## Introduction After John Milnor's 1966 examples of isospectral non-isometric tori, many mathematicians constructed different manifolds that were isospectral but not isometric. However, most of these were one-off constructions, and it was not until 1985 that Toshikazu Sunada created a framework for producing such examples. In his 1985 paper *Riemannian coverings and isospectral manifolds,* he applied to spectral geometry what he called "a geometric analogue of a routine method in number theory" and transformed the problem from one of geometry to one of group theory. Refinements of Sunada's method are what eventually led to the Gordon-Webb-Wolpert examples of isospectral non-isometric drums. Let $N$ be a Riemannian manifold with a group $G \leq \operatorname{Isom}(N)$ with two discrete subgroups $\Gamma_1$ and $\Gamma_2$ acting freely^[for discussion about expanding this to non-free actions, see Alberto Arabia, *Introduction to Isospectrality*] . Then the following are equivalent conditions that imply that the manifolds $\Gamma_1\backslash N$ and $\Gamma_2\backslash N$ are isospectral: 1. The subgroups $\Gamma_1$ and $\Gamma_2$ are almost-conjugate in $G$, i.e., for each conjugacy class $[g]$ the number of elements in $\Gamma_1\cap [g]$ is the same as the number of elements in $\Gamma_2\cap [g]$ 2. The representations $\operatorname{Ind}_{\Gamma_1}^G(\mathbf{1}_{\Gamma_1})$ and $\operatorname{Ind}_{\Gamma_2}^G(\mathbf{1}_{\Gamma_2})$ induced by the trivial representation are isomorphic The method in number-theory that Sunada was referring to is the method of constructing non-isomorphic number fields $k_1$ and $k_2$ with the same Dedekind zeta functions, and the analogues of "Riemannian coverings" and "isometry groups" are "field extensions" and "Galois groups." ## Upstairs and Downstairs Let $(M, g)$ be a Riemannian manifold with a group $G$ that acts on it via isometries. We want to define a representation of $G$ on some vector space, but for now all we have is a manifold. To begin, we can think of the Hilbert space of square-integrable functions on $M$, $L^2(M, g)$, which we can decompose into eigenspaces of $\Delta_M$, [[The Laplace-Beltrami Operator]], i.e. $L^2(M) = \bigoplus_{\lambda \in\operatorname{Spec}(\Delta_g)} L^2(M)_\lambda$Now, we can define what's called the "regular representation" of $G$ on $L^2(M)$, where for some $f\in L^2(M), g \in G$,^[the inversion is to preserve the order of representation] $g\cdot f(x)=f(g^{-1}x)$ Here's a fact about the Laplacian: [[Laplacian commutes with isometries|it commutes with isometries]]. Since $G$ acts via isometries, this means that $\Delta (g\cdot f) = g\cdot (\Delta f)$, so if $f$ is a $\lambda-$eigenvector for the Laplacian, so is $g\cdot f$ because $\Delta (g\cdot f) = g\cdot (\Delta f) = g\cdot \lambda f = \lambda g\cdot f$. This means that each of the eigenspaces $L^2(M)_\lambda$ are subrepresentations of $G$. Now, let $p\colon N \to M$ be a (Riemannian) covering map, with $\Gamma$ the set of deck transformations. If $f$ is some smooth function on $M$ to $\mathbb R$, we can pull it back to $p^* f = f\circ p \in C^\infty(N)$ by simply precomposing by $p$. Can we take a function on $N$ and push it to a function on $M$? Recall that $\Gamma$ acts on $C^\infty(N)$ via the regular representation (as above). Then following fact holds: $C^\infty(M) \cong C^\infty(N)^\Gamma$ i.e. the set of smooth functions on $M$ is precisely the set of smooth functions on $N$ invariant under the action of $\Gamma$ (so, functions $f$ such that $f(x) = f(\gamma\cdot x)$ for all $\gamma \in \Gamma$[^1]. This gives us a good way of categorizing the "upstairs" eigenfunctions in terms of the "downstairs" eigenfunctions: $\lambda \in \operatorname{Spec}(\Delta_M) \iff \lambda \in \operatorname{Spec}(\Delta_N) \text{ and } \exists f \in L^2(N)_\lambda\text{ invariant under the action of }\Gamma$Using this, we can calculate the spectrum of of $\Gamma/M$ from the spectrum of $M$. All the eigenvalues of $\Delta_M$ are the same as those in $\Delta_N$, and the multiplicity of $\lambda \in \Delta_N$ is the dimension of the invariant subspace $L^2(N)_\lambda^\Gamma$. Since $L^2(N)_\lambda$ is a representation of $G$, it is also a representation of $\Gamma$ (in fact it is exactly the restriction $\operatorname{Res}_\Gamma^GL^2(N)_\lambda$ of the regular representation of $G$ on $L^2(N)_\lambda$). Moreover, the fixed subspace $L^2(N)_\lambda^\Gamma$ is a trivial subrepresentation of $\Gamma$, whose dimension determines the spectrum of $\Delta_M$. Two subgroups $\Gamma_1$ and $\Gamma_2$ produce isospectral quotient manifolds precisely when the dimension of these subrepresentations coincide for every $L^2(N)_\lambda$ ## Some Representation Theory To analyze this further, we will need some facts from representation theory. If $V$ is a representation of a **finite** group $G$, then the **character** $\chi_V\colon G \to \mathbb C$ is the function $\chi_V(g) = \operatorname{Trace}(g)$, the trace of $g$ as a linear operator on $V$. We then have the following facts about the character of a representation^[for proofs and exposition, see section 2.1 of Fulton and Harris, *Representation Theory: A First Course*] 1. The character is a class function, i.e., if $g$ and $h$ belong to the same conjugacy class in $G$ then $\chi(g) = \chi(h)$ 2. There are exactly as many irreducible representations of $G$ as there are conjugacy classes, and the characters of irreducible representations form an orthonormal basis on the vector space of class functions (which includes the characters of arbitrary representations) under the inner product $\langle \chi_V, \chi_W\rangle = \frac1{|G|}\sum_{g\in G}\overline{\chi_V(g)}\chi_W(g)$ 3. An arbitrary representation $V$ is characterized uniquely by its character $\chi_V$ because (by complete reducibility) it can be expressed uniquely as a direct sum of irreducibles $V = \oplus_i V_i$ and so $\chi_V = \oplus_i \chi_{V_i}$ 4. The number of times an irreducible representation $R$ appears as a direct summand in any arbitrary representation $V$ is $\langle \chi_V, \chi_R\rangle$ Recall that we were trying to find the dimension of $L^2(N)_\lambda^\Gamma$, which is a trivial representation of $\Gamma$. This representation is composed only of the one-dimensional irreducible trivial representation $\mathbb 1_\Gamma$ repeated $\dim L^2(N)_\lambda^\Gamma$ times. Thus, by point 3 above, the multiplicity of $\lambda$ as an eigenvalue of $\Delta_M$ is $\dim L^2(N)_\lambda^\Gamma = \langle \chi_{\mathbb 1_\Gamma}, L^2(N)_\lambda^\Gamma\rangle = \langle \chi_{\mathbb 1_\Gamma}, \chi_{\operatorname{Res}_\Gamma^GL^2(N)_\lambda}\rangle$ where this inner product is being taken over $\Gamma$. To analyze this further, we will need the concept of induced representations. If $H$ is a subgroup of $G$ and $W$ a representation of $H$, we can construct a representation $\operatorname{Ind}_H^GW$ of $G$ with the following properties^[The actual construction, and proofs for these properties, are discussed [[Understanding Induced Representations|here]]] 1. Any $H-$equivariant map from a representation $W$ of $H$ to a representation $V$ of $G$ extends uniquely to a $G-$equivariant map from $\operatorname{Ind}_H^GW$ to $V$. This is the universal property that characterizes induced representations and can be written more concisely as $\operatorname{Hom}_H(W, \operatorname{Res}_H^GV)\cong\operatorname{Hom}_G(\operatorname{Ind}_H^GW, V)$ which expresses $\operatorname{Res}$ and $\operatorname{Ind}$ as adjoint functors between the categories of representations of $H$ and representations of $G$. 2. Using the above property, we can prove **Frobenius Reciprocity**: $\langle \chi_W, \chi_{\operatorname{Res}_H^GV}\rangle_H=\langle \chi_{\operatorname{Ind}_H^GW}, \chi_{V}\rangle_G$ 3. If $W$ is the trivial irreducible representation of $H$, then for any conjugacy class $[g] \in G$ the we have^[Fulton-Harris, section 3.3, p34] $\chi_{\operatorname{Ind}_H^GW}(g) = \frac{[G:H]}{\#[g]}\cdot\#([g]\cap H)$ Using the language of induced representations, we can now write $\dim L^2(N)_\lambda^\Gamma = \langle \chi_{\mathbb 1_\Gamma}, L^2(N)_\lambda^\Gamma\rangle_\Gamma = \langle \chi_{\mathbb 1_\Gamma}, \chi_{\operatorname{Res}_\Gamma^GL^2(N)_\lambda}\rangle_\Gamma = \langle \chi_{\operatorname{Ind}_\Gamma^G\mathbb1_\Gamma}, \chi_{L^2(N)_\lambda}\rangle_G \tag{\dagger}$ ## A Proof of Sunada's Theorem We now have all the ingredients to prove Sunada's theorem. Two quotient manifolds $\Gamma_1/N$ and $\Gamma_2/N$ are isospectral if and only if $\dim L^2(N)_\lambda^{\Gamma_1} = \dim L^2(N)_\lambda^{\Gamma_1}$. By $(\dagger)$, this condition is equivalent to $\langle \chi_{\operatorname{Ind}_{\Gamma_1}^G\mathbb1_{\Gamma_1}}, \chi_{L^2(N)_\lambda}\rangle_G = \langle \chi_{\operatorname{Ind}_{\Gamma_2}^G\mathbb1_{\Gamma_2}}, \chi_{L^2(N)_\lambda}\rangle_G$ which holds if $\Gamma_1$ and $\Gamma_2$ induce the same representations. To see the equivalence between that and being almost-conjugate, note that $\chi_{\operatorname{Ind}_{\Gamma_i}^G\mathbb 1_{\Gamma_i}}(g) = \frac{[G:\Gamma_i]}{\#[g]}\cdot\#([g]\cap \Gamma_i)$ which is the same for $\Gamma_1$ and $\Gamma_2$ because they have the same number of elements (so same index) and intersect each conjugacy class exactly the same number of times. There are other ways to prove Sunada's theorem. In ["The Sunada Method"](http://www.ams.org/conm/231/), Robert Brooks gives four approaches, including the one we sketched above. The second proof relies on [[The heat kernel and the heat trace|the heat kernel]] by lifting the heat kernel downstairs to one upstairs. A third proof relies on the [[The Laplace spectrum and the length spectrum|connection between the laplace spectrum and the length spectrum]], by directly moving closed geodesics between manifolds. This proof is only valid in situations where the length spectrum fully determines the Laplace spectrum. The fourth proof uses Berard's idea of [[Transplantation of Eigenfunctions|transplantations]] but is essentially equivalent to the one using representation theory. ## Drawbacks The main drawback of the Sunada method is that the manifolds constructed using it are are quotients of the same universal cover, and hence *locally* isometric. Another drawback is that they are not simply connected, because their fundamental groups are the $\Gamma$ that we quotient by. There is a way to extend the Sunada method, developed by Craig Sutton in [this paper](https://arxiv.org/abs/math/0301376), in which the groups $G, \Gamma_1, \Gamma_2$ are not finite but instead connected Lie groups. This leads to isospectral simply-connected non-locally-isometric manifolds. A separate way around this is to use [[The Gordon-Schueth Torus method]] to produce examples of manifolds with different local geometry. Lastly, even though Sunada's method cannot account for these examples, Hubert Pesce showed (explained [here](https://www.sciencedirect.com/science/article/pii/S1874574100800096))that the set of metrics on $M$ where isospectrality implies the existence of subgroups $\Gamma$ fulfilling Sunada's conditions is dense in the set of all metrics in $M$. [^1]: For $f$ to be invariant under the regular representation, it means that $\gamma \cdot f = f$ or $f(\gamma^{-1}x) = f(x)$ for all $\gamma \in \Gamma$. But since every element is the inverse of some other element we can readily replace $\gamma^{-1}$ by $\gamma$ in this equation