Journal of Lie Theory Volume 8 (1998) 255–277 ◯^{C}1998 Heldermann Verlag

Automorphisms and quasi-conformal mappings of Heisenberg-type groups

Paolo Emilio Barbano

Communicated by K. H. Hofmann

Abstract. The Lie algebras of trace-zero derivations of Heisenberg-type groups are explicitly characterized, along with the connected component of their groups of measure preserving automorphisms. We establish a general criterion on properties of the stabilizer of a lattice in a simply connected nilpotent Lie group and apply it to the full family of H -type Lie groups. A necessary condition for the existence of non-conformal quasi-conformal mappings on H -type groups is also given.

1. Introduction and Background

In this article we study some properties of Lie groups called of Heisenberg type which were first introduced by A. Kaplan ([10]) as a generalization of the Heisenberg group itself. We describe the structure of their automorphisms as well as some properties of their lattices.

A Heisenberg type Lie group N (or H -type group) is a connected and simply connected two-step nilpotent Lie group such that its commutator subgroup satisfies: [N, N] = Z(N) and such that on its Lie algebra 𝔑 there is a positive definite real quadratic form Q(·) = ⟨·, ·⟩ which is compatible with the natural decomposition,

𝔑 = ℨ ⊕ 𝔙  (1)

where ℨ is the center of 𝔑 and 𝔙 is its orthocomplement with respect to ⟨·, ·⟩. Here compatibility refers to Kaplan’s basic assumption that the family of operators

{ad(X) : X ∈ 𝔙 | ⟨X, X⟩ = 1}  (2)

consists of partial isometries of ker (ad(X))^{⊥} onto ℨ. Kaplan refers to this fact as property H. A. Korányi ([11]) has characterized the homogeneous left invariant Carnot-Carathéodori metrics on these manifolds. He has also shown that H -type groups can always be equipped with such a left invariant homogeneous metric. It is important to note that H -type groups, together with their solvable extensions

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag A ⋉ N by the one-dimensional groups of dilations have provided counterexamples to several conjectures in differential and spectral geometry (see, e.g. [3], [8], [19]).

The existence of the quadratic form gives rise to two other equivalent algebraic descriptions of H -type groups. Given an H -type Lie algebra 𝔑 = ℨ⊕𝔙 with m = dim(ℨ), consider the map J : ℨ → End(𝔙) defined, by:

⟨J_{Z}(V), V′⟩ = ⟨Z, [V, V′]⟩  Z ∈ ℨ, V, V′ ∈ 𝔙.  (3)

By straightforward computation one can prove that, given an orthonormal basis {e_{i}} of ℨ, the family {J_{e_{i}}} satisfies the relations of the generators of the Clifford algebra Cl(−Q, ℨ) = C_{m}. In other words J : ℨ → End(𝔙) extends to a unitary representation of C_{m} on the orthocomplement of the center 𝔙. Thus we say that 𝔙 carries a structure of so called Clifford module over C_{m} (see [1], pgg. 22 ff.): ℨ acts on 𝔙 as a set of linear transformations satisfying, for any two orthogonal e_{i}, e_{j} ∈ ℨ of unit norm:

J_{e_{i}}J_{e_{j}} + J_{e_{j}}J_{e_{i}} = 0 and J_{e_{i}}^{2} = −Id

The other description of H -type algebras can be given by using compositions of quadratic forms. Let (R, q_{R}) and (S, q_{S}) be two real quadratic spaces. Given a normalized composition of their quadratic forms μ : R × S → S, we can define a Lie algebra as follows. Consider the dual map ϕ : S × S → R identified by the linear system

⟨r, ϕ(s, s′)⟩ = ⟨μ(r, s), s′⟩,  r ∈ R, s, s′ ∈ S,  (4)

and an element r_{0} satisfying μ(r_{0}, s) = s for all s ∈ S. Let π be the orthogonal projection onto ℝr_{0}^{⊥}. If we choose r ∈ π(R) and r′ = r_{0}, the identity

⟨μ(r, s), μ(r′, s)⟩ = ⟨r, r′⟩q_{S}(s).

implies

⟨μ(r, s), s⟩ = ⟨r, ϕ(s, s)⟩ = 0.

Thus the new map [·, ·] = π ∘ ϕ(·, ·) is skew symmetric: given (r_{1}, s_{1}) and (r_{2}, s_{2}) in π(R) ⊕ S we define write:

[(r_{1}, s_{1}), (r_{2}, s_{2})] = (π ∘ ϕ(s_{1}, s_{2}), 0)  (5)

The last equations amounts to defining a two-step nilpotent Lie algebra structure on 𝔑 = π(R) ⊕ S. π(R) becomes the center and S its orthocomplement. By computation one shows that such an 𝔑 satisfies property (H) and thus it is a Lie algebra of Heisenberg type with the map J in (3) given by: J_{r}(s) = μ(r, s) (see [10]).

An H -type algebra 𝔑 is said to be irreducible if the Clifford module identified by the map J is irreducible. Given any Clifford algebra with m generators there is up to equivalence one or possibly two (when m = 3, 7 mod(8)) irreducible Clifford modules associated to it. The theory of quadratic forms provides a complete classification of all Clifford modules. A detailed account on the construction of these modules is given in [13] (Chapter I, sec. 5). The fact that dim(ℨ) determines the dimension of 𝔙 and that two inequivalent Clifford modules of the same dimension are associated to isomorphic H -type algebras allows us to classify all of them (see [1], pg. 23). A new explicit realization of irreducible H -type Lie algebras for 0 ≤ dim(ℨ) ≤ 7 is given in the next paragraph. In the general case one observes that, since Clifford modules are completely reducible, any H -type Lie algebra 𝔑 is decomposable as: 𝔑 = ℨ ⊕ 𝔙_{1} ⊕ ⋯ 𝔙_{k} where all the H -type subalgebras ℨ ⊕ 𝔙_{i} are irreducible. H -type Lie algebras arise in a very natural way: consider a simple Lie group G of real-rank one. Classification tells us that, either it belongs to one of the families SO_{0}(n, 1), SU_{0}(n, 1) and Sp(n, 1), or it is isomorphic to the exceptional group F_{4,20}. Such a group admits an Iwasawa decomposition: G = K ·A·N. Korányi ([11]) has proved that 𝔑 = Lie(N) is always of Heisenberg type: 𝔑 can be resp. ℝ^{n−1}, the classical 2(n − 1) + 1 dimensional Heisenberg group 𝔑_{1}^{n−1} or its quaternionic and octonionic analogues 𝔑_{3}^{n−1} and 𝔑_{7}.

2. Real, Complex, Quaternionic, and Cayley algebras

By making use of the quadratic forms characterization of H -type algebras we will prove a new basic result that makes explicit computations considerably easier. Fix n and i, positive integers, i ≤ 7 and let K^{n} be the quadratic space ℂ^{n}, ℍ^{n} or 𝕆^{n} where 𝕆 are the Cayley numbers. The real vector space K^{n} carries the scalar product: ⟨\underline{V} , \underline{V′}⟩ = ∑_{k} Re(\underline{V_{k}} · \underline{\overline{V′}_{k}})). Denote with K_{i}^{∗} the quadratic space obtained by restricting the standard quadratic form to an arbitrarily fixed i-dimensional ℝ-subspace of the imaginary elements in K. For each k between 1 and n we choose μ^{k} : K_{i}^{∗} × K → K to be the left or right composition. That is: μ^{k}(Z, X) = Z · X or μ^{k}(Z, X) = X · Z. Given \underline{Y} = (Y_{1}, …, Y_{n}) ∈ K^{n}, we let, with slight abuse of notation, \underline{Y_{k}} = (0, …, 0, Y_{k}, 0, …, 0), Y_{k} ∈ K. Now define the composition of quadratic forms: μ : K_{i}^{∗} × K^{n} → K^{n} by:

μ(X, \underline{Y}) = ∑_{k} μ^{k}(X, \underline{Y_{k}}), ∀X ∈ K_{i}^{∗}, \underline{Y} ∈ K^{n}.

Now set R = ℝ ⊕ K_{i}^{∗}, S = K^{n} and perform the construction discussed in the introduction. The procedure we just described equips the vector space K_{i}^{∗} ⊕ K^{n} with the structure of an H -type algebra.

In order to determine ϕ we proceed as follows: for simplicity we assume n = 1 and μ(Z, X) = Z · X; by (4), with obvious notation,

⟨μ_{i}(Z, X), X′⟩ = ⟨Z, ϕ(X, X′)⟩
Re(ZX · \overline{X′}) = Re(Z · \overline{ϕ(X, X′)}),
Re(Z · X\overline{X}′) = Re(Z · \overline{X′\overline{X}});

the last identity allows to conclude that

ϕ(X, X′) = X′ · \overline{X} ∀X, X′ ∈ K.

In case we choose the right action (μ(Z, X) = X · Z), the Lie bracket becomes: [(Z, X), (Z′, X′)] = (Im_{i}(\overline{X}X′), 0). Note here that while our conclusion is trivial in the associative cases, for K = 𝕆 it can be deduced once the observation is made that the real part of the product of Cayley numbers is in fact associative (cf. e.g. [4], pg. 14).

Proposition 2.1. Each irreducible H -type Lie algebra 𝔑 = ℨ ⊕ 𝔙 with i = dim(ℨ) ≤ 7 is isomorphic to an algebra 𝔑_{i} ≃ K_{i}^{∗} ⊕ K.

Proof. In the case when i = 0, we define 𝔑 ≃ ℝ^{n} and there is nothing to prove. Otherwise, we note that the mapping π ∘ ϕ(·, ·) can be taken as the projection onto K_{i}^{∗} of the standard Hermitian product defined on K × K. If i = 1 we obtain 𝔑 = iℝ ⊕ ℂ and so 𝔑 ≃ 𝔑_{1}, the classical Heisenberg Lie algebra.

In the remaining cases we will prove that we can construct an irreducible 𝔑_{i} by direct computation.

If i = 2, 3 we choose K = ℍ; this way we can construct two quaternionic compositions of quadratic forms: μ_{2} : ℍ_{2}^{∗} × ℍ → ℍ and μ_{3} : ℍ_{3}^{∗} × ℍ → ℍ. Both are defined by the equation:

μ_{i}(X, Y) = X · Y X ∈ ℍ_{i}^{∗}, Y ∈ ℍ, i = 2, 3

We also note that, in the case of i = 3, there is an inequivalent Clifford module corresponding to the composition defined by:

μ′_{3}(X, Y) = Y · X X ∈ ℍ^{∗}, Y ∈ ℍ.

The Lie algebras with these two (left) compositions will be denoted by 𝔑_{2} ≃ ℍ_{2}^{∗}⊕ℍ and 𝔑_{3} ≃ ℍ_{3}^{∗} ⊕ ℍ.

For i = 4, 5, 6 or 7 we choose K = 𝕆; the realizations are exactly the same as for the quaternionic cases. The algebras we get this way are 𝔑_{i} ≃ 𝕆_{i}^{∗} ⊕ 𝕆, 4 ≤ i ≤ 7.

For all the Lie algebras described above the Lie bracket is defined as follows: given two elements (Z, X) and (Z′, X′) in 𝔑_{i}, we have:

ϕ(X, X′) = X′ · \overline{X}

and therefore:

[(Z, X), (Z′, X′)] = (Im_{i}(X′ · \overline{X}), 0) ,

where Im_{i} is the projection onto the space K_{i}^{∗}. In doing so, we get the identity

⟨J_{Z}X, X′⟩ = Re(ZX · \overline{X′}) = Re(Z · \overline{Im_{i}(X′\overline{X})}) = ⟨Z, [X, X′]⟩

as required by our definition (4). To prove irreducibility one has to observe that, since J_{Z}(X) = Z · X, given an orthonormal basis Z_{1}, …, Z_{n} of K and any unit vector X ∈ K we have:

⟨Z_{s} · X, Z_{t} · X⟩ = Re(Z_{s}X · \overline{Z_{t}X})
= Re(Z_{s} · X\overline{X} · \overline{Z_{t}}) = ∥X∥^{2} Re(Z_{s}\overline{Z_{t}})
= δ_{s,t}.

Therefore the family {Z_{1}X, …, Z_{n}X} is an orthonormal basis for (K, ⟨·, ·⟩). In the case i = 2 we can consider any two orthonormal vectors Z_{1}, Z_{2} ∈ ℍ_{2}^{∗} and set Z_{3} = Z_{1} · Z_{2}, Z_{4} = Z_{1} · \overline{Z_{1}} = 1. Given any nonzero vector X ∈ ℍ, the action of ℍ_{2}^{∗} on X determines a submodule M_{X} ⊂ ℍ which, by the previous observation, contains an orthonormal basis for ℍ given by: \{\frac{1}{∥X∥}Z_{1}X, …, \frac{1}{∥X∥}Z_{4}X\}, proving irreducibility.

For i = 4 the same argument can be used by choosing any four orthonormal vectors Z_{1}, …, Z_{4} ∈ 𝕆_{4}^{∗} and by observing that for any of them, say Z_{4}, it should hold that Z_{5} = Z_{1} · Z_{4}, Z_{6} = Z_{2} · Z_{4}, Z_{7} = Z_{3} · Z_{4}, are actually orthogonal to {Z_{1}, …, Z_{4}}. To see this just consider that, for imaginary orthonormal octonions Z_{i}, Z_{j} and Z_{k}, the identity

⟨Z_{i} · Z_{j}, Z_{k}⟩ = δ_{ijk}

holds, thanks to the associativity of the real part of the octonionic product.

By setting Z_{8} = Z_{4} · \overline{Z}_{4} = 1 we conclude that, given any nonzero X ∈ 𝕆 \{\frac{1}{∥X∥}Z_{1}X, …, \frac{1}{∥X∥}Z_{8}X\} is an orthonormal basis for 𝕆.

In the cases i = 3 and i = 5, 6, 7 irreducibility follows by choosing K_{i}^{∗}’s satisfying the inclusion relation K_{i−1}^{∗} ⊂ K_{i}^{∗}.

The above constructed irreducible H -type Lie algebras with centers ℨ_{i} of real dimensions ranging from zero to seven exhaust all possibilities.

3. Automorphisms and Derivations

We start this section with some notation. If N is an H -type Lie group, M(N)_{0} denotes the connected component of its group of (Haar) measure-preserving automorphisms. If we denote with Aut(𝔑)_{0} the connected component of the (Lebesgue) measure-preserving automorphisms of 𝔑, we have that

Der_{0}(𝔑) = Lie(Aut(𝔑)_{0}) ≃ Lie(M(N)_{0})

Where Der_{0}(𝔑) is the Lie algebra of trace-zero derivations of 𝔑. Let D_{ℨ} and D_{𝔙} be the restrictions of a derivation D resp. to the center and its orthocomplement:

D_{ℨ}([X, Y]) = ad(D_{𝔙}X)(Y) + ad(X)(D_{𝔙}Y)  X, Y ∈ 𝔙  (6)

Therefore, any derivation D ∈ Der(𝔑) can be written as the sum D_{1} + D_{2} of a dilation D_{1} and an block triangular matrix D_{2}:

D_{1} = ( \begin{matrix}λ · Id & 0 \\ 0 & \frac{λ}{2} · Id \end{matrix} ) D_{2} = ( \begin{matrix}C & B \\ 0 & A \end{matrix} ) .

where C = D_{ℨ} is a square matrix acting on the center, A a square matrix acting on its orthocomplement and B a rectangular block. Note here that, due to 2-step nilpotency, the entries of B are not subject to any conditions. In this situation we see that the derivations of the type:

( \begin{matrix}0 & B \\ 0 & 0 \end{matrix} )  (7)

form a Lie subalgebra of Der(𝔑)_{0} contained in its nilradical. Let us denote the Lie subalgebra of those derivations with R′.

\underline{Definition}: For the rest of this paper the notation Aut(𝔑)′_{0} indicates the connected component of the group of (Lebesgue) measure-preserving automorphisms of 𝔑, mod the non-grading preserving elements of its nilradical.

Let T = exp(D) be an element of Aut(𝔑)′_{0}. Since it preserves the grading ℨ ⊕ 𝔙 of 𝔑, the matrices A = T_{𝔙} and C = T_{ℨ} satisfy:

⟨Z, [AX, AY]⟩ = ⟨Z, C([X, Y])⟩ X, Y ∈ 𝔙, Z ∈ ℨ

which implies by (3):

⟨A^{t}J_{Z}(AX), Y ⟩ = ⟨J_{C^{t}(Z)}(X), Y ⟩,

and therefore

A^{t} ∘ J_{Z} ∘ A = J_{C^{t}(Z)}  ∀Z ∈ ℨ  (8)

where J is the operator defining the Clifford module structure.

We immediately get the following

Lemma 3.1. Given two irreducible H -type Lie algebras 𝔑^{1} and 𝔑^{2}, with dim(Z(𝔑^{1})) = dim(Z(𝔑^{2})) mod(8), it holds: Aut(𝔑^{1})′_{0} ≃ Aut(𝔑^{2})′_{0}.

Proof. First we observe that, for the Clifford algebras in i and i+8 generators, the relation holds: C_{i+8} = C_{i}⊗C_{8} = C_{i}⊗ℝ(16), where ℝ(16) = C_{8} is the full 16dimensional matrix algebra over ℝ; hence C_{i+8} can be represented as the algebra of matrices M(16, C_{i}) with coefficients in C_{i} (see [7], pg. 57). For 1 ≤ n, m ≤ 16 we rewrite (m, n)-th entry of the matrix equation (8) for Aut(𝔑_{i+8})′_{0} as:

∑_{s,t=1}^{16} A_{s,m}^{t} · X_{s,t} · A_{t,n} = ∑_{k=1}^{16} C_{m,k} · X_{k,n},  (9)

where X_{p,q} ∈ J(ℨ_{i}) and the blocks A_{i,j}, C_{i,j} are real matrices.

If we choose X = (X_{p,q}) such that X_{p,q} = \underline{0} unless (p, q) = (m, n), (9) gives

A_{m,m}^{t} · X · A_{n,n} = C_{m,m} · X,  ∀X ∈ J(ℨ_{i})  (10)

Since A in (8) is an automorphism and therefore invertible, and J(ℨ_{i}) contains invertible elements, (10) forces the diagonal blocks of A and C to be invertible. Furthermore, if n = m we get that A_{m,m} ∈ Aut(𝔑_{i})′_{0} for all m’s. The RHS in (10) does not depend on n, which in turn implies

C_{m,m}X = A_{m,m}^{t} · X · A_{n,n} = A_{m,m} · X · A_{m.m}, ∀m, ∀X ∈ J(ℨ_{i}).

Thus A_{m,m} = A_{n,n} for all m and n. To show that A_{m,n} = \underline{0} if m ≠ n we proceed in the same way: fix a pair (m, n) with m ≠ n and pick X = (X_{p,q}) so that X_{p,q} = \underline{0} unless (p, q) = (m, s) where s ≠ n. This way we obtain:

A_{m,m}^{t} · X_{m,s} · A_{s,n} = \underline{0} ∀X ∈ J(ℨ_{i})

By the same reasoning described above the last equation forces: A_{m,n} = \underline{0} if m ≠ n and our proof is complete.

The H -type Lie algebras with center on dimension zero and one are isomorphic to, resp. ℝ^{n} and the real Heisenberg algebras 𝔑_{1}^{n}. The groups Aut(𝔑)′_{0} are isomorphic to SL(n, ℝ) and Sp(n, ℝ) (see [15]), while Aut(𝔑_{3}^{n})′_{0} ≃ Sp(1) × Sp(n) (see [16], Prop. 10.1) We study the other cases.

To establish our notation we prove the following

Lemma 3.2. 𝔑_{2} is isomorphic to the complexification 𝔑_{1}^{ℂ} of the real Heisenberg Lie algebra.

Proof. Assume 𝔑 = 𝔑_{2} ≃ ℨ ⊕ 𝔙; by the results proven in the previous section we can choose any two-dimensional subspace of ℍ^{∗} to be equal to ℨ. Given the standard basis \{1, \underline{i}, \underline{j}, \underline{k}\} of ℍ, we set ℨ = ℝ\underline{j} ⊕ ℝ\underline{k}, and define:

Im_{2}(a + b\underline{i} + c\underline{j} + d\underline{k}) = c\underline{j} + d\underline{k}.

Given any h ∈ ℍ can write: h = z_{1} +z_{2}\underline{j}, where z_{1} and z_{2} are two complex numbers. For any X, Y ∈ 𝔙 ≃ ℍ we write: X = x_{1} + x_{2}\underline{j}, Y = y_{1} + y_{2}\underline{j}, so that

[(0, X), (0, Y)] = (Im_{2}(Y · \overline{X}), 0)
= (Im_{2}((y_{1} + y_{2}\underline{j}) · (\overline{x_{1}} + \overline{x_{2}\underline{j}})), 0)
= ((y_{2}x_{1} − y_{1}x_{2})\underline{j}, 0)
= ((x_{1}y_{2} − x_{2}y_{1})\underline{j}, 0)

With obvious notation we define ϕ : 𝔑_{1}^{ℂ} → 𝔑_{2} by:

ϕ(z_{1}, z_{2}, z_{3}) = (z_{3}\underline{j}, z_{1} + z_{2}\underline{j});

by the RHS of the previous equation ϕ is an isomorphism of Lie algebras and the lemma is proven.

Proposition 3.3. Aut(𝔑_{2}^{n})′_{0} ≃ U(1) ⋉ Sp(2n, ℂ)

Proof. We consider here n = 1, the general case being completely analogous. Let ψ : ℂ^{2} × ℂ^{2} → ℂ be the complex symplectic form defined by: ψ((x_{1}, x_{2}), (y_{1}, y_{2})) = (x_{1}y_{2} −x_{2}y_{1}) so that, with the notation used for the Lemma:

[(0, \underline{X}), (0, \underline{Y})] = (ψ(\underline{X}, \underline{Y}), \underline{0}) = (x_{1}y_{2} − x_{2}y_{1}, 0, 0).

Let A and C be the restrictions of an automorphism ϕ ∈ Aut(𝔑_{2})′_{0} to, resp., 𝔙 and ℨ. First suppose C = Id; we then have that A ∈ GL(4, ℝ) satisfies:

ψ(A(\underline{X}), A(\underline{Y})) = (x_{1}y_{2} − x_{2}y_{1}) = ψ(\underline{X}, \underline{Y})
Let s · \underline{X} = (sx_{1}, sx_{2}) for any s ∈ ℂ. It holds then:
ψ(A(s · \underline{X}), A(\underline{Y})) = sψ(\underline{X}, \underline{Y})

which in turn, by the non-degeneracy of ψ, means that A is actually ℂ-linear and thus: A ∈ Sp(2, ℂ), implying that all elements of Aut(𝔑_{2})′_{0} acting trivially on the center form a complex Lie group isomorphic to Sp(2, ℂ).

Consider now the general case of an automorphism ϕ ∈ Aut(𝔑_{2})′_{0}:

[(0, A(\underline{X})), (0, A(\underline{Y}))] = C(x_{1}y_{2} − x_{2}y_{1})  (11)

Where ϕ_{𝔙} = A ∈ GL(4, ℝ) and ϕ_{ℨ} = C ∈ GL(2, ℝ). If we denote with

D the derivation in Der_{0}(𝔑_{2}) such that exp(D) = ϕ and write D = ( \begin{matrix}S & 0 \\ 0 & T \end{matrix} ),

equation (11) can be restated on the Lie algebra as:

[T (\underline{X}), \underline{Y}] + [\underline{X}, T (\underline{Y})] = S([\underline{X}, \underline{Y}]).  (12)

If we also write T in terms of 2 × 2 blocks:

T = ( \begin{matrix}A′ & B′ \\ C′ & D′ \end{matrix} ) ,

and choose complex vectors X = (x, 0), Y = (0, y), we obtain:

S(x · y) = A′(x) · y + x · D′(y).

We now let x = x_{1} + ix_{2}, y = y_{1} + iy_{2}, S = ( \begin{matrix}s_{1} & s_{2} \\ s_{3} & s_{4} \end{matrix} ) and so on for the blocks

of T; at this point we rewrite the previous identity as:

( \begin{matrix}s_{1}(x_{1}y_{1} − x_{2}y_{2}) + s_{2}(x_{1}y_{2} + x_{2}y_{1}) \\ s_{3}(x_{1}y_{1} − x_{2}y_{2}) + s_{4}(x_{1}y_{2} + x_{2}y_{1}) \end{matrix} ) =
( \begin{matrix}(a_{1} + d_{1})x_{1}y_{1} − (a_{4} + d_{4})x_{2}y_{2} + (d_{2} − a_{3})x_{1}y_{2} + (a_{2} − d_{3})x_{2}y_{1} \\ (a_{3} + d_{3})x_{1}y_{1} − (a_{2} + d_{2})x_{2}y_{2} + (d_{1} + a_{4})x_{1}y_{2} + (a_{4} + d_{1})x_{2}y_{1} \end{matrix} )

An analogous formula will be obtained by choosing X = (0, x) and Y = (y, 0) in terms of B′ and C′. At this point a long and elementary computation

shows that the matrices A′, B′, C′ and D′ have the form: ( \begin{matrix}s & t \\ −t & s \end{matrix} ) and

therefore T is a complex 2×2 matrix. The same immediately follows for S. Since the Lie algebra of a Lie group is complex if and only if the connected component of the underlying Lie group is a complex Lie group we get that C is contained in GL(1, ℂ) and therefore: C(z) = z_{C} · z where z_{C} ∈ ℂ.

Equation (11) now gives:

[A(\underline{X}), A(\underline{Y})] = z_{C} · (x_{1}y_{2} − x_{2}y_{1})

We observe that for complex numbers s, t such that s · t = z_{C}^{−1} we can define the complex linear automorphism T_{s,t} : 𝔑_{2} → 𝔑_{2} as:

T_{s,t}(z_{1}, z_{2}, z_{3}) = (s · z_{1}, t · z_{2}, st · z_{3})

By doing so it holds: T_{s,t} ∘ ϕ ∈ Sp(2, ℂ). Thus: ϕ = T_{s,t}^{−1} ∘ ϕ′ with ϕ′ ∈ Sp(2, ℂ). It is immediate to check that for any ϕ ∈ Sp(2, ℂ): T_{s,t} ∘ ϕ ∘ T_{s,t}^{−1} ∈ Sp(2, ℂ). We note that T_{s,t} is measure preserving if and only if |s · t| = 1. We conclude:

Aut(𝔑_{2})′_{0} ≃ U(1) ⋉ Sp(2, ℂ)

and the proposition is proven.

We present here a different proof of a known result for Aut(𝔑_{3})′_{0}

Proposition 3.4. Aut(𝔑_{3})′_{0} ≃ Sp(1) × Sp(1)

Proof. By using the notation as before, we get that for any \underline{X} and \underline{Y} in the orthocomplement of ℨ, the Lie bracket is given by:

[\underline{X}, \underline{Y}] = Im_{3}(\underline{Y} · \underline{\overline{X}}) = Im_{3}(y_{1}\overline{x_{1}} + \overline{x_{2}}y_{2} + (y_{2}x_{1} − x_{2}y_{1})\underline{j}).

In other terms, given a ϕ ∈ Aut(𝔑_{3})′_{0} that fixes the center, its restriction A to 𝔙 should preserve the complex symplectic form ψ discussed in Proposition 3.3 (hence A ∈ Sp(2, ℂ) is a complex transformation) and preserve the imaginary part of the form

ν((x_{1}, x_{2}), (y_{1}, y_{2})) = y_{1}\overline{x_{1}} + \overline{x_{2}}y_{2}.

Now, since A is a complex transformation, the latter is equivalent to saying that A actually preserves ν. To see this, consider the identity:

Im(ν(\underline{X}, i · \underline{Y})) = Re(ν(\underline{X}, \underline{Y}));

in other words, for any A preserving the imaginary part of ν:

Im(ν(A(\underline{X}), A(i · \underline{Y})) = Re(ν(\underline{X}, \underline{Y})),

but since A is a complex linear transformation:

Im (ν(A(\underline{X}), A(i · \underline{Y}))) = Re (ν(A(\underline{X}), A(\underline{Y}))) ,

this forces:

Re (ν(A(\underline{X}), A(\underline{Y}))) = Re (ν(\underline{X}, \underline{Y})) ,

and thus A is a complex norm-preserving transformation: A ∈ U(2) ∩ Sp(2, ℂ) ≃ Sp(1), (cf. e.g [6], pg. 80). The action of these automorphisms can be realized as right multiplication by a unit quaternion: A(X) = X · h, ∥h∥ = 1.

In the case where A acts nontrivially on the center we have:

Im_{3}(A(\underline{Y}) · \overline{A(\underline{X})}) = C(Im_{3}(\underline{Y} · \overline{\underline{X}}));

which implies, by direct computation (see appendix), that C ∈ SO(3). So if we set A′(\underline{X}) = L_{h_{C}} ∘ A = h_{C} · A(\underline{X}) for a suitable unit quaternion h_{C} we have:

Im_{3}(A′(\underline{Y}) · \overline{A′(\underline{X})}) = h_{C} · C(Im_{3}(\underline{Y} · \overline{\underline{X}})) · \overline{h_{C}} = Im_{3}(\underline{Y} · \overline{\underline{X}}); .

Hence, by ℍ-linearity of Sp(1), A(\underline{X}) = L_{h_{C}^{−1}} ∘ A′ = A′ ∘ L_{h_{C}^{−1}} . The same argument shows that Aut(𝔑_{3}^{n})′_{0} ≃ Sp(1) × Sp(n).

4. 4 ≤ dim(Z) ≤ 7

In analogy to the quaternionic algebras it turns out to be convenient to use the representation of an octonion c ∈ 𝕆 as c = h_{1} +h_{2}\underline{l} where h_{1} and h_{2} are in ℍ and \underline{l} one of the unit generators as done in ([4], pg. 15) so that, for any \underline{X} = x_{1} + x_{2}\underline{l}, \underline{Y} = y_{1} + y_{2}\underline{l}:

\underline{Y} · \overline{\underline{X}} = y_{1}\overline{x_{1}} + \overline{x_{2}}y_{2} + (y_{2}x_{1} − x_{2}y_{1})\underline{l}
= ν((x_{1}, x_{2}), (y_{1}, y_{2})) + ψ((x_{1}, x_{2}), (y_{1}, y_{2}))\underline{l};

where the bar is now the quaternionic conjugate and ψ is a so-called anti-Hermitian form (cf.[6],pg. 91). The connected component of the Lie group stabilizing a antihermitian form over the vector space ℍ^{n} is given by:

SU_{ψ}(n, ℍ) = U(n, n) ∩ O(2n, ℂ) ∩ SL(n, ℍ).

We are now able to prove the following

Proposition 4.1. Aut(𝔑_{4})′_{0} is non compact.

Proof. The convenient choice for ℨ is the vector space ℍ · \underline{l}. After making this choice, the Lie bracket becomes equivalent to:

[\underline{X}, \underline{Y}] = ψ(\underline{X}, \underline{Y}).

By a modification of the same argument exposed for 𝔑_{2} we obtain that the connected component of the group of automorphisms of 𝔑_{4} acting trivially on the center is isomorphic to a linear quaternionic group stabilizing the anti-Hermitian form ψ. We denote this group by SU_{ψ}(2, ℍ), and observe that, since it contains the measure preserving maps of the kind

ϕ_{h}(\underline{X}) = (h · x_{1}, x_{2} · h^{−1}), h ∈ GL(1, ℍ),

it is non-compact. A detailed discussion of these forms and their corresponding orthogonal groups can be found in ([5], chapter I). Finall we observe that the automorphisms acting non-trivially on the center, contain a group isomorphic to the multiplicative group GL(1, ℍ) given by the maps

M_{h}(\underline{X}) = (x_{1} · h, h^{−1} · x_{2}), h ∈ GL(1, ℍ),

Proposition 4.2. For i = 5, 6, 7 the groups Aut(𝔑_{i})′_{0} are compact.

Proof. The proof is achieved by direct computation with the help of a computer running MAPLE on a MATLAB platform.

We first write the equation on the Lie algebra:

Im (A(Y) · \overline{X} + Y · \overline{A(X)}) = C (Im(Y · \overline{X}))

as a set of 7 × 4 = 28 linear equations by choosing X · Y = e_{i} · e_{j} with i ≠ j and {e_{i}} the standard basis of 𝕆^{∗}.

The long series of conditions on the coefficients of A and C immediately forces them to be skew-symmetric (see appendix). The same result follows also for the non-irreducible case.

We now return to the general case. Let 𝔑 be a H -type Lie algebra with center ℨ of real dimension m admitting a natural decomposition

𝔑 = 𝔑_{m}^{a} = ℨ ⊕ 𝔙^{a},

where the subalgebra 𝔑^{n} = ℨ ⊕ 𝔙^{a} is defined by 𝔑_{m}^{a} = ℨ ⊕ 𝔙 ⊕ … ⊕ 𝔙. It is easy to adapt the proof of Lemma 3.1 to show that Aut(𝔑_{m}^{a})′_{0} ≃ Aut(𝔑_{m+8}^{a})′_{0}. If m = 3, 7 mod(8), the subalgebras 𝔑_{m} = ℨ⊕𝔙^{a} and 𝔑_{m} = ℨ⊕𝔙^{b} carry inequivalent Lie algebra structures. Thus Aut(𝔑)′_{0} act separately on each inequivalent component, so that if Aut(ℨ ⊕ 𝔙^{a})′_{0} = A ⋉ B_{1} and Aut(ℨ ⊕ 𝔙^{b})′_{0} = A ⋉ B_{2}, we have: Aut(ℨ ⊕ 𝔙^{a} ⊕ 𝔙^{b})′_{0} = A ⋉ (B_{1} × B_{2}), so that Aut(𝔑_{m}^{a,b})′_{0} ≃ Aut(𝔑_{m+8}^{a,b})′_{0}.

5. Stabilizers of Lattices

In this section we establish some general results on lattices of M(N), the group of (Haar) measure-preserving automorphisms of a simply connected nilpotent Lie group N and its Lie algebra, Der_{0}(𝔑). These were first studied by R. Mosak and M. Moskowitz in [15]. There they assumed the quite general connected simply connected nilpotent group had a log-lattice Γ —that is: the set Λ = log(Γ) is a group in 𝔑 = Lie(N). By Malcev’s results these lattices can always be found in H -type groups. The stabilizer of Γ in M(N)_{0} defined by:

Stab_{M(N)_{0}}(Γ) = {ϕ ∈ M(N)_{0} | ϕ(Γ) = Γ}

In [15] (Theorem 2.2.), a criterion was developed which shows when Stab_{M(N)_{0}}(Γ) is a lattice or a uniform lattice in M(N)_{0}. This criterion, established on the Lie algebra Der_{0}(𝔑) = Lie(M(N)_{0}), deals with the radical R = Rad(Der_{0}(𝔑)) and its maximal nilpotent ideal R_{n} = Rad(Der_{0}(𝔑))_{n}:

1. If R = R_{n}, then Stab_{M(N)_{0}}(Γ) is a lattice in M(N)_{0}.

2. If Der_{0}(𝔑)/R_{n} is in addition of compact type, then Stab_{M(N)_{0}}(Γ) is uniform.

The above result remains valid also in the case we replaced M(N)_{0} by any of its closed subgroups. Furthermore, the following holds:

Theorem 5.1. Let Γ be a non log-lattice in a connected simply connected nilpotent Lie group N then there is a log-lattice Γ′ ⊂ N such that Stab_{M(N)_{0}}(Γ) has finite index in Stab_{M(N)_{0}}(Γ′).

Proof. One can show ([14], Theorem 2) that there is always a log-lattice Γ_{α} containing Γ. Since the intersection of two such log-lattices, say Γ_{α} and Γ_{β}, is also a log-lattice containing Γ, we can define the minimal object in that class:

Γ_{2} = ⋂_{α} Γ_{α}.

Consider now ϕ ∈ Stab_{M(N)}(Γ); we claim that ϕ ∈ Stab_{M(N)}(Γ_{2}). Let Γ_{ϕ} be the image of Γ_{2} under the automorphism ϕ: Γ_{ϕ} = ϕ(Γ_{2}). It holds that

Γ ⊂ Γ_{ϕ} ∩ Γ_{2};

that means, by the minimality assumption: Γ_{ϕ} = ϕ(Γ_{2}) ⊃ Γ_{2} and therefore Γ_{2} = ϕ(Γ_{2}), thus:

Stab_{M(N)_{0}}(Γ) ⊂ Stab_{M(N)_{0}}(Γ_{2}).

Now, it can also be shown that Γ actually contains a log-lattice; let Γ_{1} be such a lattice: Γ_{1} ⊂ Γ. If we denote Λ_{i} = log(Γ_{i}), it follows from the construction of the Γ_{i}’s that for some K ∈ ℕ:

Λ_{1} = K · Λ_{2}.

Consider now a measure preserving automorphism ϕ ∈ Stab_{M(N)_{0}}(Γ_{2}). Its differential ϕ_{∗} will yield:

ϕ_{∗}(Λ_{1}) = ϕ_{∗}(K · Λ_{2}) = K · ϕ_{∗}(Λ_{2}) = Λ_{1}.

And therefore: Stab_{M(N)_{0}}(Γ_{1}) ⊂ Stab_{M(N)_{0}}(Γ_{2}). The same argument shows that Stab_{M(N)_{0}}(Γ_{2}) ⊂ Stab_{M(N)_{0}}(Γ_{1}). Thus:

Stab_{M(N)}(Γ_{1}) = Stab_{M(N)}(Γ_{2}).

Take now ψ ∈ Stab_{M(N)}(Γ_{1}). We can write, for any positive integer k: Γ^{k} = ψ^{k}(Γ)

Γ_{1} ⊂ Γ^{k} ⊂ Γ_{2}.

The Γ^{k} are therefore a family of subgroups contained between a group (Γ_{2}) and one of its subgroups of finite index (Γ_{1}). This implies that the number of Γ^{k} ’s has to be finite. Therefore there is a positive integer K_{ψ} such that: ψ^{K_{ψ}} ∈ Stab_{M(N)}(Γ). The set of all K_{ψ} ’s is bounded by above so if we take K_{max} = max{K_{ψ} ∀ψ ∈ Stab_{M(N)}(Γ_{1})} we get that Stab_{M(N)}(Γ) is a subgroup of index K_{max} in Stab_{M(N)}(Γ_{1}). And the theorem is proven.

We would like to thank Professor G. Prasad for suggesting the following result. We first note that since N contains a lattice and is simply connected Aut(𝔑)_{0} ≃ M(N)_{0} is the group of real points of an algebraic group, say A(N), defined over ℚ (see [15]).

Corollary 5.2. Let Γ be a lattice in a simply connected nilpotent Lie group N. Then Stab_{M(N)_{0}}(Γ) is an arithmetic subgroup of A(N).

Proof. Let ϕ : M(N)_{0} → Aut(𝔑)_{0} be an algebraic isomorphism. By the previous result there is a log-lattice Γ_{2} containing Γ such that Stab_{M(N)_{0}}(Γ) has finite index in Stab_{M(N)_{0}}(Γ_{2}). Let Λ_{2} = log Γ_{2} and observe that ϕ(Stab_{M(N)_{0}}(Γ_{2})) stabilizes the ℚ-lattice Λ_{2} and hence it is an arithmetic subgroup of A(N). Since Stab_{M(N)_{0}}(Γ) has finite index in Stab_{M(N)_{0}}(Γ_{2}), we have proven our claim.

Theorem 5.3. Let Γ be a lattice in simply connected nilpotent Lie group N; if the quotient M(N)_{0}/Rad_{n}(M(N)_{0}) is compact, Stab_{M(N)_{0}}(Γ) is a uniform lattice.

Proof. If Rad(M(N)_{0}) = Rad_{n}(M(N)_{0}) the result follows from the criterion and Theorem 5.1. So the only case to consider is that K = M(N)_{0}/Rad_{n}(M(N)_{0}) contains an abelian factor, say A:

K = K_{1} × A.

We apply our criterion to the closed subgroup M_{1}(N) = K_{1} ⋉ Rad_{n}(M(N)_{0}) and get, for any lattice Γ ⊂ N, that

Stab_{M_{1}(N)}(Γ) ⊂ M(N)

is a uniform lattice in M_{1}(N). Since Stab_{A}(Γ) is a finite set and the elements of A and K_{1} commute, that proves the Theorem.

We summarize the preceding results in the main result of this section.

Theorem 5.4. If Γ is a lattice in an irreducible group of Heisenberg-type N with 7 ≥ dim_{ℝ}(Z(N)) ≥ 5 mod(8) or dim_{ℝ}(Z(𝔑)) = 3 mod(8), Stab_{M(N)_{0}}(Γ) is a uniform lattice.

Proof. In the Lie algebras Lie(M(N_{i})_{0}) with i = 3, 7 mod(8) the nilradical coincides with the radical; the quotient Der_{0}(𝔑)/R_{n} is always of compact type and the above mentioned criterion applies directly. For the remaining two cases we should apply our extension of the result of Mosak and Moskowitz (Theorem 5.1).

6. Isometries and Quasi-conformal Mappings

In this section we study the isometries of H -type groups and show how their structure gives a necessary and sufficient condition for the existence of non-conformal quasi-conformal mappings.

As a consequence of the proven results we first notice that any trace zero derivation D of a Lie algebra of Heisenberg-type with center of dimension equal to 3, 5, 6, 7 mod(8) can be decomposed as

D = D_{K} + D_{N},

where D_{N} is nilpotent and D_{K} is in a Lie algebra of compact type.

From this it will follow that quasi-conformal mappings of certain H -type groups must be conformal.

Consider an H -type group, N, equipped with the left-invariant metric as in [2]. Let Iso(N) be its group of isometries. Then, by [9], section 3:

Iso(N) = A(N) ⋉ N  (13)

The group A(N) consists of those automorphisms of N whose differentials are isometries of the Lie algebra 𝔑:

A(N) = {ϕ ∈ Aut(N) | ϕ_{∗} ∈ Iso(𝔑)}.

We conclude that in the case of a Heisenberg-type group the Lie algebra of A(N) (or A(N)_{0}, the connected component of A(N)) satisfies:

Lie(A(N)) ⊆ \frac{Der_{0}(𝔑)}{R′}.  (14)

Our computations will be based on this latter fact. As a result we are able to deal with groups of automorphisms locally and thus avoid covering space arguments which make their appearance in previous work on the subject (see for example Pansu [16] and Riehm [18]). Since we are interested in the compactness of A(N), and since there are a finite number of connected components, we can restrict our attention to the identity component of the automorphism group.

In his paper ([16]) P. Pansu establishes a result on conformal mappings for the groups N_{3}^{n} and N_{7}^{n}. A homeomorphism T : U → U′ between open subsets of an H -type group is called λ-quasiconformal if there exists a real number λ ∈ [1, ∞) such that for all x ∈ U, ϵ > 0 and all sufficiently small r there is an R > 0 such that:

B(T x, R) ⊆ T (B(x, r)) ⊆ B(T x, (λ + ϵ)R).

A quasiconformal map ϕ is said to be conformal when λ = 1. This is equivalent to saying that ϕ is quasiconformal and D(ϕ)_{e}, the differential of ϕ at the identity, is an isometry of the Lie algebra of N times a dilation ([16], pg. 44).

Pansu proves the following result ([16], Corollary 11.2.):

Theorem - A quasiconformal homeomorphism of N_{3}^{n} (resp. of N_{7}^{n}), acting as maximal unipotent group of isometries on the hyperbolic quaternionic (resp. octonionic) symmetric space, is conformal.

Combining our results with those of Pansu we can prove a more general statement.

Theorem 6.1. A quasiconformal homeomorphism of an H -type group with center of dimension 3, 5, 6, 7 mod(8) must be conformal.

Proof. Let ϕ be the homeomorphism, N our H -type group and 𝔑 ≃ ℨ ⊕ 𝔙 its Lie algebra satisfying dim(Z(𝔑)) = 3, 5, 6, 7 mod(8). By Pansu’s differentiability theorem ([16], sec.VII) the differential exists almost everywhere. We first observe that ([2], pg.12) that its differential at the identity is a grading-preserving automorphism ^{1}

D(ϕ)_{e}(𝔙) ⊂ 𝔙

By equation (7) it is clear that the component with respect to R′ of any gradingpreserving automorphism is zero. This in turn implies, by the hypothesis and equation (14), that the corresponding derivation yields D_{ϕ} = D_{ϕ′} + D_{ϕ′′}, where

D_{ϕ′} ∈ Der_{0}(𝔑)/R′ and D_{ϕ′′} is a matrix of the type D_{ϕ′′} = ( \begin{matrix}λ · Id & 0 \\ 0 & λ/2 · Id \end{matrix} )

and therefore ϕ = exp(D(ϕ)_{e}) is a dilation times an isometry, which equivalent is to saying that the map is conformal.

7. Appendix

Given the matrices A = (x_{i,j}) ∈ ℝ(8) and B ∈ ℝ(8) we compute explicitly the set of linear equations:

A(Y) · \overline{X} + Y · \overline{A(X)} = C (Y · \overline{X})

Where X, Y are elements of the standard basis \{e_{i}\}_{1≤i≤8} = \{1, \underline{i}, \underline{j}, …\} of the Cayley numbers 𝕆.

For X = e_{1}, Y = e_{2}

( \begin{matrix}x_{1,2} + x_{2,1} \\ x_{2,2} + x_{1,1} \\ x_{3,2} + x_{4,1} \\ x_{4,2} − x_{3,1} \\ x_{5,2} + x_{6,1} \\ x_{6,2} − x_{5,1} \\ x_{7,2} − x_{8,1} \\ x_{8,2} + x_{7,1} \end{matrix} ) = C(e_{2})

for X = e_{1}, Y = e_{3}

( \begin{matrix}x_{1,3} + x_{3,1} \\ x_{2,3} − x_{4,1} \\ x_{3,3} + x_{1,1} \\ x_{4,3} + x_{2,1} \\ x_{5,3} + x_{7,1} \\ x_{6,3} + x_{8,1} \\ x_{7,3} − x_{5,1} \\ x_{8,3} − x_{6,1} \end{matrix} ) = C(e_{3})

1For the Heisenberg group this fact was proven by Korányi and Reimann ([12])

for X = e_{1}, Y = e_{4}

( \begin{matrix}x_{1,4} + x_{4,1} \\ x_{2,4} + x_{3,1} \\ x_{3,4} − x_{2,1} \\ x_{4,4} + x_{1,1} \\ x_{5,4} + x_{8,1} \\ x_{6,4} − x_{7,1} \\ x_{7,4} + x_{6,1} \\ x_{8,4} − x_{5,1} \end{matrix} ) = C(e_{4})

for X = e_{1}, Y = e_{5}

( \begin{matrix}x_{1,5} + x_{5,1} \\ x_{2,5} − x_{6,1} \\ x_{3,5} − x_{7,1} \\ x_{4,5} − x_{8,1} \\ x_{5,5} + x_{1,1} \\ x_{6,5} + x_{2,1} \\ x_{7,5} + x_{3,1} \\ x_{8,5} + x_{4,1} \end{matrix} ) = C(e_{5})

for X = e_{1}, Y = e_{6}

( \begin{matrix}x_{1,6} + x_{6,1} \\ x_{2,6} + x_{5,1} \\ x_{3,6} − x_{8,1} \\ x_{4,6} + x_{7,1} \\ x_{5,6} − x_{2,1} \\ x_{6,6} + x_{1,1} \\ x_{7,6} − x_{4,1} \\ x_{8,6} + x_{3,1} \end{matrix} ) = C(e_{6})

for X = e_{1}, Y = e_{7}

( \begin{matrix}x_{1,7} + x_{7,1} \\ x_{2,7} + x_{8,1} \\ x_{3,7} + x_{5,1} \\ x_{4,7} − x_{6,1} \\ x_{5,7} − x_{3,1} \\ x_{6,7} + x_{4,1} \\ x_{7,7} + x_{1,1} \\ x_{8,7} − x_{2,1} \end{matrix} ) = −C(e_{7})

for X = e_{1}, Y = e_{8}

( \begin{matrix}x_{1,8} + x_{8,1} \\ x_{2,8} − x_{7,1} \\ x_{3,8} + x_{6,1} \\ x_{4,8} + x_{5,1} \\ x_{5,8} − x_{4,1} \\ x_{6,8} − x_{3,1} \\ x_{7,8} + x_{2,1} \\ x_{8,8} + x_{1,1} \end{matrix} ) = C(e_{8})

for X = e_{2}, Y = e_{3}

( \begin{matrix}x_{2,3} + x_{3,2} \\ −x_{1,3} − x_{4,2} \\ −x_{4,3} + x_{1,2} \\ x_{3,3} + x_{2,2} \\ −x_{6,3} + x_{7,2} \\ x_{5,3} + x_{8,2} \\ x_{8,3} − x_{5,2} \\ −x_{7,3} − x_{6,2} \end{matrix} ) = C(e_{4})

for X = e_{2}, Y = e_{4}

( \begin{matrix}x_{2,4} + x_{4,2} \\ −x_{1,4} + x_{3,2} \\ −x_{4,4} − x_{2,2} \\ x_{3,4} + x_{1,2} \\ −x_{6,4} + x_{8,2} \\ x_{5,4} − x_{7,2} \\ x_{8,4} + x_{6,2} \\ −x_{7,4} − x_{5,2} \end{matrix} ) = −C(e_{3})

for X = e_{2}, Y = e_{5}

( \begin{matrix}x_{2,5} + x_{5,2} \\ −x_{1,5} − x_{6,2} \\ −x_{4,5} − x_{7,2} \\ x_{3,5} − x_{8,2} \\ −x_{6,5} + x_{1,2} \\ x_{5,5} + x_{2,2} \\ x_{8,5} + x_{3,2} \\ −x_{7,5} + x_{4,2} \end{matrix} ) = C(e_{6})

for X = e_{2}, Y = e_{6}

( \begin{matrix}x_{2,6} + x_{6,2} \\ −x_{1,6} + x_{5,2} \\ −x_{4,6} − x_{8,2} \\ x_{3,6} + x_{7,2} \\ −x_{6,6} − x_{2,2} \\ x_{5,6} + x_{1,2} \\ x_{8,6} − x_{4,2} \\ −x_{7,6} + x_{3,2} \end{matrix} ) = −C(e_{5})

for X = e_{2}, Y = e_{7}

( \begin{matrix}x_{2,7} + x_{7,2} \\ −x_{1,7} + x_{8,2} \\ −x_{4,7} + x_{5,2} \\ x_{3,7} − x_{6,2} \\ −x_{6,7} − x_{3,2} \\ x_{5,7} + x_{4,2} \\ x_{8,7} + x_{1,2} \\ −x_{7,7} − x_{2,2} \end{matrix} ) = −C(e_{8})

for X = e_{2}, Y = e_{8}

( \begin{matrix}x_{2,8} + x_{8,2} \\ −x_{1,8} − x_{7,2} \\ −x_{4,8} + x_{6,2} \\ x_{3,8} + x_{5,2} \\ −x_{6,8} − x_{4,2} \\ x_{5,8} − x_{3,2} \\ x_{8,8} + x_{2,2} \\ −x_{7,8} + x_{1,2} \end{matrix} ) = −C(e_{7})

for X = e_{3}, Y = e_{4}

( \begin{matrix}x_{3,4} + x_{4,3} \\ x_{4,4} + x_{3,3} \\ −x_{1,4} − x_{2,3} \\ −x_{2,4} + x_{1,3} \\ −x_{7,4} + x_{8,3} \\ −x_{8,4} − x_{7,3} \\ x_{5,4} + x_{6,3} \\ x_{6,4} − x_{5,3} \end{matrix} ) = C(e_{2})

for X = e_{3}, Y = e_{5}

( \begin{matrix}x_{3,5} + x_{5,3} \\ x_{4,5} − x_{6,3} \\ −x_{1,5} − x_{7,3} \\ −x_{2,5} − x_{8,3} \\ −x_{7,5} + x_{1,3} \\ −x_{8,5} + x_{2,3} \\ x_{5,5} + x_{3,3} \\ x_{6,5} + x_{4,3} \end{matrix} ) = −C(e_{7})

for X = e_{3}, Y = e_{6}

( \begin{matrix}x_{3,6} + x_{6,3} \\ x_{4,6} + x_{5,3} \\ −x_{1,6} − x_{8,3} \\ −x_{2,6} + x_{7,3} \\ −x_{7,6} − x_{2,3} \\ −x_{8,6} + x_{1,3} \\ x_{5,6} − x_{4,3} \\ x_{6,6} + x_{3,3} \end{matrix} ) = C(e_{8})

for X = e_{3}, Y = e_{7}

( \begin{matrix}x_{3,7} + x_{7,3} \\ x_{4,7} + x_{8,3} \\ −x_{1,7} + x_{5,3} \\ −x_{2,7} − x_{6,3} \\ −x_{7,7} − x_{3,3} \\ −x_{8,7} + x_{4,3} \\ x_{5,7} + x_{1,3} \\ x_{6,7} − x_{2,3} \end{matrix} ) = −C(e_{5})

for X = e_{3}, Y = e_{8}

( \begin{matrix}x_{3,8} + x_{8,3} \\ x_{4,8} − x_{7,3} \\ −x_{1,8} + x_{6,3} \\ −x_{2,8} + x_{5,3} \\ −x_{7,8} − x_{4,3} \\ −x_{8,8} − x_{3,3} \\ x_{5,8} + x_{2,3} \\ x_{6,8} + x_{1,3} \end{matrix} ) = −C(e_{6})

for X = e_{4}, Y = e_{5}

( \begin{matrix}x_{4,5} + x_{5,4} \\ −x_{3,5} − x_{6,4} \\ x_{2,5} − x_{7,4} \\ −x_{1,5} − x_{8,4} \\ −x_{8,5} + x_{1,4} \\ x_{7,5} + x_{2,4} \\ −x_{6,5} + x_{3,4} \\ x_{5,5} + x_{4,4} \end{matrix} ) = C(e_{8})

for X = e_{4}, Y = e_{6}

( \begin{matrix}x_{4,6} + x_{6,4} \\ −x_{3,6} + x_{5,4} \\ x_{2,6} − x_{8,4} \\ −x_{1,6} + x_{7,4} \\ −x_{8,6} − x_{2,4} \\ x_{7,6} + x_{1,4} \\ −x_{6,6} − x_{4,4} \\ x_{5,6} + x_{3,4} \end{matrix} ) = C(e_{7})

for X = e_{4}, Y = e_{7}

( \begin{matrix}x_{4,7} + x_{7,4} \\ −x_{3,7} + x_{8,4} \\ x_{2,7} + x_{5,4} \\ −x_{1,7} − x_{6,4} \\ −x_{8,7} − x_{3,4} \\ x_{7,7} + x_{4,4} \\ −x_{6,7} + x_{1,4} \\ x_{5,7} − x_{2,4} \end{matrix} ) = C(e_{6})

for X = e_{4}, Y = e_{8}

( \begin{matrix}x_{4,8} + x_{8,4} \\ −x_{3,8} − x_{7,4} \\ x_{2,8} + x_{6,4} \\ −x_{1,8} + x_{5,4} \\ −x_{8,8} − x_{4,4} \\ x_{7,8} − x_{3,4} \\ −x_{6,8} + x_{2,4} \\ x_{5,8} + x_{1,4} \end{matrix} ) = −C(e_{5})

for X = e_{5}, Y = e_{6}

( \begin{matrix}x_{5,6} + x_{6,5} \\ x_{6,6} + x_{5,5} \\ x_{7,6} − x_{8,5} \\ x_{8,6} + x_{7,5} \\ −x_{1,6} − x_{2,5} \\ −x_{2,6} + x_{1,5} \\ −x_{3,6} − x_{4,5} \\ −x_{4,6} + x_{3,5} \end{matrix} ) = C(e_{2})

for X = e_{5}, Y = e_{7}

( \begin{matrix}x_{5,7} + x_{7,5} \\ x_{6,7} + x_{8,5} \\ x_{7,7} + x_{5,5} \\ x_{8,7} − x_{6,5} \\ −x_{1,7} − x_{3,5} \\ −x_{2,7} + x_{4,5} \\ −x_{3,7} + x_{1,5} \\ −x_{4,7} − x_{2,5} \end{matrix} ) = C(e_{3})

for X = e_{5}, Y = e_{8}

( \begin{matrix}x_{5,8} + x_{8,5} \\ x_{6,8} − x_{7,5} \\ x_{7,8} + x_{6,5} \\ x_{8,8} + x_{5,5} \\ −x_{1,8} − x_{4,5} \\ −x_{2,8} − x_{3,5} \\ −x_{3,8} + x_{2,5} \\ −x_{4,8} + x_{1,5} \end{matrix} ) = C(e_{4})

for X = e_{6}, Y = e_{7}

( \begin{matrix}x_{6,7} + x_{7,6} \\ −x_{5,7} + x_{8,6} \\ x_{8,7} + x_{5,6} \\ −x_{7,7} − x_{6,6} \\ x_{2,7} − x_{3,6} \\ −x_{1,7} + x_{4,6} \\ x_{4,7} + x_{1,6} \\ −x_{3,7} − x_{2,6} \end{matrix} ) = −C(e_{4})

for X = e_{6}, Y = e_{8}

( \begin{matrix}x_{6,8} + x_{8,6} \\ −x_{5,8} − x_{7,6} \\ x_{8,8} + x_{6,6} \\ −x_{7,8} + x_{5,6} \\ x_{2,8} − x_{4,6} \\ −x_{1,8} − x_{3,6} \\ x_{4,8} + x_{2,6} \\ −x_{3,8} + x_{1,6} \end{matrix} ) = C(e_{3})

for X = e_{7}, Y = e_{8}

( \begin{matrix}x_{7,8} + x_{8,7} \\ −x_{8,8} − x_{7,7} \\ −x_{5,8} + x_{6,7} \\ x_{6,8} + x_{5,7} \\ x_{3,8} − x_{4,7} \\ −x_{4,8} − x_{3,7} \\ −x_{1,8} + x_{2,7} \\ x_{2,8} + x_{1,7} \end{matrix} ) = −C(e_{2})

We then use the results of the computation to solve the equation:

P_{5} (A(e_{j}) · \overline{e_{i}} + e_{j} · \overline{A(e_{i})}) = C (P_{5}(e_{j} · \overline{e_{i}})) = 0  (15)

Where P_{5} is the projection onto the five-dimensional subspace 𝕆_{5}^{∗} of the imaginary Cayley numbers. In doing so we get that

x_{1,1} = x_{2,2} = … = x_{8,8} = \frac{tr(A)}{8}.

As well as a set of seven linear systems of the kind (for brevity we write only one of them, the others being derived in the exact same way):

\{ \begin{matrix}x_{7,2} − x_{8,1} = x_{5,4} + x_{6,3} = −x_{3,6} − x_{4,5} = x_{1,8} − x_{2,7} \\ x_{6,3} + x_{8,1} = −x_{5,4} + x_{7,2} = −x_{2,7} + x_{4,5} = −x_{1,8} − x_{3,6} \\ x_{5,4} + x_{8,1} = −x_{6,3} + x_{7,2} = −x_{8,1} − x_{4,5} = −x_{2,7} + x_{3,6} \\ −x_{4,5} + x_{8,1} = x_{3,6} + x_{7,2} = x_{2,7} − x_{6,3} = −x_{1,8} + x_{5,4} \end{matrix}

those can be solved directly and give seven equations of the type:

x_{7,2} + x_{2,7} = x_{8,1} + x_{1,8} = x_{5,4} + x_{4,5} = x_{6,3} + x_{3,6} = −(x_{6,3} + x_{3,6})

so that, in general:

x_{i,j} = −x_{j,i} 1 ≤ i, j ≤ 8,

thus A is actually the sum of a skew-symmetric matrix and scalar multiple of Id_{ℝ(8)}. The same conclusion can reached by taking P_{6} or P_{7} instead of P_{5} since the conditions are actually redundant in (15).

References

[1] Berndt, J., F. Tricerri and L. Vanhecke, “Generalized Heisenberg Groups and Damek-Ricci Spaces,” Lecture Notes in Math. 1598, 1995.

[2] Cowling, M., A. Dooley, A. Korányi, and F. Ricci, H -type groups and Iwasawa decompositions, Advances in Math. 87 (1995), 1–41.

[3] Damek E., and F. Ricci, A class of nonsymmetric harmonic riemannian spaces, Bull. Am. Math. Soc. 27 (1991), 139–142.

[4]  Dickson, L. E., “Linear Algebras,” Hafner, New York, 1914.

[5] Dieudonné, J., “La Géométrie des Groupes Classiques, ” Springer, New York, 1963.

[6] Hein, W., “Struktur und Darstellungstheorie der klassischen Gruppen, ” Springer, New York, 1990.

[7] Gilbert, J.E., and M. A. M. Murray, “Clifford algebras and Dirac operators in harmonic analysis,” Cambridge Univ. Press, Cambridge, 1991.

[8] Gordon, C., Isospectral closed Riemannian manifolds which are not locally isometric, J. Diff. Geom. 37, (1993), 639–649.

[9] Kaplan, A., Riemannian nilmanifolds attached to Clifford modules, Geom. Dedicata 11, (1981), 127–136.

[10] —, Fundamental solutions for a class of hypoelliptic partial differential equations generated by the composition of quadratic forms, Trans. Amer. Math. Soc. 258 (1980), 147–153.

[11] Korányi, A., Geometric properties of Heisenberg-type groups, Advances in Math. 56 (1985), 28–38.

[12] Korányi, A., and H. M. Reimann, Quasiconformal mappings on the Heisenberg group, Invent. Math. 80 (1985), 309–338.

[13] Lawson, B. and M.-L. Michelsohn, “Spin Geometry, ” Princeton Univ. Press, 1989.

[14] Moore, C., Decompositions of unitary representations defined by discrete subgroups of nilpotent groups, Annals of Math. 82 (1965), 142–182.

[15] Mosak R., and M. Moskowitz, Stabilizers of Lattices in Lie Groups, J. of Lie Theory 4 (1994), 1–16.

[16] Pansu, P., Métriques de Carnot-Carathéodory et quasiisometries des espaces symetriques de rang un, Ann. of Math. 129, (1989), 1–60.

[17] Raghunathan, M. S., “Discrete Subgroups of Lie Groups,” Erg. d. Math. 68, Springer Verlag Berlin etc., 1972.

[18] Riehm, C., The automorphism group of a composition of quadratic forms, Trans. Amer. Math. Soc. 269 (1982), 403–414.

[19] Szabo, Z. T., Spectral Theory for Operator Families on Riemannian Manifolds, Proc. Symp. in pure Math., 53 (1993), 615–665.

Department of Mathematics Yale University 10 Hill House Avenue New Haven CT 06520 barbano@jules.math.yale.edu

Received February 26, 1997 and in final form June 15, 1998