Detecting Representation Isomorphism Through Twisted Spectra

Abstract

In this chapter, we give a spectral characterisation of isomorphism of induced representations. We also discuss strong isospectrality in the sense of Pesce (which, by a result of Sunada, is implied by weak conjugacy of subgroups), discuss an illustrative example of lens spaces due to Ikeda, and use the first result to give a spectral characterisation of weak conjugacy.

In this chapter, we give a spectral characterisation of isomorphism of induced representations. We also discuss strong isospectrality in the sense of Pesce (which, by a result of Sunada, is implied by weak conjugacy of subgroups), discuss an illustrative example of lens spaces due to Ikeda, and use the first result to give a spectral characterisation of weak conjugacy.

1 Spectral Detection of Isomorphism of Induced Representations

In this section, we assume again that we have a diagram (1.2)

of finite coverings. We start with a proposition that allows us to detect isomorphism of representations induced from linear characters purely from spectral data.

Proposition 4.1.1

For two linear characters and , the following are equivalent:

• (i) $\operatorname {Ind}_{H_1}^G \chi _1 \cong \operatorname {Ind}_{H_2}^G \chi _2$.

• (ii) The spectrum $\sigma _{M_i}(\overline \chi _i \otimes \operatorname {Res}_{H_i}^G \operatorname {Ind}_{H_j}^G \chi _j)$ is independent of i, j = 1, 2.

• (ii’) Condition (ii) holds for the pairs (i, j) given by (1, 1), (2, 1) and (1, 2), (2, 2).

• (iii) The multiplicity of the zero eigenvalue in $\sigma _{M_i}(\overline \chi _i \otimes \operatorname {Res}_{H_i}^G \operatorname {Ind}_{H_j}^G \chi _j)$ is independent of i, j = 1, 2.

• (iii’) Condition (iii) holds for the pairs (i, j) given by (1, 1), (2, 1) and (1, 2), (2, 2).

Remark 4.1.2

Condition (ii’) is

$$\displaystyle \begin{aligned} \left\{ \begin{array}{l} \sigma_{M_1}(\overline \chi_1 \otimes \operatorname{Res}_{H_1}^G \operatorname{Ind}_{H_1}^G \chi_1) = \sigma_{M_2}(\overline \chi_2 \otimes \operatorname{Res}_{H_2}^G \operatorname{Ind}_{H_1}^G \chi_1); \\ \sigma_{M_1}(\overline \chi_1 \otimes \operatorname{Res}_{H_1}^G \operatorname{Ind}_{H_2}^G \chi_2) = \sigma_{M_2}(\overline \chi_2 \otimes \operatorname{Res}_{H_2}^G \operatorname{Ind}_{H_2}^G \chi_2). \end{array} \right. \end{aligned}$$

In this form, the statement of the proposition is similar to a number-theoretical result of Solomatin [89] that inspired our proof below.

Proof of Proposition 4.1.1

We start by proving that (i) implies (ii). Let ρ denote any irreducible representation of G; then for any i = 1, 2, we have

$$\displaystyle \begin{aligned} \langle \operatorname{Ind}_{H_i}^G \left(\overline \chi_i \otimes \operatorname{Res}_{H_i}^G \operatorname{Ind}_{H_i}^G \chi_i\right), \rho \rangle \$= \langle \overline \chi_i \otimes \operatorname{Res}_{H_i}^G \operatorname{Ind}_{H_i}^G \chi_i, \operatorname{Res}_{H_i}^G \rho \rangle \\ \$ = \langle \overline \chi_i , \operatorname{Res}_{H_i}^G \rho \otimes \overline{\operatorname{Res}_{H_i}^G\operatorname{Ind}_{H_i}^G \chi_i} \rangle \\ \$ = \langle \overline \chi_i , \operatorname{Res}_{H_i}^G \left (\rho \otimes \overline{\operatorname{Ind}_{H_i}^G \chi_i}\right) \rangle \\ \$ = \langle \overline{{\operatorname{Ind}_{H_i}^G \chi_i}}, \rho \otimes \overline{\operatorname{Ind}_{H_i}^G \chi_i} \rangle. \end{aligned} $$

By assumption (i), this final expression is independent of i = 1, 2, and hence the same holds for the initial expression. Since this holds for any ρ, we find that

$$\displaystyle \begin{aligned} \operatorname{Ind}_{H_1}^G \left(\overline \chi_1 \otimes \operatorname{Res}_{H_1}^G \operatorname{Ind}_{H_1}^G \chi_1\right) = \operatorname{Ind}_{H_2}^G \left(\overline \chi_2 \otimes \operatorname{Res}_{H_2}^G \operatorname{Ind}_{H_2}^G \chi_2\right). \end{aligned}$$

By Lemma 3.8.1, we find

$$\displaystyle \begin{aligned} \sigma_{M_1}(\overline \chi_1 \otimes \operatorname{Res}_{H_1}^G \operatorname{Ind}_{H_1}^G \chi_1) \$ = \sigma_{M_2} (\overline \chi_2 \otimes \operatorname{Res}_{H_2}^G \operatorname{Ind}_{H_2}^G \chi_2) \\ \$ = \sigma_{M_2} (\overline \chi_2 \otimes \operatorname{Res}_{H_2}^G \operatorname{Ind}_{H_1}^G \chi_1), {} \end{aligned} $$

(4.1)

the last line again by assumption (i). This is condition (ii) for (i, j) = (1, 1) and (i, j) = (2, 1). Using assumption (i), one may replace $\operatorname {Ind}_{H_1}^G \chi _1$ by $\operatorname {Ind}_{H_2}^G \chi _2$ in formula (4.1) on one or both sides, and this shows condition (ii) for all other choices of i, j.

Passing from stronger to weaker statements, (ii) implies (ii’) and (iii), and (iii), as well as (ii’), imply (iii’). Hence we only need to prove that (iii’) implies (i). Consider, for different i, j,

$$\displaystyle \begin{aligned} a_{i,j}:= \langle \overline \chi_i \otimes \operatorname{Res}_{H_i}^G \operatorname{Ind}_{H_j}^G \chi_j, \mathbf{1} \rangle = \langle \operatorname{Ind}_{H_j}^G \chi_j, \operatorname{Ind}_{H_i}^G \chi_i \rangle, \end{aligned} $$

(4.2)

where the last equality follows by Frobenius reciprocity. Setting ψ to be the class function $\psi := \operatorname {Ind}_{H_1}^G \chi _1 - \operatorname {Ind}_{H_2}^G \chi _2$, this allows us to compute that

$$\displaystyle \begin{aligned} \langle \psi, \psi \rangle = a_{1,1}+a_{2,2} - a_{1,2} - a_{2,1}, \end{aligned}$$

and since in (iii’) we are assuming a _1,1 = a _2,1 and a _1,2 = a _2,2, it follows that 〈ψ, ψ〉 = 0, so ψ = 0, which is condition (i). □

2 Strong Isospectrality and Spectral Detection of Weak Conjugacy

Isospectrality of manifolds M ₁ and M ₂ in a diagram of the form (1.2) does not in general imply that H ₁ and H ₂ are weakly conjugate.

Example 4.2.1

Consider the situation where M = S ⁵, and M ₁ = L(11;1, 2, 3) and M ₂ = L(11;1, 2, 4) are lens spaces L(q;s ₁, s ₂, s ₃) defined as the quotient of S ⁵ by the block diagonal 6 × 6 matrix given by three 2 × 2 blocks representing planar rotations over respective angles 2πs _i∕q. In this case, the two groups H _i≅Z∕11Z are not equal as subgroups of the isometry group of S ⁵; they commute, and we can set

$$\displaystyle \begin{aligned} M_0 = (H_1 \times H_2) \backslash M. \end{aligned}$$

Ikeda has shown that M ₁ and M ₂ are isospectral for the Laplace operator on functions (see [52, p. 313], observing that since 8 = −3 mod 11, by Ikeda [52, Thm. 2.1] M ₁ is isometric to L(11;1, 2, 8), where the latter parameters are the ones used by Ikeda). However, H ₁ and H ₂ are not weakly conjugate: since G is abelian, conjugacy classes are singletons and |{g}∩ H _i| is 0 or 1 depending on whether g is in H _i or not.$\lozenge$

Sunada [90, Lemma 1] proved that weak conjugacy of H ₁ and H ₂ implies strong isospectrality, defined as follows in the sense of Pesce [79, §II].

Definition 4.2.2

Two Riemannian manifolds M ₁ and M ₂ admitting a common cover M are called strongly isospectral if the spectra of q _i∗ A acting on $L^2(M_i, q_{i\ast } E) \cong L^2(M,E)^{H_i}$ are equal for any natural operator A on M. Here, $q_i \colon M \twoheadrightarrow M_i$ are the corresponding covering maps, and an operator A as above on M is natural if G acts isometrically on the fibers of the bundle E and A commutes with the action of G.

The Laplace operators acting on k-forms (see Example 3.1.2) are natural for all k, so if H ₁ and H ₂ are weakly conjugate, then the spectra of all of these are equal (sometimes, “strong isospectrality” is used to mean that precisely these operators are isospectral, but we will follow Pesce’s definition as above).

Example 4.2.3

The lens spaces in Example 4.2.1 are isospectral for the Laplacian on functions, but not on all k-forms, cf. [60, Remark 3.8]. In fact, M ₁ and M ₂ from Example 4.2.1 are isospectral on functions, but not on 1-forms, as is explained in [53, Example 1, p. 416] (in the notation of loc. cit., $M_1=\overline {L}_2$ and M ₂ is isometric to $\overline {L}_1=L(11;1,2,5)$).

Strongly isospectral lens spaces are isometric [60, Proposition 7.2]. There exist lens spaces isospectral on k-forms for all k, while at the same time not strongly isospectral (i.e., isometric) in the sense of Definition 4.2.2; for example, M ₁ = L(49;1, 6, 15) and M ₂ = L(49;1, 6, 20). This follows from the characterisation of k-isospectrality for all k in terms of lattice norms with an extra geometric condition in [60], which permits the construction of many examples, as well as infinite families of pairs of lens spaces with this property, e.g., [60, Table 1 and Theorem 7.1]. $\lozenge$

The next Proposition 4.2.4 provides a spectral criterion that is equivalent to weak conjugacy, and is an immediate corollary of Proposition 4.1.1. It is analogous to a number theoretical result of Nagata [73].

Proposition 4.2.4

Suppose M is a connected smooth closed Riemannian manifold, G a finite group of isometries of M and H ₁ and H ₂ are two subgroups of fixed-point free isometries in G with associated quotient manifolds M ₁ := H ₁∖M and M ₂ := H ₂∖M. Then H ₁ and H ₂ are weakly conjugate if and only if the multiplicity of the zero eigenvalue in $\sigma _{M_i}(\operatorname {Res}_{H_i}^G \operatorname {Ind}_{H_j}^G \mathbf {1})$ is independent of i, j = 1, 2.

Proof

The groups H ₁ and H ₂ are weakly conjugate precisely when there is an isomorphism of permutation representations $\operatorname {Ind}_{H_1}^G \mathbf {1} \cong \operatorname {Ind}_{H_2}^G \mathbf {1}$. By Proposition 4.1.1, this is equivalent to the claim. □

One may vary the condition of twisted isospectrality using the equivalent conditions in Proposition 4.1.1. For example, using Remark 4.1.2, we deduce the following (adding some redundant information).

Corollary 4.2.5

If a diagram as in ( 1.2 ) is given, and the following twisted spectra agree:

$$\displaystyle \begin{aligned} \left\{ \begin{array}{l} \sigma_{M_1}(\operatorname{Res}_{H_1}^G \operatorname{Ind}_{H_1}^G \mathbf{1}) = \sigma_{M_2}(\operatorname{Res}_{H_2}^G \operatorname{Ind}_{H_1}^G \mathbf{1}); \\ \sigma_{M_1}(\operatorname{Res}_{H_1}^G \operatorname{Ind}_{H_2}^G \mathbf{1}) = \sigma_{M_2}(\operatorname{Res}_{H_2}^G \operatorname{Ind}_{H_2}^G \mathbf{1}), \end{array} \right. \end{aligned}$$

then the manifolds M ₁ and M ₂ are strongly isospectral.

Remark 4.2.6

Mackey’s theorem describes how, for two subgroups K ₁ and K ₂ of a group G, a representation of the form $\operatorname {Res}_{K_2}^G \operatorname {Ind}_{K_1}^G \rho$ splits into irreducibles (see, e.g., [88, Prop. 22]). In the situation of Proposition 4.2.4, with K _i ∈{H ₁, H ₂}, we find that $\operatorname {Res}_{K_2}^G \operatorname {Ind}_{K_1}^G \mathbf {1}$ splits as the direct sum of the permutation representations corresponding to the action of K ₂ on the cosets of sK ₁ s ⁻¹ ∩ K ₂ for s ∈ K ₂∖G∕K ₁, and the occurring spectra are the (multiset-)union of the spectra corresponding to these representations; for example,

$$\displaystyle \begin{aligned} \sigma_{M_i} (\operatorname{Res}_{H_i}^G \operatorname{Ind}_{H_i}^G \mathbf{1}) = \bigcup_{s \in H_i \backslash G / H_i} \sigma_{M_i}( \operatorname{Ind}_{sH_is^{-1}\cap H_i}^{H_i} \mathbf{1}) \end{aligned}$$

contains the usual Laplace spectra $\sigma _{M_i}(\Delta _{M_i})$ (setting s to be the trivial double coset).

Project

It is possible that, given an integer p ₀ satisfying 0 < p ₀ ≤ n, n-dimensional manifolds are isospectral for the Laplacian on p-forms for all p < p ₀, but not for p = p ₀, and one could say that the larger p ₀∕n, the “more strongly isospectral” the manifolds are. Can one give a geometric meaning to p ₀∕n, given two manifolds (possibly of special type)? For example, this problem is solved by Lauret [59] for lens spaces by encoding some geometric properties in a generating series.