Commuting Isometries of the Complex Hyperbolic Space

Let $H^n$ denote the complex hyperbolic space of dimension $n$. The group $U(n,1)$ acts as the group of isometries of $H^n$. In this paper we investigate when two isometries of the complex hyperbolic space commute. Along the way we determine the centralizers.


Introduction
Let H n C denote the complex hyperbolic space of dimension n. The group U (n, 1) acts as the group of isometries of H n C . In this paper we ask when two isometries of the complex hyperbolic space commute. A related problem is to determine the centralizers. For the two dimensional complex hyperbolic space, Basmajian-Miner [1, Corollaries-8. 2-8.3] determined the conditions for two isometries to commute. In arbitrary dimension, a partial answer to this question was given by Kamiya [5,]. However, the answer of Kamiya was far from being complete. In this research note, we classify the elements which commute with a given isometry, and along the way determine the centralizers.
Let V be a vector space over C of dimension n + 1. Let V be equipped with the standard complex Hermitian form of signature (n, 1). Let V − denote the set of all vectors with negative length under this Hermitian form. The n-dimensional complex hyperbolic space H n C is the image of V − in the projective space P (V). The isometry group U (n, 1) of the Hermitian space V acts as the isometries of H n C . The actual group of the isometries is P U (n, 1) = U (n, 1)/Z(U (n, 1)), where the center Z(U (n, 1)) can be identified with the circle group. Thus an isometry g of H n C lifts to a unitary transformationg in U (n, 1) and in the projective model of H n C , the fixed points of g correspond to eigenvectors ofg. In the following, we shall often forget the lift, and shall use the same symbol for an isometry, as well as its lifts. An eigenvalue λ of g ∈ U (n, 1) is called negative if it has an eigenvector in V − . In the ball model of the hyperbolic space, by Brouwer's fixed point theorem it follows that every isometry g has a fixed point on the closure H n C . An isometry g is called elliptic if it has a fixed point on H n C . It is called parabolic, resp. hyperbolic if it is non-elliptic, and has exactly one, resp. two fixed points on the boundary ∂H n C . The conjugacy classification of the isometries of H n C is known essentially from the work of Chen-Greenberg [2, Theorem 3.4.1]. It follows from the conjugacy classification that the elliptic and hyperbolic elements are semisimple, i.e. their minimal polynomial is a product of linear factors. The parabolic elements are not semisimple.
Let g be elliptic. It follows from the conjugacy classification that all the eigenvalues have norm 1, and it has a negative eigenvalue. If the negative eigenvalue has multiplicity at least 2, then it is called a boundary elliptic. Otherwise, g is a regular elliptic, i.e. all the eigenvalues are distinct. Suppose g is hyperbolic. Then it has a complex eigenvalue outside the unit circle, and one eigenvalue inside the unit circle. The other eigenvalues lie on the circle, i.e. they have norm one. If all the eigenvalues of T are real, then g is called a strictly hyperbolic. Otherwise, it is called a loxodromic. Note that a loxodromic can be written as g = g r g e , where g r is a stretch, and g e is a boundary elliptic. Suppose g is parabolic. If g is unipotent, i.e. all the eigenvalues are 1, it is called a translation. A translation g is called vertical, or non-vertical, according as the minimal polynomial is (x − 1) 2 , or (x − 1) 3 . If g is a non-unipotent parabolic, then it has the Jordan decomposition g = g s g u , where g s is semisimple, g u is unipotent in U (n, 1), and g s and g u commute. A nonunipotent parabolic is also called an ellipto-parabolic. A non-vertical translation keeps invariant a unique two dimensional totally geodesic real subspace, which is called the axis.
A parabolic element which fixes the point '∞' on the boundary, is called a Heisenberg translation. Two Heisenberg translations (τ, t), (σ, s) ∈ N n = C n−1 × R are called isotropic if Im(τ * σ) = 0. The vertical Heisenberg translations are isotropic with any Heisenberg translation. The isotropy between two non-vertical Heisenberg translations implies that the angle, cf. Goldman [3, p-36], between the axes is a non-zero real, and geometrically they are parallel. Suppose S and T are two non-vertical translations of H n C . Then they can be simultaneously conjugated to two non-vertical Heisenberg translations. The axes of S and T are parallel if the axes of the corresponding Heisenberg translations are parallel. It will be clear in Section 3 that the parallelism of the axes, or, the isotropy between two translations, is invariant under conjugation. Our main theorem is the following. (ii) S is translation with S(x) = x, S(f ix(T s )) = f ix(T s ) and S and T u are isotropic.
(iii) S is an ellipto-parabolic with fixed point x and, it has the Jordan decomposition S = S s S u such that T u (f ix(S s )) = f ix(S s ), S u (f ix(T s )) = f ix(T s ), T u and S u are isotropic, and, T s preserves each eigenspace of S s .
A useful corollary to the above theorem is the description of the centralizers, up to conjugacy. Following Kulkarni [6], two elements in a group G are said to be in the same z-class if their centralizers are conjugate in in G. The z-classes provide useful stratification of the group G, and provide better insight about the nature of the action of G on any set X, cf. Kulkarni [6]. The z-classes of the real hyperbolic space have been classified by Gongopadhyay-Kulkarni [4]. The z-classes of the isometries of the complex hyperbolic space is obtained from the above theorem.
where T s , T u are the semisimple and the unipotent part of T in its Jordan decomposition.
In Section 2, we review the models of H n C , and the Heisenberg similarity transformations. In Section 3, several properties of the Heisenberg similarities are given. Especially, we show that a unipotent isometry and an elliptic isometry commute if and only if they mutually fix their fixed point sets. This is crucial to the proof of our main theorem. In Section 4, we prove our main theorem, and, the Corollary 1.2 is established in Section 4.3.

Preliminaries
In this section we give the necessary background material on complex hyperbolic space. More extensive facts may be found in [2,3,7].
2.1. The Complex Hyperbolic Space.

The Siegel Domain
Model. This model for H n C is obtained in the same way as above except that the Hermitian form on V under consideration is The two models are related by a Cayley transformation, cf. [2,3,7]. The two Hermitian forms are referred as first and second Hermitian forms respectively. When there is no confusion, we shall forget the subscript from the Hermitian forms, and it should be clear from the context. A unitary transformation g is an automorphism of C n,1 , that is, a linear bijection such that g(z), g(w) = z, w for all z and w in C n,1 . We denote the group of all unitary transformations by U (n, 1).
The isometry group of H n C is P U (n, 1) = U (n, 1)/Z(U (n, 1)). However, since the elements of U (n, 1) are linear, it is convenient to deal with U (n, 1) rather than P U (n, 1).
If g ∈ U (n, 1). Hence in the Siegel domain model, for all z and w in C n,1 . Letting z and w vary over a basis for C n,1 , we see that J = g * Jg. From this we find g −1 = J −1 g * J. That is: where A is an (n − 1) × (n − 1) matrix, α, β, γ, δ are column vectors in C n−1 , and a, b, c, d ∈ C.

2.2.
The Heisenberg Similarity Transformations. The (2n − 1)-dimensional Heisenberg group N n is C n−1 × R with the group law ). The boundary of H n C may be identified with the one-point compactification of the Heisenberg group. We define horospherical coordinates on the complex hyperbolic space as follows.
ψ : We also define the origin o to be the point in ∂H n C with horospherical coordinates (0, 0, 0), that is ψ(o) = (0, · · · , 0, 1) t . In this way the ∂H n C is identical with N n × {0} ∪ ∞. The unitary group U (n − 1) acts on horospherical coordinates as Heisenberg rotation, whose action is given by The corresponding matrix in U (n, 1) acting on C n,1 is The fixed point set of such a map is a totally geodesic subspace of H n C passing through o and ∞.
The positive real numbers r ∈ R + act as Heisenberg dilation D r on horospherical coordinates, whose action is given by D r : (ζ, v, u) → (rζ, r 2 v, r 2 u).
As a matrix D r is given by The (τ, t) ∈ N n acts as Heisenberg translation on horospherical coordinates, whose action is given by As a matrix T (τ,t) is given by When τ = 0, T (τ,t) is a vertical Heisenberg translation. Otherwise, it is a nonvertical Heisenberg translation. It is easy to see that the subgroup T form by vertical Heisenberg translations is the center of N n and T ≈ R. A parabolic element is called screw parabolic if it is the product of a Heisenberg translation and a Heisenberg rotation. When n ≥ 3, we will show that a screw parabolic which is the product of a non-vertical Heisenberg translation and a Heisenberg rotation can not be conjugated to a screw parabolic which is the product of a vertical Heisenberg translation and Heisenberg rotation in Section 3.
Let U (n, 1) ∞ denote the isotropy group of U (n, 1) at ∞. Then where ⋉ denotes the semidirect product of groups. Thus every element g in U (n, 1) ∞ can be uniquely written as g = D r R U T (τ,t) , where R U is the elliptic part, D r the hyperbolic part and T (τ,t) the unipotent part of this Heisenberg isometry. Elements in U (n, 1) ∞ are called the Heisenberg similarity transformations. If D r is the identity in the above decomposition, g is called a Heisenberg isometry. It is easy to see that elements g fixing o and ∞ are of the form whereμλ = 1 and A ∈ U (n − 1).

Commuting elements of Heisenberg translations
Throughout this section, we use the second Hermitian form of the hyperbolic space.
Lemma 3.1. Let T, S ∈ P U (n, 1). If S commutes with T then By matrix computation, we have the following observation Using this we have the following.  Proof. Suppose that U τ = τ and T is elliptic. Then T is boundary elliptic, and by its action on horospherical coordinate there exists point (ζ, v, u) in H n C such that U (τ + ζ) = ζ, t + 2Im(ζ * τ ) = 0.
Hence we have that This implies that τ = 0 and t = 0. This contradiction implies T is parabolic.
Remark 3.4. Note that T = R U T (τ,t) with U τ = τ and R U = I n−1 can also be conjugated to the form R U T (τ1,t1) such that U τ 1 = τ 1 . We give the reason as follows. Let V 1 (U ) = {x ∈ C n−1 |U x = x}. For τ = 0, U τ = τ , by linear algebraic theory, the matrix equation (U − I)x = τ has no solution x. This is σ + τ − U σ = 0. When n ≥ 3, there exists nonzero σ / ∈ V 1 (U ). For such σ, T (σ,s) T T −1 (σ,s) is The above two conjugations can not make the unipotent part of T = R U T (τ,t) to be vertical Heisenberg translation. The above observations imply that, when n ≥ 3, we can not conjugate a screw parabolic T = R U T (τ,t) with U τ = τ = 0 to the form of R U ′ T (0,t ′ ) . By the three conjugations (3.2), (3.3), (3.4), we know that the definition of isotropic in Section 1 is well defined. We also mention that a boundary elliptic element R v T (σ,s) can be conjugated in U (n, 1) ∞ to diagonal form.
By the above, it also follows that, for n ≥ 3, there are two types of screw parabolic elements T = R U T (τ,t) with U τ = τ , τ = 0 and τ = 0, respectively.
In what follows, we concern about which element can commute with a unipotent parabolic element T . By conjugation, we may assume that T fixes ∞, that is T = R U T (τ,t) . By Lemma 3.1, if S commutes with T , then we must have If S is hyperbolic, then by T (f ix(S)) = f ix(S), we get a contradiction. This implies that S is Heisenberg isometry which can be written uniquely as S = R V T (σ,s) .  Proof. By Lemma 3.1, we only need to prove the necessity. Without loss of generality, let T = T (τ,t) and S = R V T (σ,s) with V σ = σ. The action of S on horospherical coordinate (ζ, v, u) is This implies that V τ = τ, Im(τ * σ) = 0. By Lemma 3.2, this implies that ST = T S.

Proof of the Main Theorem
In this section, we shall prove our main theorem case by case. Mainly using Lemma 3.2 and the proof of our main theorem, we shall determine the centralizers up to conjugacy. 4.1. The semisimple cases. To classify the commuting elements of the semisimple isometries, we use the first Hermitian form , . It is customary to call a subspace Suppose T is semisimple, i.e. elliptic or hyperbolic. Suppose S commutes with T . Let λ be an eigenvalue of T and let V λ be the eigenspace to λ.
Let λ and µ be two distinct eigenvalues of T .
thus ifλµ = 1, then V λ and V µ are orthogonal to each-other.
(1) Let T be elliptic. Suppose each eigenspace of T is S-invariant. We claim that S commutes with T . For, suppose the distinct eigenvalues of T are e iθj , j = 1, ..., s, where e iθ1 is assumed to be time-like and the rest are space-like. For each j, let V j be the eigenspace to e iθj . Then V has the orthogonal decomposition: V = ⊕ s j=1 V j . Let Q j denote the Hermitian form , | Vj . Since S is an isometry of V, each V j is S-invariant, hence S preserves the orthogonal decomposition and also the restricted metric on each of the summands. Thus S commutes with T if and only if S| Vj commutes with T | Vj . Let Q j denote the restriction of Q to V j . Since T | Vj = e iθj I Vj , it is a central element in U (V j , Q j ). Hence any S| Vj in U (V j , Q j ) commute with T | Vj . This establishes the claim.
(2). Suppose T is hyperbolic. Let L r , resp, L r −1 denote the one-dimensional eigenspaces corresponding to the light-like eigenvalues r, resp. r −1 . Let V r = L r + L r −1 . Let V j denote the eigenspace of T corresponding to the space-like eigenvalue e iθj . In this case V has the orthogonal decomposition: where V r = L r + L r −1 . Clearly S keeps V r invariant. For j = r, 1, ..., s, Let Q j denote the Hermitian form , | Vj . As above, we see that S| Vj commutes with T | Vj for each j = 1, ..., s. Hence S commutes with T if and only if S| Vr commutes with T | Vr . Since S keeps the eigenspaces of T invariant, it keeps the light-like lines L r and L r −1 invariant. Hence S| Vr commutes with T | Vr if and only if it is either a strictly hyperbolic or a central element in U (V r , Q r ). Consequently the assertion follows.

4.2.
Parabolic case. We shall use the Siegel domain model, i.e. the second Hermitian form to deal with the parabolic case. Let T be a parabolic isometry. Let S commute with T . Suppose S is semisimple. It can not be hyperbolic: for otherwise, T would have two fixed points on the boundary. So S must be elliptic. We assert that S can not be a regular elliptic. For otherwise, T fixes the unique fixed point of S, which is impossible. Hence S must be a boundary elliptic. By conjugation, we assume its unique fixed point to be ∞, i.e. T is a Heisenberg isometry. If T is ellipto-parabolic, it has the unique Jordan decomposition T = T s T u = T u T s , where T s and T u are its semi-simple and unipotent components, respectively. From the Jordan decomposition, it follows that an element commutes with T if and only if it commutes with both T s and T u . Now the assertions follow from Lemma 3.1, Lemma 3.5 and Lemma 3.5.
This completes the proof of the main theorem.

Elliptic elements.
Let T be elliptic. Then we have seen that V has an orthogonal decomposition: V = ⊕ s j=1 V j , where for each j, V j is the eigenspace to e iθj , and θ 1 be the negative eigenvalue. Let dim V j = r j . For each j, T | Vj is a central element in U (V j , Q j ),which we identify with U (r 1 − 1, 1), resp. U (r j ) for j = 1 resp. > 1. Hence Hyperbolic elements. Let T be hyperbolic. Then V has the orthogonal decomposition V = V r ⊕ s j=1 V j , where V r is the 2-dimensional subspace spanned by the light-like eigenvectors, and V j are the eigenspaces spanned by space-like eigenvectors to eigenvalues e iθj , j = 1, .., s. Hence Z(T ) = Z(T | Vr ) × ΠZ(T | Vj ). Let dim V j = r j . As above, we identify it with U (r j ). Note that T | Vr acts as a hyperbolic element in U (V r , Q r ) ≡ U (1, 1), hence Z(T | Vr ) = S 1 × R. Hence
By direct computation, following the approach of Chen-Greenberg [2, section-4.2], one can see that every element S in Z(T ) can be written uniquely as S = ABP , A ∈Ŝ 1 , B ∈Û (n − 1), P ∈ N n .
By direct computation, one can see that every element S in Z(T ) can be written uniquely as S = BP , where B ∈Ê a , P ∈ N . Further we see thatÊ a can be identified with the group S 1 × U (n − 1). Hence, up to conjugation in U (n, 1), We identify C n−1 = R 2(n−1) , and let , o be the usual Euclidean norm on R 2(n−1) . Then a, c o = r is a (2n − 3)-dimensional affine hyperspace orthogonal to a, and hence N/T ≈ R × R 2(n−2) ≈ R × C n−2 . Thus Hence