A NEW CONSTRUCTION OF ISOSPECTRAL

RIEMANNIAN NILMANIFOLDS WITH EXAMPLES

Ruth Gornet

Abstract. We present a new construction for obtaining pairs of higher-step isospectral Riemannian nilmanifolds and compare several resulting new examples. In particular, we present new examples of manifolds that are isospectral on functions, but not isospectral on one-forms.

§1 Introduction.

The spectrum of a closed Riemannian manifold (M, g), denoted spec(M, g), is the collection of eigenvalues with multiplicities of the associated Laplace–Beltrami operator acting on smooth functions. Two Riemannian manifolds (M, g) and (M′, g′) are said to be isospectral if spec(M, g) = spec(M′, g′). A basic question in spectral geometry is determining what geometric information is contained in the spectrum of a Riemannian manifold.

Despite considerable research in the area, only a few geometric properties are known to be spectrally determined; for example, dimension, volume, and total scalar curvature. Examples of isospectral manifolds provide us with the only means for determining properties not determined by the spectrum.

Most previous constructions for producing isospectral manifolds are based solely on representation theory. In fact, most known examples can be explained by Sunada’s method [S] or its generalizations [GW1],[DG2]. (See [B2] for a general overview.) All manifolds constructed by these methods are strongly isospectral; that is, all natural, strongly elliptic, self-adjoint operators on the manifolds are isospectral. In particular, these manifolds share the same p-form spectrum. The p-form spectrum of a manifold is the eigenvalue spectrum of the Laplace-Beltrami operator extended to act on smooth p-forms for p a positive integer.

The primary goal of this paper is the development of a new construction for producing pairs of isospectral nilmanifolds. Two-step nilmanifolds have been a rich source of examples of isospectral manifolds. Moreover, their geometry has been studied in some detail. (See [DG1],[BG],[O],[P1],[P2],[P3],[GW1],[GW2],[G1],[G2], and [E].) This new construction uses techniques from Riemannian geometry, Lie groups, and representation theory to produce

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 58G25, 22E27; Secondary 53C30.

Key words and phrases. Isospectral manifolds; Spectrum of the Laplacian on forms; Nilpotent Lie groups; Cocompact, discrete subgroups; Quasi-regular representations; Square-integrable representations.

Research at MSRI supported in part by NSF grant DMS-9022140. Research at MSRI and Texas Tech supported in part by NSF grant DMS-9409209.

pairs of isospectral nilmanifolds of arbitrary step. The higher-step examples have a much richer geometry, exhibiting many properties not previously found.

While representation theory is used as a tool, this construction differs from previous ones in that the resulting pairs of isospectral manifolds need not be isospectral on one-forms, and so do not fall under the traditional Sunada setup. This property was previously exhibited by pairs of isospectral Heisenberg manifolds constructed by Gordon and Wilson [GW2], [G2]. And for any choice of P, Ikeda [I2] has constructed examples of isospectral lens spaces that are isospectral on p-forms for p = 0, 1, ⋯, P but not isospectral on (P + 1)-forms. These are the only known examples.

The only other examples of isospectral manifolds that do not fall under the traditional Sunada construction are bounded domains of Urakwa [U], and nonlocally isometric examples of Szabo [Sz2], Gordon [G3], [G4], and Gordon–Wilson [GW3].

Recent results of Pesce [P4] now explain the Ikeda and Urakawa examples in a Sunada-like setting. This setting requires a genericity assumption that excludes nilmanifolds. Moreover, the construction presented here generalizes the method used by Gordon–Wilson to construct the Heisenberg examples.

Consequently, outside of the nonlocally isometric examples mentioned above, the construction below subsumes all known examples of isospectral manifolds that do not fall under a Sunada setup.

After establishing notation in Section 2, the new construction for producing pairs of higher-step isospectral nilmanifolds is presented in Section 3. In Section 4, we use the new construction to produce pairs of isospectral three-step nilmanifolds with the following combinations of properties.

Table I: New Examples of Isospectral Manifolds

Pair of 3-Step  ∀p Same Rep. Equiv. Isomorphic  Same  Same
Isospectral  p-form Fundamental Fundamental Length Marked Length
Nilmanifolds  Spectrum  Groups  Groups  Spectrum  Spectrum
I(7 dim)  Yes  Yes  No  No  No
II(5 dim)  Yes  Yes  Yes  Yes  No
III\IV(7\5 dim)  No  No  No  No  No
V(7 dim)  No  No  Yes  Yes  Yes

The properties listed in the above table are defined as follows. Two cocompact (i.e. Γ\G compact), discrete subgroups Γ_{1} and Γ_{2} of a Lie group G are called representation equivalent if the associated quasi-regular representations are unitarily equivalent. (See Section 2 for details.) The length spectrum of a Riemannian manifold is the set of lengths of closed geodesics, counted with multiplicity. The multiplicity of a length is defined as the number of distinct free homotopy classes in which the length occurs. (Note: other definitions of multiplicity also appear in the literature.) The pairs of isospectral manifolds above have the same lengths of closed geodesics. However, the length spectra often differ in the multiplicities that occur. The marked length spectrum takes into account not only the lengths of the closed geodesics but also the free homotopy classes in which the geodesics occur.

In Section 4 we compare the quasi-regular representations and the fundamental groups of Examples I through V. We also compare the p-form spectrum, but the calculations are left to an Appendix. The length spectrum and marked length spectrum of these examples will be examined in [Gt4].

Example I is the first example of a pair of nonisomorphic, representation equivalent, cocompact, discrete subgroups of a nilpotent Lie group. It is also the first example of a pair of representation equivalent cocompact, discrete subgroups of a solvable Lie group producing Riemannian manifolds that do not have the same length spectrum. This example has implications in representation theory on nilpotent Lie groups and motivated [Gt1] and [Gt2].

We prove in the Appendix that the manifolds in Examples III, IV and V are not isospectral on one-forms. Outside of the traditional Sunada setup, no general method is known for comparing the one-form spectrum of manifolds. The methods illustrated in the Appendix are new, as previously used techniques could not be applied to the higher-step examples. The only previous examples of manifolds that are isospectral on functions but not isospectral on p-forms for all p are the lens spaces and Heisenberg manifolds mentioned above.

Example V is the first example of a pair of Riemannian manifolds with the same marked length spectrum, but not the same spectrum on one-forms. This example contrasts with two-step results relating the marked length spectrum and the p-form spectrum [E]. This example will be studied in detail in [Gt4].

Most of the contents of this paper are contained in the author’s thesis at Washington University in St. Louis in partial fulfillment of the requirements for the degree of Doctor of Philosophy. The author wishes to express deep gratitude to her advisor, Carolyn S. Gordon, for all of her suggestions, encouragement and support.

§2 Background and Notation.

Let G be a simply connected Lie group and let Γ be a cocompact, discrete subgroup of G. A Riemannian metric g is left invariant if the left translations of G are isometries. The left invariant metric g projects to a Riemannian metric on Γ\G, which we also denote by g. Note that a left invariant metric is determined by a choice of orthonormal basis of the Lie algebra 𝔤 of G.

As G is unimodular, the Laplace–Beltrami operator of (Γ\G, g) may be written

(2.1)  Δ = − ∑_{i=1}^{n} E_{i}^{2},

where {E_{1}, ⋯, E_{n}} is an orthonormal basis of the Lie algebra 𝔤 of G.

The Laplace-Beltrami operator acting on smooth p-forms is defined by Δ = dδ +δd. Here δ is the metric adjoint of d. Equivalently δ = (−1)^{n(p+1)+1} ∗ d∗, where ∗ is the Hodge-∗ operator. We denote the p-form spectrum of a Riemannian manifold (M, g) by p-spec(M, g).

The quasi-regular representation ρ_{Γ} of G on L^{2}(Γ\G) is defined as follows: for all x in G and f in L^{2}(Γ\G),

ρ_{Γ}(x)f = f ∘ R_{x}.

Here R_{x} denotes the right action of x on Γ\G. The quasi-regular representation is known to be unitary.

We say Γ_{1} and Γ_{2} are representation equivalent if ρ_{Γ_{1}} and ρ_{Γ_{2}} are unitarily equivalent; that is, Γ_{1} and Γ_{2} are representation equivalent if there exists a unitary isomorphism T : L^{2}(Γ_{1}\G) → L^{2}(Γ_{2}\G) such that T (ρ_{Γ_{1}}(x)f) = ρ_{Γ_{2}}(x)T f for every x in G and every f in L^{2}(Γ_{1}\G).

Proposition 2.2 (Gordon–Wilson [GW1]). Let Γ_{1} and Γ_{2} be cocompact, discrete subgroups of a simply connected Lie group G. Let g be a left invariant metric on G. If Γ_{1} and Γ_{2} are representation equivalent, then

p-spec(Γ_{1}\G, g) = p-spec(Γ_{2}\G, g)

for p = 0, 1, ⋯, dim(G).

Remark. Pairs of isospectral manifolds constructed using the traditional Sunada method are of the form (Γ_{1}\M, g) and (Γ_{2}\M, g) where Γ_{1} and Γ_{2} are representation equivalent, cocompact, discrete subgroups of a group G acting by isometries on a Riemannian manifold (M, g).

For a Lie algebra 𝔤, denote by 𝔤^{(1)} the derived algebra [𝔤, 𝔤] of 𝔤. That is, 𝔤^{(1)} is the Lie subalgebra of 𝔤 generated by all elements of the form [X, Y] for X, Y in 𝔤. Inductively, define 𝔤^{(k+1)} = [𝔤, 𝔤^{(k)}]. A Lie algebra 𝔤 is said to be k-step nilpotent if 𝔤^{(k)} ≡ 0 but 𝔤^{(k−1)} ≢ 0. A Lie group G is called k-step nilpotent if its Lie algebra is.

Let G^{(k)} = exp(𝔤^{(k)}) denote the kth derived subgroup of G. We denote the center of G by Z(G) and the center of 𝔤 by 𝔷. Note that if G is k-step nilpotent, then G^{(k−1)} ⊂ Z(G).

Let exp denote the Lie algebra exponential from 𝔤 to G. The Campbell–Baker–Hausdorff formula gives us the group operation of G in terms of 𝔤. Namely, for X, Y ∈ 𝔤 :

exp(X)exp(Y) = exp(X + Y + \frac{1}{2}[X, Y] + \frac{1}{12}[X, [X, Y]] + \frac{1}{12}[Y, [Y, X]] + ⋯),

where the remaining terms are higher-order brackets. Note that for two-step nilpotent Lie groups, only the first three terms in the right-hand side are nonzero. For three-step groups, only the first five terms are nonzero. If 𝔤 is nilpotent and G is simply connected, then exp is a diffeomorphism from 𝔤 onto G. Denote its inverse by log.

Let Γ_{1} and Γ_{2} be cocompact, discrete subgroups of nilpotent Lie groups G_{1} and G_{2} respectively. Any abstract group isomorphism Φ : Γ_{1} → Γ_{2} extends uniquely to a Lie group isomorphism Φ : G_{1} → G_{2}.

Let Γ be a cocompact, discrete subgroup of a nilpotent Lie group G with left invariant metric g. The locally homogeneous space (Γ\G, g) is called a Riemannian nilmanifold. If G is an abelian Lie group, then Γ is merely a lattice of full rank in G, and in this case logΓ is also a lattice in 𝔤.

Let 𝔤_{Q} = span_{Q}{logΓ}. This is a rational Lie algebra; that is, there exists a basis of 𝔤 made up of elements of logΓ such that the structure constants are rational. A Lie subalgebra 𝔥 of 𝔤 is called rational if 𝔥 is spanned by 𝔥 ∩ 𝔤_{Q}. Note that the notion of rational depends on Γ. If H = exp(𝔥) is the connected Lie subgroup of G with rational Lie algebra 𝔥, then Γ ∩ H is a cocompact, discrete subgroup of H. The 𝔤^{(k)} are always rational Lie subalgebras of 𝔤.

The Kirillov theory of irreducible unitary representations of nilpotent groups gives us a correspondence between irreducible unitary representations of G and elements of 𝔤^{∗}, the dual of 𝔤. In particular, fix τ ∈ 𝔤^{∗}. Let 𝔥 be a rational subalgebra of 𝔤 that is maximal with respect to the property that τ([𝔥, 𝔥]) ≡ 0. The subalgebra 𝔥 is called a polarization of τ. Let H = exp(𝔥) be the connected subgroup of G with Lie algebra 𝔥.

Define a character \overline{τ} of H by

(2.3)  \overline{τ}(h) = e^{2πiτ(log(h))}

for all h in H. Define π_{τ} to be the irreducible representation of G induced by the representation \overline{τ} of H. Denote by H_{τ} the representation space of π_{τ} . Two such irreducible representations π_{τ} and π_{τ′} are unitarily equivalent if and only if τ′ = τ ∘ Ad(x) for some x in G. Here Ad(x) is the adjoint map from 𝔤 to 𝔤.

For τ in 𝔤^{∗}, the coadjoint orbit of τ is

O(τ) = {τ ∘ Ad(x) : x ∈ G}.

Hence π_{τ} and π_{τ′} are unitarily equivalent if and only if τ and τ′ lie in the same coadjoint orbit of 𝔤^{∗}.

As G is nilpotent, every irreducible representation of G is unitarily equivalent to π_{τ} for some τ ∈ 𝔤^{∗}, and the quasi-regular representation ρ_{Γ} is completely reducible. Thus the representation space L^{2}(Γ\G) is unitarily isomorphic to

L^{2}(Γ\G) ≅ ⨁_{τ∈T}m(τ)H_{τ}

for some T ⊂ 𝔤^{∗}. Here m(τ) denotes the multiplicity of H_{τ} , and we assume T contains at most one element of each coadjoint orbit of 𝔤^{∗}.

A good reference for representation theory on nilpotent Lie groups is [CG].

§3 A New Construction of Isospectral Nilmanifolds.

Let G be a simply connected, k-step nilpotent Lie group with Lie algebra 𝔤. Define Ḡ to be the simply connected, (k–1)-step nilpotent Lie group G/G^{(k−1)}. For Γ a cocompact, discrete subgroup of G, denote by Γ̄ the image of Γ under the canonical projection from G onto Ḡ. The group Γ̄ is then a cocompact, discrete subgroup of Ḡ. For a left invariant metric g on G, we associate a left invariant metric ḡ on Ḡ by restricting g to an orthogonal complement of 𝔤^{(k−1)} in 𝔤.

We call the (k–1)-step nilmanifold (Γ̄\Ḡ, ḡ) the quotient nilmanifold of (Γ\G, g). By using the definition of ḡ, one easily sees that the projection (Γ\G, g) → (Γ̄\Ḡ, ḡ) is a Riemannian submersion.

The Lie algebra 𝔤̄ of Ḡ is just 𝔤/𝔤^{(k−1)}. We denote elements of 𝔤̄ by Ū, where Ū is the image of U under the canonical projection from 𝔤 onto 𝔤̄.

Definition 3.1. Let G be a simply connected nilpotent Lie group. We say G is strictly nonsingular if the following property holds: for every z in Z(G) and every noncentral x in G there exists an element a in G such that

[x, a] = z.

Here [x, a] denotes the commutator [x, a] = xax^{−1}a^{−1}.

Equivalently, the Lie algebra 𝔤 is strictly nonsingular if for every noncentral X in 𝔤,

𝔷 ⊂ ad(X)(𝔤).

That is, for every X in 𝔤 − 𝔷 and every Z in 𝔷 there exists a vector Y in 𝔤 such that [X, Y] = Z.

Theorem 3.2. Let G be a simply connected, strictly nonsingular nilpotent Lie group with left invariant metric g. If Γ_{1} and Γ_{2} are cocompact, discrete subgroups of G such that

Γ_{1} ∩ Z(G) = Γ_{2} ∩ Z(G) and spec(Γ̄_{1}\Ḡ, ḡ) = spec(Γ̄_{2}\Ḡ, ḡ),

then

spec(Γ_{1}\G, g) = spec(Γ_{2}\G, g).

Remark. The above construction is a generalization of the construction used by Gordon and Wilson to obtain pairs of isospectral Heisenberg manifolds [GW2]. If we let the Lie group G be a simply connected, strictly nonsingular, two-step nilpotent Lie group with a one-dimensional center, then G = H_{n} for some n, where H_{n} denotes the (2n+1)-dimensional Heisenberg group.

Proof of Theorem 3.2.

We use the notation of Section 2.

For i = 1, 2, let T_{i} be a subset of 𝔤^{∗} such that

L^{2}(Γ_{i}\G) ≅ ⨁_{τ∈T_{i}}m_{i}(τ)H_{τ} .

Recall that m_{i}(τ) denotes the multiplicity of π_{τ} in the quasi-regular representation of G on L^{2}(Γ_{i}\G), and we assume that T_{i} contains at most one element of each coadjoint orbit of 𝔤^{∗}.

We decompose the index set T_{i} = T′_{i} ∪ T′′_{i} by letting

T′_{i} = {τ ∈ T_{i} : τ(𝔷) ≡ 0}, and T′′_{i} = {τ ∈ T_{i} : τ(𝔷) ≢ 0}.

We likewise decompose the representation space L^{2}(Γ_{i}\G) by letting

H′_{i} = ⨁_{τ∈T′_{i}}m_{i}(τ)H_{τ} and H′′_{i} = ⨁_{τ∈T′′_{i}}m_{i}(τ)H_{τ} .

As representation spaces L^{2}(Γ_{i}\G) = H′_{i} ⊕ H′′_{i}.

Clearly spec(Γ_{i}\G, g) = spec′(Γ_{i}\G, g) ∪ spec′′(Γ_{i}\G, g), where spec′(Γ_{i}\G, g) and spec′′(Γ_{i}\G, g) are defined as the spectrum of the Laplacian restricted to acting on H′_{i}, and H′′_{i}, respectively. The multiplicity of an eigenvalue in spec(Γ_{i}\G, g) is equal to the sum of its multiplicities in spec′(Γ_{i}\G, g) and spec′′(Γ_{i}\G, g).

Lemma 3.3. The Laplacian of (Γ_{i}\G, g) acting on H′_{i} is precisely the Laplacian of (Γ̄_{i}\Ḡ, ḡ) acting on L^{2}(Γ̄_{i}\Ḡ). Thus spec(Γ̄_{i}\Ḡ, ḡ) = spec′(Γ_{i}\G, g) for i = 1, 2.

Lemma 3.4. The representations of G on H′′_{1} and H′′_{2} are unitarily equivalent, hence spec′′(Γ_{1}\G, g) = spec′′(Γ_{2}\G, g).

Theorem 3.2 now follows.

The proof of Lemma 3.3 is essentially an extension of the first part of the proof used by Gordon–Wilson to construct pairs of isospectral Heisenberg manifolds. (See [GW2], Theorem 4.1) The details are included here for completeness and because of a difference in notation.

Proof of Lemma 3.3.

Let {Z_{1}, Z_{2}, ⋯, Z_{T} } be an orthonormal basis of 𝔷 = 𝔤^{(k−1)}. Extend it to {E_{1}, E_{2}, ⋯, E_{N}, Z_{1}, Z_{2}, ⋯, Z_{T} }, an orthonormal basis of 𝔤. By (2.1), the Laplace-Beltrami operator of (G, g) is

Δ = − ∑_{n=1}^{N} E_{n}^{2} − ∑_{k=1}^{K} Z_{k}^{2}.

View functions in L^{2}(Γ_{i}\G) as left Γ_{i}-invariant functions of G. The subspace H′_{i} is then those functions in L^{2}(Γ_{i}\G) that are independent of the center, which correspond to functions in L^{2}(Γ̄_{i}\Ḡ) in a natural way. So when we restrict Δ to H′_{i}, we have

Δ = − ∑_{n=1}^{N} E_{n}^{2},

which corresponds to the Laplacian of (Γ̄_{i}\Ḡ, ḡ).

The Laplacian of (Γ_{i}\G, g) acting on H′_{i} is then precisely the Laplacian of (Γ̄_{i}\Ḡ, ḡ) acting on L^{2}(Γ̄_{i}\Ḡ), so spec(Γ̄_{i}\Ḡ, ḡ) = spec′(Γ_{i}\G, g), as desired.

Before proving Lemma 3.4, we must introduce some of the theory of square integrable representations of nilpotent Lie groups.

Definition 3.5. Let G be a locally compact, unimodular group with center Z(G). We say that an irreducible unitary representation π of G on a Hilbert space H is square integrable if there are nonzero vectors x_{1} and x_{2} in H such that

∫_{G/Z(G)} |(π(s)x_{1}, x_{2})|^{2}d\overline{μ}(\overline{s}) < ∞.

Here d\overline{μ}(\overline{s}) denotes integration over G/Z(G) with respect to a choice of Haar measure \overline{μ} on G/Z(G). As the center acts trivially, the integrand may be viewed as a function of G/Z(G).

Let 𝔷^{⊥} be the subalgebra of 𝔤^{∗} defined by 𝔷^{⊥} = {μ ∈ 𝔤^{∗} : μ(𝔷) ≡ 0}. Note that 𝔷^{⊥} ≅ (𝔤/𝔷)^{∗}. For τ ∈ 𝔤^{∗}, let b_{τ} denote the skew-symmetric, bilinear form on 𝔤/𝔷 defined by b_{τ} (X̄, Ȳ) = τ([X, Y]) for all X̄, Ȳ in 𝔤/𝔷. Here X and Y are any elements of 𝔤 that project onto X̄ and Ȳ respectively.

Theorem 3.6 (Moore–Wolf [MW]). For a linear functional τ in 𝔤^{∗} with coadjoint orbit O(τ) and corresponding irreducible unitary representation π_{τ} , the following three conditions are equivalent:

(1) π_{τ} is square integrable.

(2) O(τ) = τ + 𝔷^{⊥}.

(3) b_{τ} is nondegenerate on 𝔤/𝔷.

Proof of Lemma 3.4.

Fix τ ∈ T′′_{i}. Let Z ∈ 𝔷 be such that τ(Z) ≠ 0. By strict nonsingularity, for all noncentral X ∈ 𝔤 there exists Y ∈ 𝔤 such that [X, Y] = Z. Hence b_{τ} (X̄, Ȳ) = τ([X, Y]) ≠ 0. So b_{τ} is nondegenerate. By Theorem 3.6, π_{τ} is square integrable. Note that b_{τ} nondegenerate implies that N = dim(𝔤/𝔷) is even.

Recall from Section 2 that π_{τ} is independent of the choice of τ in O(τ). By Theorem 3.6, as π_{τ} is square integrable, the coadjoint orbit O(τ) is uniquely determined by the restriction of τ to the center. We may thus assume τ ∈ 𝔷^{∗}.

Let α be a volume form on G/Z(G). That is, let α be a fixed, alternating, N-linear form over 𝔤̄ = 𝔤/𝔷. As b_{τ} is nondegenerate, b_{τ}^{( \frac{1}{2} N)} = b_{τ} ∧⋯∧b_{τ} is also a volume form on G/Z(G) and hence a scalar multiple of α. Define P_{α}(τ) by

P_{α}(τ)α = b_{τ} ∧ ⋯ ∧ b_{τ} .

The polynomial P_{α} is homogeneous of degree \frac{1}{2}N on 𝔤^{∗} and depends only on the choice of volume form α.

Let 𝔷^{∗} denote the dual of 𝔷. Let L = Γ_{1} ∩ Z(G) = Γ_{2} ∩ Z(G) and let L^{∗} ⊂ 𝔷^{∗} be the dual lattice of L.

We now use the following occurrence and multiplicity condition, also due to Moore and Wolf.

Theorem 3.7 (Moore–Wolf [MW]). Let G be a nilpotent Lie group and Γ a cocompact, discrete subgroup of G. Let L = Γ ∩ Z(G). Fix a volume form α_{Γ} on 𝔤/𝔷 so that Γ̄\Ḡ has volume one. Let τ be a nonzero element of 𝔷^{∗} such that π_{τ} is square integrable. The representation π_{τ} occurs in the quasi-regular representation of G on L^{2}(Γ\G) if and only if τ ∈ L^{∗}. Moreover, its multiplicity m(τ) is |P_{α_{Γ}}(τ)|.

By Theorem 3.7, the square integrable representation π_{τ} occurs in the quasi-regular representation of G on L^{2}(Γ_{i}\G) if and only if τ is contained in L^{∗}. Thus for i = 1, 2 the coadjoint orbits represented in T′′_{i} correspond to the elements of L^{∗}. We may assume T′′_{1} = T′′_{2}.

As the Riemannian metrics of (Γ̄_{1}\Ḡ, ḡ) and (Γ̄_{2}\Ḡ, ḡ) arise from the same left invariant metric ḡ on Ḡ, we know that the Riemannian volume forms of (Γ̄_{1}\Ḡ, ḡ) and (Γ̄_{2}\Ḡ, ḡ) arise from the same left invariant volume form on Ḡ. We will denote this volume form and its projections onto (Γ̄_{1}\Ḡ, ḡ) and (Γ̄_{2}\Ḡ, ḡ) by Ω.

Let α_{Γ_{1}} and α_{Γ_{2}} be as in Theorem 3.7. The volume forms α_{Γ_{i}} are then scalar multiples of Ω. For i = 1, 2, let α_{Γ_{i}} = p_{i}Ω.

So

∫_{Γ̄_{1}\Ḡ} α_{Γ_{1}} = 1 = ∫_{Γ̄_{2}\Ḡ} α_{Γ_{2}}

p_{1} ∫_{Γ̄_{1}\Ḡ} Ω = p_{2} ∫_{Γ̄_{2}\Ḡ} Ω
p_{1}Vol (Γ̄_{1}\Ḡ, ḡ) = p_{2}Vol (Γ̄_{2}\Ḡ, ḡ)

By hypothesis, spec(Γ̄_{1}\Ḡ, ḡ) = spec(Γ̄_{2}\Ḡ, ḡ), and the spectrum of the Laplace-Beltrami operator is known to determine the volume of a closed manifold. Thus Vol (Γ̄_{1}\Ḡ, ḡ) = Vol (Γ̄_{2}\Ḡ, ḡ), which implies p_{1} = p_{2}, and so α_{Γ_{1}} = α_{Γ_{2}}.

As the definition of P_{α_{Γ_{i}}} depends only on the volume form α_{Γ_{i}}, we must have

P_{α_{Γ_{1}}} (τ) = P_{α_{Γ_{2}}} (τ)

for all τ in T′′_{1} = T′′_{2}. Hence m_{1}(τ) = m_{2}(τ) for all τ in T′′_{1} = T′′_{2}.

Thus the representations of G on H′′_{1} and H′′_{2} are unitarily equivalent, and by Proposition 2.2

spec′′(Γ_{1}\G, g) = spec′′(Γ_{2}\G, g),

as desired.

The proofs of Lemma 3.4 and Theorem 3.2 are now complete.

Corollary 3.8. Let G be a simply connected, strictly nonsingular nilpotent Lie group. Two cocompact, discrete subgroups Γ_{1} and Γ_{2} are representation equivalent subgroups of G if and only if Γ̄_{1} and Γ̄_{2} are representation equivalent subgroups of Ḡ and Γ_{1} ∩ Z(G) = Γ_{2} ∩ Z(G).

Proof of Corollary 3.8.

Let Γ_{1} and Γ_{2} be cocompact, discrete subgroups of G. As in the proof of Theorem 3.2, we decompose the representation spaces of ρ_{Γ_{1}} and ρ_{Γ_{2}} as

L^{2}(Γ_{i}\G) = H′_{i} ⊕ H′′_{i},

for i = 1, 2.

If Γ_{1} and Γ_{2} are representation equivalent, then the square integrable representations occurring in the quasi-regular representations must correspond. We showed in the proof of Lemma 3.4 that the square integrable representations occurring in ρ_{Γ_{1}} and ρ_{Γ_{2}} are precisely the irreducible representations appearing in H′′_{i}. Thus Γ_{1} and Γ_{2} are representation equivalent subgroups of G if and only if the representations of G on H′_{1} and H′_{2} are unitarily equivalent and the representations of G on H′′_{1} and H′′_{2} are unitarily equivalent.

For i = 1, 2 the irreducible components of the representation of G on H′_{i} correspond to the linear functionals in T_{i} that are zero on the center of 𝔤. As these functionals may be viewed as functionals on 𝔤̄, we may likewise view the irreducible components as representations of Ḡ. Hence the representations of G on H′_{1} and H′_{2} may be viewed as the quasi-regular representations of Ḡ on L^{2}(Γ̄_{1}\Ḡ) and L^{2}(Γ̄_{2}\Ḡ), respectively. Thus Γ_{1} and Γ_{2} are representation equivalent subgroups of G if and only if Γ̄_{1} and Γ̄_{2} are representation equivalent subgroups of Ḡ and the representations of G on H′′_{1} and H′′_{2} are unitarily equivalent.

Theorem 3.7 tells us that the irreducible representations occurring in H′′_{i} correspond to the elements of the dual lattice of Γ_{i} ∩ Z(G). So if the representations of G on H′′_{1} and H′′_{2} are unitarily equivalent, then the duals of Γ_{1} ∩ Z(G) and Γ_{2} ∩ Z(G) must coincide. Hence Γ_{1} ∩ Z(G) = Γ_{2} ∩ Z(G). The proof of the forward direction is now complete.

The reverse direction follows from Proposition 2.2 applied to the quotient nilmanifolds and Lemma 3.4.

§4 New Examples of Isospectral Nilmanifolds.

Using Theorem 3.2, we construct and compare five new pairs of isospectral nilmanifolds. A summary of the properties of these examples is listed in Section 1, Table I. In this Section we compare the quasi-regular representations and the fundamental groups of these examples. The p-form spectra of these examples are also compared, but the calculations are left to the Appendix. The methods used in the Appendix are new, as previously used techniques could not be applied to compare the p-form spectrum of these higher-step examples. The length spectra and marked length spectra of these examples will be studied in [Gt4].

With the exception of the column comparing the representation equivalence of the fundamental groups, all of the properties listed in Table I are geometric invariants. Hence a “No” in any one of these columns demonstrates that an Example is nontrivial.

Before proceeding, we need the following.

Definition 4.1. Let Φ be a Lie group automorphism of G. Let Γ be a cocompact, discrete subgroup of G. We say Φ is an almost inner automorphism if for all elements x of G there exists a_{x} in G such that Φ(x) = a_{x}xa_{x}^{−1}.

Theorem 4.2 (Gordon–Wilson [GW1]). Let G be a nilpotent Lie group and let Γ be a cocompact, discrete subgroup of G. If Φ is an almost inner automorphism of G, then Γ and Φ(Γ) are representation equivalent.

Example I.

Consider the simply connected, strictly nonsingular, three-step nilpotent Lie group G with Lie algebra

𝔤 = span_{R}{X_{1}, X_{2}, Y_{1}, Y_{2}, Z_{1}, Z_{2}, W }

and Lie brackets

[X_{1}, Y_{1}] = [X_{2}, Y_{2}] = Z_{1}
[X_{1}, Y_{2}] = Z_{2}
[X_{1}, Z_{1}] = [X_{2}, Z_{2}] = [Y_{1}, Y_{2}] = W

and all other basis brackets zero.

Let Γ_{1} be the cocompact, discrete subgroup of G generated by

{exp(2X_{1}), exp(2X_{2}), exp(Y_{1}), exp(Y_{2}), exp(Z_{1}), exp(Z_{2}), exp(W)}.

Let Γ_{2} be the cocompact, discrete subgroup of G generated by

\{exp(2X_{1}), exp(2X_{2}), exp(Y_{1}), exp(Y_{2} + \frac{1}{2}Z_{2}), exp(Z_{1}), exp(Z_{2}), exp(W)\}.

Note that Γ_{1} ∩ Z(G) = Γ_{2} ∩ Z(G) = {exp(jW) : j ∈ Z}.

Now Γ̄_{2} = Φ(Γ̄_{1}) where Φ is the almost inner automorphism of Ḡ given on the Lie algebra level by

X̄_{1} → X̄_{1},
X̄_{2} → X̄_{2},
Ȳ_{1} → Ȳ_{1},
Ȳ_{2} → Ȳ_{2} + \frac{1}{2}Z̄_{2},
Z̄_{1} → Z̄_{1},
Z̄_{2} → Z̄_{2}.

By Theorem 4.2, Γ̄_{1} and Γ̄_{2} are representation equivalent subgroups of Ḡ. By Corollary 3.8, Γ_{1} and Γ_{2} are representation equivalent subgroups of G.

By Proposition 2.2, for any choice of left invariant metric g of G, !x we have p-spec(Γ_{1}\G, g) = p-spec(Γ_{2}\G, g), for p = 0, 1, ⋯, 7.

Proposition 4.3 ([Gt1]). The subgroups Γ_{1} and Γ_{2} are not isomorphic as groups.

Remark. The author previously established the representation equivalence of Γ_{1} and Γ_{2} in [Gt1] by using a direct calculation. This example was presented in [Gt1] as the first example of a pair of nonisomorphic, representation equivalent subgroups of a solvable Lie group. Note that a nilpotent Lie group is necessarily solvable. Also, this example was presented in [Gt2] as the first example of a pair of representation equivalent subgroups of a solvable Lie group producing nilmanifolds with unequal length spectra. Contrast this example with what must happen in the two-step case.

Definition 4.4. Let G be a two-step nilpotent Lie group and let Γ be a cocompact, discrete subgroup of G. We call the automorphism Φ of G a Γ-equivalence if for all γ in Γ there exists a_{γ} in G and γ′_{γ} in Γ ∩ G^{(1)} such that Φ(γ) = a_{γ}γa_{γ}^{−1}γ′_{γ}.

Theorem 4.5 ([Gt1], [Gt2]). Let G be a two-step nilpotent Lie group. Let Γ_{1} and Γ_{2} be cocompact, discrete subgroups of G. The subgroups Γ_{1} and Γ_{2} are representation equivalent if and only if there exists Φ, a Γ_{1}-equivalence of G, such that Φ(Γ_{1}) = Γ_{2}. Thus if Γ_{1} and Γ_{2} are representation equivalent, they are necessarily isomorphic. In addition, if Γ_{1} and Γ_{2} are representation equivalent, then (Γ_{1}\G, g) and (Γ_{2}\G, g) have the same length spectrum for any choice of left invariant metric g of G.

Example II.

Consider the simply connected, strictly nonsingular, three-step nilpotent Lie group G with Lie algebra

𝔤 = span_{R}{X_{1}, Y_{1}, Y_{2}, Z, W }

and Lie brackets

[X_{1}, Y_{1}] = Z
[X_{1}, Z] = [Y_{1}, Y_{2}] = W

and all other basis brackets zero.

Let Γ_{1} be the cocompact, discrete subgroup of G generated by

{exp(2X_{1}), exp(Y_{1}), exp(Y_{2}), exp(Z), exp(W)}.

Let Γ_{2} be the cocompact, discrete subgroup of G generated by

\{exp(2X_{1}), exp(Y_{1} + \frac{1}{2}Z), exp(Y_{2}), exp(Z), exp(W)\}.

Note that Γ_{1} ∩ Z(G) = Γ_{2} ∩ Z(G) = {exp(jW) : j ∈ Z}.

Now Γ̄_{2} = Φ(Γ̄_{1}) where Φ is the inner automorphism of Ḡ given on the Lie algebra level by

X̄_{1} → X̄_{1},
Ȳ_{1} → Ȳ_{1} + \frac{1}{2}Z̄,
Ȳ_{2} → Ȳ_{2},
Z̄ → Z̄.

Note that an inner automorphism is necessarily almost inner.

By Theorem 4.2, Γ̄_{1} and Γ̄_{2} are representation equivalent subgroups of Ḡ. By Corollary 3.8, Γ_{1} and Γ_{2} are representation equivalent subgroups of G.

By Proposition 2.2, p-spec(Γ_{1}\G, g) = p-spec(Γ_{2}\G, g) for p = 0, 1, ⋯, 5, for any choice of left invariant metric g of G.

Here Γ_{1} and Γ_{2} are isomorphic. Indeed, a simple calculation shows that the isomorphism Ψ given on the Lie algebra level by

X_{1} → X_{1} + \frac{1}{2}Y_{2},
Y_{1} → Y_{1} + \frac{1}{2}Z,
Y_{2} → Y_{2},
Z → Z,
W → W,

is an isomorphism of G such that Ψ(Γ_{1}) = Γ_{2}.

Proposition 4.6. No isomorphism between Γ_{1} and Γ_{2} will project to a Γ̄_{1}-equivalence of Ḡ.

Remark. Example I illustrates that in the higher-step case, the representation equivalence of Γ_{1} and Γ_{2} and the isomorphism class of Γ_{1} and Γ_{2} need not be related. Example II shows that, in contrast to Theorem 4.5, even in the case where Γ_{1} and Γ_{2} are isomorphic, knowing the isomorphisms between Γ_{1} and Γ_{2} is not enough to use Corollary 3.8 to establish whether or not Γ_{1} and Γ_{2} are representation equivalent.

Proof of Proposition 4.6.

Let Ψ be an isomorphism from Γ_{1} to Γ_{2}. Extend it to the Lie group isomorphism Ψ : G → G such that Ψ(Γ_{1}) = Γ_{2}.

On the Lie algebra level, any such isomorphism must preserve the following ideals of 𝔤 :

(1) 𝔤^{(2)} = span_{R}{W },

(2) 𝔤^{(1)} = span_{R}{Z, W },

(3) 𝔄 = span_{R}{Y_{2}, Z, W }, and

(4) ℭ = span_{R}{Y_{1}, Y_{2}, Z, W }, the centralizer of 𝔤^{(1)} in 𝔤. To see (3), note that ad(U)(𝔤) ⊂ 𝔤^{(2)} if and only if U ∈ 𝔄.

Note that the generators of Γ_{1} and Γ_{2} presented above are canonical in the sense that every element of Γ_{1} may be expressed uniquely as

exp(2n_{1}X_{1})exp(m_{1}Y_{1})exp(m_{2}Y_{2})exp(kZ)exp(jW),

for integers n_{1}, m_{1}, m_{2}, k, j. Similarly for Γ_{2}.

As Ψ(Γ_{1}) = Γ_{2}, generators of Γ_{1} must go to generators of Γ_{2}, and these generators must be expressible in terms of the canonical generators of Γ_{2}, given above.

Combining this fact with properties (1) through (4), we obtain:

Ψ_{∗}(W) = ±W, by (1). Ψ_{∗}(Z) = ±Z + h_{0}W using (2). Ψ_{∗}(Y_{2}) = ±Y_{2} mod 𝔤^{(1)}, using (3). Ψ_{∗}(Y_{1}) = ±(Y_{1} + \frac{1}{2}Z) + h_{1}Y_{2} + h_{2}Z mod 𝔤^{(2)} using (4). Ψ_{∗}(X_{1}) = ±X_{1} + \frac{1}{2}h_{3}Y_{1} + \frac{1}{2}h_{4}Y_{2} mod 𝔤^{(1)},

where h_{0}, h_{1}, h_{2}, h_{3} and h_{4} are integers.

Finally, we use the fact that Ψ_{∗} is a Lie algebra isomorphism. By examining the W coefficient of Ψ_{∗}([X_{1}, Y_{1}]) = [Ψ_{∗}(X_{1}), Ψ_{∗}(Y_{1})] we have the equation

h_{0} = ±\frac{1}{2} + \frac{1}{2}h_{1}h_{3} ± h_{2} ± \frac{1}{2}h_{4}.

Thus either h_{3} ≠ 0 or h_{4} ≠ 0.

As Ȳ_{1} and Ȳ_{2} are not in [X̄_{1}, 𝔤̄] and not in 𝔤̄^{(1)}, we see that the projection of Ψ cannot possibly be a Γ̄_{1}-equivalence.

Example III.

Consider again the seven-dimensional Lie group G presented in Example I.

We again let Γ_{1} be the cocompact, discrete subgroup of G generated by

{exp(2X_{1}), exp(2X_{2}), exp(Y_{1}), exp(Y_{2}), exp(Z_{1}), exp(Z_{2}), exp(W)},

and let Γ_{2} be the cocompact, discrete subgroup of G generated by

{exp(X_{1}), exp(X_{2}), exp(2Y_{1}), exp(2Y_{2}), exp(Z_{1}), exp(Z_{2}), exp(W)}.

Note that Γ_{1} ∩ Z(G) = Γ_{2} ∩ Z(G) = {exp(jW) : j ∈ Z}.

Let g be the left invariant metric on G defined by letting

{X_{1}, X_{2}, Y_{1}, Y_{2}, Z_{1}, Z_{2}, W }

be an orthonormal basis of 𝔤.

Now Γ̄_{2} = Φ(Γ̄_{1}) where Φ is the automorphism of Ḡ given on the Lie algebra level by

X̄_{1} → Ȳ_{2},
X̄_{2} → Ȳ_{1},
Ȳ_{1} → X̄_{2},
Ȳ_{2} → X̄_{1},
Z̄_{1} → −Z̄_{1},
Z̄_{2} → −Z̄_{2}.

The automorphism Φ is also an isometry of (Ḡ, ḡ), and an isometry must preserve the spectrum. Thus spec(Γ̄_{1}\Ḡ, ḡ) = spec(Γ̄_{2}\Ḡ, ḡ). By Theorem 3.2, spec(Γ_{1}\G, g) = spec(Γ_{2}\G, g).

Proposition 4.7. The manifolds (Γ_{1}\G, g) and (Γ_{2}\G, g) are not isospectral on one-forms.

The proof of Proposition 4.7 is left to the Appendix. Note that Proposition 4.7 and Proposition 2.2 together imply that Γ_{1} and Γ_{2} are not representation equivalent subgroups of G.

Proposition 4.8. The subgroups Γ_{1} and Γ_{2} are not isomorphic as groups.

Proof of Proposition 4.8.

If there existed a group isomorphism between Γ_{1} and Γ_{2}, it would extend to a Lie group automorphism Ψ of G such that Ψ(Γ_{1}) = Γ_{2}.

Each of the following ideals of 𝔤 must be preserved by the Lie algebra automorphism Ψ_{∗} :

(1) 𝔤^{(2)} = 𝔷 = span_{R}{W },

(2) 𝔤^{(1)} = span_{R}{Z_{1}, Z_{2}, W },

(3) ℭ = span_{R}{Y_{1}, Y_{2}, Z_{1}, Z_{2}, W }, the centralizer of 𝔤^{(1)} in 𝔤, and

(4) 𝔄 = span_{R}{X_{2}, Y_{1}, Z_{1}, Z_{2}, W }. To see (4), note that the image of ad(U) has dimension less than three if and only if U ∈ 𝔄.

Now Ψ(Γ_{1}) = Γ_{2}. Consequently, generators of Γ_{1} must go to generators of Γ_{2}, and these generators must be expressible in terms of the canonical generators of Γ_{2}, given above. Combining this fact with properties (1) through (4), we obtain:

A NEW CONSTRUCTION OF ISOSPECTRAL NILMANIFOLDS 17

Ψ_{∗}(Y_{1}) = ±2Y_{1} mod 𝔤^{(1)} by (3) and (4). Ψ_{∗}(Y_{2}) = ±2Y_{2} mod span_{R}{Y_{1}, Z_{1}, Z_{2}, W } by (3). Ψ_{∗}(W) = ±W by (1).

But then Ψ_{∗}([Y_{1}, Y_{2}]) ≠ [Ψ_{∗}(Y_{1}), Ψ_{∗}(Y_{2})].

Example IV.

Consider again the five-dimensional Lie group G presented in Example II. We again let Γ_{1} be the cocompact, discrete subgroup of G generated by

{exp(2X_{1}), exp(Y_{1}), exp(Y_{2}), exp(Z), exp(W)},

and let Γ_{2} be the cocompact, discrete subgroup of G generated by

{exp(X_{1}), exp(2Y_{1}), exp(Y_{2}), exp(Z), exp(W)}.

Note that Γ_{1} ∩ Z(G) = Γ_{2} ∩ Z(G) = {exp(jW) : j ∈ Z}.

Let g be the left invariant metric on G defined by letting

{X_{1}, Y_{1}, Y_{2}, Z, W }

be an orthonormal basis of 𝔤.

Now Γ̄_{2} = Φ(Γ̄_{1}) where Φ is the automorphism of Ḡ given on the Lie alegebra level by

X̄_{1} → Ȳ_{1},
Ȳ_{1} → X̄_{1},
Ȳ_{2} → Ȳ_{2},
Z̄ → −Z̄.

The automorphism Φ is clearly an isometry of (Ḡ, ḡ), and an isometry preserves the spectrum. Thus spec(Γ̄_{1}\Ḡ, ḡ) = spec(Γ̄_{2}\Ḡ, ḡ). By Theorem 3.2, spec(Γ_{1}\G, g) = spec(Γ_{2}\G, g). Proposition 4.9. The manifolds (Γ_{1}\G, g) and (Γ_{2}\G, g) are not isospectral on one-forms.

The proof of Proposition 4.9 is left to the Appendix.

Proposition 4.10. The subgroups Γ_{1} and Γ_{2} are not isomorphic as groups.

Remark. The combination of properties exhibited by Examples III and IV are similar to properties exhibited by pairs of isospectral Heisenberg manifolds constructed by Gordon and Wilson [GW2, G2].

Proof of Proposition 4.10.

If there existed a group isomorphism between Γ_{1} and Γ_{2}, it would extend to a Lie group automorphism Ψ of G such that Ψ(Γ_{1}) = Γ_{2}.

As before, generators of Γ_{1} must go to generators of Γ_{2}, and these generators must be expressible in terms of the canonical generators of Γ_{2}, given above. Combining this with properties (1) through (4) from the proof of Proposition 4.6, we have

Ψ_{∗}(Y_{1}) = ±2Y_{1} mod span_{R}{Y_{2}, Z, W }.

Ψ_{∗}(Y_{2}) = ±Y_{2} mod 𝔤^{(1)}.

Ψ_{∗}(W) = ±W. But then Ψ_{∗}([Y_{1}, Y_{2}]) ≠ [Ψ_{∗}(Y_{1}), Ψ_{∗}(Y_{2})].

Example V.

Consider again the seven-dimensional Lie group G presented in Example I.

We fix a left invariant metric on G by letting {E_{1}, E_{2}, E_{3}, E_{4}, E_{5}, E_{6}, E_{7}} be an orthonormal basis of 𝔤 where

E_{1} = X_{1} − \frac{1}{2}X_{2} − \frac{1}{4}Y_{2},
E_{2} = X_{2} − \frac{1}{4}Y_{1},
E_{3} = Y_{1},
E_{4} = Y_{1} + Y_{2},
E_{5} = Z_{1},
E_{6} = \frac{1}{2}Z_{1} + Z_{2},
E_{7} = W.

Let Φ be the automorphism of G defined on the Lie algebra level by

X_{1} → −X_{1} + X_{2} + \frac{1}{4}Y_{1} + \frac{1}{2}Y_{2},
X_{2} → X_{2} − \frac{1}{2}Y_{1} + \frac{1}{4}Z_{1},
Y_{1} → −Y_{1},
Y_{2} → 2Y_{1} + Y_{2} + Z_{2},
Z_{1} → Z_{1} + \frac{1}{2}W,
Z_{2} → −Z_{1} − Z_{2} + \frac{1}{4}W,
W → −W.

A straightforward calculation shows that Φ_{∗}([U, V]) = [Φ_{∗}(U), Φ_{∗}(V)] for all U, V in 𝔤. Thus Φ is indeed a Lie group automorphism.

Let Γ_{1} be the cocompact, discrete subgroup of G generated by

{exp(2X_{1}), exp(2X_{2}), exp(Y_{1}), exp(Y_{2}), exp(Z_{1}), exp(Z_{2}), exp(W)},

and let Γ_{2} = Φ(Γ_{1}). Note that Γ_{1} ∩ Z(G) = Γ_{2} ∩ Z(G) = {exp(jW) : j ∈ Z}.

Let Φ̄ be the projection of Φ onto Ḡ. Then Φ̄ factors as Φ̄ = Ψ_{1} ∘ Ψ_{2} where Ψ_{1} is the automorphism of Ḡ given on the Lie algebra level by

X̄_{1} → −X̄_{1} + X̄_{2} + \frac{1}{4}Ȳ_{1} + \frac{1}{2}Ȳ_{2},
X̄_{2} → X̄_{2} − \frac{1}{2}Ȳ_{1},
Ȳ_{1} → −Ȳ_{1},
Ȳ_{2} → 2Ȳ_{1} + Ȳ_{2},
Z̄_{1} → Z̄_{1},
Z̄_{2} → −Z̄_{1} − Z̄_{2},

and Ψ_{2} is the automorphism of Ḡ given on the Lie algebra level by

X̄_{1} → X̄_{1},
X̄_{2} → X̄_{2} + \frac{1}{4}Z̄_{1},
Ȳ_{1} → Ȳ_{1},
Ȳ_{2} → Ȳ_{2} − Z̄_{1} − Z̄_{2},
Z̄_{1} → Z̄_{1},
Z̄_{2} → Z̄_{2}.

By rewriting Ψ_{1} in terms of the orthonormal basis {Ē_{1}, Ē_{2}, Ē_{3}, Ē_{4}, Ē_{5}, Ē_{6}} of 𝔤̄, one easily sees that Ψ_{1}(Ē_{i}) = ±Ē_{i} for i = 1, …, 6. Thus the automorphism Ψ_{1} is also an isometry of Ḡ and must preserve the spectrum. A simple calculation shows that Ψ_{2} is an almost inner automorphism of Ḡ, which by Theorem 4.2 also preserves the spectrum. Thus spec(Γ̄_{1}\Ḡ, ḡ) = spec(Γ̄_{2}\Ḡ, ḡ). By Theorem 3.2, spec(Γ_{1}\G, g) = spec(Γ_{2}\G, g).

Proposition 4.11. The manifolds (Γ_{1}\G, g) and (Γ_{2}\G, g) are not isospectral on oneforms.

The proof of Proposition 4.11 is left to the Appendix.

Remark. We will show in [Gt4] that the automorphism Φ marks the length spectrum of these examples. This is the first example of a pair of manifolds with the same marked length spectrum but not the same spectrum on one-forms.

Appendix: Comparing the p-form spectrum of nilmanifolds.

In this appendix, we show that the pairs of isospectral manifolds in Examples III, IV, and V are not isospectral on one-forms.

Recall that on smooth p-forms, the Laplace-Beltrami operator is defined as

Δ = dδ + δd.

Here δ is the metric adjoint of d. Equivalently, δ = (−1)^{n(p+1)+1} ∗ d∗ where ∗ is the Hodge-∗ operator. Let E^{p}(M) denote the exterior algebra of smooth differential p-forms on M. Then for f ∈ C^{∞}(M) and τ ∈ E^{p}(M), Gordon and Wilson [GW1] showed that

(A.1)  Δ(fτ) = (Δf)τ + f(Δτ) − 2∇_{grad f}τ.

For G a simply connected Lie group with cocompact, discrete subgroup Γ, view E^{p}(Γ\G) as

E^{p}(Γ\G) = C^{∞}(Γ\G) ⊗ Λ^{p}(𝔤^{∗}).

Here elements of Λ^{p}(𝔤^{∗}) are viewed both as left invariant p-forms of G and also as elements of E^{p}(Γ\G).

Proposition 4.7. The nilmanifolds (Γ_{1}\G, g) and (Γ_{2}\G, g) as presented in Example III are not isospectral on one-forms.

Outline of Proof.

Step 1: We decompose 1-spec(Γ_{i}\G, g) into four components:

1-spec^{I}(Γ_{i}\G, g) ∪ 1-spec^{II}(Γ_{i}\G, g) ∪ 1-spec^{III}(Γ_{i}\G, g) ∪ 1-spec^{IV} (Γ_{i}\G, g).

The multiplicity of an eigenvalue in 1-spec(Γ_{i}\G, g) is the sum of its multiplicities in each of the four components.

Step 2: Using representation theory, we show

1-spec^{IV} (Γ_{1}\G, g) = 1-spec^{IV} (Γ_{2}\G, g)

and

1-spec^{III}(Γ_{1}\G, g) = 1-spec^{III}(Γ_{2}\G, g).

Step 3: We show that the multiplicity of every eigenvalue in 1-spec^{II}(Γ_{i}\G, g) is congruent to 0 modulo 4.

Step 4: Finally, we show that the eigenvalue π^{2} + 1 does not occur in 1-spec^{I}(Γ_{1}\G, g) but occurs with multiplicity 2 in 1-spec^{I}(Γ_{2}\G, g).

The result now follows.

Proof of Proposition 4.7.

Step 1.

Using the notation of Section 2, for i = 1, 2, let T_{i} be a subset of 𝔤^{∗} such that

L^{2}(Γ_{i}\G) ≅ ⨁_{τ∈T_{i}}m_{i}(τ)H_{τ} .

Let T_{i} = T_{i}^{I} ∪ T_{i}^{II} ∪ T_{i}^{III} ∪ T_{i}^{IV} where

T_{i}^{I} = {τ ∈ T_{i} : τ(Z_{1}) = τ(Z_{2}) = τ(W) = 0},
T_{i}^{II} = {τ ∈ T_{i} : τ(Z_{2}) ≠ 0, τ(Z_{1}) = τ(W) = 0},
T_{i}^{III} = {τ ∈ T_{i} : τ(Z_{1}) ≠ 0, τ(W) = 0},
T_{i}^{IV} = {τ ∈ T_{i} : τ(W) ≠ 0}.

Let

H_{i}^{I} = ⨁_{τ∈T_{i}^{I}}m_{i}(τ)H_{τ} ,  H_{i}^{II} = ⨁_{τ∈T_{i}^{II}}m_{i}(τ)H_{τ} ,

H_{i}^{III} = ⨁_{τ∈T_{i}^{III}}m_{i}(τ)H_{τ} ,  H_{i}^{IV} = ⨁_{τ∈T_{i}^{IV}} m_{i}(τ)H_{τ} .

As representation spaces,

L^{2}(Γ_{i}\G) = H_{i}^{I} ⊕ H_{i}^{II} ⊕ H_{i}^{III} ⊕ H_{i}^{IV} .

Decompose 1-spec(Γ_{i}\G, g) as

1-spec^{I}(Γ_{i}\G, g) ∪ 1-spec^{II}(Γ_{i}\G, g) ∪ 1-spec^{III}(Γ_{i}\G, g) ∪ 1-spec^{IV} (Γ_{i}\G, g),

where 1-spec^{I}(Γ_{i}\G, g) is defined as the spectrum of the Laplacian acting on H_{i}^{I} ⊗ Λ^{1}(𝔤^{∗}). Define 1-spec^{II}(Γ_{i}\G, g), 1-spec^{III}(Γ_{i}\G, g), and 1-spec^{IV} (Γ_{i}\G, g) similarly. The multiplicity of an eigenvalue in 1-spec(Γ_{i}\G, g) is equal to the sum of its multiplicities in each of the four components.

Step 2.

By Lemma 3.4, the representations of G on H_{1}^{IV} and H_{2}^{IV} are unitarily equivalent and

1-spec^{IV} (Γ_{1}\G, g) = 1-spec^{IV} (Γ_{2}\G, g).

We now show that the representations of G on H_{1}^{III} and H_{2}^{III} are unitarily equivalent.

The irreducible representations of G corresponding to elements of 𝔤^{∗} that are zero on the center may be viewed as irreducible representations of Ḡ = G/Z(G). It is easy to see that such representations of G are unitarily equivalent if and only if the corresponding representations of Ḡ are unitarily equivalent.

For all τ ∈ T_{i}^{III}, τ(𝔷) = 0. We may thus use the following Proposition to calculate the representations of G on H_{1}^{III} and H_{2}^{III}.

Proposition A.2 (see Pesce [P2]). Let N be a simply connected, two-step nilpotent Lie group with Γ a cocompact, discrete subgroup of N. Let τ ∈ 𝔑^{∗}. Define 𝔑_{τ} = {Y ∈ 𝔑 : τ([Y, 𝔑]) ≡ 0}. Then the irreducible representation π_{τ} appears in the quasi-regular representation of N on L^{2}(Γ\N) if and only if τ(logΓ ∩ 𝔑_{τ}) ⊂ Z. The multiplicity of π_{τ} is one if τ(𝔑^{(1)}) ≡ 0. Define a nondegenerate, skew-symmetric bilinear form on 𝔑/𝔑_{τ} by B_{τ} (U, V) = τ([U, V]) for all U, V ∈ 𝔑/𝔑_{τ} . Then if τ(𝔑^{(1)}) ≢ 0, the multiplicity of π_{τ} is equal to \sqrt{detB_{τ}} , where the determinant is calculated with respect to any basis of L_{τ} = logΓ/(logΓ ∩ 𝔑_{τ}).

Remark. The occurrence condition above actually follows directly from a more general occurrence and multiplicity theorem due independently to Richardson [R] and Howe [H].

Let {α_{1}, α_{2}, β_{1}, β_{2}, ζ_{1}, ζ_{2}, ω} be the dual basis to the orthonormal basis {X_{1}, X_{2}, Y_{1}, Y_{2}, Z_{1}, Z_{2}, W } of 𝔤.

If τ ∈ T_{i}^{III}, then

τ = A_{1}α_{1} + A_{2}α_{2} + B_{1}β_{1} + B_{2}β_{2} + C_{1}ζ_{1} + C_{2}ζ_{2}

for some A_{1}, A_{2}, B_{1}, B_{2}, C_{1}, C_{2} ∈ R with C_{1} ≠ 0.

Now 𝔤̄_{τ} = span_{R}{Z̄_{1}, Z̄_{2}}. Hence

logΓ̄_{1} ∩ 𝔤̄_{τ} = logΓ̄_{2} ∩ 𝔤̄_{τ}
= log(exp(ZZ̄_{1})exp(ZZ̄_{2}))
= span_{Z}{Z̄_{1}, Z̄_{2}}.

So τ(logΓ̄_{i} ∩ 𝔤̄_{τ}) ⊂ Z if and only if C_{1} ∈ Z and C_{2} ∈ Z. By Proposition A.2, we see that τ ∈ T_{i}^{III} if and only if C_{1} ∈ Z and C_{2} ∈ Z. Moreover, distinct values of C_{1} and C_{2} determine distinct coadjoint orbits of 𝔤^{∗}. As these conditions are the same for both T_{1}^{III} and T_{2}^{III}, we may assume T_{1}^{III} = T_{2}^{III}.

We now calculate multiplicities.

A basis for L_{1}_{τ} = logΓ̄_{1}/(logΓ̄_{1} ∩ 𝔤̄_{τ}) is {2X̄_{1}, 2X̄_{2}, Ȳ_{1}, Ȳ_{2}}. A basis for L_{2}_{τ} = logΓ̄_{2}/(logΓ̄_{2} ∩ 𝔤̄_{τ}) is \{X̄_{1}, X̄_{2}, 2Ȳ_{1}, 2Ȳ_{2}\}. In both cases, \sqrt{detB_{τ}} = 4C_{1}^{2}. Thus for τ ∈ T_{1}^{III} = T_{2}^{III}, the multiplicities m_{1}(τ) and m_{2}(τ) are equal. Hence the representations of Ḡ are unitarily equivalent, so the representations of G on H_{1}^{III} and H_{2}^{III} are unitarily equivalent.

By Proposition 2.2,

1-spec^{III}(Γ_{1}\G, g) = 1-spec^{III}(Γ_{2}\G, g),

as desired.

Step 3.

We now show that for any eigenvalue in 1-spec^{II}(Γ_{i}\G, g), its multiplicity in 1-spec^{II}(Γ_{i}\G, g) is always congruent to 0 modulo 4.

We first compute the multiplicity of the irreducible representations occurring here, using the same technique as in Step 2.

For all τ ∈ T_{i}^{II}, τ = A_{1}α_{1} + A_{2}α_{2} + B_{1}β_{1} + B_{2}β_{2} + C_{2}ζ_{2} for some A_{1}, A_{2}, B_{1}, B_{2}, C_{2} ∈ R with C_{2} ≠ 0. We may again use Proposition A.2.

Now 𝔤̄_{τ} = span_{R}{X̄_{2}, Ȳ_{1}, Z̄_{1}, Z̄_{2}}. Hence

\begin{aligned}logΓ̄_{1} ∩ 𝔤̄_{τ} & = log(exp(2ZX̄_{2})exp(ZȲ_{1})exp(ZZ̄_{1})exp(ZZ̄_{2})) \\ & = span_{Z}\{2X̄_{2}, Ȳ_{1}, Z̄_{1}, Z̄_{2}\}. \end{aligned}

So τ(logΓ̄_{1} ∩ 𝔤̄_{τ}) ⊂ Z if and only if A_{2} ∈ \frac{1}{2}Z, B_{1} ∈ Z and C_{2} ∈ Z.

However,

\begin{aligned}logΓ̄_{2} ∩ 𝔤̄_{τ} & = log(exp(ZX̄_{2})exp(2ZȲ_{1})exp(ZZ̄_{1})exp(ZZ̄_{2})) \\ & = span_{Z}\{X̄_{2}, 2Ȳ_{1}, Z̄_{1}, Z̄_{2}\}. \end{aligned}

So τ(logΓ̄_{2} ∩ 𝔤̄_{τ}) ⊂ Z if and only if A_{2} ∈ Z, B_{1} ∈ \frac{1}{2}Z and C_{2} ∈ Z.

We now calculate m_{i}(τ). A basis for L_{1}_{τ} = logΓ̄_{1}/(logΓ̄_{1} ∩ 𝔤̄_{τ}) is {2X̄_{1}, Ȳ_{2}}. A basis for L_{2}_{τ} = logΓ̄_{2}/(logΓ̄_{2} ∩ 𝔤̄_{τ}) is \{X̄_{1}, 2Ȳ_{2}\}. In both cases \sqrt{detB_{τ}} = 2|C_{2}|.

Thus if τ is in T_{i}^{II}, the irreducible representation π_{τ} occurs in the representation of G on H_{i}^{II} with multiplicity 2|C_{2}|. As the integer C_{2} ≠ 0, the multiplicity must be even. Hence any eigenvalue of Δ acting on H_{i}^{II} ⊗ Λ^{1}(𝔤^{∗}) must occur in 1-spec^{II}(Γ_{i}\G, g) with multiplicity congruent to 0 modulo 2.

We now use the following.

Proposition A.3 (Gordon–Wilson, [GW1]). Let G be a simply connected Lie group with left invariant metric g, and let Γ_{1} and Γ_{2} be cocompact, discrete subgroups of G. Let H_{1} and H_{2} be invariant subspaces of ρ_{Γ_{1}} and ρ_{Γ_{2}}, respectively. Denote by p-spec′(Γ_{i}\G, g) the spectrum of Δ restricted to acting on H_{i} ⊗ Λ^{p}(𝔤^{∗}). Assume there exists an automorphism Φ of G such that

(1) Φ is also an isometry of (G, g), and

(2) ρ_{Γ_{1}} restricted to H_{1} and ρ_{Γ_{2}} ∘ Φ restricted to H_{2} are unitarily equivalent, then

p-spec′(Γ_{1}\G, g) = p-spec′(Γ_{2}\G, g).

Here ρ_{Γ_{2}} ∘ Φ is defined by ((ρ_{Γ_{2}} ∘ Φ)(x))f = f ∘ R_{Φ(x)} for x in G and f in L^{2}(Γ_{1}\G).

Let Φ be the Lie group automorphism of G defined on the Lie algebra level by

X_{1} → −X_{1}
X_{2} → X_{2}
Y_{1} → −Y_{1}
Y_{2} → Y_{2}
Z_{1} → Z_{1}
Z_{2} → −Z_{2}
W → −W

The automorphism Φ is also an isometry of (G, g). Note that τ ∘ Φ_{∗} = −A_{1}α_{1} + A_{2}α_{2} − B_{1}β_{1} + B_{2}β_{2} − C_{2}ζ_{2}. Clearly, if τ satisfies Condition (∗) or (∗∗), then so does τ ∘ Φ_{∗}. A straightforward calculation shows that since C_{2} ≠ 0, the functionals τ and τ ∘ Φ_{∗} are not in the same coadjoint orbit of 𝔤^{∗}, so π_{τ} and π_{τ∘Φ_{∗}} are not unitarily equivalent. Thus if π_{τ} occurs in H_{i}^{II} with multiplicity 2|C_{2}| then so does π_{τ∘Φ_{∗}}, also with multiplicity 2|C_{2}|.

Note that by (2.3) π_{τ∘Φ_{∗}} = π_{τ} ∘ Φ. Using Proposition A.3, any eigenvalue of Δ acting on H_{τ} ⊗Λ^{1}(𝔤^{∗}) must also occur as an eigenvalue of Δ acting on H_{τ∘Φ_{∗}} ⊗Λ^{1}(𝔤^{∗}). And each of the representation spaces H_{τ} and H_{τ∘Φ_{∗}} occurs in H_{i}^{II} with multiplicity 2|C_{2}|. Consequently, the multiplicity of any eigenvalue in spec^{II}(Γ_{i}\G, g) is a multiple of 4|C_{2}|, which is clearly congruent to 0 modulo 4, as desired.

Step 4.

We now show that the eigenvalue π^{2} + 1 does not occur in 1-spec^{I}(Γ_{1}\G, g) but occurs with multiplicity 2 in 1-spec^{I}(Γ_{2}\G, g).

For τ ∈ T_{1}^{I} or τ ∈ T_{2}^{I}, τ(𝔤^{(1)}) ≡ 0. We again use Proposition A.2 to calculate the irreducible representations occurring here. We write τ = A_{1}α_{1} + A_{2}α_{2} + B_{1}β_{1} + B_{2}β_{2} for some A_{1}, A_{2}, B_{1}, B_{2} ∈ R.

Now 𝔤̄_{τ} = 𝔤̄, so τ(logΓ̄_{i} ∩ 𝔤̄_{τ}) ⊂ Z if and only if τ(logΓ̄_{i}) ⊂ Z.

Thus, τ ∈ T_{1}^{I} if and only if

(∗)  A_{1}, A_{2} ∈ \frac{1}{2}Z and B_{1}, B_{2} ∈ Z,

and τ ∈ T_{2}^{I} if and only if

(∗∗)  A_{1}, A_{2} ∈ Z and B_{1}, B_{2} ∈ \frac{1}{2}Z.

Let H_{τ} be the associated representation space of π_{τ} . Then H_{τ} may be viewed as the one-dimensional subspace of L^{2}(Γ_{i}\G) generated by (see Section 2):

\begin{matrix}F_{τ} (exp(x_{1}X_{1}) & exp(x_{2}X_{2})exp(y_{1}Y_{1})exp(y_{2}Y_{2})exp(z_{1}Z_{1})exp(z_{2}Z_{2})exp(wW)) \\ = exp\{2πiτ(x_{1}X_{1} + x_{2}X_{2} + y_{1}Y_{1} + y_{2}Y_{2})\}. \end{matrix}

That is, H_{τ} = CF_{τ} .

We now calculate Δ acting on H_{τ} ⊗ Λ^{1}(𝔤^{∗}). Note that if we let CΛ^{1}(𝔤^{∗}) denote Λ^{1}(𝔤^{∗}) with complex coefficients, then

H_{τ} ⊗ Λ^{1}(𝔤^{∗}) = F_{τ} ⊗ CΛ^{1}(𝔤^{∗}).

Let F_{τ} ⊗ μ ∈ F_{τ} ⊗ CΛ^{1}(𝔤^{∗}). Then μ = a_{1}α_{1} + a_{2}α_{2} + b_{1}β_{1} + b_{2}β_{2} + z_{1}ζ_{1} + z_{2}ζ_{2} + wω for some a_{1}, a_{2}, b_{1}, b_{2}, z_{1}, z_{2}, w ∈ C.

26 RUTH GORNET

As F_{τ} is independent of z_{1}, z_{2}, and w we have

ΔF_{τ} = −X_{1}^{2}F_{τ} − X_{2}^{2}F_{τ} − Y_{1}^{2}F_{τ} − Y_{2}^{2}F_{τ}
= 4π^{2}(A_{1}^{2} + A_{2}^{2} + B_{1}^{2} + B_{2}^{2})F_{τ}
= 4π^{2}S^{2}F_{τ}

where S^{2} = A_{1}^{2} + A_{2}^{2} + B_{1}^{2} + B_{2}^{2}.

Let ∗ denote the Hodge-∗ operator. One easily sees that d ∗ μ = 0 for all μ ∈ 𝔤^{∗}. Hence δμ = ± ∗ d ∗ μ = 0 for all μ ∈ 𝔤^{∗}. Consequently Δ = δd on Λ^{1}(𝔤^{∗}).

For μ ∈ Λ^{1}(𝔤^{∗}), dμ(U, V) = −μ([U, V]) for all U, V ∈ 𝔤. Using this fact together with the definition of δ as the metric adjoint of d, one easily computes that Δα_{1} = Δα_{2} = Δβ_{1} = Δβ_{2} = 0, Δζ_{1} = 2ζ_{1}, Δζ_{2} = ζ_{2}, and Δω = 3ω.

To calculate Δ(F_{τ} ⊗ μ), it remains to calculate

∇_{gradF_{τ}} μ = 2πiF_{τ} (A_{1}∇_{X_{1}}μ + A_{2}∇_{X_{2}}μ + B_{1}∇_{Y_{1}}μ + B_{2}∇_{Y_{2}}μ).

For Lie algebras with a left invariant metric, the covariant derivatives can be calculated via the following equation:

< ∇_{U}V, U′ >= \frac{1}{2} < [U′, U], V > +\frac{1}{2} < [U′, V], U > +\frac{1}{2} < [U, V], U′ >,

for U, V, U′ left invariant vector fields of 𝔤. A simple calculation shows that ∇_{U}(V^{♭}) = (∇_{U}V)^{♭}, where V^{♭} denotes the dual of V in 𝔤^{∗} with respect to our choice of orthonormal basis; that is, V^{♭}(U) =< V, U > for all U in 𝔤^{∗}.

We thus obtain the following chart:

\begin{matrix}∇_{U} μ & α_{1} & α_{2} & β_{1} & β_{2} & ζ_{1} & ζ_{2} & ω \\ X_{1} & 0 & 0 & \frac{1}{2} ζ_{1} & \frac{1}{2} ζ_{2} & − \frac{1}{2} β_{1} + \frac{1}{2} ω & − \frac{1}{2} β_{2} & − \frac{1}{2} ζ_{1} \\ X_{2} & 0 & 0 & 0 & \frac{1}{2} ζ_{1} & − \frac{1}{2} β_{2} & \frac{1}{2} ω & − \frac{1}{2} ζ_{2} \\ Y_{1} & − \frac{1}{2} ζ_{1} & 0 & 0 & \frac{1}{2} ω & \frac{1}{2} α_{1} & 0 & − \frac{1}{2} β_{2} \\ Y_{2} & − \frac{1}{2} ζ_{2} & − \frac{1}{2} ζ_{1} & − \frac{1}{2} ω & 0 & \frac{1}{2} α_{2} & \frac{1}{2} α_{1} & \frac{1}{2} β_{1} \end{matrix}

Using (A.1) and the above information, a straightforward calculation shows that if we let E_{τ} =

( \begin{matrix}4π^{2}S^{2} & 0 & 0 & 0 & −2πiB_{1} & −2πiB_{2} & 0 \\ 0 & 4π^{2}S^{2} & 0 & 0 & −2πiB_{2} & 0 & 0 \\ 0 & 0 & 4π^{2}S^{2} & 0 & 2πiA_{1} & 0 & −2πiB_{2} \\ 0 & 0 & 0 & 4π^{2}S^{2} & 2πiA_{2} & 2πiA_{1} & 2πiB_{1} \\ 2πiB_{1} & 2πiB_{2} & −2πiA_{1} & −2πiA_{2} & 4π^{2}S^{2} + 2 & 0 & 2πiA_{1} \\ 2πiB_{2} & 0 & 0 & −2πiA_{1} & 0 & 4π^{2}S^{2} + 1 & 2πiA_{2} \\ 0 & 0 & 2πiB_{2} & −2πiB_{1} & −2πiA_{1} & −2πiA_{2} & 4π^{2}S^{2} + 3 \end{matrix} ) ,

then Δ(F_{τ} ⊗ μ) = λ(F_{τ} ⊗ μ) if and only if λ is an eigenvalue of the matrix E_{τ} .

We now calculate necessary conditions on τ = A_{1}α_{1} + A_{2}α_{2} + B_{1}β_{1} + B_{2}β_{2} so that π^{2} + 1 is an eigenvalue of E_{τ} . As τ ∈ T_{1}^{I} or τ ∈ T_{2}^{I}, we know A_{1}, A_{2}, B_{1}, B_{2} ∈ Q. If det(E_{τ} − (π^{2} + 1)I_{7}) = 0, then π is the root of a polynomial with rational coefficients. However π is transcendental. Thus the coefficients of the powers of π must be zero.

A straightforward calculation shows that π^{14} is the highest power of π occurring in the polynomial, and the coefficient of π^{14} is equal to (4S^{2} − 1)^{7}. Thus if π^{2} + 1 is an eigenvalue of E_{τ} , then S^{2} = \frac{1}{4}. Recall S^{2} = A_{1}^{2} + A_{2}^{2} + B_{1}^{2} + B_{2}^{2}.

For τ in T_{1}^{I}, S^{2} = \frac{1}{4} if and only if (see (∗)) τ = ±\frac{1}{2}α_{1} or τ = ±\frac{1}{2}α_{2}. And for τ in T_{2}^{I}, S^{2} = \frac{1}{4} if and only if (see (∗∗)) τ = ±\frac{1}{2}β_{1} or τ = ±\frac{1}{2}β_{2}.

For τ = ±\frac{1}{2}α_{1}, τ = ±\frac{1}{2}α_{2}, or τ = ±\frac{1}{2}β_{2}, a simple calculation shows that det(E_{τ} − (π^{2} + 1)I_{7}) ≠ 0. Thus, the eigenvalue π^{2} + 1 does not arise from the Laplacian acting on H_{± \frac{1}{2} α_{1}} ⊗ Λ^{1}(𝔤^{∗}), H_{± \frac{1}{2} α_{2}} ⊗ Λ^{1}(𝔤^{∗}), or H_{± \frac{1}{2} β_{2}} ⊗ Λ^{1}(𝔤^{∗}), and

π^{2} + 1 ∉ 1-spec^{I}(Γ_{1}\G, g).

However for τ = ±\frac{1}{2}β_{1}, det(E_{τ} − (π^{2} + 1)I_{7}) = 0. Thus π^{2} + 1 is an eigenvalue for the Laplacian acting on H_{± \frac{1}{2} β_{1}} ⊗ Λ^{1}(𝔤^{∗}). Indeed, the eigenspace of π^{2} + 1 in H^{I} ⊗ Λ^{1}(𝔤^{∗}) is

span_{C}\{F_{\frac{1}{2} β_{1}} ⊗ ζ_{2} , F_{− \frac{1}{2} β_{1}} ⊗ ζ_{2}\},

which has dimension 2.

Thus π^{2} + 1 ∈ 1-spec^{I}(Γ_{2}\G, g) with multiplicity 2, as desired.

The proof of Proposition 4.7 is now complete.

Proposition 4.9. The nilmanifolds (Γ_{1}\G, g) and (Γ_{2}\G, g) as presented in Example IV are not isospectral on one-forms.

Proof of Proposition 4.9.

Again using the notation of Section 2, for i = 1, 2, let T_{i} be a subset of 𝔤^{∗} such that

L^{2}(Γ_{i}\G) ≅ ⨁_{τ∈T_{i}}m_{i}(τ)H_{τ} .

Let T_{i} = T′_{i} ∪ T′′_{i} ∪ T′′′_{i}, where

T′_{i} = {τ ∈ T_{i} : τ(𝔤^{(1)}) ≡ 0},
T′′_{i} = {τ ∈ T_{i} : τ(𝔤^{(2)}) ≡ 0, τ(𝔤^{(1)}) ≢ 0},
T′′′_{i} = {τ ∈ T_{i} : τ(𝔤^{(2)}) ≢ 0}.

As in the proof of Proposition 4.7, decompose the representation spaces and spectrum accordingly.

By Lemma 3.4, the representations of G on H′′′_{1} and H′′′_{2} are unitarily equivalent and

1-spec′′′(Γ_{1}\G, g) = 1-spec′′′(Γ_{2}\G, g).

A calculation almost identical to Step 2 in the proof of Proposition 4.7 shows that the representations of G on H′′_{1} and H′′_{2} are unitarily equivalent. Thus

1-spec′′(Γ_{1}\G, g) = 1-spec′′(Γ_{2}\G, g).

It remains to show that 1-spec′(Γ_{1}\G, g) ≠ 1-spec′(Γ_{2}\G, g). The proof of this corresponds to Step 4 in the proof of Proposition 4.7.

Let {α_{1}, β_{1}, β_{2}, γ, ω} be the dual to the orthonormal basis {X_{1}, Y_{1}, Y_{2}, Z, W } given in Example IV. For τ ∈ T′_{i}, τ = A_{1}α_{1} + B_{1}β_{1} + B_{2}β_{2} for some A_{1}, B_{1}, B_{2} ∈ R. We again use Proposition A.2.

Now 𝔤̄_{τ} = 𝔤̄. Hence τ(logΓ̄_{i} ∩ 𝔤̄_{τ}) ⊂ Z if and only if τ(logΓ̄_{i}) ⊂ Z.

Thus, τ ∈ T_{1}^{I} if and only if

(∗)  A_{1} ∈ \frac{1}{2}Z and B_{1}, B_{2} ∈ Z,

and, τ ∈ T_{2}^{I} if and only if

(∗∗)  A_{1}, B_{2} ∈ Z and B_{1} ∈ \frac{1}{2}Z.

Let H_{τ} be the associated representation space of π_{τ} . As in the proof of Proposition 4.7, H_{τ} ⊗ Λ^{1}(𝔤^{∗}) = F_{τ} ⊗ CΛ^{1}(𝔤^{∗}) where

\begin{aligned}F_{τ} (exp(x_{1}X_{1})exp(y_{1} & Y_{1})exp(y_{2}Y_{2})exp(zZ)exp(wW)) \\ & = exp\{2πiτ(x_{1}X_{1} + y_{1}Y_{1} + y_{2}Y_{2})\}. \end{aligned}

Let F_{τ} ⊗μ ∈ F_{τ} ⊗CΛ^{1}(𝔤^{∗}) with μ = a_{1}α_{1}+b_{1}β_{1}+b_{2}β_{2}+zζ +wω for some a_{1}, b_{1}, b_{2}, z, w ∈ C.

Then ΔF_{τ} = 4π^{2}S^{2}F_{τ} where S^{2} = A_{1}^{2} + B_{1}^{2} + B_{2}^{2}. And Δα_{1} = Δβ_{1} = Δβ_{2} = 0, Δζ = ζ and Δω = 2ω. We also obtain the following chart:

\begin{matrix}∇_{U} μ & α_{1} & β_{1} & β_{2} & ζ & ω \\ X_{1} & 0 & \frac{1}{2} ζ & 0 & − \frac{1}{2} β_{1} + \frac{1}{2} ω & − \frac{1}{2} ζ \\ Y_{1} & − \frac{1}{2} ζ & 0 & \frac{1}{2} ω & \frac{1}{2} α_{1} & − \frac{1}{2} β_{2} \\ Y_{2} & 0 & − \frac{1}{2} ω & 0 & 0 & \frac{1}{2} β_{1} \end{matrix}

Using (A.1) and the above information, if we let

E_{τ} = ( \begin{matrix}4π^{2}S^{2} & 0 & 0 & −2πiB_{1} & 0 \\ 0 & 4π^{2}S^{2} & 0 & 2πiA_{1} & −2πiB_{2} \\ 0 & 0 & 4π^{2}S^{2} & 0 & 2πiB_{1} \\ 2πiB_{1} & −2πiA_{1} & 0 & 4π^{2}S^{2} + 1 & 2πiA_{1} \\ 0 & 2πiB_{2} & −2πiB_{1} & −2πiA_{1} & 4π^{2}S^{2} + 2 \end{matrix} ) ,

then Δ(F_{τ} ⊗ μ) = λ(F_{τ} ⊗ μ) if and only if λ is an eigenvalue of the matrix E_{τ} .

We calculate necessary conditions on τ = A_{1}α_{1} + B_{1}β_{1} + B_{2}β_{2} so that π^{2} + 1 is an eigenvalue of E_{τ} . As we assumed τ ∈ T′_{1} or τ ∈ T′_{2}, we know A_{1}, B_{1}, B_{2} ∈ Q. If det(E_{τ} − (π^{2} + 1)I_{5}) = 0, then π is the root of a polynomial with rational coefficients. However π is transcendental. Thus the coefficients of the powers of π must be zero.

A straightforward calculation shows that π^{10} is the highest power of π occurring in the polynomial, and the coefficient of π^{10} is equal to (4S^{2} − 1)^{5}. Thus if π^{2} + 1 is an eigenvalue of E_{τ} , then S^{2} = \frac{1}{4}. Recall S^{2} = A_{1}^{2} + B_{1}^{2} + B_{2}^{2}.

For τ in T′_{1}, S^{2} = \frac{1}{4} if and only if (see (∗)) τ = ±\frac{1}{2}α_{1}. And for τ in T′_{2}, S^{2} = \frac{1}{4} if and only if (see (∗∗)) τ = ±\frac{1}{2}β_{1}.

For τ = ±\frac{1}{2}α_{1}, det(E_{τ} − (π^{2} + 1)I_{5}) = 0. The eigenspace of π^{2} + 1 in H′ ⊗ Λ^{1}(𝔤^{∗}) is

span_{C}\{F_{\frac{1}{2} α_{1}} ⊗ (πiβ_{1} + ζ + πiω), F_{− \frac{1}{2} α_{1}} ⊗ (πiβ_{1} − ζ + πiω)\},

which has dimension 2. Thus π^{2} + 1 ∈ 1-spec′(Γ_{1}\G, g) with multiplicity two.

However, for τ = ±\frac{1}{2}β_{1}, det(E_{τ} − (π^{2} + 1)I_{5}) ≠ 0. Thus π^{2} + 1 cannot occur in 1-spec′(Γ_{2}\G, g) and

1-spec′(Γ_{1}\G, g) ≠ 1-spec′(Γ_{2}\G, g),

as desired.

The proof of Proposition 4.9 is now complete.

Proposition 4.11. The nilmanifolds (Γ_{1}\G, g) and (Γ_{2}\G, g) as presented in Example V are not isospectral on one-forms.

Proof of Proposition 4.11.

Using the notation of Section 2, for i = 1, 2, let T_{i} be a subset of 𝔤^{∗} such that

L^{2}(Γ_{i}\G) ≅ ⨁_{τ∈T_{i}}m_{i}(τ)H_{τ} .

Let T_{i} = T_{i}^{I} ∪ T_{i}^{II} ∪ T_{i}^{III} ∪ T_{i}^{IV} be defined as in the proof of Proposition 4.7, and decompose the representation spaces and spectrum accordingly.

By Lemma 3.4, the representations of G on H_{1}^{IV} and H_{2}^{IV} are unitarily equivalent and

1-spec^{IV} (Γ_{1}\G, g) = 1-spec^{IV} (Γ_{2}\G, g).

A calculation almost identical to that in Step 2 in the proof of Proposition 4.7 shows that the representations of G on H_{1}^{III} and H_{2}^{III} are unitarily equivalent. Thus

1-spec^{III}(Γ_{1}\G, g) = 1-spec^{III}(Γ_{2}\G, g).

A calculation almost identical to that in Step 3 in the proof of Proposition 4.7 shows that the representations of G on H_{1}^{II} and H_{2}^{II} are unitarily equivalent. Thus

1-spec^{II}(Γ_{1}\G, g) = 1-spec^{II}(Γ_{2}\G, g).

It remains to show that 1-spec^{I}(Γ_{1}\G, g) ≠ 1-spec^{I}(Γ_{1}\G, g).

For τ ∈ T_{1}^{I} or τ ∈ T_{2}^{I}, τ(𝔤^{(1)}) ≡ 0. We again use Proposition A.2 to calculate the irreducible representations occurring here. Let {ϵ_{1}, ⋯, ϵ_{7}} be the dual to the orthonormal basis {E_{1}, ⋯, E_{7}} given in Example V. Then τ = A_{1}ϵ_{1} + A_{2}ϵ_{2} + A_{3}ϵ_{3} + A_{4}ϵ_{4} for some A_{1}, ⋯, A_{4} ∈ R.

Now 𝔤̄_{τ} = 𝔤̄. Hence τ(logΓ̄_{i} ∩ 𝔤̄_{τ}) ⊂ Z if and only if τ(logΓ̄_{i}) ⊂ Z.

Thus, τ ∈ T_{1}^{I} if and only if

(∗)  2A_{1} + A_{2} − \frac{1}{4}A_{3} + \frac{1}{2}A_{4} ∈ Z, 2A_{2} + \frac{1}{2}A_{3} ∈ Z, and A_{3}, A_{4} ∈ Z.

And τ ∈ T_{2}^{I} if and only if

(∗∗)  2A_{1} + A_{2} + \frac{1}{4}A_{3} + \frac{1}{2}A_{4} ∈ Z, 2A_{2} + \frac{1}{2}A_{3} ∈ Z, and A_{3}, A_{4} ∈ Z.

Note that the only distinction between the two conditions is in the sign of \frac{1}{4}A_{3}.

Let H_{τ} be the associated representation space of π_{τ} . As in the proof of Proposition 4.7, H_{τ} ⊗ Λ^{1}(𝔤^{∗}) = F_{τ} ⊗ CΛ^{1}(𝔤^{∗}) where

\begin{matrix}F_{τ} (exp(e_{1}E_{1}) & exp(e_{2}E_{2})exp(e_{3}E_{3})exp(e_{4}E_{4})exp(e_{5}E_{5})exp(e_{6}E_{6})exp(e_{7}E_{7})) \\ = exp\{2πiτ(e_{1}E_{1} + e_{2}E_{2} + e_{3}E_{3} + e_{4}E_{4})\}. \end{matrix}

Let F_{τ} ⊗ μ ∈ F_{τ} ⊗ CΛ^{1}(𝔤^{∗}) with μ = a_{1}ϵ_{1} + a_{2}ϵ_{2} + a_{3}ϵ_{3} + a_{4}ϵ_{4} + a_{5}ϵ_{5} + a_{6}ϵ_{6} + a_{7}ϵ_{7} for some a_{i} ∈ C, for i = 1, …, 7. As in the proof of Proposition 4.7, ΔF_{τ} = 4π^{2}S^{2}F_{τ} where S^{2} = A_{1}^{2} + A_{2}^{2} + A_{3}^{2} + A_{4}^{2}. Also, Δϵ_{1} = Δϵ_{2} = Δϵ_{3} = Δϵ_{4} = 0, Δϵ_{5} = 2ϵ_{5}, Δϵ_{6} = ϵ_{6} + \frac{1}{4}ϵ_{7}, and Δϵ_{7} = \frac{1}{4}ϵ_{6} + (3 + \frac{1}{256} + \frac{3}{16})ϵ_{7}. And we obtain the following chart:

\begin{matrix}∇_{U} μ & ϵ_{1} & ϵ_{2} & ϵ_{3} & ϵ_{4} & ϵ_{5} & ϵ_{6} & ϵ_{7} \\ E_{1} & 0 & − \frac{1}{32} ϵ_{7} & \frac{1}{2} ϵ_{5} + \frac{1}{8} ϵ_{7} & \frac{1}{2} ϵ_{6} + \frac{1}{8} ϵ_{7} & − \frac{1}{2} ϵ_{3} + \frac{1}{2} ϵ_{7} & − \frac{1}{2} ϵ_{4} & \frac{1}{32} ϵ_{2} − \frac{1}{8} ϵ_{3} − \frac{1}{8} ϵ_{4} − \frac{1}{2} ϵ_{5} \\ E_{2} & \frac{1}{32} ϵ_{7} & 0 & 0 & \frac{1}{2} ϵ_{5} − \frac{1}{8} ϵ_{7} & − \frac{1}{2} ϵ_{4} & \frac{1}{2} ϵ_{7} & − \frac{1}{32} ϵ_{1} + \frac{1}{8} ϵ_{4} − \frac{1}{2} ϵ_{6} \\ E_{3} & − \frac{1}{2} ϵ_{5} − \frac{1}{8} ϵ_{7} & 0 & 0 & \frac{1}{2} ϵ_{7} & \frac{1}{2} ϵ_{1} & 0 & \frac{1}{8} ϵ_{1} − \frac{1}{2} ϵ_{4} \\ E_{4} & − \frac{1}{2} ϵ_{6} − \frac{1}{8} ϵ_{7} & − \frac{1}{2} ϵ_{5} + \frac{1}{8} ϵ_{7} & − \frac{1}{2} ϵ_{7} & 0 & \frac{1}{2} ϵ_{2} & \frac{1}{2} ϵ_{1} & \frac{1}{8} ϵ_{1} − \frac{1}{8} ϵ_{2} + \frac{1}{2} ϵ_{3} \end{matrix}

Using (A.1) and the above information, a straightforward calculation shows that if we let E_{τ} be the skew-Hermitian matrix defined by

( \begin{matrix}4π^{2}S^{2} & 0 & 0 & 0 & 2πiA_{3} & 2πiA_{4} & πi(− \frac{1}{8} A_{2} + \frac{1}{2} A_{3} + \frac{1}{2} A_{4}) \\ & 4π^{2}S^{2} & 0 & 0 & 2πiA_{4} & 0 & πi( \frac{1}{8} A_{1} − \frac{1}{2} A_{4}) \\ & & 4π^{2}S^{2} & 0 & −2πiA_{1} & 0 & πi(− \frac{1}{2} A_{1} + 2A_{4}) \\ & & & 4π^{2}S^{2} & −2πiA_{2} & −2πiA_{1} & πi(− \frac{1}{2} A_{1} + \frac{1}{2} A_{2} − 2A_{3}) \\ & & & & 4π^{2}S^{2} + 2 & 0 & −2πiA_{1} \\ & & & & & 4π^{2}S^{2} + 1 & −2πiA_{2} + \frac{1}{4} \\ & & & & & & 4π^{2}S^{2} + \frac{817}{256} \end{matrix} )

then Δ(F_{τ} ⊗ μ) = λ(F_{τ} ⊗ μ) if and only if λ is an eigenvalue of the matrix E_{τ} .

Let

λ = \frac{17}{4}π^{2} + 1 + \sqrt{\frac{17}{4}π^{2} + 1}.

We now calculate necessary conditions on τ = A_{1}ϵ_{1} + A_{2}ϵ_{2} + A_{3}ϵ_{3} + A_{4}ϵ_{4} so that λ is an eigenvalue of E_{τ} . As τ ∈ T_{1}^{I} or τ ∈ T_{2}^{I}, we know A_{1}, A_{2}, A_{3}, A_{4} ∈ Q.

By a computation using Maple or Mathematica, if det(E_{τ} −λI_{7}) = 0, then π is the root of a polynomial with rational coefficients. However π is transcendental. Thus the coefficients of the powers of π must be zero.

A straightforward calculation shows that π^{14} is the highest power of π occurring in the polynomial, and the coefficient of π^{14} is equal to (4S^{2} − \frac{17}{4})^{7}. Thus if λ is an eigenvalue of E_{τ} , then S^{2} = \frac{17}{16}. Recall that S^{2} = A_{1}^{2} + A_{2}^{2} + A_{3}^{2} + A_{4}^{2}.

For τ in T_{1}^{I}, S^{2} = \frac{17}{16} if and only if (see (∗)) τ = ±(\frac{1}{4}ϵ_{2} + ϵ_{3}) or τ = ±\frac{1}{4}ϵ_{1} ± ϵ_{4}. And for τ in T_{2}^{I}, S^{2} = \frac{17}{16} if and only if (see (∗∗)) τ = ±(\frac{1}{4}ϵ_{2} − ϵ_{3}) or τ = ±\frac{1}{4}ϵ_{1} ± ϵ_{4}. Note that the only difference is in the sign of ϵ_{3}.

For τ = ±(\frac{1}{4}ϵ_{2} + ϵ_{3}) or τ = ±\frac{1}{4}ϵ_{1} ± ϵ_{4}, a calculation using Maple or Mathematica shows

that det(E_{τ} − λI_{7}) ≠ 0. Thus \frac{17}{4}π^{2} + 1 + \sqrt{\frac{17}{4}π^{2} + 1} ∉ 1-spec^{I}(Γ_{1}\G, g).

However, for τ = ±(\frac{1}{4}ϵ_{2} − ϵ_{3}), det(E_{τ} − λI_{7}) = 0. Thus \frac{17}{4}π^{2} + 1 + \sqrt{\frac{17}{4}π^{2} + 1} is an

eigenvalue for the Laplacian acting on H_{±( \frac{1}{4} ϵ_{2}−ϵ_{3})} ⊗ Λ^{1}(𝔤^{∗}), and \frac{17}{4}π^{2} + 1 + \sqrt{\frac{17}{4}π^{2} + 1} ∈ 1-spec^{I}(Γ_{2}\G, g).

Consequently

1-spec^{I}(Γ_{1}\G, g) ≠ 1-spec^{I}(Γ_{2}\G, g),

as desired.

The proof of Proposition 4.11 is now complete.

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Ruth Gornet; Texas Tech University; Department of Mathematics; Lubbock, Texas 79409-1042; email: gornet@math.ttu.edu September 1994