DIVISION ALGEBRAS AND NON-COMMENSURABLE ISOSPECTRAL MANIFOLDS

ALEXANDER LUBOTZKY, BETH SAMUELS, AND UZI VISHNE

Abstract. A. Reid [?] showed that if Γ_{1} and Γ_{2} are arithmetic lattices in G = PGL_{2}(ℝ) or in PGL_{2}(ℂ) which give rise to isospectral manifolds, then Γ_{1} and Γ_{2} are commensurable (after conjugation). We show that for d ≥ 3 and S = PGL_{d}(ℝ)/PO_{d}(ℝ), or S = PGL_{d}(ℂ)/PU_{d}(ℂ), the situation is quite different: there are arbitrarily large finite families of isospectral non-commensurable compact manifolds covered by S.

The constructions are based on the arithmetic groups obtained from division algebras with the same ramification points but different invariants.

1. Introduction

Let X be a compact Riemannian manifold and Δ = Δ_{X} its Laplacian. The spectrum of X, spec(X), is the multiset of eigenvalues of Δ acting on L^{2}(X). This is a discrete subset of ℝ. If Y is another such object, we say that X and Y are isospectral (or, sometimes, cospectral) if spec(X) = spec(Y). The problem of finding isospectral non-isomorphic objects has a long history, starting with Marc Kac’s seminal paper [?], “Can one hear the shape of a drum?” (see [?] and the references therein).

The most powerful and general method of obtaining such pairs is due to Sunada [?]: he showed that if X_{0} covers X with a finite Galois group H, i.e. X = X_{0}/H, and if H_{1} and H_{2} are subgroups of H such that for every conjugacy class C ⊂ H, |C ∩ H_{1}| = |C ∩ H_{2}|, then X_{1} = X_{0}/H_{1} and X_{2} = X_{0}/H_{2} are isospectral. Of course, one still has to determine whether X_{1} and X_{2} are isomorphic (for example, X_{1} and X_{2} will be isomorphic if H_{1} is conjugate to H_{2}, but not only in such a case).

Date: April 13, 2005.

2 ALEXANDER LUBOTZKY, BETH SAMUELS, AND UZI VISHNE

Sunada’s method has been implemented in many situations (e.g. [?], [?]), and, in particular, has led to the solution of the original problem of M. Kac (see [?]).

A common feature to all the isospectral pairs, X_{1} and X_{2}, constructed by Sunada’s method, is that they are commensurable, that is, X_{1} and X_{2} have a common finite cover X_{0}. So, while X_{1} and X_{2} are not isomorphic, they are ‘virtually isomorphic’.

Sunada implemented his method to obtain non-isomorphic isospectral Riemann surfaces. He gave examples of torsion free cocompact lattices Γ⊲Γ_{0} in G = PGL_{2}(ℝ) with Γ_{0}/Γ ≅ H and H_{1} = Γ_{1}/Γ, and H_{2} = Γ_{2}/Γ, as above. His method implies that the G-representa-tions L^{2}(Γ_{1}\G) and L^{2}(Γ_{2}\G) are isomorphic, which demonstrates that Γ_{1}\ℍ^{2} is isospectral to Γ_{2}\ℍ^{2}. Here ℍ^{2} = PGL_{2}(ℝ)/K is the associated symmetric space, and K is the maximal compact subgroup PO_{2}(ℝ) of G.

Vigneras was the first to present examples of isospectral Riemann surfaces in [?], where she used the theory of quaternion algebras. Her examples are also commensurable to each other. In fact, it is still an open problem whether every two isospectral Riemann surfaces are commensurable, or equivalently whether every two cocompact lattices Γ_{1} and Γ_{2} in G = PGL_{2}(ℝ), such that Γ_{1}\G/K and Γ_{2}\G/K are isospectral, are commensurable (after conjugation, i.e. for some g ∈ G, [Γ_{2} :g^{−1}Γ_{1}g ∩ Γ_{2}] < ∞). But, Alan Reid [?] showed that if Γ_{1} and Γ_{2} are such arithmetic lattices, then indeed Γ_{1} and Γ_{2} are commensurable. He also showed a similar result for G = PGL_{2}(ℂ).

Our main result shows that the situation is quite different for G = PGL_{d}(ℝ) or PGL_{d}(ℂ) if d ≥ 3.

Theorem 1. Let F = ℝ or ℂ, G = PGL_{d}(F) where d ≥ 3, K ≤ G a maximal compact subgroup, and S = G/K the associated symmetric space. Then for every m ∈ ℕ, there exists a family of m torsion free cocompact arithmetic lattices {Γ_{i}}_{i=1,...,m} in G, such that Γ_{i}\S are isospectral and not commensurable.

Taking K = PO_{d}(ℝ) in the real case and K = PU_{d}(ℂ) in the complex case, the quotients Γ_{i}\G/K are isospectral compact manifolds

DIVISION ALGEBRAS AND ISOSPECTRALITY 3

covered by the symmetric space S = PGL_{d}(F)/K which is equal to SL_{d}(ℝ)/SO_{d}(ℝ) or SL_{d}(ℂ)/SU_{d}(ℂ), respectively. The covering map S→Γ_{i}\S is a local isomorphism, since the Γ_{i} are torsion free.

This result should be compared with Spatzier’s construction [?] of isospectral locally symmetric spaces of high rank. However, his isospectral examples are always commensurable to each other.

Before outlining the proof, let us make some remarks putting our results in a more general framework (see [?] for more details). There are isospectral compact manifolds whose universal covers are not isometric, but whether this is possible for locally symmetric manifolds seems to be still open.

Given a symmetric space X, the isospectral families of manifolds covered by X are finite, although they can be as large as one wishes. Isospectral manifolds have the same volume. It will be interesting to bound the size of such isospectral families as a function of the volume. See [?] and the references therein for a count of all manifolds covered by X as a function of the volume.

A remark and a question in a different direction: many of the isospectral manifolds are obtained by Sunada’s method (see [?, Section 4.5]) for a discussion). Our examples are not, but they share with Sunada pairs the property that they come from lattices which are ‘representation theoretical isospectral’, i.e. L^{2}(Γ_{1}\G) and L^{2}(Γ_{2}\G) are isomorphic as G-modules (see below). In some cases it is known that geometric isospectrality and representation isospectrality are equivalent (see [?], [?] and [?]). But for general simple Lie groups this seems to be an open problem.

We can now outline the proof of Theorem ??. The following proposition is well known, see for example [?] and [?].

Proposition 2. Let G be a semisimple group, K ≤ G a maximal compact subgroup, and Γ_{1}, Γ_{2} ≤ G discrete cocompact subgroups. If

(1)  L^{2}(Γ_{1}\G) ≅ L^{2}(Γ_{2}\G)

4 ALEXANDER LUBOTZKY, BETH SAMUELS, AND UZI VISHNE

as (right) G-representations, then Γ_{1}\G/K and Γ_{2}\G/K are isospectral quotients of G/K (indeed, they are even strongly isospectral, i.e. with respect to the higher dimensional Laplacians).

To prove Theorem ?? for G = PGL_{d}(F) (F = ℝ or ℂ), we choose the discrete subgroups Γ_{i} to be arithmetic lattices of inner forms, as follows. Let k be a global field and {k_{ν}} the completions with respect to its valuations. Recall that Br(k_{ν}) ≅ ℚ/ℤ for a non-archimedean valuation, while Br(ℝ) ≅ \frac{1}{2}ℤ/ℤ and Br(ℂ) = 0, where Br(·) denotes the Brauer group of a field. For a division algebra D, [D] ∈ Br(k), the value associated to D⊗_{k}k_{ν} is called the ‘local invariant’ at ν. By Albert-Brauer-Hasse-Noether theorem, [D] ↦ ([D⊗_{k}k_{ν}])_{ν} defines an injective map for the Brauer groups

(2)  Br(k) ⟶ ⊕ Br(k_{ν}),

and the image of this map is composed of vectors whose sum is zero. Over local (and thus also global) fields, the degree of a division algebra is equal to its exponent (which is the order of its equivalence class in the Brauer group).

Fix d > 1 and let T = {θ_{1}, …, θ_{t}} be a finite non-empty set of non-archimedean valuations of k. Let a_{1}, …, a_{t}, b_{1}, …, b_{t} ∈ ℕ be such that for every j = 1, …, t, 0 ≤ a_{j}, b_{j} < d, (a_{j}, d) = (b_{j}, d) = 1, and ∑ a_{j} ≡ ∑ b_{j} ≡ 0 (mod d). Let D_{1} (resp. D_{2}) be the unique division algebra of degree d (i.e. dimension d^{2}) over k which ramifies exactly at T, and whose invariant at k_{θ_{j}} is a_{j}/d (resp. b_{j}/d). Thus D_{1} and D_{2} split at every ν ∉ T. Let G′_{i} be the algebraic group D_{i}^{×}/Z^{×} (i = 1, 2) where Z is the center, and G = PGL_{d}. Since G′_{i}(k_{ν}) splits for every ν ∉ T, the groups G′_{1}(𝔸_{T^{c}}) and G′_{2}(𝔸_{T^{c}}) are equal where 𝔸_{T^{c}} = ∏′_{ν∉T} k_{ν}. We identify these groups with G_{0} = G(𝔸_{T^{c}}). Here, 𝔸 is the ring of adèles and ∏′ denotes the restricted product, namely the vectors (x_{ν}) such that ν(x_{ν}) ≥ 0 for almost every ν.

A basic result is

Theorem 3. Let G_{0} = PGL_{d}(𝔸_{T^{c}}) be the group defined above, and let Δ_{i} = G′_{i}(k) (i = 1, 2), cocompact lattices in G_{0}. The spaces L^{2}(Δ_{1}\G_{0}) and L^{2}(Δ_{2}\G_{0}) are isomorphic as G_{0}-representations.

DIVISION ALGEBRAS AND ISOSPECTRALITY 5

The result follows by comparing trace formulas. This was done explicitly for the case where d is prime in [?, Theorem 1.12] and in [?, Theorem 54]. It seems that the general case is also known to experts, but we were unable to locate a reference. For the sake of completeness we show in Section ?? how the result can be deduced (for arbitrary d) from the global Jacquet-Langlands correspondence, as described in [?]. (The proof there is given for the characteristic zero case, and we refer the reader to [?, Remark 1.6] for some details about the positive characteristic case).

We should stress that in the current paper, we are using only the characteristic zero case. We prefer to give here the more general formulation, preparing for a subsequent paper [?], in which we use analogous ideas over a local field of positive characteristic to construct isospectral simplicial complexes and isospectral Cayley graphs of some finite simple groups.

One deduces the isospectrality in Theorem ?? by applying strong approximation to Theorem ??. This is done in Section ??. To prove the non-commensurability, we first show (Theorem ??) that two division k-algebras D_{1} and D_{2} give rise to commensurable arithmetic lattices in PGL_{d}(F) iff D_{1}(k) and D_{2}(k) are isomorphic or anti-isomorphic (as rings, rather than as k-algebras). We then analyze when two such division algebras are isomorphic (or anti-isomorphic) as rings, and show that we can produce as many examples as we need of non-isomorphic division rings. This is done in Section ??.

Finally, we mention that our work is very much in the spirit of the paper of Vigneras [?], who used a similar idea to find isospectral Riemann surfaces. However, as mentioned above, her examples, which are lattices in PGL_{2}(ℝ), are commensurable, as they should be by Reid’s theorem. The case d = 2 differs from those of d ≥ 3 in that a global division algebra of degree 2 is determined by its ramification points, while for d ≥ 3 there are non-isomorphic division algebras with the same ramification points.

We thank G. Margulis for helpful discussions, A.S. Rapinchuk for his help with Theorem ??, and D. Goldstein and R. Guralnick for simplifying the proof of Lemma ??. The authors also acknowledge support of grants by the NSF and the U.S.-Israel Binational Science Foundation.

2. Isospectrality

Although for the proof of Theorem ?? we may assume characteristic zero, we are stating and proving the results of this section for positive characteristic as well. Therefore, we obtain Theorem ?? together with:

Theorem 4. Let F be a local field of positive characteristic, G = PGL_{d}(F) where d ≥ 3, K a maximal compact subgroup and B_{d}(F) = G/K the associated Bruhat-Tits building. Then for every m ∈ ℕ there exists a family of m torsion free cocompact arithmetic lattices {Γ_{i}}_{i=1,...,m} in G, such that the finite complexes Γ_{i}\B_{d}(F) are isospectral and not commensurable.

Notice that Theorems ?? and ?? cover the archimedean local fields, and the non-archimedean local fields of positive characteristic. Our methods do not apply to non-archimedean local fields F of zero characteristic, where there are no cocompact lattices of inner type in PGL_{d}(F) (d ≥ 3). There are cocompact lattices of outer type in these groups; however we leave open the following problem:

Question 5. Let F be a non-archimedean local field of zero characteristic, G = PGL_{d}(F) where d ≥ 3, K a maximal compact subgroup and B_{d}(F) = G/K the associated Bruhat-Tits building. Do there exist torsion free cocompact arithmetic lattices {Γ_{i}}_{i=1,2} in G, such that the finite complexes Γ_{i}\B_{d}(F) are isospectral and not commensurable?

Let k be a global field, and assume it has at most one archimedean valuation. Let ν_{0} denote the archimedean valuation if char k = 0, and a valuation of degree 1 if k is a function field (for k = 𝔽_{q}(t), ν_{0} is either the minus-degree valuation or is determined by a linear prime). Let F = k_{ν_{0}}. (For F = ℝ or F = ℂ, we take k = ℚ or k = ℚ[i], respectively).

Let V denote the set of all valuations of k other than ν_{0}. For a valuation ν ∈ V, O_{ν} is the ring of integers in the completion k_{ν}, and P_{ν} is the valuation ideal.

DIVISION ALGEBRAS AND ISOSPECTRALITY 7

Let T ⊆ V be a finite subset, and define D_{i} and G′_{i} as in the introduction (i = 1, 2). Let G denote the algebraic group PGL_{d}. Notice that ν_{0} ∉ T, so in particular G′_{i}(F) = G(F) (i = 1, 2). Let 𝔸_{T} denote the direct product ∏_{θ∈T} k_{θ}, so that 𝔸 = 𝔸_{T} × 𝔸_{T^{c}} is the ring of adèles over k.

Let R_{0} = {x ∈ k: ν(x) ≥ 0 for every ν ≠ ν_{0}}. (In the cases k = ℚ or k = ℚ[i], with ν_{0} as the archimedean valuation, we obtain R_{0} = ℤ or R_{0} = ℤ[i], respectively.) Then, for every θ ∈ T, we can choose a uniformizer ϖ_{θ} such that ν(ϖ_{θ}) = 0 for every ν ∉ {ν_{0}, θ} (so in particular, ϖ_{θ} ∈ R_{0}). In this situation,

(3)  R_{0,T} = {x ∈ k: ν(x) ≥ 0 for every ν ∉ T ∪ {ν_{0}}}

is equal to R_{0}[{ϖ_{θ}^{−1}}_{θ∈T}]. Fix i ∈ {1, 2}. Since division algebras over global fields are cyclic, there is a Galois field extensions k_{i}/k of dimension d contained in D_{i}(k). If ϕ ∈ Gal(k_{i}/k) is a generator, there is an element b_{i} ∈ k^{×} such that D_{i} = k_{i}[z | zaz^{−1} = ϕ(a), z^{d} = b_{i}]. Taking an integral basis, one finds an order O_{i} ⊆ k_{i} which is a cyclic extension (of rings) of R_{0,T}. The constant b_{i} can be chosen to be in the multiplicative group generated by the uniformizers ϖ_{θ}. Thus, b_{i} is invertible in R_{0,T}. So O_{i}[z] (with the relations as above) is an Azumaya algebra of rank d over the center R_{0,T}, and D_{i} = k⊗_{R_{0,T}} O_{i}[z]. The group of invertible elements in O_{i}[z] (again with the above relations), modulo its center, is G′_{i}(R_{0,T}).

If an algebraic group G′ is defined over a ring R and 0 ≠ I⊲R is an ideal, we have the principal congruence subgroups

(4)  G′(R, I) = Ker(G′(R)→G′(R/I)).

The congruence subgroups G′_{i}(R_{0,T} , I) are discrete in G(F), and cocompact if T ≠ ∅ [?, Theorem 5.5].

Let 0 ≠ I⊲R_{0,T}. Define a function r: V−(T ∪ {ν_{0}})→ℕ ∪ {0} by setting

(5)  r_{ν} = min {ν(a) : a ∈ I},

and let

(6)  U^{(r)} = ∏_{ν∈V−(T ∪{ν_{0}})} G(O_{ν}, P_{ν}^{r_{ν}}),

an open compact subgroup of G(𝔸_{T^{c}−{ν_{0}}}). For almost every ν, r_{ν} = 0, and then P_{ν}^{0} = O_{ν} and G(O_{ν}, P_{ν}^{r_{ν}}) = G(O_{ν}) by definition. Let

(7)  U_{i} = U^{(r)}G′_{i}(𝔸_{T}),

an open compact subgroup of G′_{i}(𝔸).

Lemma 6. The quotient G′_{i}(𝔸)/(G′_{i}(k)G(F)U_{i}) is a d-torsion finite abelian group, which is independent of i.

Proof. Recall the reduced norm map D_{i}(k)^{×}→k^{×}, which coincides with the determinant over splitting fields of k. Let G_{i} denote the simply connected cover of G′_{i}, i.e. G_{i} is the subgroup of elements of reduced norm 1 in D_{i}^{×}. For every field k_{1} ⊇ k, the image of the covering map Ψ : G_{i}(k_{1})→G′_{i}(k_{1}) is co-abelian in G′_{i}(k_{1}). The strong approximation theorem [?] (see also [?, Subsection 3.2]) applies to G_{i}, and so G_{i}(k)G_{i}(F)U = G_{i}(𝔸) where

U = ∏_{ν∈V−(T ∪{ν_{0}})} G_{i}(O_{ν}, P_{ν}^{r_{ν}}) × G_{i}(𝔸_{T}),

and G_{i}(O_{ν_{0}})U is open compact in G_{i}(𝔸).

The reduced norms G′_{i}(k_{ν})→k_{ν}^{×}/k_{ν}^{×}^{d} is onto whenever G′_{i} splits at k_{ν} (as G′_{i}(k_{ν}) = PGL_{d}(k_{ν})), and also whenever k_{ν} is non-archimedean (since over local fields the norm map in a division algebra is onto, [?, Chapter 17]). In the second case, the restriction to G′_{i}(O_{ν})→O_{ν}^{×}/O_{ν}^{×}^{d} is also onto. Thus, the norm map G′_{i}(𝔸)→𝔸^{×}/𝔸^{×}^{d} is onto, and induces an isomorphism Φ : G′_{i}(𝔸)/Ψ(G_{i}(𝔸))→𝔸^{×}/𝔸^{×}^{d}.

Since Ψ maps G_{i}(k), G_{i}(F) and U into G′_{i}(k), G(F) and U_{i}, respectively, we have Ψ(G_{i}(𝔸)) = Ψ(G_{i}(k)G_{i}(F)U) ⊆ G′_{i}(k)G(F)U_{i}. We need to compute the image of G′_{i}(k)G(F)U_{i} under Φ. For a valuation ring O with maximal ideal P and r ≥ 0, let O^{(r)} denote the subgroup {a ∈ O^{×}: a ≡ 1 (mod P^{r})}. Now, modulo d powers, the determinant maps G(F) to F^{×} and G(O_{ν}, P_{ν}^{r_{ν}}) to O_{ν}^{(r_{ν})}. It follows that Φ takes G′_{i}(k) to k^{×}𝔸^{×}^{d}, G(F) to F^{×}𝔸^{×}^{d}, and U_{i} to the product (∏_{ν∈V−T} O_{ν}^{(r_{ν})} × ∏_{θ∈T} k_{θ}^{×})𝔸^{×}^{d}. We can now compute that

G′_{i}(𝔸)/(G′_{i}(k)G(F)U_{i}) ≅ 𝔸^{×}/(k^{×}F^{×}𝔸^{×}^{d} ∏_{ν∈V−(T ∪{ν_{0}})} O_{ν}^{(r_{ν})} ∏_{θ∈T} k_{θ}^{×})
≅ 𝔸_{T^{c}}^{×}/(k^{×}F^{×}𝔸_{T^{c}}^{×}^{d} ∏_{ν∈V−(T ∪{ν_{0}})} O_{ν}^{(r_{ν})}).

Recall that k_{ν}^{×} ≅ ℤ × O_{ν}^{×}, where the ℤ summand is the value group. Dividing the numerator and denominator in the last expression by ∏_{ν∈V−(T ∪{ν_{0}})} O_{ν}^{×}^{d}, the quotient is

∏′k_{ν}^{×}/(k^{×}∏′O_{ν}^{(r_{ν})}k_{ν}^{×}^{d}) ≅

≅ ∏′(ℤ × O_{ν}^{×})/(k^{×}∏′(dℤ × O_{ν}^{(r_{ν})}O_{ν}^{×}))
≅ ∏′(ℤ × O_{ν}^{×}/O_{ν}^{×}^{d})/(k^{×}∏′(dℤ × O_{ν}^{(r_{ν})}O_{ν}^{×}/O_{ν}^{×}^{d}))
≅ (∏′ℤ∏O_{ν}^{×}/O_{ν}^{×}^{d})/(k^{×}∏′dℤ∏O_{ν}^{(r_{ν})}O_{ν}^{×}/O_{ν}^{×}^{d})
≅ (∏′(ℤ/d × O_{ν}^{×}/O_{ν}^{(r_{ν})}O_{ν}^{×}^{d}))/k^{×},

where all products are over ν ∈ V − (T ∪ {ν_{0}}) and ∏′ denotes the restricted product (such that ν(a_{ν}) ≥ 0 almost always). The quotients O_{ν}^{×}/O_{ν}^{(r_{ν})} are always finite, and r_{ν} = 0 for almost all ν, hence O_{ν}^{×}/O_{ν}^{(r_{ν})}O_{ν}^{×}^{d} = 0 for all but finitely many valuation. In the last expression, k^{×} embeds in the ν component by a ↦ (ν(a), aϖ_{ν}^{−ν(a)}O_{ν}^{(r_{ν})}O_{ν}^{×}^{d}), and in particular the components with O_{ν}^{×}/O_{ν}^{(r_{ν})}O_{ν}^{×}^{d} = 0 vanish in the quotient. Therefore we have a finite product of finite groups, and the result follows. □

Proposition 7. Let 0 ≠ I⊲R_{0,T}. There is an isomorphism

(8)  L^{2}(G′_{1}(R_{0,T} , I)\G(F)) ≅ L^{2}(G′_{2}(R_{0,T} , I)\G(F)),

as representations of G(F) = PGL_{d}(F).

10 ALEXANDER LUBOTZKY, BETH SAMUELS, AND UZI VISHNE

Proof. We prove this statement by transferring it to its adèlic analogue. Let U_{i} be the group defined in Equation (??). Then G(O_{ν_{0}})U_{i} is open compact in G′_{i}(𝔸), and

(9)  G′_{i}(R_{0,T} , I) = G′_{i}(k) ∩ G(F)U_{i},

the intersection taken in G′_{i}(𝔸) and then projected to the G(F) component. Therefore G′_{i}(R_{0,T} , I)\G(F) is related to G′_{i}(k)\G′_{i}(𝔸)/U_{i}.

Since G_{0} = G(𝔸_{T^{c}}) by definition, G′_{i}(𝔸)/U_{i} = G′_{i}(𝔸_{T^{c}})/U^{(r)} = G_{0}/U^{(r)}, and

G′_{i}(k)\G′_{i}(𝔸)/U_{i} ≅ G′_{i}(k)\G_{0}/U^{(r)}.

Consequently,

L^{2}(G′_{i}(k)\G_{0})^{U^{(r)}} ≅ L^{2}(G′_{i}(k)\G_{0}/U^{(r)}) ≅ L^{2}(G′_{i}(k)\G′_{i}(𝔸)/U_{i}).

Now let H_{i} = G′_{i}(k)G(F)U_{i}. From Equation (??) it follows that

G′_{i}(R_{0,T} , I)\G(F) ≅ G′_{i}(k)\H_{i}/U_{i}.

By the lemma, C = G′_{i}(𝔸)/H_{i} is finite (d-torsion) abelian, which is independent of i. Since G(F) commutes with U_{i} and is contained in H_{i}, we have the decomposition

L^{2}(G′_{i}(k)\G′_{i}(𝔸)/U_{i}) ≅ ⊕_{|C|} L^{2}(G′_{i}(k)\H_{i}/U_{i})

as G(F)-representations. Thus

L^{2}(G′_{i}(k)\G_{0})^{U^{(r)}} ≅ ⊕_{|C|}L^{2}(G′_{i}(k)\H_{i}/U_{i}) ≅ ⊕_{|C|}L^{2}(G′_{i}(R_{0,T} , I)\G(F)).

By Theorem ??, L^{2}(G′_{1}(k)\G_{0}) ≅ L^{2}(G′_{2}(k)\G_{0}) as representation spaces, so the result follows. □

Now, let 0 ≠ I⊲R_{0,T}, and take Γ_{i} = G′_{i}(R_{0,T} , I). By Proposition ?? and Proposition ??, we see that Γ_{1}\G(F)/K and Γ_{2}\G(F)/K are isospectral, where K is a fixed maximal compact subgroup of G(F). If I is small enough, then Γ_{1} and Γ_{2} are torsion free, and so the projection G(F)/K→Γ_{i}\G(F)/K is a local isomorphism.

Remark 8. When T is fixed, the choice of U^{(r)} determines the family of quotients. When U^{(r)}′ ⊆ U^{(r)}, the quotients corresponding to U^{(r)} are covered by those corresponding to U^{(r)}′. In this sense, we find not only families of isospectral structures, but infinite “inverse limits” of such families.

3. Non-commensurability

As in the previous section, we state and prove the results of this section for arbitrary characteristic, proving Theorems ?? and ?? together.

To show that Γ_{i}\G(F)/K are not commensurable, we need the following theorem. Recall that the maximal compact subgroup K of PGL_{d}(F) is taken to be K = PO_{d}(ℝ) if k = ℝ, K = PU_{d}(ℂ) if k = ℂ, and K = PGL_{d}(O) (where O is the ring of integers of F = k_{ν_{0}}) if char k > 0. Let D_{1} and D_{2} be two central division algebras over k, as in the previous section, and G′_{i} the corresponding algebraic groups. Let T_{1} and T_{2} be the ramification points of D_{1} and D_{2} (this time not necessarily equal), and let R_{0,T_{i}} be the subrings of k defined in Equation (??) (for T_{i} rather than T). Note that if σ: D_{1}(k)→D_{2}(k) is an isomorphism or anti-isomorphism of rings, then σ induces an automorphism of the center k which acts on the set of (non-archimedean) valuations of k. This maps T_{1} to T_{2}. For Theorem ?? we only need the implication (1) ⟹ (3) of the next theorem, however we give the full picture, which seems to be of independent interest.

Theorem 9. Let S = G(F)/K be the building or symmetric space corresponding to G(F). The following are equivalent:

1. There exists finite index torsion-free subgroups Ω_{1} of G′_{1}(R_{0,T_{1}}) and Ω_{2} of G′_{2}(R_{0,T_{2}}) such that the manifolds or complexes Ω_{1}\S and Ω_{2}\S are isomorphic.

2. For every finite index torsion-free subgroup Ω_{1} ≤ G′_{1}(R_{0,T_{1}}), there exists a finite index torsion-free subgroup Ω_{2} ≤ G′_{2}(R_{0,T_{2}}) such that Ω_{1}\S and Ω_{2}\S are isomorphic.

3. There is a ring isomorphism or anti-isomorphism σ: D_{1}(k)→D_{2}(k), such that the restriction of σ to the center k fixes ν_{0} and maps T_{1} to T_{2}.

Proof. The second assertion trivially implies the first. We prove (1) ⟹ (3). Since S is the universal cover of Ω_{i}\S and Ω_{i} is the fundamental group of Ω_{i}\S, the given isomorphism lifts to an automorphism

12 ALEXANDER LUBOTZKY, BETH SAMUELS, AND UZI VISHNE

ψ: S→S such that ψΩ_{1}ψ^{−1} = Ω_{2} in Isom(S), the group of isometries of S.

By Cartan’s theorem [?, Chapter IV] in characteristic zero and Tits’ theorem [?, Corollaries 5.9 and 5.10] in positive characteristic, Isom(S) = the continuous automorphisms of PGL_{d}(F).

It is well known that Aut(PGL_{d}(F)) is a semidirect product of the subgroup generated by (continuous) field automorphisms and inner automorphisms, by the subgroup of order 2 generated by the ‘diagram automorphism’ a ↦ (a^{t})^{−1}. Replacing D_{2} with D_{2}^{op} if necessary, we may ignore the last option. Therefore ψ extends to an automorphism of the matrix algebra M_{d}(F).

Recall that Ω_{i} ⊆ G′_{i}(R_{0,T_{i}}) ⊆ G′_{i}(k) ⊆ PGL_{d}(F). The quotient PGL_{d}(F)/PSL_{d}(F) is abelian torsion, and Ω_{i} are finitely generated (as the fundamental groups of compact objects). Therefore, we can replace the Ω_{i} by finite index subgroups, which will be contained in PSL_{d}(F), still satisfying the conjugation property above. Moreover, as Aut(PGL_{d}(F)) = Aut(PSL_{d}(F)) = Aut(SL_{d}(F)) = Aut(GL_{d}(F)), we can lift Ω_{i} to subgroups Ω_{i}^{(1)} of SL_{d}(F), with ψ ∈ Aut(GL_{d}(F)) mapping Ω_{1}^{(1)} to Ω_{2}^{(1)}.

Let H′_{i} denote the group of elements of norm one in D_{i}^{×}; in particular, H′_{i}(F) = SL_{d}(F). Then Ω_{i}^{(1)} ⊆ H′_{i}(R_{0,T_{i}}) ⊆ H′_{i}(k) ⊆ SL_{d}(F). Let N_{i} be the commensurator of Ω_{i}^{(1)} in GL_{d}(F). We claim that N_{i} is D_{i}^{×}(k) times the scalar matrices. Indeed, an element a ∈ N_{i} conjugates a finite index subgroup A of Ω_{i}^{(1)} into Ω_{i}^{(1)}. Since Ω_{i}^{(1)} is Zariski dense, so is A. Therefore, the F̄-subalgebra of M_{d}(F̄) generated by A is equal to M_{d}(F̄) (where F̄ is the algebraic closure). This algebra is the scalar extension of the k-subalgebra generated by A, which is therefore a d^{2}- dimensional subalgebra of D_{i}(k) (as A ⊆ D_{i}^{×}(k)) and thus equal to D_{i}(k). Therefore, conjugation by a is an automorphism of D_{i}(k). But by Skolem-Noether there is an element of D_{i}^{×}(k) inducing the same automorphism, so up to scalar matrices, a ∈ D_{i}^{×}(k).

Since ψ maps Ω_{1}^{(1)} to Ω_{2}^{(1)}, it maps N_{1} to N_{2}. It also maps the commutator subgroup of N_{1}, which is the commutator subgroup of D_{1}^{×}(k), to that of D_{2}^{×}(k). It follows that ψ maps the division subring generated by commutators of D_{1}(k) to the division subring generated

DIVISION ALGEBRAS AND ISOSPECTRALITY 13

by commutators of D_{2}(k), but by Lemma ?? below, these are D_{1}(k) and D_{2}(k), respectively. Hence we proved that ψ extends to an isomorphism (of rings) from D_{1}(k) to D_{2}(k).

The restriction to k of this isomorphism preserves ν_{0} since it is induced by a continuous automorphism of F = k_{ν_{0}}. Also, it takes the ramification points T_{1} of D_{1} to the ramification points T_{2} of D_{2} (note that the opposite algebra has the same set of ramification points).

We remark that in characteristic zero we could bypass the argument with the commensurator, by observing that up to complex conjugation, ψ must be an algebraic map and hence maps G′_{1}(k) to G′_{2}(k).

To show that (3) ⟹ (2), we first assume D_{2}(k) is isomorphic to D_{1}(k). By tensoring with F, we obtain an automorphism of D_{i}(F) = M_{d}(F), which is continuous by assumption, and so induces an automorphism of S. Moreover it maps the set T_{1} to T_{2}, inducing an isomorphism from the subring R_{0,T_{1}} to R_{0,T_{2}}. This extends to an isomorphism from G′_{1}(R_{0,T_{1}}) to G′_{2}(R_{0,T_{2}}). Now, if Ω_{1} ≤ G′_{1}(R_{0,T_{1}}) ≤ GL_{d}(F) is any finite index torsion-free subgroup, then its image under the automorphism, Ω_{2}, is a finite index torsion-free subgroup of G′_{2}(R_{0,T_{2}}), with Ω_{1}\S isomorphic to Ω_{2}\S.

If D_{2}(k) is isomorphic to D_{1}(k)^{op}, then by tensoring with F we obtain an anti-automorphism of M_{d}(F), and the argument follows throughout.

□

Lemma 10. Let D be a non-commutative division ring. Then the smallest division subring of D containing all multiplicative commutators is D itself.

Proof. First note that a division subring of D containing all commutators is invariant under conjugations (as xyx^{−1} = [x, y]y). By the Cartan-Brauer-Hua theorem [?, p. 186] such a subring is either central or equal to D.

Assume all commutators in D are central, and let x ∈ D be a noncentral element. Then the field generated by x over Cent(D) is invariant and therefore equal to D, so D is commutative, contrary to assumption. □

14 ALEXANDER LUBOTZKY, BETH SAMUELS, AND UZI VISHNE

Theorem ?? gives the condition for two manifolds to be commensurable when they are defined from two division k-algebras. To complete the proof of Theorem ??, we need to construct arbitrarily many division algebras with the same set of ramification points, which are not isomorphic as rings.

The classification of division algebras over k by their invariants is crucial for that matter, so we add some (well known) details on this topic (see [?, Chapsters 17–18]). Let D be a division algebra, with center k, a global field. Let T denote the set of valuations θ of k such that D⊗k_{θ} is not a matrix algebra, where k_{θ} is the completion with respect to θ. Then T is a finite set. Recall that by Grunwald-Wang theorem [?, Chapter 10], there exists a cyclic field extension k_{1} of dimension d over k, which is unramified with respect to every valuation θ ∈ T. From Albert-Brauer-Hasse-Noether theorem it then follows that D contains an isomorphic copy of k_{1}.

Let ν be a non-archimedean valuation of k. Let ϕ denote the Frobenius automorphism of k_{1}/k (i.e. the generator of Gal(k_{1}/k) which induces the Frobenius automorphism on the residue field of k_{1} over that of k). By Skolem-Noether theorem, there is an element z ∈ D such that conjugation by z induces ϕ on k_{1}. Then z^{d} commutes with k_{1}[z] = D, so z^{d} is central, i.e. z^{d} ∈ k. D = k_{1}[z | zaz^{−1} = ϕ(a), z^{d} = b ∈ k]. If k_{2} ⊆ D is an isomorphic copy of k_{1}, then again by Skolem-Noether there is an element w ∈ D such that wk_{1}w^{−1} = k_{2}. Now wzw^{−1} induces ϕ (or more precisely its isomorphic image) on k_{2}, and the d-power of this element is wz^{d}w^{−1} = z^{d}. Of course, z is not unique; if z′ is another element inducing ϕ on k_{1}, then z′z^{−1} commutes with k_{1}. So z′ = az for some a ∈ k_{1}, since k_{1} is its own centralizer. Then, (z′)^{d} = N_{k_{1}/k}(a)z^{d}. However, ν(N_{k_{1}/k}(a)) is divisible by d since k_{1}/k is unramified, so ν(z^{d}) (mod d) is well defined. This number, divided by d, is called the local invariant of D at ν (the invariant is viewed as an element of ℚ/ℤ). The completion k_{ν} is a splitting field of D iff the invariant is zero.

We remark that D is determined (up to isomorphism as a k-algebra) by its local invariants. For every finitely supported function f : V ∪ {ν_{0}}→{0, 1/d, …, (d − 1)/d} ⊂ ℚ/ℤ such that ∑ f(ν) ≡ 0 (mod 1) and f(ν_{0}) = 0, there is a central simple k-algebra of degree d which is matrices over F = k_{ν_{0}} and with f(ν) as the local invariant at ν.

Proposition 11. Let D and D′ be k-central division algebras which are ramified at places (ν_{1}, …, ν_{t}) with invariants (a_{1}, …, a_{t}) and (ν′_{1}, …, ν′_{t′}) with invariants (a′_{1}, …, a′_{t′}), respectively. Then D and D′ are isomorphic as rings if and only if t = t′ and there exists a k-automorphism σ such that (after reordering) ν′_{i} = ν_{i} ∘ σ, and a_{i} = a′_{i}.

Moreover, if σ̄ is a fixed isomorphism D→D′ extending σ, then every isomorphism from D to D′ is equal to σ̄ composed with an inner automorphism.

Proof. Assume D′ and D are isomorphic as rings. The ring isomorphism induces an isomorphism of the centers, which is an automorphism σ of k. Let k_{1} denote an cyclic maximal subfield of D as before. If conjugation by z induces ϕ on k_{1}, then conjugation by its image z′ induces an automorphism ϕ′ on the image k′_{1} of k_{1}, and of course (z′)^{d} = σ(z^{d}). Therefore, the invariant of D′ at ν ∘ σ^{−1} is (ν ∘ σ^{−1})((z′)^{d}) = ν(z^{d}), namely the invariant of D at the valuation ν. In the other direction, the given σ extends to an isomorphism of the rings translating the ramification data of D to that of D′.

Finally, if two isomorphisms ψ, ψ′: D→D′ agree on the center of D, then ψ^{−1} ∘ ψ′ is an automorphism of D as a central division algebra. By Skolem-Noether, it is an inner automorphism. □

Corollary 12. Let D_{1} and D_{2} be k-division algebras of degree d with the same ramification points, such that for every automorphism of k, the vector a = (a_{1}, …, a_{t}) of ramified invariants of D_{1} is not permuted to the vector of ramified invariants b = (b_{1}, …, b_{t}) of D_{2}, nor to −b.

Let Ω_{i} ⊆ G′_{i}(R_{0,T}) be finite index subgroups (where G′_{i} are defined as before). Then Ω_{1}\S and Ω_{2}\S are not commensurable (where S = G(F)/K). Equivalently, Ω_{1} and Ω_{2} are not commensurable (after conjugation) as subgroups of Aut(PGL_{d}(F)) (and in particular not as subgroups of PGL_{d}(F)).

16 ALEXANDER LUBOTZKY, BETH SAMUELS, AND UZI VISHNE

Proof. Obviously the two claims are equivalent, for if Ω_{0} = Ω_{1} ∩gΩ_{2}g^{−1} is a finite index subgroup, then Ω_{0}\S ≅ gΩ_{0}g^{−1}\S is a joint cover of finite index, and vice versa.

If such a joint cover exists, then D_{2} must be isomorphic or antiisomorphic to D_{1} as rings by Theorem ??, and this is prohibited by the previous proposition. □

Before we continue, we first prove a Lemma.

Lemma 13. A global field k has infinitely many orbits of valuations under the action of its automorphism group.

Proof. Indeed, in zero characteristic there are infinitely many valuations (covering the p-adic valuations for p the rational primes). Since the Galois group over ℚ is finite, there are finitely many valuations in each orbit.

In the case of a function field, k is finite over 𝔽_{q}(t) where 𝔽_{q} is the maximal finite subfield. The degree of a valuation is the dimension of its residue field over 𝔽_{q}. If p ∈ 𝔽_{q}[t] is a prime, then the degree of the p-adic valuation on 𝔽_{q}(t) is equal to the degree of p (as a polynomial), and serves as a lower bound for the degree of an extension of this valuation to k. Therefore, the degrees of valuations of k are unbounded. However, the automorphism group, when acting on the valuations, does not change the degree. □

We can now finish the proof of Theorems ?? and ??. Given the number m, let t be an even integer such that 2^{t}/t ≥ 2m. Choose a set T ⊆ V composed of t non-archimedean valuations of k, which belong to different orbits of the automorphism group of k. For every e = (e_{1}, …, e_{t}) with e_{i} prime to d which sum to zero modulo d, let D_{e} be the division algebra over k with Brauer-Hasse-Noether invariants \frac{e_{1}}{d} , …, \frac{e_{t}}{d} in the valuations of T, which is unramified outside of T. Since t is even we can choose half of the invariants to be 1 and half to be −1 (1 ≢ −1 as d > 2). There are at least \binom{t}{t/2} ≥ 2^{t}/t ≥ 2m options. In each case, the sum of e_{i} is zero modulo d. Furthermore we set e_{1} = 1, so there are at least m vectors e which are not only different, but also not opposite to each other.

DIVISION ALGEBRAS AND ISOSPECTRALITY 17

Since our valuations in T are from different orbits and we have chosen distinct invariants, there is no automorphism σ of k which permutes the ramified valuations and their invariants. This guarantees that the D_{e} are not isomorphic to each other (nor their opposites) as rings. Having chosen the division algebras, take U = U^{(r)} an open compact subgroup of G(𝔸_{T^{c}−{ν_{0}}}) as in Equation (??). Let G′_{e} denote the k-algebraic group D_{e}^{×}/Z^{×} (where Z is the center), and take Γ_{e} = G′_{e}(R_{0,T} , I) = G′_{e}(k) ∩ G(F)UG′_{e}(𝔸_{T}) (see Equation (??)). We saw above that X_{e} = Γ_{e}\G(F)/K are all isospectral, where K is a fixed maximal compact subgroup of G(F). As a congruence subgroup, Γ_{e} is of finite index in G′_{e}(R_{0,T}), so the X_{e} are non-commensurable by Theorem ??.

4. Proof of Theorem ??

Let D be a division algebra of degree d over the global field k, and let G′ and G denote the algebraic groups D^{×}/Z^{×} and PGL_{d}, respectively. Let T denote the set of ramification points of D, and assume as before that D⊗k_{θ} is a division algebra for each θ ∈ T (so the invariants of D at T are prime to d). Now let θ ∈ T. The local Jacquet-Langlands correspondence is a bijective correspondence, which maps every irreducible, unitary representation ρ′ of G′(k_{θ}) to an irreducible, unitary squareintegrable representation ρ = JL_{θ}(ρ′) of G(k_{θ}) (see [?, p. 29] or [?] for details). Every such representation of G(k_{θ}) is either super-cuspidal (i.e. the matrix coefficients are compactly supported modulo the center), or is induced from a super-cuspidal representation of a subgroup of smaller rank: ρ = Sp_{s}(ψ) where ψ is a super-cuspidal representation of PGL_{d/s}(k_{θ}), and Sp_{s} is the construction of a ‘special representation’. See [?] or [?, Subsection 2.5]. We note that Sp_{1}(ψ) = ψ.

Let 𝔸 denote the ring of adèles over k. The automorphic representations of G(𝔸) are the irreducible sub-representations of L^{2}(G(k)\G(𝔸)) (as right G(𝔸)-modules), and likewise for G′(𝔸). Such a representation decomposes as a tensor product of local components π_{ν} (π = ⊗π_{ν}), which are representations of G(k_{ν}) (respectively G′(k_{ν})) for the various valuations ν of k. Moreover, π is determined by all but finitely many local components (this is ‘strong multiplicity one’ for PGL_{d}.) .

18 ALEXANDER LUBOTZKY, BETH SAMUELS, AND UZI VISHNE

The global Jacquet-Langlands correspondence maps an irreducible automorphic representation π′ = ⊗π′_{ν} of G′(𝔸) to an irreducible automorphic representation π = JL(π′) = ⊗JL(π′)_{ν} of G(𝔸) which occurs in the discrete spectrum (see [?, p. 195]). If ν ∉ T, then JL(π′)_{ν} ≅ π′_{ν}. For the valuations θ ∈ T, the local correspondence takes (π′)_{θ} to JL_{θ}((π′)_{θ}) = Sp_{s}(ψ) for a suitable s | d and ψ super-cuspidal. Now in the global map, the local component JL(π′)_{θ} is isomorphic to either Sp_{s}(ψ) or to a certain representation which we denote by C_{s}(ψ), which is never square-integrable. In both cases, ψ and s are uniquely determined by π′_{θ}.

By part 3 of [?, Theorem VI.1.1], the image of the global JL consists of the automorphic representations π in the discrete spectrum, such that for each θ ∈ T, the local component π_{θ} is either of the form Sp_{s}(ψ) or C_{s}(ψ), a condition which depends on T but is independent of the invariants of D. Therefore the image of JL only depends on the set T, and not on the invariants.

Let G_{0} = G′(𝔸_{T^{c}}) and Δ = G′(k) ⊆ G_{0} (with the diagonal embedding). Recall that G_{0} = G(𝔸_{T^{c}}), since G′(k_{ν}) = G(k_{ν}) for every ν ∉ T. First, we show that the only finite-dimensional sub-re-presentation of L^{2}(Δ\G_{0}) is the trivial one. It is well known that all finite-dimensional sub-representations are one-dimensional, and so they come from the quotient PGL_{d}(𝔸_{T^{c}})/PSL_{d}(𝔸_{T^{c}}). However, since T is not empty, PGL_{d}(k) · PSL_{d}(𝔸_{T^{c}}) = PGL_{d}(𝔸_{T^{c}}), and so all such representations are trivial as in Lemma ??.

Let us now define a map JL^{0}, which takes an irreducible infinitedimensional sub-representation π′′ of L^{2}(Δ\G_{0}) to an irreducible infinitedimensional sub-representation π of L^{2}(G(k)\G(𝔸)), which occurs in the discrete spectrum. JL^{0}(π′′) is defined as follows: Since G_{0} = G′(𝔸)/G′(𝔸_{T}) and L^{2}(G′(k)\G′(𝔸)/G′(𝔸_{T})) ≅ L^{2}(G′(k)\G′(𝔸))^{G′(𝔸_{T})} ⊆ L^{2}(G′(k)\G′(𝔸)), π′′ lifts to a sub-representation π′ of L^{2}(G′(k)\G′(𝔸)) on which G′(𝔸_{T}) acts trivially. Namely, (π′)_{θ} is trivial for every θ ∈ T. By the global Jacquet-Langlands correspondence, there is an automorphic representation π = JL(π′) of G(𝔸) such that (π′)_{ν} ≅ π_{ν} for every ν ∉ T. We set JL^{0}(π′′) = π. Now, for θ ∈ T, JL_{θ}((π′)_{θ}) = JL_{θ}(1) = Sp_{d}(δ) is the Steinberg representation, for δ = |·|^{(1−d)/2}. Moreover

DIVISION ALGEBRAS AND ISOSPECTRALITY 19

C_{d}(δ) is the trivial representation, and hence π_{θ} is either trivial or Steinberg. But if π is infinite-dimensional, π_{θ} cannot be trivial. Therefore, it is the Steinberg representation.

The strong multiplicity one theorem for D^{×} (see part 4 of [?, Theorem VI.1.1]) implies that π′ has multiplicity one in L^{2}(G′(k)\G′(𝔸)) and hence π′′ has multiplicity one in L^{2}(Δ\G_{0}). Moreover, it implies that JL^{0}(π′′) determines π′′. By part 3 of [?, Thmeorem VI.1.1] and the discussion above, the image of JL^{0} is composed of the discrete automorphic representations π of PGL_{d}(𝔸) such that for every θ ∈ T, the local component π_{θ} is a Steinberg representation. In particular, the image depends on T but not on the specific invariants of D.

Theorem ?? follows immediately. Since L^{2}(Δ_{1}\G_{0}) and L^{2}(Δ_{2}\G_{0}) have the same isomorphic images via the JL^{0} map in L^{2}(G(k)\G(𝔸)), they have isomorphic irreducible sub-representations, each appearing with multiplicity one.

We should remark that [?] deals with representations of GL_{d} or D^{×} with a fixed central character (and not PGL_{d} and D^{×}/Z^{×}), but this does not change the argument as we can take the character to be trivial.

5. Generalization of Theorem ?? and Proposition ??

In this section we generalize Theorem ?? and its main corollary. The generalizations are not needed here, but we use them in a subsequent paper on isospectral Cayley graphs [?].

As before, let k be a global field, T a fixed finite set of valuations, and D_{1}, D_{2} central division algebras of degree d over k, which are unramified outside of T, and remain division algebras over the completions k_{θ} for θ ∈ T. Let G′_{i} denote the multiplicative group of D_{i} modulo center. Recall that G′_{i}(𝔸_{T^{c}}) can be identified with G_{0} = G(𝔸_{T^{c}}). Let J_{i} = G′_{i}(𝔸_{T}), so that G′_{i}(𝔸) = G′_{i}(𝔸_{T^{c}})J_{i} = G_{0}J_{i}. Again Δ_{i} = G′_{i}(k) is embedded diagonally in G′_{i}(𝔸).

Theorem 14. Let J_{i}^{0} be finite index subgroups of J_{i} (i = 1, 2), with [J_{1} :J_{1}^{0}] = [J_{2} :J_{2}^{0}].

Then L^{2}(Δ_{1}\G′_{1}(𝔸))^{J_{1}^{0}} and L^{2}(Δ_{2}\G′_{2}(𝔸))^{J_{2}^{0}} (the spaces of J_{i}^{0}-invariant functions on Δ_{i}\G′_{i}(𝔸)) are isomorphic as G_{0}-representations.

Since L^{2}(Δ_{i}\G′_{i}(𝔸))^{J_{i}} = L^{2}(Δ_{i}\G′_{i}(𝔸_{T^{c}})), we obtain Theorem ?? as a special case by taking J_{i}^{0} = J_{i}. In fact, Theorem ?? follows at once from Theorem ??: the natural projection

Δ_{i}\G′_{i}(𝔸)→Δ_{i}\G′_{i}(𝔸_{T^{c}})

has fiber J_{i} over every point. Therefore

L^{2}(Δ_{i}\G′_{i}(𝔸))^{J_{i}^{0}} = (L^{2}(Δ_{i}\G′_{i}(𝔸))^{J_{i}})^{[J_{i}:J_{i}^{0}]}
= L^{2}(Δ_{i}\G′_{i}(𝔸_{T^{c}}))^{[J_{i}:J_{i}^{0}]}

where the first equality (for G_{0}-modules) follows from the fact that J_{i}^{0} and G_{0} commute elementwise. As the indices are equal by assumption, this completes the proof of Theorem ??.

Fix a valuation ν_{0} ∉ T, and let F = k_{ν_{0}} be the completion with respect to this valuation. Recall the rings R_{0,T} defined in Equation (??), and

(10)  R_{0} = {x ∈ k: ν(x) ≥ 0 for every ν ≠ ν_{0}}.

Of course, R_{0} ⊂ R_{0,T}, and every ideal I⊲R_{0,T} induces the ideal I ∩ R_{0} of R_{0}. On the other hand not every ideal of R_{0} has this form (for example when I⊲R_{0} contains an element a ∈ I with θ(a) = 0 for some θ ∈ T).

Fix embeddings G′_{i}(k) ↪ GL_{t}(k) (same t for both i = 1, 2) such that over the completions k_{θ}, θ ∈ T, G′_{i}(k_{θ}) = G′_{i}(O_{θ}). This is possible since G′(k_{θ}) is compact for every θ in the finite set T. Set G′_{i}(R_{0}) = G′_{i}(k) ∩ GL_{t}(R_{0}), and define the congruence subgroups G′_{i}(R_{0}, I) as in Equation (??). We obtain the following extension of Proposition ??.

Proposition 15. Let 0 ≠ I⊲R_{0,T} be an ideal, and let J_{i}^{0} be finite index subgroups of J_{i} = G′_{i}(A_{T}) with [J_{1} :J_{1}^{0}] = [J_{2} :J_{2}^{0}]. Let Λ_{i} = G′(k) ∩ G(F)U^{(r)}J_{i}^{0} be the corresponding discrete subgroup, where U^{(r)} is defined as in (??). Then

L^{2}(Λ_{1}\G(F)) ≅ L^{2}(Λ_{2}\G(F))

as representations of G(F) = PGL_{d}(F).

Proof. Let Δ_{i} = G′_{i}(k) as before. Theorem ?? implies that

L^{2}(Δ_{1}\G′_{1}(𝔸))^{U^{(r)}J_{1}^{0}} ≅ L^{2}(Δ_{2}\G′_{2}(𝔸))^{U^{(r)}J_{2}^{0}},

so

L^{2}(Δ_{1}\G′_{1}(𝔸)/U^{(r)}J_{1}^{0}) ≅ L^{2}(Δ_{2}\G′_{2}(𝔸)/U^{(r)}J_{2}^{0}).

On the other hand, combining Lemma ?? and the fact that [J_{i}:J_{i}^{0}] is independent of i, we see that

G′_{i}(𝔸)/Δ_{i}G(F)U^{(r)}J_{i}^{0}

is independent of i. As in the proof of Proposition ??, this implies that

L^{2}(Δ_{i}\Δ_{i}G(F)U^{(r)}J_{i}^{0}/U^{(r)}J_{i}^{0})

is independent of i. But this space is isomorphic to L^{2}(Λ_{i}\G(F)) by the choice of Λ_{i}. □

This proposition generalizes Proposition ??, providing isospectrality for principal congruence subgroups with respect to ideals of R_{0}, rather than only ideals of R_{0,T}. (However it also covers non-principal congruence subgroups).

Corollary 16. Let 0 ≠ I⊲R_{0}. There is an isomorphism

L^{2}(G′_{1}(R_{0}, I)\G(F)) ≅ L^{2}(G′_{2}(R_{0}, I)\G(F)),

as representations of G(F) = PGL_{d}(F).

Proof. Define a function r: V−{ν_{0}}→ℕ ∪ {0} as in Equation (??), and let U^{(r)} be the group defined in Equation (??). Take J_{i}^{0} = ∏_{θ∈T} G′_{i}(O_{θ}, P_{θ}^{r_{θ}}), so that G′_{i}(R_{0}, I) = G′_{i}(k) ∩ G(F)U^{(r)}J_{i}^{0}. From the detailed description of the quotients G′(k_{θ})/G′_{i}(O_{θ}, P_{θ}^{r_{θ}}) in [?, Proposition 1.5], one sees that the index of J_{i}^{0} in G′_{i}(𝔸_{T}) does not depend on i, so the proposition applies. □

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Inst. of Math., Hebrew University, Givat Ram, Jerusalem 91904, Israel

E-mail address: alexlub@math.huji.ac.il

Dept. of Math., Yale University, 10 Hillhouse Av., New Haven CT 06520, USA

E-mail address: beth.samuels@yale.edu

Dept. of Math., Bar-Ilan University, Ramat Gan 52900, Israel

E-mail address: vishne@math.biu.ac.il