proceed by induction on d . d “
1: If d “
1, then H ⊳ G and therefore H is normalised by EU p R, ∆ q . It fol-lows from the previous theorem that EU pp R, ∆ q , U p H qq ď H as desired (we use the19onvention I “ R ). d Ñ d `
1: Suppose H ⊳ d ` G , i.e. H “ H ⊳ H ⊳ ¨ ¨ ¨ ⊳ H d ⊳ H d ` “ G forsome subgroups H , . . . , H d of G . Write U p H q “ p I, Ω q and U p H q “ p J, Σ q . By theinduction assumption we haveEU pp R, ∆ q , U p H q ˚ J k q ď H ď CU pp R, ∆ q , U p H qq (12)where k “ d ´ ´
1. Since H ⊳ H , it follows that H is normalised by EU pp R, ∆ q ,U p H q ˚ J k q . Hence EU pp R, ∆ q , U p H q ˚ J p k ` q q ď H, by the previous theorem. It follows from (12) that H ď CU pp R, ∆ q , U p H qq whence U p H q Ď U p H q . Therefore I Ď J and thus we obtainEU pp R, ∆ q , U p H q ˚ I p k ` q q ď H. One checks easily that 12 p k ` q “ d ` ´ ´ U n ` p R, ∆ q where n ě In this section n denotes an integer greater than or equal to 4 and p R, ∆ q a Hermitianform ring where R is commutative. We start by proving an analogue of Lemma 37. Lemma 44.
Let p I, Ω q be an odd form ideal and H a subgroup of U n ` p R, ∆ q nor-malised by EU n ` p I, Ω q . Let σ P H and r, s, t P Θ hb such that r ‰ ˘ s and t ‰ ˘ r, ˘ s .Then(i) T ij p σ s, ´ t ¯ σ sr a q P H for any i, j P Θ hb , i ‰ ˘ j, a P I and(ii) T ij p σ rs ¯ σ rr a q P H for any i, j P Θ hb , i ‰ ˘ j, a P I .Proof. Let a , . . . , a P I . Set τ : “ T rt p σ ss ¯ σ sr ¯ a q T st p´ σ sr ¯ σ sr ¯ a q T r, ´ s p´ λ p ǫ p t q´ ǫ p s qq{ σ s, ´ t ¯ σ sr a q¨¨ T r p , λ p ǫ p t q´ ǫ p r qq{ σ s, ´ t ¯ σ ss a ´ λ p´ ǫ p t q´ ǫ p r qq{ σ ss ¯ σ s, ´ t ¯ a q and τ : “ T rt p σ rs ¯ σ rr a q T st p´ σ rr ¯ σ rr a q T r, ´ s p´ λ p ǫ p t q´ ǫ p s qq{ σ r, ´ t ¯ σ rr ¯ a q¨¨ T r p , λ p ǫ p t q´ ǫ p r qq{ σ r, ´ t ¯ σ rs ¯ a ´ λ p´ ǫ p t q´ ǫ p r qq{ σ rs ¯ σ r, ´ t a q . Clearly τ , τ P EU n ` p I, Ω q . Set ξ : “ στ ´ σ ´ and ξ : “ στ ´ σ ´ . One checkseasily that p στ ´ q s ˚ “ σ s ˚ and p τ ´ σ ´ q ˚ , ´ s “ σ , ´ s . Hence p ξ q s ˚ “ e s ˚ and p ξ q ˚ , ´ s “ ˚ , ´ s . Similarly p στ ´ q r ˚ “ σ r ˚ and p τ ´ σ ´ q ˚ , ´ r “ σ , ´ r . Hence p ξ q r ˚ “ e r ˚ and p ξ q ˚ , ´ r “ e ˚ , ´ r . A straightforward computation shows that p τ , ξ q T ´ s,t p a q ,T ´ s,r p a q ,T tu p´ λ p ǫ p s q´ ǫ p t qq{ a q ÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÑ p T ´ s,u p σ s, ´ t ¯ σ sr a a a a q , e q for any u ‰ ˘ s, ˘ t , and p τ , ξ q T ´ s,u p a q ,T ´ s,r p a q ,T uv p´ λ p ǫ p s q´ ǫ p t qq{ a q ÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÑ p T ´ s,v p σ s, ´ t ¯ σ sr a a a a q , e q for any u ‰ ˘ r, ˘ s, ˘ t and v ‰ ˘ s, ˘ u . It follows from Lemma 36 that H containsthe matrices T ´ s,u p σ s, ´ t ¯ σ sr a a a a q where u ‰ ˘ s . It is now an easy exercise toobtain (i) (see the proof of Lemma 35).On the other hand one checks easily that p τ , ξ q T sr p a q ,T tu p a q ,T vs p´ a q ÝÝÝÝÝÝÝÝÝÝÝÝÝÝÑ p T vu p σ rs ¯ σ rr a a a a q , e q for any u ‰ ˘ r, ˘ s, ˘ t and v ‰ r, ˘ s, ´ t, ˘ u . It follows from Lemma 36 that T vu p σ rs ¯ σ rr a a a a q P H for any u ‰ ˘ r, ˘ s, ˘ t and v ‰ r, ˘ s, ´ t, ˘ u. (13)Choose p, q P Θ hb such that p ‰ ˘ q and p, q ‰ ˘ r, ˘ s . By applying (13) with t “ ˘ p and u “ ˘ q resp. t “ ˘ q and u “ ˘ p one obtains T ˘ p, ˘ q p σ rs ¯ σ rr a a a a q , T ˘ q, ˘ p p σ rs ¯ σ rr a a a a q P H. (14)It is an easy exercise to deduce (ii) from (14) (see the proof of Lemma 35).Lemmas 45 and 46 below are analogues of Lemmas 38 and 39, respectively. Lemma 45.
Let p I, Ω q be an odd form ideal and H a subgroup of U n ` p R, ∆ q nor-malised by EU n ` p I, Ω q . Then T ij p σ rs a q P H for any i, j P Θ hb , i ‰ ˘ j , σ P H , r, s P Θ hb , r ‰ ˘ s and a P I .Proof. Choose a t P Θ hb such that t ‰ ˘ r, ˘ s . Let a P I and set τ : “ r σ ´ , T tr p´ ¯ σ rs ¯ a qs P H. Let J be the involution invariant ideal generated by the set t a σ rs ¯ σ rr , a σ rs ¯ σ r, ˘ t u .Clearly τ tt “ ´ σ tt ¯ σ rs ¯ a σ rt ` σ t, ´ r λ p ǫ p r q´ ǫ p t qq{ σ rs a σ ´ t,t “ ´ σ tt ¯ σ rs ¯ a σ rt looomooon P J ` λ ´ ǫ p t q ¯ σ r, ´ t σ rs a loooomoooon P J σ ´ t,t ” J and τ tr “ ´ σ tt ¯ σ rs ¯ a σ rr ` σ t, ´ r λ p ǫ p r q´ ǫ p t qq{ σ rs a σ ´ t,r ` τ tt ¯ σ rs ¯ a “ σ tt ¯ σ rs ¯ a σ rr looomooon P J ` λ ´ ǫ p t q ¯ σ r, ´ t σ rs a loooomoooon P J σ ´ t,r ` τ tt ¯ σ rs ¯ a ” ¯ σ rs ¯ a mod J
21y Lemma 14. Hence τ tt ¯ τ tr ” σ rs a mod J . By Lemma 44(ii) we have T ij p τ tr ¯ τ tt a q P H for any i, j P Θ hb , i ‰ ˘ j, a P I whence T ij p τ tt ¯ τ tr a q P H for any i, j P Θ hb , i ‰ ˘ j, a P I (because of Relation (S1) in Lemma 18). Since τ tt ¯ τ tr a “ σ rs a a ` xa for some x P J , we obtain the assertion of the lemma in view of Lemma 44. Lemma 46.
Let p I, Ω q be an odd form ideal and H a subgroup of U n ` p R, ∆ q nor-malised by EU n ` p I, Ω q . Let p J, Σ q “ U p H q . Then EU n ` p J I , Ω JI min q ď H .Proof. Let σ P U n ` p R, ∆ q and i, j, r, s P Θ hb such that i ‰ ˘ j and r ‰ ˘ s . Fur-thermore, let x, y P J p ∆ q . The previous lemma implies that(i) T ij p σ rs a q P H for any a P I ,(ii) T ij p σ r, ´ r a q P H for any a P I ,(iii) T ij p σ r xa q P H for any a P I ,(iv) T ij p ¯ xµσ s a q P H for any a P I ,(v) T ij pp σ rr ´ σ ss q a q P H for any a P I ,(vi) T ij pp σ rr ´ σ ´ s, ´ s q a q P H for any a P I and(vii) T ij p ¯ xµ p σ ´ σ ss q ya q P H for any a P I ,cf. the proof of Lemma 39. It follows that EU n ` p J I , Ω JI min q ď H in view ofRelations (S5) and (SE2) in Lemma 18. Theorem 47.
Let p I, Ω q be an odd form ideal and H a subgroup of U n ` p R, ∆ q normalised by EU n ` p I, Ω q . Then EU n ` p U p H q ˚ I q ď H ď CU n ` pp R, ∆ q , U p H qq . Proof.
The inclusion H Ď CU n ` pp R, ∆ q , U p H qq holds by Proposition 32. It remainsto show the inclusion EU n ` p U p H q ˚ I q Ď H . Because of Lemma 46 it suffices toshow (i) and (ii) below.(i) T i pp Q p σ ˚ s q ˝ a q ´ ǫ p i q q P H for any σ P H , i, s P Θ hb and a P I .(ii) T i pppp Q p σ ˚ q . ´ p , qq ˝ y . ` p y, z q . ´ p y, z q ˝ σ ss q ˝ a q ´ ǫ p i q q P H for any σ P H , i, s P Θ hb , p y, z q P ∆ and a P I .The proof of (i) and (ii) above is essentially the same as the proof of (i) and (ii) inthe proof of Theorem 41 (of course one uses Lemmas 45 and 46 instead of Lemmas38 and 39, respectively). Theorem 48.
Let p I, Ω q be an odd form ideal and H a subgroup of U n ` p R, ∆ q normalised by EU n ` pp R, ∆ q , p I, Ω qq . Then EU n ` pp R, ∆ q , U p H q ˚ I q ď H ď CU n ` pp R, ∆ q , U p H qq . roof. See the proof of Theorem 42.
Theorem 49.
Suppose that H ⊳ d G where d is a positive integer and G a subgroupof U n ` p R, ∆ q containing EU n ` p R, ∆ q . Suppose U p H q “ p I, Ω q . Then EU n ` pp R, ∆ q , U p H q ˚ I k q ď H ď CU n ` pp R, ∆ q , U p H qq where k “ d ´ ´ .Proof. See the proof of Theorem 43.
Remark 50.
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