Spectral theory of a class of nilmanifolds attached to Clifford modules

We determine the spectrum of the sub-Laplacian on pseudo H-type nilmanifolds and present pairs of isospectral but non-homeomorphic nilmanifolds with respect to the sub-Laplacian. We observe that these pairs are also isospectral with respect to the Laplacian. More generally, our method allows us to construct an arbitrary number of isospectral but mutually non-homeomorphic nilmanifolds. Finally, we present two nilmanifolds of different dimensions such that the short time heat trace expansions of the corresponding sub-Laplace operators coincide up to a term which vanishes to infinite order as time tends to zero.


Introduction
In 1966 Mark Kac's famous paper [22] asked the question "Can one hear the shape of a drum?". This work can be regarded as the beginning of a central topic of spectral geometry although the problem itself traces back to Hermann Weyl' s work at the beginning of the 20th century. Especially in the multi-dimensional situation, a negative answer to the above question was expected early on. Therefore, an important task was to construct isospectral but nonisometric or non-diffeomorphic or even non-homeomorphic manifolds. Such examples allow to determine geometric properties that are not determined by the spectrum. In high dimensions the first example of such manifolds was given by Milnor even earlier in 1964. In [32] a pair of 16-dimensional flat tori have been constructed which are isospectral but non-isometric. Nowadays, a general construction method by Sunada [37] and a wide range of examples are known, cf. [16,17,19,20,27]. In particular, they include lens spaces, spherical space forms or Heisenberg manifolds. Generalizing the last example the isospectrality problem may be considered for quotients \G of nilpotent Lie groups G of step k ≥ 2 by a lattice . In the following we will call such manifolds k-step nilmanifolds. Via an adaptation of representation theoretical methods due to C.S. Gordon and N.E. Wilson, pairs of isospectral nilmanifolds ( 1 \G, 2 \G) of step k ≥ 3 were constructed in [17]. Different from the known examples based on Sunada's theorem these manifolds need not to be isospectral for the Laplacian on 1-forms.
In the realm of Riemannian geometry it remains an interesting problem to construct isospectral but non-homeomorphic manifolds in a systematic way. Moreover, by restricting M. Kac's question to specific sub-classes of smooth manifolds (e.g. spheres or certain nilmanifolds) or by considering the spectrum of geometric operators different from the Laplacian one is led to new classification problems.
In the present paper we consider Kac's question for a class of subriemannian manifolds M carrying a geometrically defined second order sub-elliptic differential operator, called sub-Laplacian. More precisely, in our setup M is assumed to be a nilmanifold of step 2 whose covering simply connected nilpotent Lie group is of pseudo H -type. Such groups are generalizations of the well known Heisenberg type groups introduced by A. Kaplan in [23]. Their Lie algebras are called of pseudo H -type as well and were first considered in [10]. Pseudo H -type Lie algebras are constructed from Clifford algebras C r ,s of signature (r , s) and their (admissible) modules, cf. Sect. 4 or [10,[13][14][15] for a definition and more details. We also recall that the existence of lattices in pseudo H -type Lie groups G has been proven in [13]. With respect to a standard (integral) lattice we can therefore consider compact left-coset spaces \G.
Based on an explicit heat trace formula for the sub-Laplacian combined with the recent classification of pseudo H -type algebras in [14,15] we can give the negative answer to Kac's question in this non-standard setting and present a list of new examples. The subriemannian structure we deal with naturally extends to a Riemannian structure and we may as well consider the corresponding Laplacian on M. As it turns out in our examples the difference D := − sub is a "sum-of-squares-operator", i.e. it can be expressed in the form with globally defined vector fields Z k on M. Moreover, the operators D and sub commute and therefore and sub commute as well. As a consequence we have obtained new exam-ples of isospectral, non-homeomorphic manifolds in the usual sense, i.e. with respect to the Riemannian structure. Different from previously studied isospectral, non-isometric quotients ( 1 \G, 2 \G) for which the covering simply connected Lie group G is fixed and the lattice varies we note that in our construction also the group G is varying in the definition of both manifolds. Furthermore, by choosing the space dimension suitably high our method not only allows us to select pairs but any given number of isospectral, non-homeomorphic nilmanifolds. To our best knowledge these are the first examples of this kind.
Before stating the results more in detail we review some definitions. By a subriemannian manifold we understand a triple (M, H, ·, · ) where M is a smooth manifold (orientable and without boundary), H is a bracket generating subbundle in the tangent bundle T M and ·, · denotes a family of inner products on H which smoothly vary with the base point. Recall that H is called bracket generating if vector fields taking values in H together with a finite number of their iterated brackets span the tangent space at any point of M.
Based on the bracket generating condition one can assign to each point q ∈ M a flag of vector spaces which exhausts the full tangent space (see [1] for details). The subriemannian structure is called equi-regular, if k(q) and the dimensions dim H j q for j = 1, . . . , k(q) are independent of q ∈ M. Under this assumption and based on the Popp measure construction one intrinsically can define a sub-elliptic operator sub on M which generalizes the Laplace-Beltrami operator in Riemannian geometry (cf. [1]). We call this operator sub-Laplacian whereas it is called intrinsic hypoelliptic Laplacian in [1]. In case of a left-invariant subriemannian structure on a unimodular Lie group, which includes the case of a nilpotent Lie group, this operator is known to be a sum-of-squares of vector fields, [1,Proposition 12]. The same remains true if we descend sub to the quotient of G by a cocompact discrete subgroup (lattice) and consider the (intrinsic) sub-Laplacian on the left-coset space M = \G.
The manifolds M in this paper are equipped with a bracket generating subbundle H which is trivial as a vector bundle and a metric on H which naturally extends to a Riemannian metric on M. More precisely, there is a globally defined frame {X i } of H which is orthonormal at any point and skew-symmetric with respect to a naturally chosen volume form such that 1 : As is well-known the bracket generating property (also called Hörmander condition) implies that sub is a sub-elliptic operator (i.e. it satisfies an "a priori estimate with a loss of derivative", cf. [18]). Clearly, this property does not depend on the chosen Riemannian metric. As a consequence it can be shown that the sub-Laplacian on a closed manifold M has a compact resolvent and spectrum only consisting of eigenvalues with finite multiplicities (see [18]). Hence we can define the notion of isospectrality of two given subriemannian manifolds by replacing the spectrum of the Laplacian with the spectrum of the sub-Laplacian. The asymptotic distribution of eigenvalues for classes of self-adjoint operators with double characteristics acting on compact manifolds or an asymptotic expansion of the heat kernel for "sum-of-squares operators" with first order term satisfying Hörmander's condition have been obtained in [7,8,21,[29][30][31]36], respectively. 1 In oder to simplify the heat kernel expression we have chosen the factor 1 2 in front of the sum.
In the special case where M is a 2-step nilmanifold it was shown in [3] that the heat trace of the sub-Laplacian admits an asymptotic expansion similar to the heat trace expansion of the Laplacian on a torus.
In order to detect isospectral (subriemannian) nilmanifolds we first need to determine the spectrum of the sub-Laplacian \G sub on a 2-step nilmanifold M = \G. Based on an explicit expression of the heat kernel for sub on the covering group G in [6,9,12] a formula for the heat trace of \G sub descended from G to M = \G was obtained in [4]. In case of a pseudo H -type group G this trace formula simplifies further and in principle can be used to explicitly calculate the spectrum of sub on M. However, we need not to perform the full calculation. In order to identify isospectral manifolds it is sufficient to compare the corresponding trace formulas.
In a second step we need to classify non-homeomorphic nilmanifolds 1 \G 1 and 2 \G 2 of the same dimension. First, we reduce this task to a classification of pseudo H -type Lie algebras up to isomorphisms (cf. Corollary 7.2). Then we apply the very recent classification results in [14,15].
The paper is organized as follows: in Sect. 2 we introduce the sub-Laplacian on a general 2-step nilpotent Lie group G and we recall an explicit integral expression of its heat kernel known as Beals-Gaveau-Greiner formula, cf. [5,6,9].
Assuming the existence of a lattice in G we decompose the sub-Laplacian sub on the compact nilmanifold M = \G into an infinite sum of elliptic operators acting on line bundles in Sect. 3. Via this method we obtain a decomposition of the heat trace of \G sub into the heat traces of its component elliptic operators, cf. [4], and we present a trace formula for the sub-Laplacian on M.
In Sect. 4 we recall the notion of pseudo H -type Lie algebras and groups following [10,13,14]. We discuss the existence and some basic properties of integral lattices for such groups. These will play a role in our construction in Sect. 7.
In Sect. 5 we study the eigenvalues of a matrix-valued function which encodes the structure constants of the pseudo H -type Lie algebra. These data are essential in the calculation of the heat kernel of the sub-Laplacian in Sect. 2 and the trace formula in Sect. 3. Based on the trace formula we give a criterion for isospectrality of two pseudo H -type nilmanifolds in Sect. 6 (Theorem 6.3).
The last sections contain our main results. We use the classification of pseudo H -type Lie algebras in [14,15] to construct finite families of isospectral, non-homeomorphic pseudo H -type nilmanifolds. Finally, we present two nilmanifolds of different dimensions such that the short time heat trace expansions of the corresponding sub-Laplace operators coincide up to a term vanishing to infinite order as time tends to zero.

Heat kernel on two step nilpotent Lie groups
We recall the integral form of the heat kernel for a sub-Laplacian on simply connected two step nilpotent Lie groups given in [5,6], see also [9,12].

Sub-Laplacian on two step nilpotent groups
Let G be a simply connected two step nilpotent Lie group with Lie algebra N . We assume that and its complement, respectively. Moreover, we assume that N is equipped with an inner product with respect to which {X i , Z k } becomes an orthonormal basis. Hence the Lie algebra N is decomposed into an orthogonal sum The expansion of Lie brackets

Remark 2.1
Throughout the paper we identify the group G with R N × R d via the above coordinates, i.e.
Then the exponential map exp : N ∼ = → G is the identity. Via the Baker-Campbell Hausdorff formula and this identification we can express the group product * on G ∼ = N in the form More explicitly and with respect to the above coordinates one has: Let X i denote the left-invariant vector field on G corresponding to X i ∈ N and consider the sub-Laplacian Based on (2.1) the operator G sub is known to be sub-elliptic [18] and essentially selfadjoint in L 2 (G) with respect to the Haar measure and considered on compactly supported smooth functions C ∞ 0 (G), cf. [34,35].

Beals-Gaveau-Greiner formula
Next we recall the integral expression of the kernel function K (t, g, h) ∈ C ∞ (R + × G × G) of the heat operator e −t G sub , (2.5) where G is a general 2-step nilpotent Lie group as above. The existence of a smooth kernel has been shown in [34,35] and since the sub-Laplacian G sub is a left-invariant operator it follows that K is a convolution kernel, i.e.
In [5,6,9,12] an integral expression of k G is given explicitly. Below we will calculate the spectrum of the sub-Laplacian on a class of nilmanifolds by using this expression. Recall that in the integrand of k G two functions (action and volume function) appear. The integration is taken over a space which can be interpreted as the characteristic variety of the sub-Laplacian. Here we will neither present the details of this structure nor a proof of the next theorem. Theorem 2.2 (Beals-Gaveau-Greiner formula, [5,9]) The integral kernel of the heat operator (2.5) has the form: where the functions f = f (τ, g) ∈ C ∞ (R d ×G) and W (τ ) ∈ C ∞ (R d ) are given as follows: where z, z = d k=1 z k z k denotes the Euclidean inner product on R d . Remark 2.3 Later on we will use the notation •, • r ,s for a non-degenerate indefinite scalar product with the signature (r , s) such that •, • = •, • d,0 .
We call f = f (τ, x, z) and W (τ )dτ the complex action function and the volume form, respectively. Recall that f is constructed by the complex Hamilton-Jacobi method, and the volume function W (τ ) is sometimes referred to as van Vleck determinant. It is the Jacobian of the correspondence between the space of initial conditions and boundary conditions when we solve the Hamilton equation associated to the symbol of the sub-Laplacian. The solution can be interpreted as the bi-characteristic flow in the subriemannian setting. We recall that the volume function satisfies a transport equation.

Lattices and decomposition of a sub-Laplacian
Based on Theorem 2.2 we describe the heat kernel of the sub-Laplacian descended to the quotient space \G (left coset space) by a lattice . Such a space is called a (compact) 2-step nilmanifold. In the following we assume that there exists a lattice (cocompact discrete subgroup) in G. We recall Malćev's Theorem: (Malćev,[26,33]) A nilpotent Lie group G possesses a lattice , i.e. \G is compact, if and only if there exists a basis {Y i } in its Lie algebra g such that the structure constants {α k i j } defined by are all rational numbers.

Torus bundle and a family of elliptic operators
We recall a heat trace formula which previously has been obtained in [4,Theorem 4.2]. Our analysis is essential based on this formula and in order to keep the paper self-contained we now repeat the main steps of the calculation. Let be a lattice in a simply connected 2-step nilpotent Lie group G ∼ = R N × R d . The quotient space \G can be equipped with a subriemannian structure naturally inherited from that of G. Its sub-Laplacian, which we now denote by \G sub , is the operator descended from the sub-Laplacian G sub on G. For an element g ∈ G we will denote by [g] ∈ \G the corresponding class in the quotient space. Then, the heat kernel Assuming the existence of a lattice in G we can decompose the sub-Laplacian into a family of differential operators acting on invariant subspaces according to a torus bundle structure of \G. Next, we present some details and give the heat kernel expression for each component elliptic operator.
Let A ∼ = R d be the center of the group G where as before the identification is done with respect to the fixed orthonormal basis {Z k } of A. We obtain a principal bundle with the with the sum in the Lie algebra. Let n be an element in the "dual lattice" [ ∩ A] * of ∩ A, that is, n is a linear functional on A with the property that We may express n in the form n = d k=1 n k Z k with integer coefficients n k ∈ Z such that Then, the function space C ∞ ( \G) is decomposed via a Fourier series expansion: √ −1 n,λ is a unitary character corresponding to a dual element n ∈ [ ∩ A] * . So, we decompose The subspace F (n) can be seen as a space of smooth sections of a line bundle E (n) on the base space ( + A)\G ∼ = ( / ∩ A)\(G/A) associated to the character χ n . The sub-Laplacian leaves invariant each subspace F (n) and therefore it can be interpreted as a differential operator D (n) acting on the line bundle E (n) . Since the subbundle spanned by the (left)-invariant vector fields { X i | i = 1, . . . , N } defines a connection, i.e., its linear span is equivariant and transversal to the structure group action by A/( ∩ A), each operator D (n) is elliptic. Hence the sub-Laplacian \G sub can be seen as an infinite sum of elliptic operators on the torus * A\G.
As a consequence we obtain a decomposition of the operator trace: Recall that {Z k | k = 1, . . . , d} denotes an orthonormal basis of the center [N , N ] of N . As before we write Z k , k = 1, . . . , d for the corresponding left-invariant vector fields on the group G. We equip G with a left-invariant Riemannian metric defined by assuming that the frame [ X 1 , . . . , X N , Z 1 , . . . , Z d ] is orthonormal at any point of G. Then the corresponding Laplacian has the form The action of the difference G − G sub on the subspace F (n) for each dual element n ∈ [ ∩ A] * is given as follows:

Heat trace of the component operators
Next we give an expression of the heat trace of each operator D (n) . Recall that the heat kernel K \G of \G sub is given by (3.1). Let F and F ∩A be a fundamental domain for the lattice in G and ∩ A in the Euclidean space A, respectively. Then the integral is the kernel function for the heat operator e −tD (n) , that is it satisfies where θ ∈ A. Let M = {μ i } be a set of complete representatives of the coset space /( ∩A), then the trace of the heat operator e −tD (n) is given as follows: [4]) For each n in the dual lattice [ ∩ A] * and with the heat kernel K G of the sub-Laplacian on G: Here denotes the set of representatives of the quotient group A . Applying Theorem 2.2 we can give a more concrete expression of the formula in Proposition 3.3. For this purpose and for the sake of simplicity, we assume that the structure constants c k i j in (2.2) are of the form with a common positive integer p 0 ≥ 1 and integers q k i j . Then we fix a lattice where n k ∈ Z we have where the function ϕ t (τ, μ) in the integrand is given by: In the following we write ϕ t (τ, μ) for the Fourier transform of ϕ t with respect to the τvariable. Then With a suitable set of linear independent vectors a 1 (n), . . . , Here b(n) ≤ N and b(n) = N if and only if n = 0. Hence Theorem 3.4 For each n in the dual lattice [ ∩ A] * and with the above notation: In particular, it holds: Proof It suffices to show the second equation in (3.4). Note that the defining equation The last statement follows from (3.2) and (3.4). in [23,24] and have been first introduced in [10]. An extensive analysis of the structure and classification of pseudo H -type groups and their algebras can be found in the recent papers [13][14][15]. For completeness we recall the relevant definitions: We write R r ,s for the Euclidean space R r +s equipped with the non-degenerate scalar product Consider the quadratic form q r ,s (x) = x, x r ,s and let C r ,s denote the Clifford algebra generated by (R r ,s , q r ,s ) [25]. We call a C r ,s -module V admissible, if there is a nondegenerate bilinear form (= scalar product) •, • V on V satisfying the following conditions: (a) There is a Clifford module action J : Moreover, from (a) and (b) one concludes: We write {J , V , •, • V } for an admissible module of the Clifford algebra C r ,s with the module action J = J z and the scalar product •, • V .

Remark 4.1
The existence of an admissible C r ,s -module V has been shown in [10]. If s = 0 then an admissible module V needs not to be irreducible. More precisely, five cases are possible which all are present in the classification. If C r ,s has, up to equivalence, only one irreducible representation (J , V ), then either V or the sum V ⊕ V is admissible. In the case where C r ,s has two non-equivalent irreducible representations (J (i) , V i ), i = 1, 2, then either V i for i = 1, 2 both are admissible, or only V 1 ⊕ V 2 is admissible, or V 1 ⊕ V 1 and V 2 ⊕ V 2 simultaneously are admissible. These cases are complementary to each other (cf. [10,[13][14][15]).
In the case s = 0 the situation is simpler. Every irreducible module V is admissible with respect to an inner product (i.e. •, • V is positive definite). Originally such cases have been defined and studied by Kaplan in [23].
In the following, we call a vector X ∈ V positive (resp. negative) if the scalar product X , X V is positive (resp. negative) and null vector if X , X V = 0. A similar notation is used for vectors Z ∈ R r ,s . If s > 0, then an admissible module V with scalar product •, • V has positive and negative subspaces of the same dimension N with respect to the above scalar product •, • V , cf. [10]. In particular, dim V = 2N is even.
Moreover, V decomposes into the orthogonal sum of minimal dimensional admissible modules. In fact, since the scalar product restricted to such an invariant subspace is nondegenerate the orthogonal complement is also an admissible module. (1) The 2-step nilpotent Lie algebra V ⊕ ⊥ R r ,s with center R r ,s and Lie brackets defined via the relation will be denoted by N r ,s (V ). We write G r ,s (V ) for the corresponding simply connected Lie group and call it a pseudo H -type group, cf. [10,13]. (2) If V is of minimal dimension among all admissible modules, then we call V minimal admissible and we shortly write N r ,s := N r ,s (V ) and G r ,s := G r ,s (V ).

Remark 4.3
Note that minimal admissible modules are cyclic and the nilpotent Lie algebra N r ,s is unique up to isomorphisms, even if the Clifford algebra C r ,s admits two nonequivalent irreducible representations (cf. [13]).
We fix an orthonormal basis {Z k } r +s k=1 in R r ,s , i.e. we assume that: j = 1, . . . , s), and Theorem 4.4 (cf. [11,13]) Assume that s > 0. Then there exists an orthonormal basis For each k, the operator J Z k maps X i to some X j or − X j with j = i.

Remark 4.6
An interesting problem, which we will postpone to a future work, consists in a classification of integral bases up to isomorphisms. Consider an orthonormal basis {Z k } r +s k=1 of R r ,s in the above sense. If V is a minimal admissible C r ,s -module, then we can define a finite subgroup G in GL(V ) generated by {J Z k : k = 1, . . . , r +s}. Consider the commutative subgroup: By "A > 0" we mean that A maps positive (resp. negative) vectors in V to positive (resp. negative) vectors. Such groups are partially ordered with respect to the inclusion and we assume that S is a maximal element. Further, we assume that v ∈ V is a common eigenvector of elements in S. Necessarily, v is not a null vector, i.e. v, v V = 0. Consider We conjecture that a suitable choice of the common eigenvector v leads to an integral basis Conversely, let {X i , Z j } be an integral basis and put ±B := {±X : = 1, . . . , m = dim V }. Then each J Z i defines a bijective map J Z i : ±B → ±B and elements in the group G act on ±B. We obtain a subgroup S as above from this basis by defining A ∈ G : A(X 1 ) = X 1 =: S ⊂ G.
We conjecture that every maximal subgroup S defines an integral basis. A classification of integral bases up to isomorphisms is left as an interesting problem, which we postpone to a future study.
From now on we assume that {X i , Z k } is an integral basis of N r ,s (V ).

Corollary 4.7 If there exists i
Hence any basis vector X i is mapped to some X j or −X j by at most one operator J Z k .
Proof If k ≤ r then J Z k maps positive to positive and negative to negative elements. Similarly, if k > r , then J Z k maps positive to negative and negative to positive elements. Therefore, under the above assumption only the cases k, ≤ r or k, > r are possible.
Let us assume k = such that ±X i = J Z k J Z (X i ). By the previous remark we have This equation contradicts the existence of the eigenvalue 1 or −1 of J Z k J Z .

Corollary 4.8 If we put [X i , X j ] = c k i j Z k , then c k i j can be non-zero for at most one k. If c k i j is non-zero then it equals ±1.
Proof The statement follows from Corollary 4.7 and

Definition 4.9 From an integral basis
In the following we call r ,s (V ) a standard integral lattice in N r ,s (V ). If N r ,s is constructed from a minimal admissible module V (cf. Definition 4.2), then we write r ,s := r ,s (V ).

Remark 4.10
A standard integral lattice is not unique. A complete classification will be subject of another work. For particular cases the construction of r ,s is found in [13].
In the following two sections we consider the sub-Laplacian  descended from (4.5) to the nilmanifold r ,s (V )\G r ,s (V ). Based on the sub-ellipticity of (4.6) it is known that the spectrum of the sub-Laplacian only consists of eigenvalues with finite multiplicities. In principle our trace formula in Theorem 3.4 can be used to obtain the spectrum of (4.6). However, we will not calculate the eigenvalues and multiplicities explicitly since a comparison of heat traces is sufficient to decide isospectrality.

The structure constants of pseudo H-type groups
In the case of pseudo H -type groups we calculate the characteristic polynomial of the matrix (z) in (2.3) in the case where s > 0 in Definition 4.2 of the pseudo H -type group G r ,s (V ). Recall that this matrix is an essential ingredient for the integral expression of the heat kernel in Theorem 2.2.
Throughout this section we assume that s > 0 so that we can use the integral basis in can be written in form of a matrix with respect to the basis {X i } of V : Here A T (μ) denotes the transposed matrix of A(μ) ∈ R(N ). Moreover, the identity J 2 z = − z, z r ,s yields additional relations between the component matrices A, B, C and D which are collected in the next lemma.
Let L be a linear map on V ∼ = R 2N . The same notation is used for its matrix representation with respect to the basis {X i }. Let ·, · denote the Euclidean inner product on R 2N and fix x, y ∈ V . We calculate the matrix representation of the transpose L * of L with respect to the scalar product ·, · V . Consider the matrix

Then we have
which implies that L * = τ L T τ . A direct calculation shows that the skew-symmetric matrix (z) in (2.3) is related to the above matrix representation of J z as follows: In order to determine the eigenvalues of the matrix ( √ −1z) we employ the relation Hence we have According to Lemma 5.1 one has B T B = ν 2 = B B T and B T AB Together with the skew-symmetry of (z): Therefore:
If z = 0 then the matrix (z) has the eigenvalue zero only when μ = ν . In this case the matrices B(ν) and D(μ) are non-singular so that the dimension of the solution space (z) · x = 0 is N (= half the dimension of V ).

Proposition 5.5
Assume that z = 0 and μ = ν . The kernel of (z) is given by Finally we determine the dimension of the eigenspaces corresponding to the above eigenvalues ±( μ ± ν ). Proposition 5.6 Let z = 0. Then the dimensions of the eigenspaces E λ of ( √ −1z) with respect to the eigenvalue λ = ±( μ ± ν ) are given as follows: Multiplying the first equation by A( √ −1μ) and the second equation by B( On the left hand side we use Lemma 5.1 and deduce the following two equations Adding these identities and using Lemma 5.1, (ii) gives Together with the Eq. (5.2) we find that B( √ −1ν)y = ν x. This shows that Since B(ν) is non-singular for ν = 0 the vector y is uniquely determined by x. Conversely, the eigenvector x of the matrix A( √ −1μ) with the eigenvalue − μ = 0, determines the eigenvector of the matrix ( √ −1z) by putting y = ν B( √ −1ν) −1 x.
Since A(μ) is skew-symmetric for any real vector μ and A(μ) 2 = − μ 2 , the dimension of the eigenspace of the matrix A( √ −1μ) with respect to the eigenvalue μ is half the size of the matrix A, i.e. it equals N 2 . The remaining eigenvalues can be treated similarly and therefore (i) follows. (ii): Under the assumption of (ii) and by applying the relations in Lemma 5.1 we have is equivalent to B( √ −1ν)y = ν x, which can be uniquely solved for any given x ∈ R N . The case λ = − ν is treated in the same way and (ii) follows. (iii): If μ = 0 and ν = 0, then C = B = 0 and with λ = μ we have the equation The matrix ( √ −1z) is Hermitian and therefore can be diagonalized. From the expression of det ( (z) + λ) 2 , we deduce that the eigenspaces have dimension N . The case λ = − μ can be treated similarly.

Remark 5.8
In the case s = 0 we can find an admissible module V with respect to an inner product (= positive definite scalar product) and so we obtain: Therefore, the statement of Corollary 5.7 remains valid even in this case: Recall that A ∼ = R r ,s denotes the center of the group G r ,s (V ). However, our notation will not indicate the dependence on the parameter (r , s).

Remark 6.1
If s = 0, then the matrix (n) is always non-singular for n = 0. In this case we find: Furthermore, from the heat trace formula (3.2) we conclude that the eigenvalues of the sub-Laplacian on r ,0 (V )\G r ,0 (V ) are given by • λ l = 2π 2 l 2 for l ∈ Z 2N . • β n,m = 4π n (2m + N ) for n ∈ Z r \{0} and m ∈ N with multiplicity 4 N n N m+N −1 N −1 .
Since λ 2 l ∈ π 4 Z and β 2 n,m ∈ π 2 Z, we can distinguish between these different numbers, i.e. knowing the eigenvalues, we can extract the dimension of the admissible module V . We conclude that if two manifolds r ,0 (V )\G r ,0 (V ) and r ,0 (V )\G r ,0 (V ) are isospectral with respect to the sub-Laplacian, then they have the same dimension.
Based on the dependence of the above heat traces on the parameters μ , ν and N we conclude: Corollary 6.2 r ,s \G r ,s and s,r \G s,r are isospectral with respect to the Laplacian and the sub-Laplacian, if the dimensions of their admissible modules coincide.
The pseudo H -type algebra N r ,s does not depend on the chosen minimal admissible module, cf. [13]. Moreover, in the above determination we do not explicitly use the assumption that the admissible module is minimal. This implies: Theorem 6.3 If V is a sum of k minimal admissible modules, then the heat trace in each of the cases, (6.1), (6.2) and (6.3) above is the k-th power of the corresponding heat trace for the manifold r ,s \G r ,s . Let U be an admissible module of C s,r with dim V = dim U , then the two nilmanifolds r ,s (V )\G r ,s (V ) and s,r (U )\G s,r (U ) are isospectral.

Isospectral, but non-homeomorphic nilmanifolds
By applying Theorem 6.3 and the classification of pseudo H -type Lie algebras in [14,15] we detect finite families of isospectral but mutually non-homeomorphic pseudo H -type nilmanifolds. If the module V in the construction of the pseudo H -type Lie algebra is minimal admissible, then pairs of non-isomorphic pseudo H -type Lie algebras N r ,s N s,r of the same dimension dim N r ,s = dim N s,r are known (see [14,15] or Table 7.1). By choosing integral lattices r ,s and s,r in the corresponding Lie groups G r ,s and G s,r , respectively, we first detect pairs Here ∼ isosp means isospectral with respect to the sub-Laplacian and homeo indicates that two manifolds are non-homeomorphic. First, we explain the method of detecting nonhomeomorphic nilmanifolds.
If there is a homeomorphism between the nilmanifolds N r ,s := r ,s (V )\G r ,s (V ) and N s,r = s,r (U )\G s,r (U ), then their fundamental groups and are isomorphic. We can apply the following general fact from [33]: Proposition 7.1 Any isomorphism between lattices in simply connected nilpotent Lie groups can be extended to an isomorphism between the whole groups.
From these observations we conclude: Corollary 7.2 If the nilmanifolds N r ,s and N s,r are homeomorphic, then N r ,s (V ) and N s,r (U ) have to be isomorphic as Lie algebras.

Pairs of non-homeomorphic, isospectral nilmanifolds via minimal admissible modules
To obtain pairs (M 1 = N r ,s , M 2 = N s,r ) of nilmanifolds with (7.2) we determine pairs (r , s) such that dim N r ,s = dim N s,r , but N r ,s N s,r , i.e. both Lie algebras are not isomorphic. The classification of pseudo H -type algebras in Table 1 constructed from minimal admissible modules was obtained in [14,15]. This table gives us only information about the cases 0 ≤ r , s ≤ 8. For the remaining cases we use the following periodicity (see also [2]):  Table 1 we see that for (r , s) ∈ {(3, 1), (3,2), (3, 7)} both nilmanifolds N r ,s , N s,r have the same dimension but they are non-homeomorphic.
Hence we obtain the following result: Notation: If V r ,s min denotes a minimal admissible C r ,s -module, then the letter 'd' = double (or 'h'=half, respectively) at the position (r , s) means that dim V r ,s min = 2 dim V s,r min (or dim V r ,s min = 1/2 dim V s,r min , respectively). The symbol ' ∼ =' indicates that the Lie algebras N r ,s and N s,r are isomorphic while ' ' means that they are non-isomorphic

Finite families of non-homeomorphic, isospectral nilmanifolds
In case the module V in the construction of the Lie algebra N r ,s (V ) is not minimal admissible we can use the classification result in [15, Theorem 4.1.2 and Theorem 4.1.3] to determine families {M 1 , . . . , M k } of a given length k ∈ N of isospectral, mutually non-homeomorphic nilmanifolds, i.e. (7.2) holds. First, we fix the pair (r , s) and study the Lie algebra N r ,s (U ), constructed from different admissible modules. In the general case the classification of isomorphic pseudo H -type Lie algebras is more subtle and to state the result we need to introduce some notation from [15]. Note that for any given minimal admissible module {J , V , •, • V } also the module {J , V , − •, • V } is minimal admissible. The upper index in the notation V r ,s;± min;± indicate that the scalar product of the two minimal admissible modules V r ,s;+ min;± and V r ,s;− min;± differ by a sign. then we obtain non-isomorphic Lie algebras N r ,s (U ( p 1 , q 1 )) and N r ,s (U ( p 2 , q 2 )) of the same dimension R · dim V r ,s;+ min if simultaneously ( p 1 , q 1 ) = ( p 2 , q 2 ) and ( p 1 , q 1 ) = (q 2 , p 2 ).
We fix an integer k and determine all pairs of integers ( p i , q i ) with the properties: With such pairs we define: From Theorem 7.5 and the above remark we conclude that the Lie algebras N r ,s (U i ) and N r ,s (U j ) are mutually non-isomorphic for r ≡ 3 mod 4, s ≡ 1, 2, 3 mod 4. In order to present a concrete family of nilmanifolds with the required properties we choose r = 3 and s = 1 such that dim V 3,1 min = 8. Let k be even and choose m ∈ N such that k = 2m. The pairs ( p i , q i ) with (a)-(c) are of the form {(i, k − i) | 0 ≤ i ≤ m} and we conclude: The following m + 1 nilmanifolds are isospectral, but mutually non-homeomorphic with respect to the (sub)-Laplacian.

Remark 7.7
For any given integer m ∈ N and by using the above method, we can construct m + 1 nilmanifolds of the common dimension 4 + 16m which are isospectral but mutually non-homeomorphic. In particular, one obtains a pair of such manifolds of the (minimal) dimension 4 + 16 × 1 = 20. Note that via the first method (i.e. Corollary 7.4) we can find a pair of such nilmanifolds of dimension 12.
To minimize the dimension of the constructed family of nilmanifolds we should use a third method which is based on [15,Theorem 4.2], which treat the case r ≡ 3 mod 4 and s ≡ 0 mod 4. In this situation there are two non-equivalent irreducible representations and we use the lower index ± in the notation below to distinguish the minimal admissible modules corresponding to each irreducible modules (or to each sum of irreducible modules, cf. Remark 4.1).
Next, we choose r = 3, s = 0 and fix m ∈ N. Then we consider the following family of m + 1 admissible modules: We obtain a family of m + 1 mutually non-homeomorphic, isospectral nilmanifolds of common dimension 3 + 8m.
where V d = 2π d 2 / ( d 2 ) denotes the volume of the (d − 1)-dimensional unit sphere. Now we use the following power series expansion for |x| < 1: A change of variables in the integral is applied to obtain: The infinite sum is called multiple Hurwitz zeta function and previously has been studied in the literature: Hence: The left hand side can be calculated from the spectral data (more precisely, from the heat trace expansion in Theorem 8.1). Hence the problem reduces to the question, whether for each k ∈ N the assignment: is injective. (b) Consider two isospectral compact nilmanifolds M j = j \G j where j = 1, 2. Assume that both are equipped with a left-invariant subriemannian structure as described in this paper. Is it true that dim G 1 = dim G 2 (see Remark 6.1)? (c) The distribution of eigenvalues for classes of hypoelliptic operators with double characteristics on compact manifolds and under additional conditions is well-studied (e.g. see the work by Menikoff and Sjöstrand [29][30][31]), and Melrose [28]. Moreover, in the case of "sum-of-squares operators" satisfying Hörmander's bracket generating condition the asymptotic of the heat kernel at small times was found by Ben Arous, Léandre (see [7,8]) in a form which encodes geometric data of an induced subriemannian structure (such as the Carnot Carathéodory metric). However, not much seems to be known on the precise growth order or coefficient of the second term in the expansion of the eigenvalue counting function for the sub-Laplacian on compact nilmanifolds. Based on a classification of lattices and the explicit spectral data such question in the case of Heisenberg manifolds has been discussed in [36]. In generalizing R. Strichartz's result one may study the following problem: Let M := \G denote a compact nilmanifold (e.g. modelled over a pseudo H -type Lie group). Determine the growth order or even the coefficient of the second term in the eigenvalue counting function for the corresponding sub-Laplacian.
Acknowledgements Open Access funding provided by Projekt DEAL.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Appendix
In the appendix we present the dimensions of minimal admissible modules for some basic cases in Table 2. These data are taken from [14,15] which we refer to for more details and notations. The remaining cases can be obtained by (4,4), (8,0) and (0, 8)-periodicities with respect to the signature (r , s), respectively. In particular, the table indicates the cases in which two non-equivalent minimal admissible modules exist. However, it is known that pseudo H -type algebras constructed from two non-equivalent minimal admissible modules are isomorphic.