Spectra, Group Representations and Twisted Laplacians

Abstract

In this chapter, we review basic notions about spectra, group representations, and twisted Laplace operators. We first recall how to define the spectrum and the spectral zeta function for a general symmetric second order elliptic differential operator acting on smooth sections of a Hermitian line bundle. We prove that the non-zero spectrum (i.e., the spectral zeta function) determines the entire spectrum on an odd-dimensional manifold, but also give an example showing that this is not always true for even-dimensional manifolds; the example is obstructed by the non-vanishing of some topological genus. After setting up some notation from representation theory, we discuss G-sets and weak conjugacy (“Gaßmann equivalence”) of subgroups of a group, explaining the interrelations. In the final sections, we introduce twisted Laplacians, corresponding to unitary representations of the fundamental group. After this, we focus on the case of a twisted Laplacian arising from a finite Galois cover of manifolds and we relate the spectrum on the top manifold to that of the induced representation on the bottom manifold. We relate the multiplicity of zero in the spectrum to the multiplicity of the trivial representation in the given representation, and finally we show that, contrary to the general case, the multiplicity of zero in the spectrum of a twisted Laplacian is determined from the non-zero spectrum, provided one also knows the usual Laplace spectrum of the manifold.

We’re gonna do the twist and it goes like this

— “Let’s twist again”, written by Kal Mann and Dave Appell

In this chapter, we review basic notions about spectra, group representations, and twisted Laplace operators. We first recall how to define the spectrum and the spectral zeta function for a general symmetric second order elliptic differential operator acting on smooth sections of a Hermitian line bundle. We prove that the non-zero spectrum (i.e., the spectral zeta function) determines the entire spectrum on an odd-dimensional manifold, but also give an example showing that this is not always true for even-dimensional manifolds; the example is obstructed by the non-vanishing of some topological genus. After setting up some notation from representation theory, we discuss G-sets and weak conjugacy (“Gaßmann equivalence”) of subgroups of a group, explaining the interrelations. In the final sections, we introduce twisted Laplacians, corresponding to unitary representations of the fundamental group. After this, we focus on the case of a twisted Laplacian arising from a finite Galois cover of manifolds and we relate the spectrum on the top manifold to that of the induced representation on the bottom manifold. We relate the multiplicity of zero in the spectrum to the multiplicity of the trivial representation in the given representation, and finally we show that, contrary to the general case, the multiplicity of zero in the spectrum of a twisted Laplacian is determined from the non-zero spectrum, provided one also knows the usual Laplace spectrum of the manifold.

As basic background references for this chapter, we use [54, 83, 88].

1 Spectrum and Spectral Zeta Function

Let M = (M, g) denote a connected closed oriented smooth Riemannian manifold with Riemannian metric g. Let E denote a Hermitian bundle on M and A a symmetric second order elliptic differential operator acting on smooth sections C ^∞(M, E) of E with non-negative eigenvalues. The operator extends to the corresponding space L ²(M, E) of L ²-sections where it has a dense domain. The spectrum σ _M(A) of A (or σ(A) is M is fixed) is the multiset of eigenvalues of A, where the multiplicities of the elements in the set are given by the multiplicities of the eigenvalues.

We make the following convenient notational conventions: if S ₁ and S ₂ are multisets, we let S ₁ ∪ S ₂ denote the multiset consisting of elements of S ₁ or S ₂, where the multiplicity of an element is the sum of the multiplicities of that element in S ₁ and S ₂, and for a multiset S and an integer n, we mean by nS the multiset of elements of S where all multiplicities are multiplied by n.

Example 3.1.1

To (M, g) is associated a Laplace(–Beltrami) operator Δ_M, acting on the space C ^∞(M) of smooth functions on M, given in local coordinates (x ⁱ) as

$$\displaystyle \begin{aligned} \Delta_M(f):=- \det(g)^{-1/2} \sum_{i,j} \frac{d}{dx^i} \left( \det(g)^{1/2} g^{ij} \frac{d}{dx^j} f \right). \end{aligned}$$

An intrinsic definition follows from the more general next example. $\lozenge$

Example 3.1.2

More generally, for every k ≥ 0, there is such a Laplace operator $\Delta ^k_M$ acting on (the space of) k-forms $C^\infty (M, \bigwedge ^k T^* M)$, defined as follows. If $d^k \colon C^\infty (M,\bigwedge ^k T^* M) \rightarrow C^\infty (M,\bigwedge ^{k+1} T^* M)$ denotes the exterior derivative on k-forms, and $\delta ^{k+1} \colon C^\infty (M,\bigwedge ^{k+1} T^* M) \rightarrow C^\infty (M,\bigwedge ^{k} T^* M)$ the adjoint of d ^k for the inner product induced by the metric g, then $\Delta _M^k = \delta ^{k+1} d^k + d^{k-1} \delta ^k$ acting on $C^\infty (M,\bigwedge ^k T^* M)$ with d ⁻¹ = 0 by convention.

For k = 0, $\Delta _M^k$ equals $\Delta _M^0 = \delta ^1 d = \Delta _M$ as defined in Example 3.1.1. The kernel of $\Delta _M^k$ consists of so-called harmonic k-forms, which, by a theorem of Hodge, is identified with $\operatorname {H}_{\mathrm {dR}}^k(M)$, the k-th de Rham cohomology group of M. De Rham’s theorem identifies $\operatorname {H}_{\mathrm {dR}}^k(M) \cong \operatorname {H}^k(M,\mathbf {R})$ with the usual real-valued (singular) cohomology of M. Thus, the multiplicity of 0 in the spectrum of Δ^k equals the k-th Betti number of M:

$$\displaystyle \begin{aligned} \mathrm{mult}_0(\sigma_M(\Delta_M^k)) = \dim_{\mathbf{R}} \operatorname{H}^k(M,\mathbf{R}). \end{aligned} $$

(3.1)

We refer to [83, Chapter 1] for details. $\lozenge$

The spectral zeta function of A as above is defined as

$$\displaystyle \begin{aligned} \zeta_{M,A}(s) = \zeta_A(s):= \sum_{0 \neq \lambda \in \sigma(A)} \lambda^{-s} \end{aligned}$$

with the sum not involving the zero eigenvalues. The function can be meromorphically continued to the entire complex plane [83, Thm. 5.2]. Since ζ _A(s) is a (generalised) Dirichlet series, the identity theorem for such Dirichlet series [46, Thm. 6] implies that it is determined by its values on a countable set with an accumulation point, e.g., by its values at all sufficiently large integers.

2 Spectrum Versus Spectral Zeta Function

We will formulate all results using the spectrum, rather than the spectral zeta function. For odd-dimensional manifolds, these give exactly the same information, as the following proposition shows.

Proposition 3.2.1

If M is an odd-dimensional manifold, the multiset σ _M(A) and the function ζ _M,A(s) mutually determine each other.

Proof

It is clear that the function ζ _A(s) determines σ(A) −{0}, so we only need to show that if M is of odd dimension, the multiplicity of zero in the spectrum is also determined by ζ _A; this multiplicity is $\dim \ker A$, which equals − ζ _A(0) if M has odd dimension (see [83, Thm. 5.2]). □

The result in Proposition 3.2.1 does not hold in general if M is of even dimension n, as the example in the following proposition shows.

Proposition 3.2.2

There exists a 4-dimensional manifold M (in fact, M may be chosen as a complex quartic surface) and two second order bundle operators Δ ^± on M such that $\zeta _{\Delta ^+}=\zeta _{\Delta ^-}$ but σ( Δ ⁺) ≠ σ( Δ ⁻).

Proof

The basic idea is that two commuting operators have the same non-zero spectrum, but that the difference of the dimensions of their kernels can have a topological interpretation as an index, that might be non-vanishing. The proof will use the theory of spin manifolds and the index theorem for the Dirac operator, for which we refer to [11] or [39].

Suppose M is an even dimensional spin manifold with Dirac operator

$$\displaystyle \begin{aligned} D = \left(\begin{matrix} 0 \$ D^+ \\ D^- \$ 0 \end{matrix} \right), \end{aligned}$$

where the spinor bundle is decomposed into eigenspaces for the chirality operator as S ⁺ ⊕ S ⁻, with D ⁺: S ⁺ → S ⁻ and D ⁻: S ⁻→ S ⁺, D ⁻ = (D ⁺)^∗ adjoint to D ⁺. Then the second order operators

$$\displaystyle \begin{aligned} \Delta^{\pm} := D^{\mp} D^{\pm} \end{aligned}$$

have the same non-zero spectrum (this is true in general for the non-zero spectrum of the products AB and BA of two operators A and B); hence $\zeta _{\Delta ^+}=\zeta _{\Delta ^-}$.

On the other hand,

$$\displaystyle \begin{aligned} \ker \Delta^+ = \ker (D^+)^* D^+ = \ker D^+ \end{aligned}$$

since if Δ⁺ φ = (D ⁺)^∗ D ⁺ φ = 0, then

$$\displaystyle \begin{aligned} || D^+ \varphi||{}^2 = \langle D^+ \varphi, D^+ \varphi \rangle = \langle \varphi, (D^+)^* D^+ \varphi \rangle = 0. \end{aligned}$$

Hence with m ^± := mult₀(σ( Δ^±)) the multiplicity of 0 in the spectrum of Δ^±, we find that

$$\displaystyle \begin{aligned} m^+ - m^- \$ = \dim \ker \Delta^+ - \dim \ker \Delta^- = \dim \ker D^+ - \dim \ker D^- \\ \$= \mathrm{index}\ D = \int_M \widehat A(M) \end{aligned} $$

is, by the Atiyah–Singer index theorem, the $\widehat A$-genus of M, which may be non-zero (compare [11, 3.4, 4.1]). For example, if M is a complex quartic surface (of real dimension 4) in CP ³, then $\mathrm {index}\ D = \int \widehat A(M)= 2$ (see, e.g., [39, p. 727]). This shows that σ( Δ⁺) ≠ σ( Δ⁻). □

Remark 3.2.3

If one is willing to consider orbifolds instead of manifolds, there exist two-dimensional orbifolds that are isospectral for the Laplace operator acting on 1-forms (cf. Example 3.1.2), as shown by the following example of Gordon and Rossetti. Consider the quotient of the standard flat torus Z ²∖R ² by the involutions induced by the following maps on R ² (with coordinates (x, y)):

• (i) (x, y)↦(x, −y), leading to the cylinder C;

• (ii) (x, y)↦(y, x), producing the Möbius strip M;

• (iii) (x, y)↦(−x, −y), leading to the pillow orbifold $\mathscr O$.

Then the non-zero spectra of the Laplace operators acting on the space of 1-forms on C, M and $\mathscr O$ agree. However, 0 is not an eigenvalue on 1-forms for C and M, whereas it is for $\mathscr O$ [42, Example 2.5 and Theorem 3.1].

Interestingly, our main results, such as Theorem 1.2.1 and Proposition 1.2.5, are formulated in their strongest possible form using precisely the multiplicity of zero in the spectrum of certain twisted Laplacians. For these operators, it turns out that also in even dimension this multiplicity (and hence the zeta function) is fixed by the non-zero spectrum of the usual and the twisted Laplacian, cf. Proposition 3.10.1 below.

3 Group Representations

If G is a finite group, let denote the group of linear characters of G, and let $\operatorname {Irr}(G)$ denote the set of inequivalent irreducible unitary representations of G. We consider complex representations as C[G]-modules $\mathscr {M}$ or group homomorphisms $\rho \colon G \rightarrow \operatorname {Aut}(V) \cong \mathrm {GL}(N,\mathbf {C})$ with V = C ^N and freely mix these concepts, writing expressions such as “$\mathscr {M} \cong \rho$”. By further slight abuse of notation, if ρ ₁ and ρ ₂ are representations of G, we write

$$\displaystyle \begin{aligned} \langle \rho_1, \rho_2 \rangle = \langle \operatorname{tr}(\rho_1(-)), \operatorname{tr}(\rho_2(-) \rangle = \frac{1}{|G|} \sum_{g \in G} \operatorname{tr}(\rho_1(g))\overline{\operatorname{tr}(\rho_2(g))} \end{aligned}$$

for the inner product of the corresponding characters in the space of class functions. The multiplicity of an irreducible representation $\rho ' \in \operatorname {Irr}(G)$ in the decomposition into irreducibles of a general representation ρ of G is then 〈ρ, ρ′〉.

The regular representation ρ _G,reg corresponds to the C[G]-module C[G]. It decomposes as

$$\displaystyle \begin{aligned} \rho_{G,\mathrm{reg}} = \bigoplus\limits_{\rho_i \in \operatorname{Irr}(G)} \dim(\rho_i) \rho_i. \end{aligned}$$

If H is a subgroup of G and ρ a representation of H, then $\operatorname {Ind}_H^G \rho$ denotes the representation induced by ρ from H to G: if ρ corresponds to the C[H]-module V , then $\operatorname {Ind}_H^G \rho$ corresponds to the C[G] module W := C[G] ⊗_C[H] V . In coordinates, this means the following: since G permutes the cosets of H in G, if we choose coset representatives

$$\displaystyle \begin{aligned} G/H = \{ g_1H = H, \dots g_nH \}, \end{aligned}$$

then for any g ∈ G we have

$$\displaystyle \begin{aligned} gg_i = g_{g(i)} h_{g,i} \end{aligned}$$

for some h _g,i ∈ H and some permutation i↦g(i) of {1, …, n}. If we write

$$\displaystyle \begin{aligned} W = V^{G/H} = \bigoplus_{i=1}^n g_i V \end{aligned}$$

using g _i as placeholder, then with v _i ∈ V , we have

$$\displaystyle \begin{aligned} \operatorname{Ind}_H^G \rho(g) \left( \sum_i g_i v_i \right) = \sum_i g_{g(i)} \rho(h_{g,i})(v_i). \end{aligned} $$

(3.2)

Let $\operatorname {Res}_H^G \rho$ denotes the restriction of ρ, a representation of G, from G to H. If ρ is a representation, $\overline \rho$ denotes the complex conjugate representation. Recall the standard calculation rules

$$\displaystyle \begin{aligned} \langle \rho_1 \otimes \rho_2, \rho_3 \rangle = \langle \rho_1, \rho_3 \otimes \overline{\rho_2} \rangle \mbox{ and } \langle \operatorname{Ind}_H^G \rho_1, \rho_2 \rangle = \langle \rho_1, \operatorname{Res}_H^G \rho_2 \rangle, \end{aligned}$$

(the latter is known as “Frobenius reciprocity”).

More generally, the above theory applies mutatis mutandis, replacing C by an algebraically closed field of characteristic coprime to the order |G| of the group G. For a non-algebraically closed field K of characteristic coprime to |G|, an irreducible representation might decompose over the algebraic closure into a sum of irreducible (Galois-conjugate) representations, but the above theory remains valid, with the caveat that a rational character is not always the character of a rational representation, but a multiple is. If the characteristic of K divides |G|, the category of K[G]-modules is not semisimple, so complementary modules for submodules do not always exist. For us, it will be important that, in general, the regular representation is defined over Q, and induction and restriction turn K-representations into K-representations.

4 G-Sets

If G is a group, a G-set is a set that admits a left G-action. An example is the left cosets G∕H of a subgroup H with the action of left multiplication by G. A morphism of G-sets is a G-equivariant map of the sets. We say a G-set is transitive if G acts transitively on it. If X is a transitive G-set and H the stabiliser of any point in X, then X is isomorphism to G∕H as G-set.

5 (Weak) Conjugacy

We let 1 = 1 _G denote the trivial representation of a group G. If, as before, {g ₁, …, g _n} is a set of representatives for the (left) H-cosets in G, then $\operatorname {Ind}_H^G \mathbf {1}$ is the permutation representation (i.e., the action of each g ∈ G is given by a permutation matrix, a matrix having exactly one non-zero entry 1 in each row and column) given by the action of G on the vector space

$$\displaystyle \begin{aligned} \mathbf{C}[G/H]:=\bigoplus\limits_{i=1}^n \mathbf{C} g_i H \end{aligned}$$

spanned by the cosets of H in G. We have the following.

Proposition 3.5.1

Let H ₁ and H ₂ denote two subgroups of a finite group G.

• (i) The following properties are equivalent:

  • (a) The representations $\operatorname {Ind}_{H_1}^G \mathbf {1} \cong \operatorname {Ind}_{H_2}^G \mathbf {1}$ are isomorphic.

  • (b) Each conjugacy class c of G intersects H ₁ and H ₂ in the same number of elements.

  • (c) There exists a set-theoretic bijection ψ: H ₁ → H ₂ such that h ₁ and ψ(h ₁) are conjugate in G for any h ₁ ∈ H ₁.

  If any of these holds, we say H ₁ and H ₂ are weakly conjugate in G.

• (ii) The stronger property that the groups H ₁ and H ₂ are conjugate in G is equivalent to the cosets G∕H ₁ and G∕H ₂ being isomorphic as G-sets.

Weak conjugacy is sometimes called “almost conjugacy”, and also known as “Gaßmann equivalence” in number theory, cf. [77].

Proof

• (i) Representation isomorphism is the same as isomorphism of characters, and the character of the representation $\operatorname {Ind}_{H}^G \mathbf {1}$ is

  $$\displaystyle \begin{aligned} \psi(g)=|[g]\cap H| \cdot |{C_G(g)}|/{|H|},\end{aligned} $$

  (3.3)

  where [g] is the conjugacy class of g and C _G(g) is the centraliser of g in G (compare, e.g., [18, §1]). The final equivalent statement is proven in [24, Lemma 2].

• (ii) The existence of a G-isomorphism ϕ : G∕H ₁ → G∕H ₂ implies that the G-stabiliser H ₁ of the coset eH ₁ equals the G-stabiliser gH ₂ g ⁻¹ of some coset gH ₂ = ϕ(eH ₁), and hence H ₁ = gH ₂ g ⁻¹. Conversely, if $H_2 = g_0^{-1}H_1g_0$, the map ϕ : G∕H ₁ → G∕H ₂, ϕ(gH ₁) = gg ₀ H ₂ is a well-defined isomorphism of G-sets. □

Remark 3.5.2

If we have a diagram (1.1) and M ₁ and M ₂ have the same Laplace spectrum (viz., the same spectral zeta function) and the same dimension n, then they have the same volume (from Weyl’s law, or, equivalently, from the value of the residue of their zeta functions at s = n∕2). Since the covering degrees deg(p _i) of p _i are vol(M _i)∕vol(M ₀), these are also equal, and hence in diagram (1.2), we find from |H _i| = |G|∕deg(p _i) that |H ₁| = |H ₂|. So in this case, there is always a set-theoretic bijection ψ: H ₁ → H ₂.

6 Twisted Laplacian

Suppose $\rho \colon \pi _1(M) \rightarrow \operatorname {U}(N,\mathbf {C})$ is a unitary representation of the fundamental group π ₁(M) of M. Let $\Pi \colon \widetilde M \rightarrow M$ denote the universal covering of M, and set $E_\rho := \widetilde M \times _\rho {\mathbf {C}}^{{N}}$, where the subscript ρ indicates equivalence classes for the relation

$$\displaystyle \begin{aligned} (z,v) \sim (\gamma z,\rho(\gamma)^{-1} v) \end{aligned}$$

for any $z \in \widetilde M, v\in {\mathbf {C}}^{{N}}$ and γ ∈ π ₁(M). Now E _ρ is a flat vector bundle of rank N over M, whose global sections f ∈ C ^∞(M, E _ρ) correspond bijectively to smooth ρ-equivariant vector-valued functions $\overrightarrow {f} \colon \widetilde M \rightarrow {\mathbf {C}}^{{N}}$, i.e., functions with $\overrightarrow {f}(\gamma z) = \rho (\gamma )\overrightarrow {f}(z)$. The twisted Laplacian

$$\displaystyle \begin{aligned} \Delta_{M,\rho}=\Delta_\rho \colon C^\infty(M,E_\rho) \rightarrow C^\infty(M,E_\rho) \end{aligned}$$

is defined as

$$\displaystyle \begin{aligned} \overrightarrow{\Delta_\rho(f)}:=\Delta_{\widetilde M} \overrightarrow{f}(z). \end{aligned}$$

Henceforth, we will also denote the spectrum σ _M( Δ_ρ) simply by σ _M(ρ) or by σ(ρ) if the underlying manifold M is fixed.

Notice that when ρ = ρ ₁ ⊕ ρ ₂, then Δ_ρ admits a block decomposition $\Delta _{\rho _1} \oplus \Delta _{\rho _2}$ on $E_\rho \cong E_{\rho _1} \oplus E_{\rho _2}$, and thus, the spectrum satisfies (as multisets)

$$\displaystyle \begin{aligned} \sigma_M(\rho_1 \oplus \rho_2) = \sigma_M(\rho_1) \cup \sigma_M(\rho_2). \end{aligned}$$

7 Twisted Laplacians on Finite Covers

In case ρ factors through a finite group G, there is no need to use the universal covering. Let M′→ M denote a (fixed-point free) G-cover and $\rho : G \to \operatorname {U}(N,\mathbf {C})$ a unitary representation. The vector space C ^∞(M, E _ρ) is canonically isomorphic to the vector space of smooth ρ-equivariant vector-valued functions on M′ given by

$$\displaystyle \begin{aligned} C^\infty_\rho(M',{\mathbf{C}}^{{N}}) := \{ \overrightarrow{f} \in C^\infty(M',{\mathbf{C}}^{{N}}) \mid \overrightarrow{f}(\gamma x) = \rho(\gamma) \overrightarrow{f}(x), \forall\, x \in M', \gamma \in G \}. \end{aligned}$$

In this case,

$$\displaystyle \begin{aligned} \overrightarrow{\Delta_\rho f} = \Delta_{M'} \overrightarrow{f}, \end{aligned} $$

(3.4)

where $\overrightarrow {f}$ is the ρ-equivariant function in $C^\infty _\rho (M',{\mathbf {C}}^{{N}})$ corresponding to f. Note that $\Delta _{M'} \overrightarrow {f}$ is again a ρ-equivariant function in $C^\infty _\rho (M',{\mathbf {C}}^{{N}})$ and therefore represents an element in C ^∞(M, E _ρ).

8 Twisted Laplacians for Induced Representations

The following lemma is stated in [91, Lemma 1]; we write a proof using our notation.

Lemma 3.8.1

If M → M ₁ → M ₀ is a tower of finite Riemannian coverings and M → M ₀ is Galois with group G, M → M ₁ with group H, and $\rho \colon H \rightarrow \operatorname {U}(N, \mathbf {C})$ a representation, then

$$\displaystyle \begin{aligned} \sigma_{M_0}(\operatorname{Ind}_{H}^G \rho) = \sigma_{M_1}(\rho). \end{aligned}$$

Proof

Write $\rho ^*:=\operatorname {Ind}_H^G \rho \colon G \rightarrow \operatorname {U}(Nn,\mathbf {C})$ and let

$$\displaystyle \begin{aligned} G/H=\{g_1H=H,\dots,g_{{n}}H\} \end{aligned}$$

denote representatives for the distinct cosets of H in G. Define two maps

by

$$\displaystyle \begin{aligned} \Phi(\overrightarrow{f})(x) =(\overrightarrow{f}(g_1^{-1}(x)),\dots, \overrightarrow{f}(g_{{n}}^{-1}(x))) \end{aligned}$$

and

$$\displaystyle \begin{aligned} \Psi(\overrightarrow{F})=\Psi((\overrightarrow{f_1},\dots,\overrightarrow{f_{{n}}})): = \overrightarrow{f_1}. \end{aligned}$$

Recall that by definition

$$\displaystyle \begin{aligned} \rho^*(g)((\overrightarrow{f_i}(x))_{i=1}^{{n}}) =(\rho(g_{g(i)}^{-1} g g_i) \overrightarrow{f}_{g(i)}(x) )_{i=1}^{{n}} \end{aligned}$$

where g(i) is given by

$$\displaystyle \begin{aligned} g g_i H = g_{g(i)} H. \end{aligned}$$

This allows one to check that Φ and Ψ are well defined and mutually inverse bijections. Recall that $\overrightarrow {\Delta _\rho (f)}=\Delta _M(\overrightarrow {f})$ and $\overrightarrow {\Delta _{\rho ^*}(F)}=\Delta _M(\overrightarrow {F})$ with Δ_M applied componentwise. Since the g _i are isometries, Φ is a unitary operator in L ² and Δ_M Φ =  Φ Δ_M, so that we have the intertwining

$$\displaystyle \begin{aligned} \Phi \circ \Delta_\rho = \Delta_{\rho^*} \circ \Phi, \end{aligned}$$

and the equality of spectra follows. □

As is shown in [91], the main theorem of Sunada [90] follows easily from this.

Theorem 3.8.2 (Sunada’s Theorem [90, Theorem 1])

If we have a diagram of the form (1.2) and H ₁ and H ₂ are weakly conjugate in G, then M ₁ and M ₂ are isospectral.

Proof

We apply Lemma 3.8.1 to the trivial representation ρ = 1 for M → M _i → M ₀ with i = 1 and i = 2. Since H ₁ and H ₂ are weakly conjugate, $\operatorname {Ind}_{H_1}^G \mathbf {1} \cong \operatorname {Ind}_{H_2}^G \mathbf {1}$ (cf. Proposition 3.5.1), and so

$$\displaystyle \begin{aligned} \sigma_{M_1} = \sigma_{M_1}(\mathbf{1}) = \sigma_{M_0}(\operatorname{Ind}_{H_1}^G \mathbf{1}) = \sigma_{M_0}(\operatorname{Ind}_{H_2}^G \mathbf{1}) = \sigma_{M_2}(\mathbf{1}) = \sigma_{M_2}, \end{aligned}$$

finishing the proof. □

Remark 3.8.3

The original proof used a trace formula [90, Lemma 1], that is now hidden in the computations with induced representations and their characters. The trace formula proof has the advantage to apply to all “natural” operators alike, such as the Laplace operators on k-forms (cf. Example 3.1.2); see Sect. 4.2 for more on natural operators and strong isospectrality. For another exposition in the style of the original argument and background information, see [23, Chapter 11].

Remark 3.8.4

The converse of the theorem is not true: isospectral manifolds fitting into a diagram of the form (1.2) do not necessarily have H ₁ and H ₂ weakly conjugate in G. See Corollary 4.2.5 for a spectral characterisation of weak conjugacy using twisted Laplacians.

Remark 3.8.5

The main application of Theorem 3.8.2 is to the construction of isospectral, non-isometric manifolds. For this, one needs to realise a diagram of manifolds as in (1.2) and guarantee that M ₁ and M ₂ are not isometric, for example, by making sure that $\operatorname {H}_1(M_1)$ and $\operatorname {H}_1(M_2)$ are distinct.

9 Multiplicity of Zero in Twisted Laplace Spectra

Decomposing a general representation $\rho : G \to \operatorname {U}(N,\mathbf {C})$ into irreducibles as $\rho = \bigoplus \langle \rho _i,\rho \rangle \rho _i$, we have

$$\displaystyle \begin{aligned} \sigma_M(\Delta_{\rho}) = \bigcup \langle \rho_i,\rho \rangle \sigma_M(\Delta_{\rho_i}). \end{aligned} $$

(3.5)

Applied to the regular representation, we find a relation between the spectra of the usual Laplacian on the cover M′ and of the twisted Laplacians on the original manifold M, as follows:

$$\displaystyle \begin{aligned} \sigma_{M'}(\Delta_{M'}) =\sigma_{M}(\Delta_{\rho_{G,\mathrm{reg}}}) = \bigcup \dim(\rho_i) \sigma_M(\Delta_{\rho_i}). \end{aligned} $$

(3.6)

(The first equality follows from Lemma 3.8.1 since $\operatorname {Ind}_{\{1\}}^G \mathbf {1} = \mathbf {C}[G] = \rho _{G,\mathrm {reg}}$). We see in particular that the eigenvalues of any twisted Laplacian are also eigenvalues of the usual Laplace operator of the corresponding cover.

Lemma 3.9.1

Let G be a finite group acting by fixed-point free isometries on a closed connected Riemannian manifold M′ with quotient M = G∖M′. If ρ is any unitary representation of G, then the multiplicity 〈ρ, 1〉 of the trivial representation in the decomposition of ρ into irreducibles equals $\dim \ker \Delta _\rho$ , the multiplicity of the zero eigenvalue in σ _M(ρ).

Proof (First Proof of Lemma 3.9.1 )

Since M and M′ are connected, the multiplicity of zero in $\sigma _{M'}(\Delta _{M'})$ and σ _M( Δ_M) is one. It follows from the decomposition of multisets (3.6) that for any irreducible representation ρ _i ≠ 1 of G, $\sigma _M(\Delta _{\rho _i})$ does not contain zero. If we now decompose ρ as a sum of irreducibles, the decomposition of multisets (3.5) implies that the multiplicity of zero in σ _M( Δ_ρ) is indeed the multiplicity with which 1 occurs in ρ. □

We can also give a “direct” proof, as follows.

Proof (Second Proof of Lemma 3.9.1 )

Let $\rho : G \to \operatorname {U}(N,\mathbf {C})$. A function

$$\displaystyle \begin{aligned} f \in \ker \Delta_\rho \subseteq C^\infty(M,E_\rho) \end{aligned}$$

corresponds to a function $\overrightarrow {f}$ on M′ with $\Delta _{M'} \overrightarrow {f} = 0$ and $\overrightarrow {f}(\gamma z) = \rho (\gamma ) \overrightarrow {f}(z)$ for all γ ∈ Γ. Since M′ is closed and connected, this implies that $\overrightarrow {f}=\overrightarrow {f}_0$ is a constant vector in C ^N, and the equivariance condition translates into

$$\displaystyle \begin{aligned} (\rho(\gamma)-1)\overrightarrow{f}_0 = 0 {{\quad \text{for all }\gamma \in G}}. \end{aligned}$$

Hence each such linearly independent vector $\overrightarrow {f_0} \in {\mathbf {C}}^N$ can be used to split off a one-dimensional invariant subspace in ρ, and we find the result. □

10 Spectrum Versus Spectral Zeta Function for Twisted Laplacians

In Proposition 3.2.2 we showed that, in general, on an even-dimensional manifold, knowledge of the spectrum is stronger than that of the spectral zeta function, i.e., of the non-zero spectrum. The operators in Proposition 3.2.2 were not twisted Laplacians. For twisted Laplace operators, the situation is better, as the following proposition shows.

Proposition 3.10.1

Let G be a finite group acting by fixed-point free isometries on a closed connected n-dimensional Riemannian manifold M′ with quotient M = G∖M′. If ρ is any unitary representation of G, then on the one hand the pair of multisets σ _M( Δ _ρ) and σ _M( Δ), and on the other hand the pair of zeta functions $\zeta _{M,\Delta _\rho }(s)$ and ζ _M,Δ(s) mutually determine each other. In fact, the multiplicity of zero in σ( Δ _ρ) is given by

$$\displaystyle \begin{aligned} \dim \ker \Delta_\rho = (\zeta_{\Delta} (0)+1) \left. \frac{\zeta_{\Delta_\rho}(s)}{\zeta_{\Delta}(s)}\right \rvert_{s=\frac{n}{2}} - \zeta_{\Delta_\rho} (0). \end{aligned} $$

(3.7)

Remark 3.10.2

As an illustration consider the easy situation when |G| = 1, so M′ = M and ρ ∼ N 1 for some N. Then σ _M( Δ_ρ) = Nσ _M( Δ^N) = Nσ _M, $\zeta _{\Delta _\rho } = N \zeta _{\Delta }$ and mult₀(σ( Δ^N)) = N is equal to $(\zeta _\Delta (0)+1)\left . \frac {\zeta _{\Delta _\rho }(s)}{\zeta _{\Delta }(s)}\right \rvert _{s=\frac {n}{2}} -\zeta _{\Delta _\rho }(0) = 2N-N.$

Proof

It suffices to prove that ζ _Δ and $\zeta _{\Delta _\rho }$ determine the multiplicity of zero in the spectrum (i.e, the dimension of the kernel) of Δ_ρ. If the dimension n of M is odd, this follows from the stronger Proposition 3.2.1. For even n and a general operator A as in Sect. 3.1,

$$\displaystyle \begin{aligned} \zeta_A(0) = - \dim \ker A + (4 \pi)^{-n/2} \int_M u_{n/2}(A), \end{aligned} $$

(3.8)

where $\operatorname {tr}(e^{-tA}) \sim (4 \pi t)^{-n/2} \sum _{k=0}^\infty \int _M u_k(A) t^k$ is the asymptotic expansion of the heat kernel of A as t↓0 [83, Thm. 5.2].

We apply this in our situation, with

$$\displaystyle \begin{aligned} A=\Delta_\rho = \bigoplus\limits_{i=1}^N \Delta_{M'} \end{aligned}$$

acting on $C^\infty (M,E_\rho ) = C_\rho ^\infty (M',{\mathbf {C}}^N)$. Denote by 1_N the identity matrix of size N × N. Recall that the principal symbol p( Δ_M) of a Laplace operator Δ_M on a Riemannian manifold (M, g) is determined by the metric tensor g (more accurately, it is the quadratic form on the cotangent bundle dual to g). Since M′→ M is a Riemannian covering, the metric tensor of M pulls back to that of M′, and hence the principal symbol of $\Delta _{M'}$ is the same as that of Δ_M. Therefore, Δ_ρ is a “Laplace-style” operator in the sense of [40, 1.2]: it has (matrix) principal symbol the diagonal matrix p( Δ_ρ) = p( Δ_M) ⋅ 1_N. Such operators have a description depending on a connection on the bundle and an endomorphism of the bundle as in [39, Lemma 1.2.1], and in our situation, for E _ρ, the bundle connection is flat (curvature Ω ≡ 0) and the endomorphism e is trivial.

The coefficients u _k( Δ_ρ)(x) (as a function of x ∈ M) are of the form

$$\displaystyle \begin{aligned} u_k(\Delta_\rho)(x)=\operatorname{tr}_{E_{\rho,x}}(e_k(\Delta_\rho)(x)), \end{aligned}$$

where $\operatorname {tr}_{E_{\rho ,x}}$ denotes the fiberwise trace in the fibers E _ρ,x≅C ^N, and where e _k( Δ_ρ) is a linear combination with universal coefficients (independent of the dimension n of the manifold and the rank N of the bundle) of covariant derivatives of $\underline {R}\cdot 1_N$ (where $\underline {R}$ is the covariant Riemann curvature tensor of M), the bundle curvature Ω and the endomorphism e [39, §3.1.8–3.1.9]. Since the latter two are identically zero in our situation, we can write e _k( Δ_ρ) = P _k ⋅ 1_N with P _k only depending on the covariant derivatives of $\underline {R}$, in particular, not depending on ρ. We conclude that

$$\displaystyle \begin{aligned} (4 \pi)^{-n/2} \int_M u_{n/2} (\Delta_\rho)(x) = (4 \pi)^{-n/2} \int_M \operatorname{tr}_{E_{\rho,x}}(P_{n/2}(x) \cdot 1_N) = N U, \end{aligned}$$

where U = (4π)^−n∕2∫_M P _n∕2 is independent of ρ.

Therefore, applying (3.8) to Δ and Δ_ρ, we find

$$\displaystyle \begin{aligned} \dim \ker \Delta_\rho = N U - \zeta_{\Delta_\rho} (0) = N(\zeta_{\Delta} (0)+1) - \zeta_{\Delta_\rho} (0). \end{aligned} $$

(3.9)

We can compute the rank N in terms of the first coefficients in the asymptotic expansions: using e ₀( Δ_ρ) = 1_N, we find N =∫_M u ₀( Δ_ρ)∕∫_M u ₀( Δ). On the other hand, $\zeta _{\Delta _\rho }(s)$ has a simple pole at s = n∕2 with residue Γ(n∕2)⁻¹∫_M u ₀( Δ_ρ) (see, e.g., the proof of [83, Thm. 5.2]), so that the function $\zeta _{\Delta _\rho }(s)/\zeta _{\Delta }(s)$ is holomorphic at s = n∕2 and takes value N there:

$$\displaystyle \begin{aligned} N = \left. \frac{\zeta_{\Delta_\rho}(s)}{\zeta_{\Delta}(s)}\right \rvert_{s=\frac{n}{2}}. \end{aligned} $$

(3.10)

Combining Eqs. (3.9) and (3.10) gives the desired expression (3.7) for the multiplicity of zero in the spectrum in terms of spectral zeta functions only. □

Remark 3.10.3

• (i) The above argument also shows that for a twisted Laplacian Δ_ρ corresponding to a unitary representation on a fixed manifold M, the value

  $$\displaystyle \begin{aligned} \dim \ker \Delta_\rho + \zeta_{\Delta_\rho}(0) \end{aligned}$$

  only depends on the dimension of the representation ρ.

• (ii) Weyl’s law for Δ_ρ says that if N( Δ_ρ, X) denotes its number of eigenvalues ≤ X, then

  $$\displaystyle \begin{aligned} \lim_{X \rightarrow +\infty} \frac{\mathrm N(\Delta_\rho,X)}{X^{n/2}} = N \cdot \frac{\mathrm{vol}(M)}{(4 \pi)^{n/2} \Gamma\left(\frac{n}{2}+1\right)}, \end{aligned}$$

  so that on a fixed manifold, the dimension N of the representation ρ can be read off from the asymptotics of the spectra of Δ_ρ and Δ_M:

  $$\displaystyle \begin{aligned} N= \lim_{X \rightarrow +\infty} \frac{\mathrm N(\Delta_\rho,X)}{\mathrm N(\Delta_M,X)};\end{aligned} $$

  (3.11)

  formulas (3.10) and (3.11) are equivalent through Karamata’s version of the Tauberian theorem (compare [11, pp. 91–92]).

Open Problem

Find a “geometric” formula for the difference in multiplicities of zero for two operators A and B on a manifold of the general type considered here that have identical non-zero spectrum.

Open Problem

Is the multiplicity of zero in the spectrum of a twisted Laplacian determined by the non-zero spectrum of the twisted Laplacian alone, without assuming knowledge of the spectrum of the usual Laplacian?

Open Problem

Study how disjoint the spectra of the different $\Delta _{\rho _i}$ in (3.6) are, similar to the question how disjoint zeros of number theoretic L-series are (cf. [84]): the so-called grand simplicity hypothesis says that the imaginary parts of the zeros of all Dirichlet L-series for primitive characters are linearly independent over Q. From the above decomposition results, it is clear that if ρ′ is a subrepresentation of ρ, then σ(ρ′) ⊆ σ(ρ) (this even holds for infinite amenable groups if ρ′ is weakly contained in ρ, cf. [92]); here, we are asking for a kind of converse result.