Reversible Quaternionic Hyperbolic Isometries
aa r X i v : . [ m a t h . G T ] N ov REVERSIBLE QUATERNIONIC HYPERBOLIC ISOMETRIES
SUSHIL BHUNIA AND KRISHNENDU GONGOPADHYAY
Abstract.
Let G be a group. An element g ∈ G is called reversible if it isconjugate to g − within G , and called strongly reversible if it is conjugate to g − by an order two element of G . Let H n H be the n -dimensional quaternionichyperbolic space. Let PSp( n,
1) be the isometry group of H n H . In this paper, weclassify reversible and strongly reversible elements in Sp( n ) and Sp( n, n,
1) are strongly reversible. Introduction
Let G be a group. An element g ∈ G is called reversible if g − = xgx − for some x ∈ G . The terminology ‘real’ has also been used extensively in the literature to referthe reversible elements, for example, see [Ell77], [Ell83], [EFN84], [FZ82], [KN87b],and [KN87a], [ST05], [ST08], [TZ05], [Won66]. A non-trivial element g ∈ G is calledan involution if g = 1. An element g ∈ G is called strongly reversible if g − = xgx − for some x ∈ G with x = 1. Clearly, a strongly reversible element is reversible. Anelement g is strongly reversible if and only if g can be written as a product of twoinvolutions. Every element of a conjugacy class which contains a reversible (resp.strongly reversible) element is reversible (resp. strongly reversible), i.e., reversibility(resp. strongly reversibility) is a conjugacy invariant.Reversible and strongly reversible group elements have been studied in manycontexts. In classical group theory, a theorem of Frobenius and Schur states that if G is finite, the number of real-valued complex irreducible characters of G equals thenumber of reversible conjugacy classes of G . This has influenced a lot of researchon reversibility from an algebraic point of view. However, the origin of research Date : November 15, 2019.2010
Mathematics Subject Classification.
Primary 51M10; Secondary 15B33, 20E45.
Key words and phrases. hyperbolic space, quaternions, reversible elements, real elements,involutions, strongly reversible, strongly real.Bhunia is supported by a SERB-NPDF (No. PDF/2017/001049) during the course of this work.Gongopadhyay acknowledges partial support from SERB-DST MATRICS project:MTR/2017/000355. on ‘reversibility’ can be traced back to works in classical dynamics and classicalgeometry. We refer the reader to the book by O’Farrel and Short [OS15] for anextensive account of reversibility in geometry and dynamics.The motivation of the present work comes from the investigations related to thereversibility in classical geometries. Let PO( n,
1) denote the full isometry group ofthe n -dimensional real hyperbolic space H n R and let PO o ( n,
1) denote the identitycomponent, which is the group of orientation preserving isometries of H n R . It is well-known that every element of PO( n,
1) is strongly reversible, e.g. [OS15, Theorem6.11]. The reversible elements in PO o ( n,
1) have been classified in [Gon11], andin [Sho08] using a different approach, and also see [LOS07]. We refer to [OS15,Chapter 6] for an extensive treatment of reversibility in Euclidean, spherical, and realhyperbolic geometries. It follows from these works that an element g in PO o ( n, n,
1) and SU( n,
1) actas the holomorphic isometries of the n -dimensional complex hyperbolic space H n C .Reversible elements in these groups were classified in [GP13]. It follows from thiswork that an element g in U( n,
1) is reversible if and only if it is strongly reversible.An element g in the full isometry group PU( n,
1) is reversible (resp. strongly re-versible) if and only if a certain lift of g in U( n,
1) is reversible (resp. stronglyreversible), for more details, see [GP13, Theorem 4.5].The investigation of strong reversibility in PO o ( n,
1) and PU( n,
1) is related tothe broader problem of finding the involution length in the respective groups. Therelation between strong reversibility and involution length in any group G is that:every element of G is strongly reversible if and only if the involution length of G is 2.The involution length of PO o ( n,
1) is 2 or 3 depending explicitly on the congruenceclass of n mod 4, which is obtained by Basmajian and Maskit in [BM12, Theorem1.1]. Precise involution length for PU( n,
1) is not known for arbitrary n . Theinvolution length of PU(2 ,
1) is 4 (see [PW17, Theorem 1]) and when n ≥ n,
1) is at most 8 (see [PW17, Theorem 2]).Even though the decompositions of isometries into involutions in the above groupsare known to some extent, not much has been known for their quaternionic coun-terpart Sp( n, n, H n H bethe n -dimensional quaternionic hyperbolic space. Let Isom( H n H ) denote the isometrygroup which consists of all the isometries of H n H , which is isomorphic to PSp( n, EVERSIBLE QUATERNIONIC HYPERBOLIC ISOMETRIES 3 for n > n = 1, we consider the index two subgroupPSp(1 ,
1) of Isom( H H ), which is the connected component of Isom( H H ). In this pa-per, we give a complete classification of reversible and strongly reversible elementsin Sp( n, Theorem 1.1.
Every element of
Sp( n, is reversible. However, in contrast to the real and complex hyperbolic isometries, reversibilitydoes not imply strong reversibility in Sp( n, n,
1) are not strongly reversible. Let Sp( n ) denote the maximal compactsubgroup of Sp( n, Theorem 1.2.
An element g ∈ Sp( n ) is strongly reversible if and only if everyeigenvalue class of g is either ± or of even multiplicity. The involution length of Sp( n ) is known to be at most 6, for example, see [OS15,Theorem 5.9]. As a corollary to the above theorem, we improve the upper boundof the involution length of Sp( n ). The only involutions in Sp(1) are ±
1. For n ≥ Corollary 1.3.
For n > , every element in Sp( n ) can be expressed as a product offour (resp. five) involutions if n is even (resp. n is odd). We apply the above theorem to give the following characterization of strongly re-versible elements in Sp( n, n,
1) are classified into two types as elliptic and hyperbolic .An element g ∈ Sp( n,
1) which is not semisimple is called parabolic . However, it hasa Jordan decomposition g = g s g u , where g s is elliptic, hence semisimple, and g u isunipotent. In particular, if a parabolic isometry is unipotent, then it has all eigen-values 1. A unipotent parabolic with minimal polynomial ( x − (resp. ( x − )is called a vertical translation (resp. a non-vertical translation ). For details, see thenext section. For terminologies of the next theorem, we refer to Definition 2.4. Theorem 1.4.
Suppose g is an element of Sp( n, . (1) Let g be hyperbolic. Then g is strongly reversible if and only if every posi-tive eigenvalue class of g is either ± or of even multiplicity, and the nulleigenvalues of g are real numbers. SUSHIL BHUNIA AND KRISHNENDU GONGOPADHYAY (2)
Let g be elliptic. Then g is strongly reversible if and only if the eigenvalueof negative or indefinite type of g is or − and every positive eigenvalueclass of g is either ± or of even multiplicity. (3) Let g be a vertical translation. Then g is not strongly reversible. (4) Let g be a non-vertical translation. Then g is strongly reversible. (5) Let g = g s g u be non-unipotent parabolic. Then g is strongly reversible if andonly if the null eigenvalue of g is or − , the minimal polynomial of g u is ( x − , and every positive eigenvalue class of g is either ± or of evenmultiplicity. However, when we project to the full isometry group, we have strong reversibilityfor every element.
Theorem 1.5.
Every element of
PSp( n, is a product of two involutions.Structure of the paper. In Section 2, we cover the preliminaries. In Section 3, weexplore the reversible and strongly reversible elements in Sp( n ) and PSp( n ). InSection 4, we give a complete description of the reversible and strongly reversibleelements in Sp( n,
1) and we prove one of our main Theorem 1.1 of this paper. Lasttwo sections are devoted to the proof of Theorem 1.4 and Theorem 1.5 respectively.2.
Preliminaries
In this section, we fix some notations and terminologies which will be usedthroughout this paper. Let H = R ⊕ R i ⊕ R j ⊕ R ij be the real quaternions. Weidentify the subspace R ⊕ R i of H with the standard complex plane in H .2.1. Matrices over quaternions.
Let V be an n -dimensional right vector spaceover H . Let T be a right linear transformation of V . Then T is represented by an n × n matrix over H . Invertible linear maps of V are represented by invertible n × n quaternionic matrices. The group of all such linear maps is denoted by GL( n, H ).For more details on linear algebra over quaternions, see [Rod14]. In the following,we briefly recall the notions that will be used later on.Let T ∈ GL( n, H ) and v ∈ V, λ ∈ H × , are such that T ( v ) = vλ , then for µ ∈ H × we have T ( vµ ) = ( vµ ) µ − λµ. Therefore eigenvalues of T occur in similarity classes and if v is a λ -eigenvector,then vµ ∈ v H is a µ − λµ -eigenvector. Thus the eigenvalues are no more conjugacyinvariants for T , but the similarity classes of eigenvalues are conjugacy invariant. EVERSIBLE QUATERNIONIC HYPERBOLIC ISOMETRIES 5
Note that each similarity class of eigenvalues contains a unique pair of complexconjugate numbers. Often we shall refer them as ‘eigenvalues’, though it should beunderstood that our reference is towards their similarity classes. In places wherewe need to distinguish between the similarity class and a representative, we shalldenote the similarity class of an eigenvalue representative λ by [ λ ].2.2. The group
Sp( n, . In this section, we are following Chen-Greenberg [CG74].Let V := H n, = H n +1 be an ( n + 1)-dimensional right vector space over H equippedwith a H -Hermitian formΦ( z, w ) = − ¯ z w + ¯ z w + · · · + ¯ z n w n , where z = ( z , z , . . . , z n ) , w = ( w , w , . . . , w n ) ∈ H n +1 . Therefore the matrixrepresentation of Φ with respect to the standard basis { e , e , . . . , e n } of H n +1 is J = diag( − , , . . . , symplectic group of signature ( n,
1) isSp( n,
1) = { g ∈ GL( n + 1 , H ) | t ¯ gJ g = J } . If we restrict the H -Hermitian form Φ on the orthogonal complement of the one-dimensional subspace e H , then the linear transformations preserving the restrictedform is the following groupSp( n ) := { g ∈ GL( n, H ) | t ¯ gg = I n } , which is a compact subgroup of GL( n, H ). Restricting the positive-definite quater-nionic Hermitian form over the standard complex subspace C n ⊂ H n , we get a copyof the compact unitary group as a subgroup of Sp( n ), and we denote it by U( n ).Thus U( n ) := { g ∈ GL( n, C ) ⊂ GL( n, H ) | t ¯ gg = I n } . Remark 2.1.
Let P ∈ M( n, H ), then P = A + Bj , where A, B ∈ M( n, C ). Let ϕ : M( n, H ) ֒ → M(2 n, C )defined by ϕ ( P ) = ϕ ( A + Bj ) = A B − ¯ B ¯ A ! . Observe that the above map ϕ is aninjective group homomorphism from GL( n, H ) to GL(2 n, C ). Observe that, • ϕ ( I n j ) = I n − I n ! =: β . • ϕ ( P ∗ ) = ϕ ( P ) ∗ . • P is symplectic if and only if φ ( P ) is unitary. SUSHIL BHUNIA AND KRISHNENDU GONGOPADHYAY
So, we have Sp( n ) ∼ = U(2 n ) ∩ Sp(2 n, C ) , where Sp(2 n, C ) := { g ∈ GL(2 n, C ) | t gβg = β } . Here U(2 n ) and Sp( n ) are compactgroups and defined over R . The following definition will be used later. Definition 2.2.
Let g ∈ Sp( n ). Let distinct eigenvalues of g be represented by e iθ , e iθ , . . . , e iθ m , m ≤ n . The right vector space H n has the following orthogonaldecomposition into eigenspaces: H n = V θ ⊕ V θ ⊕ · · · ⊕ V θ m , where V θ l = { v ∈ H n | gv = ve iθ l } for 1 ≤ l ≤ m . We define multiplicity of e iθ l := dim (V θ l ). Equivalently, it is the repetition of the eigenvalue e iθ l in a diagonalform, up to conjugacy, of g .2.3. Hyperbolic space H n H . DefineV := { v ∈ V | Φ( v, v ) = 0 } , V + := { v ∈ V | Φ( v, v ) > } , V − := { v ∈ V | Φ( v, v ) < } . Let P (V) be the quaternionic projective space, i.e., P (V) = V r { } / ∼ , where u ∼ v if there exists λ ∈ H × such that u = vλ . Here P (V) is equipped withthe quotient topology and the quotient map is Π : V r { } → P (V). The n -dimensional quaternionic hyperbolic space is defined to be H n H := Π(V − ). Theboundary ∂ H n H in P (V) is Π(V ). The isometry group Sp( n,
1) acts by isometrieson H n H . The actual group of isometries of H n H is PSp( n,
1) = Sp( n, / Z (Sp( n, Z (Sp( n, {± I n +1 } is the center. Thus an isometry g of H n H lifts to a sym-plectic transformation ˜ g ∈ Sp( n, g correspond to eigenvectorsof ˜ g . Definition 2.3.
A subspace W of V is called space-like , light-like , or indefinite if theHermitian form restricted to W is positive-definite, degenerate, or non-degeneratebut indefinite respectively. If W is an indefinite subspace of V, then the orthogonalcomplement W ⊥ is space-like. Definition 2.4.
An eigenvalue λ of an element g ∈ Sp( n,
1) is called positive type (resp. negative type ) if every eigenvector in V λ is in V + (resp. V − ). The eigenvalue λ is called null if the eigenspace V λ is degenerate. The eigenvalue λ is called indefinitetype if the eigenspace V λ contains vectors in V + and V − . EVERSIBLE QUATERNIONIC HYPERBOLIC ISOMETRIES 7
Accordingly, a similarity class of eigenvalues [ λ ] is null , positive or negative ac-cording to its representative λ is null, positive or negative respectively.2.4. Cayley transform.
If we change the standard basis by { b e , b e , . . . , b e n } , where b e := e − e √ , b e := e + e √ and b e l := e l for 2 ≤ l ≤ n , then we get a change of basismatrix, which is the following P := / √ / √ − / √ / √ I n − . Then define C := P − , which is called the Cayley transform . Now b J := t P J P = − − I n − . Therefore the corresponding H -Hermitian form is b Φ( z, w ) = − (¯ z w + ¯ z w ) + ¯ z w + · · · + ¯ z n w n . The corresponding symplectic group is c Sp( n,
1) = { g ∈ GL( n + 1 , H ) | t ¯ g b J g = b J } . Therefore we have c Sp( n,
1) = P − Sp( n, P , since b J = t P J P . The Hermitian form b Φ gives the
Siegel domain model of the quaternionic hyperbolic space.2.5.
Classification of isometries.
The non-identity elements of Sp( n,
1) can beclassified into three disjoint classes depending on their fixed points. An isometry g is called elliptic if it has a fixed point in H n H . It is called parabolic if it has exactlyone fixed point on the boundary ∂ H n H , and is called hyperbolic (or loxodromic ) if ithas exactly two fixed points on the boundary ∂ H n H . Notice that, if two elements areconjugate, then they have the same (elliptic, parabolic, or hyperbolic) type. Lemma 2.5. (Chen-Greenberg )[CG74, Theorem 3.4.1](1)
Two elliptic elements are conjugate if and only if they have the same negativeclass of eigenvalues, and the same positive classes of eigenvalues. (2)
Two loxodromic elements are conjugate if and only if they have the samesimilarity classes of eigenvalues. (3)
Two parabolic elements are conjugate if and only if their semisimple (elliptic)and unipotent components are conjugate.
SUSHIL BHUNIA AND KRISHNENDU GONGOPADHYAY
The following follows from the conjugacy classification in c Sp( n, Lemma 2.6.
Let g be a parabolic element in c Sp( n, . Then, up to conjugacy, g isone of the following forms: (1) g = λu V B ! , where B ∈ Sp( n − , and u V = s ! ∈ c Sp(1 , with s + ¯ s = 0 , and | λ | = 1 . (2) g = λu NV B ! , where B ∈ Sp( n − , and u NV = s aa ∈ c Sp(2 , where a, s ∈ H × with s + ¯ s = | a | , and | λ | = 1 . Reversible and Strong Reversible Elements in
PSp( n )3.1. Reversible elements in
Sp( n ) . Part (1) of the following result is known, see[OS15, Theorem 5.7]. We will give a shorter and different proof that will imply part(2) of the theorem that seems unavailable in the literature.
Proposition 3.1. (1)
Every element of
Sp( n ) is reversible. (2) Every element of
PSp( n ) is strongly reversible.Proof. We know that every element of Sp( n ) is semisimple. So up to conjugacy wecan write g = diag( λ , λ , . . . , λ n ) , where λ k ∈ C with | λ k | = 1 for all k = 1 , , . . . , n . Now observe that jλ k = ¯ λ k j forall k = 1 , , . . . , n . Therefore g − = hgh − , where h = diag( j, j, . . . , j ) ∈ Sp( n ). Hence every element of Sp( n ) is reversible.Now observe that h = − I n , so every element of PSp( n ) is strongly reversible. (cid:3) In [OS15, Chapter 5], O’Farrell and Short pose the problem of characterizingstrongly reversible elements in Sp( n ). The following result answers this problem.But before proving it, we recall the concept of projective points from [GK19, Section3]. EVERSIBLE QUATERNIONIC HYPERBOLIC ISOMETRIES 9
Projective points.
Let g ∈ Sp( n ). Let λ ∈ H \ R be a chosen eigenvalue of g in the similarity class [ λ ]. We may assume that [ λ ] has multiplicity one, i.e., the[ λ ]-eigenspace has dimension one. Thus, we can identify the eigenspace V [ λ ] with H .Consider the λ - eigenset : S λ = { x ∈ V | gx = xλ } . Note that this set is the complexline x Z H ( λ ) in H . Note that for q ∈ H \ R , S qλq − = S λ q .Identify H with C . Two non-zero quaternions q and q are equivalent if q = q c , c ∈ C \
0. This equivalence relation projects H to the one-dimensional complexprojective space CP . Thus, V [ λ ] = ⊔ q ∈ CP S qλq − . The CP associated to [ λ ] in thisway will be called the [ λ ]-projective sphere.3.3. Proof of Theorem 1.2.
Suppose g = diag ( λ , λ , . . . , λ n ) is strongly re-versible. Then g − = hgh − for some h ∈ Sp( n ) such that h = I n . Then we havethe following orthogonal decomposition: H n = V [ λ ] ⊕ V [ λ ] ⊕ · · · ⊕ V [ λ n ] , where dimV [ λ l ] = 1 for all l = 1 , , . . . , n . Note that V [ λ l ] = ⊔ q ∈ CP S qλ l q − . Now, if v is an eigenvector of g in the λ l -eigenset S λ l , then g − ( h ( v )) = hg ( v ) = h ( v ) λ l , i.e., g ( h ( v )) = h ( v ) λ − l . Therefore, h ( v ) is an eigenvector of g in the λ − l -eigenset S λ − l . Thus, either h maps V [ λ l ] to V [ λ t ] with λ t = λ l for some t , or, h preserves V [ λ l ] and permutes the S λ l ’s inside V [ λ l ] . In the first case [ λ ] is of even multiplicity. Inthe second case, h is an involution on V [ λ l ] , and acts as an orientation-preservinginvolution on the [ λ l ]-projective sphere. Hence, h | V [ λl ] is ±
1, that implies λ l = ± g is conjugate toan element ˜ g of U( n ) that has the property that if λ = ± λ − . Consequently, it is strongly reversible in U( n ). Hence g is strongly reversiblein Sp( n ).This completes the proof.3.4. Proof of Corollary 1.3.
Let g ∈ Sp( n ). Up to conjugacy, we can choose g tobe the diagonal element g = λ λ . . . λ n , where λ l ’s are complex numbers with | λ l | = 1. First, we prove this result when n iseven, say n = 2 m ( m ≥ g = α α , where(3.1) α = diag( λ , λ , λ λ λ , λ λ λ , . . . , λ λ · · · λ k +1 , λ λ · · · λ k +1 , . . . )(3.2) α = diag(1 , λ λ , λ λ , . . . , λ λ · · · λ k , λ λ · · · λ k , . . . ) . Since λ j and ¯ λ j belong to the same similarity class, by the previous theorem, we seethat both α and α are strongly reversible in Sp( n ). Thus, g can be written as aproduct of four involutions.When n is odd, say n = 2 m + 1 = 2( m −
1) + 3 ( m ≥ g is semisimple,then the right vector space H n has an orthogonal decomposition into g -invariantsubspaces: H n = U ⊕ W with dim U = 3 and dim W = n −
3. Then g | U canbe considered as an element in Sp(3) and g | W as an element in Sp( n −
3) and g = g | U ⊕ g | W . Now from a result of Djokovi´c and Malzan, see [DM79, Theorem3], the involution length of Sp(3) is at most five, i.e., g | U can be written as aproduct of five involutions, say g | U = r r r r r , where r l = 1 for l = 1 , , , , g | W ∈ Sp(2( m − g | W = s s s s , where s k = 1 for k = 1 , , ,
4. Hence g = ( r ⊕ s )( r ⊕ s )( r ⊕ s )( r ⊕ s )( r ⊕ I m − ), where r k ⊕ s k are involutions.Therefore the involution length of Sp(2 m + 1) is at most 5.This completes the proof.4. Reversible Elements in
Sp( n, c Sp( n, Lemma 4.1 (Vertical translation) . Let u V := s ! ∈ c Sp(1 , , where s ∈ Im( H ) .Then u V is reversible but not strongly reversible.Proof. Let s = s i + s j + s ij ∈ Im( H ), then s = q ( ri ) q − , where r = p s + s + s and q = ( r + s ) − s j + s ij (see [CG74, Lemma 1.2.2]).Therefore we have s ! = x x ! ri ! x x ! − , EVERSIBLE QUATERNIONIC HYPERBOLIC ISOMETRIES 11 where x = q | q | (observe that x x ! ∈ c Sp(1 ,
1) as | x | = 1). Now we have ri ! − = j j ! ri ! j j ! − . Therefore we have u − V = h V u V h − V , where h V = xjx − xjx − ! . Hence u V is reversible.Now if there exists an h = a bc d ! ∈ c Sp(1 ,
1) such that λi ! − = a bc d ! λi ! a bc d ! − , then h has the following form h = a c ¯ a − ! , where ¯ ac + ¯ ca = 0 and iai = ¯ a − . If possible suppose that h is an involution, i.e., h = I , then a = ±
1. This is a contradiction, since a / ∈ R . Hence u V is not stronglyreversible. (cid:3) Remark 4.2.
Observe that in the above proof h V = − I . Lemma 4.3.
Let g = λ λs λ ! ∈ c Sp(1 , be an arbitrary parabolic element, where λ, s ∈ C with | λ | = 1 and Re( s ) = 0 . Then g is reversible.Proof. Without loss of generality, we can assume s = ri with r ∈ R × and λ = e iθ .Now pick h = j j ! ∈ c Sp(1 , h gh − = g − . (cid:3) Remark 4.4.
Observe that in the above proof h = − I . Lemma 4.5 (Non-vertical translation) . Let u NV = s aa ∈ c Sp(2 , , where s, a ∈ H × with s + ¯ s = | a | . Then the element u NV is strongly reversible. Inparticular, u NV is reversible.Proof. Choose an involution h NV in c Sp(2 ,
1) such that h NV = | d | d − ¯ d − , where d is chosen so that s + da is a real number. By direct computation, one cancheck that h NV u NV h − NV = u − NV . This completes the proof. (cid:3) Lemma 4.6.
Let g = λ λs λ λ ¯ aλa λ ∈ c Sp(2 , be an arbitrary parabolic element,where s, a ∈ R ( λ ) , | λ | = 1 and s + ¯ s = | a | . Then g is reversible.Proof. Without loss of generality we may assume that a, s, λ ∈ C with λ = e iθ .Now pick h = j ij j
00 0 − ij ∈ c Sp(2 , h gh − = g − . (cid:3) Remark 4.7.
Observe that in the above proof h = − I . Lemma 4.8.
Let g be a hyperbolic element in c Sp(1 , . Then g is reversible. Further, g is strongly reversible if and only if both eigenvalues of g are real numbers.Proof. Let g = re iθ r − e iθ ! , where r > ≤ θ ≤ π . Then g − = hgh − ,where h = jj ! ∈ c Sp(1 , g is reversible.For the second part, let g be a strongly reversible element. Then g − = hgh − forsome h ∈ c Sp(1 ,
1) with h = I . Suppose the eigenvalues of g are not real numbers. EVERSIBLE QUATERNIONIC HYPERBOLIC ISOMETRIES 13
Then we can assume θ = 0 , π . Now let h = a bc d ! . Then we get a = r e iθ ae iθ (4.1) b = e iθ be iθ (4.2) c = e iθ ce iθ (4.3) d = r − e iθ de iθ . (4.4)From Equation (4.1) we get a = a + a i + a j + a ij = r e iθ ( a + a i + a j + a ij ) e iθ ,which implies that a = 0 = a as r = 1. If a = 0 (resp. a = 0), then 1 = r e iθ ,which implies that θ = 0 or π/ π . But θ = 0 , π so θ = π/
2, which implies that r = − a = 0. Similarly, from Equation (4.4), we get d = 0.From Equation (4.3), we have c = c + c i + c j + c ij = e iθ ( c + c i + c j + c ij ) e iθ which implies that c = 0 = c as θ = 0 , π .Again, we have bc = 1 = cb = ¯ cb since h = I and h ∈ c Sp(1 , c = ¯ c ,i.e., c ∈ R \ { } . This is a contradiction to the fact that c = c j + c ij . Thereforeboth eigenvalues of g are real numbers.Conversely, if all the eigenvalues of g are real numbers, then θ = 0 or π , i.e., g = r r − ! or − r − r − ! . By direct computation, we get g − = hgh − , where h = ! ∈ c Sp(1 ,
1) such that h = I . Therefore, g is strongly reversible. (cid:3) Proof of Theorem 1.1. (i)
Let g be elliptic in Sp( n, g is semisimple with eigenvalues of norm 1. It has n similarity classesof positive eigenvalues (which may not all be distinct) and one similarityclass of negative eigenvalue (which may coincide with one of the positiveclasses), for details, see [CG74, Lemma 3.2.1 and Proposition 3.2.1]. So, upto conjugacy, we can assume g = diag( λ , λ , . . . , λ n ) , where λ k ∈ C with | λ k | = 1 for k = 0 , , . . . , n . Since jλ k = λ k j for λ k ∈ C then we have g − = hgh − , where h = diag( j, j, . . . , j ) ∈ Sp( n, n,
1) is reversible. Note that h = − I n +1 in Sp( n, g ∈ c Sp( n, Let g be hyperbolic . Let λ be the (null) eigenvalue class of g with | λ | > g -invariant orthogonal subspaces: V =U ⊕ W, where U is the direct sum of the one-dimensional eigenspaces V λ andV λ − and W is the space-like orthogonal complement to U. The Hermitianform restricted to U has the signature (1 , g | U can be consideredas a transformation in c Sp(1 ,
1) and g | W as an element in Sp( n − g = g | U ⊕ g | W . Now it follows from Lemma 4.8 and Proposition 3.1 that g − = hgh − , where h = h h ! with h = jj ! and h = I n − j = diag( j, j, . . . , j ).Note that h = − I n +1 in c Sp( n, Let g be a translation . Again from conjugation classification, there are ex-actly two conjugacy classes of unipotent parabolic elements. One is thevertical translation, denoted by U V = u V I n − ! , with minimal poly-nomial ( x − . The other one is the non-vertical translation, denotedby U NV = u NV I n − ! , whose minimal polynomial is ( x − . There-fore we have U − V = H V U V H − V , where H V = h V I n − ! and h V is de-fined in Lemma 4.1. Also, we have U − NV = H NV U NV H − NV , where H NV = h NV I n − ! and h NV is described in Lemma 4.5. Therefore unipotentelements are reversible.(iv) Let g be parabolic . From the conjugation classification, see [CG74, Propo-sition 3.4.1], we know that g ∈ c Sp( n,
1) has the Jordan decomposition g = g s g u = g u g s , where g s is a unique elliptic element and g u is a uniqueunipotent parabolic element. If g is parabolic, then H n, has a g -invariantorthogonal decomposition: H n, = U ⊕ W, where dim U = 2 or 3, g | U is EVERSIBLE QUATERNIONIC HYPERBOLIC ISOMETRIES 15 indecomposable, i.e., U cannot be written as a sum of g -invariant subspaces,and g | W acts on W as an element of Sp( n −
1) or Sp( n − λ represents the null eigenvalue of g , then g has minimal polynomial ( x − λ ) l ,where l = 2 or 3. Then, up to conjugacy, g = g g ! , where g ∈ c Sp(1 ,
1) or c Sp(2 ,
1) and g ∈ Sp( n −
1) or Sp( n − g − = h ′ gh ′− , where h ′ = h h ! or h ′ = h h ! with h as inLemma 4.3, h as in Lemma 4.6 and h as in Proposition 3.1. Note that h ′ = − I n +1 in c Sp( n, Proof of Theorem 1.4 (1) Suppose g is hyperbolic. Let λ be the (null) eigenvalue class of T with | λ | > g -invariant orthogonal subspaces: V = U ⊕ W,where U is the direct sum of the one-dimensional eigenspaces V λ and V λ − and Wis the space-like orthogonal complement to U. The Hermitian form restricted to Uhas the signature (1 , g | U can be considered as a transformation in c Sp(1 , g | W as an element in Sp( n − g is elliptic. Then g has a negative or indefinite type eigenvalue λ .Let L λ be a one-dimensional subspace spanned by the corresponding eigenvector.In the orthogonal complement L ⊥ λ , g restricts to an element in Sp( n ). Thus, up toconjugacy, we may consider g as: g = λ g ! , where g ∈ Sp( n ). It is easy to seethat the only strongly reversible elements of Sp(1) are 1 and −
1. The result nowfollows from Theorem 1.2.(3) Follows from Lemma 4.1, and (4) follows from Lemma 4.5.(5) Suppose g is parabolic. Then g has the Jordan decomposition g = g s g u , where g s is semisimple, g u is unipotent, and g s g u = g u g s . If g is strongly reversible,then clearly g s and g u are strongly reversible. The null eigenvalue λ of g will be anegative eigenvalue for g s , and hence the assertion necessarily follows from (2) andthe unipotent cases. Conversely, suppose the hypothesis holds. If g is parabolic, then H n, has a g -invariant orthogonal decomposition: H n, = U ⊕ W, where, dim U = 2 or 3, g | U isindecomposable, i.e., U cannot be written as a sum of g -invariant subspaces, and g | W acts on W as an element of Sp( n −
1) or Sp( n − λ representsthe null eigenvalue of g , then g has minimal polynomial ( x − λ ) l , where l = 2 or 3.The given hypothesis implies that g | U and g | W are strongly reversible. Hence g isstrongly reversible.This completes the proof.6. Proof of Theorem 1.5
From the proof of Theorem 1.1, we see that for g elliptic or hyperbolic, g = hg − h − , where h = − I n +1 in Sp( n, h = I n +1 in PSp( n, g isstrongly reversible in PSp( n, g a vertical translation, choose H ′ V = h V I n − j ! in c Sp( n, h V isdefined in Lemma 4.1. Hence H ′ V = − I n +1 projects to the identity element. Wehave already seen in Lemma 4.5 that a non-vertical translation is strongly-reversible.Consequently, every translation is strongly reversible in PSp( n, n,
1) follows from thelast part of the proof of Theorem 1.1, and Lemma 4.3 and Lemma 4.6.This completes the proof.
Acknowledgement:
The authors would like to thank Sagar B. Kalane of IISERMohali for many helpful discussions. The authors thank the referee for many helpfulcomments and suggestions.
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