RAMANUJAN COMPLEXES OF TYPE Ã_{d}

ALEXANDER LUBOTZKY, BETH SAMUELS, AND UZI VISHNE

With love and admiration to Hillel Fürstenberg, a teacher and friend

Abstract. We define and construct Ramanujan complexes. These are simplicial complexes which are higher dimensional analogues of Ramanujan graphs (constructed in [?]). They are obtained as quotients of the buildings of type Ã_{d−1} associated with PGL_{d}(F) where F is a local field of positive characteristic.

1. Introduction

A finite k-regular graph X is called a Ramanujan graph if for every eigenvalue λ of the adjacency matrix A = A_{X} of X either λ = ±k or |λ| ≤ 2\sqrt{k − 1}. This term was defined in [?] where some explicit constructions of such graphs were presented, see also [?], [?], [?].

These graphs were obtained as quotients of the k-regular tree T = T_{k}, for k = q + 1, q a prime power, divided by the action of congruence subgroups of G = PGL_{2}(F). Here F is a non-archimedean local field with residue field of order q, and the tree T is the Bruhat-Tits building associated with G, which is a building of type Ã_{1}. The (proved) Ramanujan conjecture for GL_{2} was an essential ingredient in the proof that the graphs are indeed Ramanujan, see [?].

The number 2\sqrt{k − 1} plays a special role in the definition of Ramanujan graphs because of the Alon-Boppana theorem (see [?]), which proves that this is the best possible bound for an infinite family of kregular graphs. A conceptual explanation was given by Greenberg [?], [?, Thm. 4.2.7] (see also [?]): for a connected graph X, let ρ(X) denote the norm of the adjacency operator A on L^{2}(X) (so ρ(T_{k}) = 2\sqrt{k − 1} ); then, Greenberg showed that no upper bound on the non-trivial eigenvalues of finite quotients of X is better then ρ(X).

These considerations motivated Cartwright, Solé and Żuk [?] to suggest a generalization of the notion of Ramanujan graphs from finite quotients of T_{k} — which is an Ã_{1} building — to the simplicial complexes obtained as finite quotients of B = B_{d}(F), the Bruhat-Tits building of type Ã_{d−1} associated with the group G = PGL_{d}(F). The vertices B^{0} of the building are labelled by a ‘color’ function ϱ: B^{0}→ℤ/dℤ, and we may look at the d − 1 colored adjacency operators A_{k}, k = 1, …, d − 1 on L^{2}(B^{0}), called the Hecke operators. They are defined by

Date: Submitted Sep. 1, 2003; revised Feb. 19, 2004.

Partially supported by grants from NSF and BSF (U.S.-Israel).

(1.1)  (A_{k}f)(x) = ∑ f(y)

where the summation is over the neighbors y of x such that ϱ(y) − ϱ(x) = k in ℤ/dℤ.

These operators A_{k} are bounded, normal, and commute with each other. Thus, they have a simultaneous spectral decomposition, and the spectrum 𝔖_{d} of (A_{1}, …, A_{d−1}) on L^{2}(B^{0}) was computed explicitly as a subset of ℂ^{d−1} (see Subsection ?? below). This set is, of course, contained in the Cartesian product 𝔖_{d,1} × ⋯ × 𝔖_{d,d−1}, where 𝔖_{d,k} is the spectrum of A_{k}, but it is not equal to the product.

Definition 1.1 (following [?]). A finite quotient X of B is called a Ramanujan complex if the eigenvalues of every non-trivial simultanenous eigenvector v ∈ L^{2}(X), A_{k}v = λ_{k}v, satisfy (λ_{1}, …, λ_{d−1}) ∈ 𝔖_{d}.

(See Subsection ?? for more detailed explanations, and in particular for a description of the trivial eigenvalues. See also [?] for a definition and construction of Ramanujan complexes which are not simplicial).

Cartwright et al. [?] also suggested a way of obtaining such Ramanujan complexes: assume F is a local field of positive characteristic; let Γ be a cocompact arithmetic lattice of G = PGL_{d}(F) of inner type, and Γ(I) a congruence subgroup of Γ. They conjectured that the quotients Γ(I)\B are Ramanujan complexes. The work of Lafforgue in the last few years, which proved the Ramanujan conjecture for GL_{d} in characteristic p (an extension of Drienfeld’s work for GL_{2} in characteristic p and of Deligne’s for GL_{2} in characteristic zero) provided hope that these combinatorial applications could be deduced.

The current work, which started from the challenge to prove the conjecture in [?], shows that for general d, the story is more subtle. It turns out that most of these quotients are indeed Ramanujan, but not all. To describe our results, let us first introduce some notation.

Let k be a global field of characteristic p > 0, and D, a division algebra of degree d over k. Denote by G′ the k-algebraic group D^{×}/k^{×}, and fix a suitable embedding of G′ as a linear group (see Section ??). Let T be the finite set of valuations of k for which D does not split. We assume that for every ν ∈ T, D_{ν} = D⊗_{k}k_{ν} is a division algebra. Let ν_{0} be a valuation of k which is not in T, and F = k_{ν_{0}}. Let

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

Then Γ = G′(R_{0}) is a discrete subgroup of G′(F), and the latter is isomorphic to G(F) = PGL_{d}(F), as F splits D. By general results, Γ is in fact a cocompact lattice in G(F) — an “arithmetic lattice of inner type”. Let B = B_{d}(F) be the Bruhat-Tits building of G(F), then B^{0} ≅ G(F)/K, where K = G(O) is a maximal compact subgroup (O is the ring of integers in F). G(F) acts on B by left translation.

For 0 ≠ I⊲R_{0} an ideal (note that R_{0} is a principal ideal domain), we have the principal congruence subgroup

(1.3)  Γ(I) = G′(R_{0}, I) = Ker(G′(R_{0})→G′(R_{0}/I)).

In the following two theorems we assume the global Jacquet-Lang-lands correspondence for function fields, see Remark ?? below regarding this assumption.

Theorem 1.2. If d is prime, then for every 0 ≠ I⊲R_{0}, Γ(I)\B is a Ramanujan complex.

So for d prime, the Cartwright-Solé-Żuk conjecture is indeed true. On the other hand, for general d:

Theorem 1.3. (a) For every d, if I is prime to some valuation θ ∈ T, i.e. θ(a) = 0 for some θ ∈ T and some a ∈ I, then Γ(I)\B is a Ramanujan complex.

(b) If d is not a prime, then there exist (infinitely many) ideals I such that Γ(I)\B are not Ramanujan.

Theorem ?? may suggest that in positive characteristic, if d is a prime, then every finite quotient of B is Ramanujan. We do not know if this is indeed the case (which would be truly remarkable), but at least in the zero characteristic analog there are counter examples. Indeed, in Section ?? we show that if E is a non-archimedean local field of characteristic zero, then congruence quotients of B = B_{d}(E) can be non-Ramanujan for every d ≥ 4. This happens if Γ is taken to be an arithmetic group of outer type.

Theorem 1.4. Let E be a non-archimedean local field of characteristic zero, and assume d ≥ 4. Then B_{d}(E) has infinitely many non-Ramanujan quotients.

For a discussion of the case d = 3, see [?]. The proof of Theorem ?? and ??(a) follows in principle the line of proof for Ramanujan graphs, as in [?]. The problem is transferred to representation theory.

Proposition 1.5. Let Γ be a cocompact lattice in G(F) = PGL_{d}(F). Then Γ\B is a Ramanujan complex iff every irreducible spherical infinitedimensional sub-representation of L^{2}(Γ\G(F)) is tempered.

The strategy now is to start with an irreducible sub-representation ρ of L^{2}(Γ(I)\G′(F)). By Strong Approximation, one can show (see Subsection ?? below) that ρ is a local factor of an adèlic automorphic representation π′ = ⊗π′_{ν} in L^{2}(G′(k)\G′(𝔸)) such that π′_{ν_{0}} = ρ, where 𝔸 is the ring of adèles of k.

We can view π′ as an automorphic representation of D^{×}(𝔸). Then, the Jacquet-Langlands correspondence associates with π′ an automorphic representation π = ⊗π_{ν} in L^{2}(GL_{d}(k)\GL_{d}(𝔸)), such that π_{ν_{0}} = π′_{ν_{0}}. We then appeal to the work of Lafforgue, who proved that if π is cuspidal, then π_{ν} is tempered for every unramified ν, and in particular π_{ν_{0}} = ρ is tempered.

Now, the cuspidality issue is exactly what distinguishes between the cases where d is a prime and where d is a composite number. If d is prime, then all infinite-dimensional irreducible sub-representations of L^{2}(GL_{d}(k)\GL_{d}(𝔸)) are cuspidal (and the others are one-dimensional, and are responsible for the “trivial” eigenvalues, see Subsection ??). Thus Theorem ?? can be proved.

On the other hand, when d is not a prime, there is a “residual spectrum” and π may be there, in which case π_{ν_{0}} is not tempered. Theorem ?? (both parts (a) and (b)) is proved by a careful analysis of the image of the Jacquet-Langlands map, as described in [?].

The proof of Theorem ?? is different. We apply the method of Burger-Li-Sarnak [?],[?] who showed how the existence of large “extended arithmetic subgroups” in Γ(I) can affect the spectrum. For arithmetic lattices associated to Hermitian forms (unlike the case of inner type), such “large” subgroups do exist, but anisotropic Hermitian forms (with enough variables) exist only if char(F) = 0.

Remark 1.6. The global Jacquet-Langlands correspondence is proved in the literature for fields of characteristic zero (see Theorem ?? below and [?, Thm. VI.1.1]). It is likely that the theorem is valid in exactly the same formulation in positive characteristic, and it seems (to some experts we consulted) that a proof can be worked out using existing knowledge. So far, this task has not been carried out. We hope that our work will give some additional motivation to complete this gap in the literature.

W. Li [?] managed to prove the existence of Ramanujan complexes of type Ã_{d} in positive characteristic, avoiding the use of the Jacquet-Lang-lands correspondence, and in fact also not using Lafforgue’s theorem, appealing to [?] instead. In order to apply this method, one needs the division algebra to be ramified in at least four places, and therefore it does not cover the case of algebras ramified in two places. This case is crucial for our next work, [?], in which we give an explicit construction of Ramanujan complexes. On the other hand, we have to assume that in the ramification points the algebra is completely ramified, while Li requires this assumption in only two prime places.

We recently learned that Alireza Sarveniazi [?] has also given a construction of Ramanujan complexes.

The paper is organized as follows: in Section ?? we describe briefly the building B, the operators A_{k}, the local representation theory, and, in particular, we prove Proposition ?? above. In Section ?? we show how strong approximation enables one to pass from the local theory to the global one. In Section ?? we survey the global theory: Lafforgue’s theorem, the residual spectrum, and the Jacquet-Langlands correspondence. After the preparations we prove Theorems ?? and ?? in Section ??, and Theorem ??, in Section ??.

Much of the material of Sections ??–?? is well known to experts, but since we expect (and hope) the paper will have readers outside representation theory and automorphic forms, we tried to present the material in a suitable way for non-experts.

Acknowledgements. We are indebted to M. Harris, R. Howe, D. Kazhdan, E. Lapid, S. Miller, E. Sayag, I. Piatetski-Shapiro, T. Steger, M.F. Vigneras, and especially to J. Rogawski and J. Cogdell for many helpful discussions while working on the project. This work was done while the first named author visited and the third named author held a post-doc position at Yale, whose hospitality and support are gratefully acknowledged. We also thank the NSF and the BSF US-Israel for their support.

2. Affine buildings and representations of the local group

In this section, F is a non-Archimedean local field of arbitrary characteristic, O its ring of integers, and ϖ ∈ O a uniformizer. Let ν_{0} : F →ℤ denote the valuation of F.

2.1. Affine buildings of type Ã_{d−1}. Recall that a complex is a structure composed of i-cells, where the 0-cells are called vertices, and every i-cell is a set of i + 1 vertices. A complex is simplicial if every subset of a cell is also a cell.

We will now describe the affine building B = B_{d}(F) associated to PGL_{d}(F), which is an (infinite) simplicial complex. Consider the Olattices of full rank in F^{d}. We define an equivalence relation on lattices by setting L ∼ sL for every s ∈ F^{×}. Since F^{×}/O^{×} is the infinite cyclic group generated by ϖ, an equivalent definition is that L ∼ ϖ^{i}L for every i ∈ ℤ.

By B^{i} we denote the set of i-cells of B. The vertices B^{0} are the equivalence classes of lattices. There is an edge (1-cell)(x, x′), from x = [L] to x′ = [L′] ∈ B^{0}, if ϖL ⊆ L′ ⊆ L. Notice that this is a symmetric relation, since then ϖL′ ⊆ ϖL ⊆ L′. The quotient L/ϖL is a vector space of dimension d over the field O/ϖO ≅ 𝔽_{q}.

As i-cells of B we take the complete subgraphs of size i + 1 of B^{0}. It immediately follows that B has (d−1)-cells (corresponding to maximal flags in quotients L/ϖL). It also follows that there are no higher dimensional cells. We call L_{0} = O^{d} ⊆ F^{d} the standard lattice.

For every lattice L, there is some i such that ϖ^{i}L ⊆ L_{0} (it then follows that every two lattices of maximal rank are commensurable). We define a color function ϱ: B^{0}→ℤ/d, by

(2.1)  ϱ(L) = log_{q} [L_{0} :ϖ^{i}L]

for i large enough; the color is well defined since log_{q} [ϖ^{i}L:ϖ^{i+1}L] = d. In a similar way, the color of an ordered edge (x, y) ∈ B^{1} is defined to be ϱ(x) − ϱ(y) (mod d).

The group GL_{d}(F) acts on lattices by its action on bases; the scalar matrices carry a lattice to an equivalent lattice, so G = PGL_{d}(F) acts (transitively) on the vertices of B. Since the action of GL_{d}(F) preserves inclusion of lattices, G respects the structure of B, and in particular the color of edges. Note that GL_{d}(F) does not preserve the color of vertices, but SL_{d}(F) does.

The stabilizer of [L_{0}] is the maximal compact subgroup K = PGL_{d}(O). We can thus identify B^{0} with G/K, where G acts by multiplication from the left. A coset gK ∈ G/K corresponds to the lattice generated by the columns of g (so [L_{0}] corresponds to the identity matrix). The color of gK can then be computed from the determinant of g:

det(g) ≡ ϖ^{ϱ(gK)} (mod F^{×}^{d}),

where F^{×}^{d} is the subgroup of d-powers in F^{×}.

Let ω_{k} = diag(()ϖ, …, ϖ, 1, …, 1), where det(ω_{k}) = ϖ^{k}. The lattice corresponding to ω_{k}K is obviously a neighbor of color k of [L_{0}]. Let Ω_{k} be the set of neighbors of color k of [L_{0}]. Then K acts (as a subgroup of G) transitively on Ω_{k}, so that Kω_{k}K = ∪yK, where the union is over yK ∈ Ω_{k}. Multiplying from the left by an arbitrary g ∈ G, we see that the neighbors forming an edge of color k with gK are {gyK}_{yK∈Ω_{k}}.

It follows that the operators A_{k} (defined in Equation (??)) act on functions f : G/K→ℂ by

(A_{k}f)(gK) = ∑_{yK∈Ω_{k}} f(gyK) = ∑_{yK∈Ω_{k}} ∫_{yK} f(gx)dx = ∫_{Kω_{k}K} f(gx)dx,

the integrals are normalized so that ∫_{K} dx = 1. See [?] and [?] for details.

2.2. Spherical representations. In this section let K = GL_{d}(O), which is a maximal compact subgroup of G = GL_{d}(F).

As in [?], we study the spectrum of the operators A_{k} via representations of GL_{d}(F). An irreducible admissible representation of G is called H-spherical if the representation space has an H-fixed vector, where H ≤ G is a subgroup. The K-spherical representations are simply called spherical. (A representation is smooth if every v ∈ V is fixed under some open compact subgroup, and admissible if, moreover, the spaces fixed by each open compact subgroup are finite dimensional).

The Hecke operators A_{k} of the preceding subsection (defined in the same way, as functions of G/K for G = GL_{d}(F) rather than G = PGL_{d}(F)) generate the Hecke algebra H(G, K) of all bi-K-invariant compactly supported functions on G, with multiplication defined by

(A ∗ A′)(g) = ∫_{G} A(x)A′(x^{−1}g)dx.

The A_{k} commute with each other, and freely generate H(G, K) (cf. [?, Sec. V]).

Let ρ: G→ End(V) be an admissible representation; the Hecke algebra acts on the representation space (see [?, Eq. (9)]) by

(2.2)  A · v = ∫_{G} A(x)(ρ(x))(v) dx,

which is an integration over a compact set since A is compactly supported. It projects V to the K-fixed subspace V^{K} (which is finite dimensional as the representation is admissible). Moreover, if V is an irreducible G-module, then V^{K} is an irreducible H(G, K)-module. Since H(G, K) is commutative and finitely generated, V^{K} is one-dimensional in this case, and consequently, every v ∈ V^{K} is an eigenvector of all the A_{k}.

We describe how spherical representations are parameterized by dtuples of complex numbers, called the Satake parameters. For details, the reader is referred to [?].

Let B denote a Borel subgroup of G (e.g. the upper triangular matrices), U its unipotent radical, and T ≅ B/U ≅ (F^{×})^{d} a maximal torus of B. We then have B = UT and G = BK = UT K.

Recall that F^{×}/O^{×} = ⟨ϖ⟩. A character χ: F^{×}→ℂ^{×} is spherical if it is trivial on the maximal compact subgroup of F^{×}, namely O^{×}. Such a character is, thus, determined by z = χ(ϖ), which is an arbitrary complex number. The character is called unitary iff z ∈ S^{1} = {w ∈ ℂ: |w| = 1}.

Every character χ: T →ℂ^{×} can be written as χ(diag(()t_{1}, …, t_{d})) = χ_{1}(t_{1}) … χ_{d}(t_{d}), for characters χ_{i}: F^{×}→ℂ^{×}. χ is said to be unramified if the χ_{i} are spherical. Since T ≅ B/U, χ extends to a character of B. The symmetric group S_{d} acts on the characters by permuting the χ_{i}.

The unitary induction of representations from B to G is defined using the modular function

(2.3)  Δ(b) = |a_{1}|^{d−1} |a_{2}|^{d−3} … |a_{d}|^{1−d} ,  b ∈ B

where a_{1}, a_{2}, …, a_{d} are the entries on the diagonal of b (in that order), and |·|_{F} is the absolute value function of F, normalized so that |x| = q^{−ν_{0}(x)} where q = |O/ϖO|. The induced representation I_{χ} = Ind_{B}^{G}(χ), is the space of locally constant functions f : G→ℂ such that

f(bg) = Δ^{1/2}(b)χ(b)f(g), b ∈ B, g ∈ G

with the action of G from the right (by g ·f(x) = f(xg)). The inclusion of the modular function Δ guarantees that if χ is unitary, then there is an inner product ⟨f, f′⟩ = ∫_{K} f(x)\overline{f′(x)}dx on I_{χ}, for which the action of G is unitary (these are called the spherical principal series representations). However, the space I_{χ} still can be unitary even if χ is not unitary (these are called spherical complementary series representations), see Subsection ??.

We remark that I_{χ} need not be irreducible. Two spaces I_{χ} and I_{χ′} are isomorphic iff χ′ = wχ for some w ∈ S_{d} ([?, Subsec. 3.3], [?, Sec. 2.6]).

Notice that |·| is spherical, so the modular function Δ is an unramified character. If g ∈ B∩K then g is upper triangular, with its diagonal entries invertible in O. Since G = BK and χ is unramified, it follows that

(2.4) f_{χ}(bk) := Δ^{1/2}(bk)χ(bk) = Δ^{1/2}(b)χ(b),  b ∈ B, k ∈ K,

is a well defined K-fixed function (unique in I_{χ}), which makes the induced representation ρ: G→ End(I_{χ}) spherical. By definition, ρ is determined by the numbers z_{i} = χ_{i}(ϖ) = χ(diag(()1, …, 1, ϖ, 1, …, 1)), called the Satake parameters of χ, where χ_{i} are the diagonal components of χ, which is a sub-representation of Δ^{−1/2}ρ|_{B}. The representations which are well defined on PGL_{d}(F) are those with z_{1} ⋯ z_{d} = 1 (since they need to be trivial on the center of GL_{d}(F)).

Let σ_{k}(z_{1}, …, z_{d}) be the kth elementary symmetric function, i.e. σ_{k}(z_{1}, …, z_{d}) = ∑_{i_{1}<···<i_{k}} z_{i_{1}} … z_{i_{k}}.

Proposition 2.1. The function f_{χ} is an eigenfunction of the A_{k}, A_{k}f_{χ} = λ_{k}f_{χ}, where λ_{k} = q^{k(d−k)/2}σ_{k}(z_{1}, …, z_{d}).

Proof. Since H(G, K) acts on I_{χ} and preserves the K-fixed subspace ⟨f_{χ}⟩, f_{χ} is an eigenvector of the A_{k}.

It is enough to compute A_{k}f_{χ} at the point g = 1 (noting that f_{χ}(1) = 1). For every subset C ⊆ {1, …, d} of size k, let Ω_{k,C} be the set of upper triangular matrices m such that m_{ii} = ϖ if i ∈ C, m_{ii} = 1 if i ∉ C, m_{ij} is in some fixed lifting of O/ϖO to O if i ∈ C and j ∉ C, and m_{ij} = 0 otherwise. For example, for d = 4 and k = 2 the sets are

(\begin{matrix}ϖ & 0 & ∗ & ∗ \\ 0 & ϖ & ∗ & ∗ \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}), (\begin{matrix}ϖ & ∗ & 0 & ∗ \\ 0 & 1 & 0 & 0 \\ 0 & 0 & ϖ & ∗ \\ 0 & 0 & 0 & 1 \end{matrix}), (\begin{matrix}ϖ & ∗ & ∗ & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & ϖ \end{matrix}), (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & ϖ & 0 & ∗ \\ 0 & 0 & ϖ & ∗ \\ 0 & 0 & 0 & 1 \end{matrix}), (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & ϖ & ∗ & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & ϖ \end{matrix}), (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & ϖ & 0 \\ 0 & 0 & 0 & ϖ \end{matrix}).

There is a one-to-one correspondence between neighbors yK ∈ Ω_{k} of [L_{0}] and subspaces of co-dimension k of L_{0}/ϖL_{0}, so using the correspondence between matrices and lattices mentioned above, Ω_{k} = ∪_{C}Ω_{k,C}.

Fix a subset C, and let s = ∑_{i∈C} i. The number of matrices in Ω_{k,C} is q^{(d−k+1)+···+d−s}, while Δ^{1/2}(y) = Π_{i∈C} |ϖ|^{(d+1)/2−i} = q^{−k(d+1)/2+s} for every yK ∈ Ω_{k,C}. It follows that the sum of Δ^{1/2}(y)χ(y) over yK ∈ Ω_{k,C} is q^{k(d−k)/2}χ(y) = q^{k(d−k)/2}Π_{i∈C}z_{i}, so by summing over all C we obtain

(A_{k}f_{χ})(1) = ∑_{yK∈Ω_{k}} Δ^{1/2}(y)χ(y) = q^{k(d−k)/2}σ_{k}(z_{1}, …, z_{d}).

□

Now let ρ: G→ End(V) be an irreducible spherical representation of G = GL_{d}(F) with a unique (up to scalar multiples) K-invariant vector v_{0} ∈ V. Let V̂ be the representation contragredient to V. The space V̂^{K} is dual to V^{K} and thus one-dimensional. Choose v̂_{0} ∈ V̂^{K} such that ⟨v_{0}, v̂_{0}⟩ = 1 (where ⟨ , ⟩ is the action of V̂ on V). Define the bi-K-invariant function ψ(g) = ⟨ρ(g)v_{0}, v̂_{0}⟩ (so that ψ(1) = 1). ψ(g) is called a spherical function. As explained earlier, v_{0} is an eigenvector of H(G, K). The action of H(G, K) on V^{K} = ⟨v_{0}⟩ defines a homomorphism ω: H(G, K)→ℂ by

(2.5)  A · v_{0} = ω(A)v_{0}.

The action of G (from the right)on the space of functions {f : G→ℂ} induces an action of the Hecke algebra on this space, and by Equation (??) and the definition of ψ, we find that A · ψ = ω(A)ψ. Using Equation (??) one can check that

(2.6)  ψ(g) = ω(1_{KgK})/μ(KgK),

where 1_{KgK} is the characteristic function of KgK ⊆ G, and μ is the normalized measure.

Proposition 2.2. Using the notation as above, if ρ_{1} and ρ_{2} are irreducible spherical representations, then ρ_{1} ≅ ρ_{2} iff ψ_{1} = ψ_{2} iff ω_{1} = ω_{2}.

Proof. Let v_{1_{0}} and v_{2_{0}} be the (unique) K-fixed vectors of ρ_{1} and ρ_{2}. The equivalence of ψ_{1} = ψ_{2} and ω_{1} = ω_{2} follows at once from Equation (??). If ρ_{1} ≅ ρ_{2} then it is obvious that ψ_{1} = ⟨ρ_{1}(·)v_{1_{0}}, v̂_{1_{0}}⟩ = ⟨ρ_{2}(·)v_{2_{0}}, v̂_{2_{0}}⟩ = ψ_{2}. In the other direction, assume ψ_{1} = ψ_{2}, and let V_{i} be the representation space of ρ_{i}. Define a map from V_{i} to ⟨ψ_{i}G⟩, the representation spanned by ψ_{i}, by sending v ∈ V_{i} to the function g ↦ ⟨ρ(g)v, v̂_{i_{0}}⟩, for i = 1, 2. This is easily seen to be a non-zero homomorphism (since v_{i_{0}} ↦ ψ_{i} ≠ 0), which is an isomorphism since V_{i} is irreducible. Then ρ_{1} ≅ ⟨ψ_{1}G⟩ = ⟨ψ_{2}G⟩ ≅ ρ_{2}. □

As a G-module, I_{χ} has finite composition length, so it has only finitely many irreducible subquotients.

Proposition 2.3 ([?]). Every irreducible spherical representation of GL_{d}(F) is isomorphic to a subquotient of I_{χ} for some unramified character χ, which is unique up to permutation.

Proof. Let ρ: G→ End(V) be an irreducible spherical representation of G with v_{0} as its K-invariant vector. Let ω: H(G, K)→ℂ be its corresponding homomorphism, defined by Equation (??).

It can be shown [?, Cor. 4.2] that every such homomorphism is of the form

ω_{χ}(A) = ∫_{G} A(x)f_{χ}(x) dx

for some unramified character χ: T →ℂ (unique up to permutation) where f_{χ} is defined in Equation (??). Then

(A · f_{χ})(1) = ∫_{G} A(x)f_{χ}(x)dx = ω_{χ}(A) = ω_{χ}(A) · f_{χ}(1),

and since f_{χ} is an eigenvector, A·f_{χ} = ω_{χ}(A)f_{χ} for every A ∈ H(G, K). Let W be an irreducible subquotient of I_{χ} in which f_{χ} has a nonzero image. By the previous proposition ρ is isomorphic to W, since ω_{χ} = ω. □

Thus, every spherical representation is determined by the Satake parameters z_{i} = χ_{i}(ϖ) = χ(diag(()1, …, 1, ϖ, 1, …, 1)), for some unramified χ, uniquely determined up to permutation.

Proposition 2.4. Let f : G/K→ℂ be a simultaneous eigenvector of A_{1}, …, A_{d−1}. Then there is an unramified character χ such that f_{χ} has the same eigenvalues.

Proof. Consider f : G→ℂ, which is invariant with respect to K. Let ⟨fG⟩ denote the linear span of the G-orbits of f, where G acts from the right. Taking this space modulo a maximal sub-module not containing f, we obtain an irreducible spherical representation, where f is a (unique) K-fixed vector. By the previous proposition, it is isomorphic to a subquotient of I_{χ} for an unramified character χ, where f_{χ} is the unique K-fixed vector. By Proposition ??, since the two representation spaces are isomorphic, they induce the same homomorphism ω: H(G, K)→ℂ, namely A_{k} · f = ω(A_{k})f and A_{k} · f_{χ} = ω(A_{k})f_{χ}. □

Let ρ: G→ End(V) be a unitary representation, and ⟨ , ⟩ the inner product defined on V. The functions of the form

ρ_{v,w}: g ↦ ⟨ρ(g)v, w⟩

where v, w ∈ V are called the matrix coefficients of ρ. Notice that if V has a K-fixed vector v_{0} and ⟨v_{0}, v_{0}⟩ = 1, then ρ_{v_{0},v_{0}} is a spherical function. In the special case of I_{χ}, ρ_{f_{χ},f_{χ}}(g) = ∫_{K} f_{χ}(xg) dx.

If V is irreducible then fixing w ≠ 0, the map v ↦ ρ_{v,w} is an isomorphism of representations (where G acts on the space of functions from the right).

A representation is called tempered, if for some 0 ≠ v, w ∈ V, ρ_{v,w} ∈ L^{2+ϵ}(G) for every ϵ > 0. The following equivalence is well known:

Proposition 2.5. An irreducible spherical unitary representation is tempered iff its Satake parameters have absolute value 1.

2.3. Ramanujan complexes and the spectrum of A_{k}. Let Γ be a cocompact lattice of G = PGL_{d}(F). Then Γ acts on B = G/K by left translation, and Γ\B is a finite complex. The color function defined on B^{0} (Equation (??)) may not be preserved by the map B→Γ\B. However, the colors defined on B^{1} by ϱ(x, y) = ϱ(x)−ϱ(y) (mod d) are preserved, since they are determined by the index of (a representative of) y as a sublattice in (a representative of) x.

Since the Hecke algebra H(G, K) acts on G from the right, and Γ is acting from the left, the operators A_{k} on L^{2}(B) = L^{2}(G/K) induce colored adjacency operators on Γ\B.

It should be noted that if Γ is torsion free, then γx ≠ x for any γ ≠ 1 and any x ∈ B^{0}, so the underlying graph of Γ\B is simple. Every cocompact lattice has a finite index torsion free subgroup.

The trivial eigenvectors appear in L^{2}(Γ\B) but not in L^{2}(B), since the former complex is finite. The trivial eigenvectors can be constructed as follows. The trivial representation of G is obviously spherical. Taking χ = Δ^{−1/2}, we see that f_{χ}(g) = 1 for every g (see Equation (??)), and the action of G on the subspace ℂf_{χ} ⊆ I_{χ} is trivial. The Satake parameters of the trivial representation are thus z_{i} = Δ^{−1/2}(diag(()1, …, 1, ϖ, 1, …, 1)) = q^{−(d−2i+1)/2}. More generally, since G(F)/PSL_{d}(F)K ≅ F^{×}/F^{×}^{d}, G has d one-dimensional spherical representations. Fixing ζ such that ζ^{d} = 1,

(2.7)  χ(g) = Δ^{−1/2}(g)ζ^{ν_{0}(det(g))}

corresponds to a one-dimensional representation, with the K-fixed vector f_{χ}(g) = ζ^{ν_{0}(det(g))} (since det(k) ∈ O^{×} for every k ∈ K). The Satake parameters in this case are ζq^{−(d−1)/2}, …, ζq^{(d−1)/2}, and the eigenvalues are

ζ^{k} · q^{k(d−k)/2}σ_{k}(q^{−(d−1)/2}, q^{−(d−3)/2}, …, q^{(d−1)/2}).

Let t = [Γ:Γ ∩ PSL_{d}(F)]. By Equation (??), the trivial eigenvector f_{χ} is well defined on Γ\B iff ζ^{d/t} = 1, and the respective d/t eigenvectors give rise to the trivial eigenvalues. Since G is infinite, these f_{χ} do not belong to L^{2}(B).

Let 𝔖_{d,k} ⊆ ℂ denote the spectrum of the operator A_{k} acting on L^{2}(B).

Definition 2.6. The complex Γ\B is pseudo-Ramanujan if for each k = 1, …, d − 1, the non-trivial eigenvalues of A_{k} acting on L^{2}(Γ\B) belong to 𝔖_{d,k}.

Let 𝔖_{d} ⊆ ℂ^{d−1} denote the simultaneous spectrum of (A_{1}, …, A_{d−1}) acting on the space L^{2}(B), namely, the set of (λ_{1}, …, λ_{d−1}) ∈ ℂ^{d−1} for which there exist a sequence of unit vectors v_{n} ∈ L^{2}(B) such that lim_{n→∞}(A_{k}v_{n} − λ_{k}v_{n}) = 0 for every k = 1, …, d − 1.

Definition 2.7. The complex Γ\B is Ramanujan if for every nontrivial simultaneous eigenvector f ∈ L^{2}(Γ\B) of the A_{k}, the eigenvalues (λ_{1}, …, λ_{d−1}) belong to 𝔖_{d}.

Since the A_{k} commute, every eigenvalue of A_{k} can be obtained by a simultaneous eigenvector. Hence, a Ramanujan complex is pseudo-Ramanujan. On the other hand, 𝔖_{d} is not the Cartesian product of the 𝔖_{d,k}. For example, inverting the direction of edges in B carries A_{k} to A_{d−k}, so the operators A_{k} and A_{d−k} are adjoint to each other. In particular for every (λ_{1}, …, λ_{d}) ∈ 𝔖_{d} we have that λ_{d−k} = λ̄_{k}.

Remark 2.8. The spectrum 𝔖_{d,k} of A_{k} on L^{2}(B) is equal to the projection of 𝔖_{d} on the kth component.

Remark 2.9. If d = 2 or d = 3, then Γ\B is Ramanujan iff it is pseudo-Ramanujan (indeed, for d = 2 the definitions coincide, and for d = 3, A_{2} is the adjoint operator of A_{1}).

Let S = {(z_{1}, …, z_{d}) : |z_{i}| = 1, z_{1} ⋯ z_{d} = 1} and σ: S→ℂ^{d−1} be the map defined by (z_{1}, …, z_{d}) ↦ (λ_{1}, …, λ_{d−1}), where

λ_{k} = q^{k(d−k)/2}σ_{k}(z_{1}, …, z_{d}).

The theorem below is proved in [?]. For completeness, we sketch the proof, following ideas from [?] (where the result was proved for d = 3). First, we will need an easy calculus lemma:

Lemma 2.10. Let (a_{n}), (b_{n}) be positive series. If lim sup(a_{n}b_{n}^{2+ϵ}) ≤ 1 for every ϵ > 0 and {a_{n}} is bounded, then lim sup(a_{n}b_{n}^{2}) ≤ 1.

Proof. Otherwise let C > 1 be an upper bound of {a_{n}} and p = lim sup(a_{n}b_{n}^{2}) > 1, and take ϵ < 2 log(p)/ log(C). Then

p = lim sup(an^{\frac{ϵ}{2+ϵ}} an^{\frac{2}{2+ϵ}} b_{n}^{2}) ≤ C^{\frac{ϵ}{2+ϵ}} lim sup(a_{n}b_{n}^{2+ϵ})^{\frac{2}{2+ϵ}} < C^{ϵ/2},

a contradiction. □

Theorem 2.11. The simultaneous spectrum 𝔖_{d} is equal to σ(S).

Proof. Let \underline{z} = (z_{1}, …, z_{d}) ∈ S. Then the corresponding character χ is unitary, and the irreducible subquotient generated by f_{χ} of the induced representation I_{χ}, is tempered (Proposition ??). Thus, the corresponding spherical function ψ_{χ} is in L^{2+ϵ}(G) for every ϵ > 0. We already saw that ψ_{χ} is an eigenvector, however it does not belong to L^{2}(G). In order to show that σ(\underline{z}) is in the spectrum, we twist ψ_{χ} to elements of L^{2}(G) which are ”almost” eigenvectors, and their almosteigenvalues converge to σ(\underline{z}).

By Proposition ?? (and since ψ_{χ} is the spherical function associated to f_{χ}), A_{k}ψ_{χ} = λ_{k}ψ_{χ} where λ_{k} = q^{k(d−k)/2}σ_{k}(z_{1}, …, z_{d}). For every vertex x ∈ B^{0} = G/K, let w(x) denote the distance (in B^{1}) of x from the origin [L_{0}].

Recall [?, V.(2.2)] that every double coset in K\G/K has a unique representative of the form diag(()ϖ^{ℓ_{1}}, …, ϖ^{ℓ_{d}}) where ℓ_{1} ≥ ⋯ ≥ ℓ_{d−1} ≥ ℓ_{d} = 0; we call this representative of KgK the type of gK, and note that the number of vertices of this type is μ(KgK). Its distance from [L_{0}] is equal to ℓ_{1}, so there are \binom{n+d−2}{d−2} < (n + d)^{d} types of distance n.

For δ > 0, define a function ψ_{χ}^{δ} on B^{0} by ψ_{χ}^{δ}(x) = (1 − δ)^{w(x)}ψ_{χ}(x). For each n, let g_{n} denote the type of the vertex of distance n for which μ(Kg_{n}K) |ψ_{χ}(g_{n}K)|^{2} is maximal.

To see that ψ_{χ}^{δ} ∈ L^{2}(B^{0}), compute that

∑_{x∈B^{0}}(1 − δ)^{2w(x)} |ψ_{χ}(x)|^{2} = ∑_{n=0}^{∞} (1 − δ)^{2n} ∑_{w(x)=n} |ψ_{χ}(x)|^{2}

≤ ∑_{n=0}^{∞} (1 − δ)^{2n}(n + d)^{d}μ(Kg_{n}K) |ψ_{χ}(g_{n}K)|^{2},

and the convergence follows from the root test once we show that lim sup (μ(Kg_{n}K) |ψ_{χ}(g_{n}K)|^{2})^{1/n} ≤ 1. But since ψ_{χ} ∈ L^{2+ϵ}(B^{0}) for every ϵ > 0, we have

lim sup (μ(Kg_{n}K) |ψ_{χ}(g_{n}K)|^{2+ϵ})^{1/n} ≤ 1,

and the result follows from μ(Kg_{n}K)^{1/n} ≤ q^{d} by the lemma.

By the definition of A_{k}, A_{k}ψ_{χ}^{δ}(x) is a sum of ψ_{χ}^{δ}(y) for neighbors y of x, and the distance of neighbors satisfies |w(y) − w(x)| ≤ 1. Since A_{k}ψ_{χ}−λ_{k}ψ_{χ} = 0, it follows that for some constant c, ||A_{k}ψ_{χ}^{δ} − λ_{k}ψ_{χ}^{δ}|| ≤ cδ||ψ_{χ}^{δ}|| for every \frac{1}{2} > δ > 0, showing that (λ_{1}, …, λ_{d−1}) ∈ 𝔖_{d}.

Now let (λ_{1}, …, λ_{d−1}) ∈ 𝔖_{d}, and let z_{1}, …, z_{d} ∈ ℂ be numbers satisfying q^{k(d−k)/2}σ_{k}(z_{1}, …, z_{d}) = λ_{k}, with the added property that z_{1} … z_{d} = 1 (the z_{i} are unique up to order). We need to show that (z_{1}, …, z_{d}) ∈ S, implying (λ_{1}, …, λ_{d−1}) ∈ σ(S).

Let v_{n} ∈ L^{2}(B^{0}) be unit vectors such that A_{k}v_{n} − λ_{k}v_{n}→0, for all k, and define a homomorphism ω: H(G, K)→ℂ by ||Av_{n} − ω(A)v_{n}||→0 (here we use the fact that the A_{k} are bounded and generate H(G, K)). Then ω is continuous in the norm of the operators on L^{2}(B^{0}), and in particular |ω(A)| ≤ ||A|| for every A ∈ H(G, K) (otherwise take ϵ such that |ω(A)| > ||A||+ϵ, then (\frac{A}{||A||+ϵ})^{n} converges to zero but ω((\frac{A}{||A||+ϵ})^{n}) does not).

For ℓ ≥ 1, let H_{ℓ} ∈ H(G, K) be the characteristic function of Kdiag(()ϖ^{dℓ}, 1, …, 1)K. We show that while ω(H_{ℓ}) is a certain combination of z_{r}^{−dℓ}, the bound ||H_{ℓ}|| is polynomial in ℓ, thus implying that |z_{r}| ≥ 1 for every r.

The vector ψ_{1} associated to the trivial character χ = 1 is strictly positive (since f_{1}(x) > 0 for every x ∈ G/K and ψ_{1}(x) = ∫_{K} f_{1}(kx)dk), so if H_{ℓ}ψ_{1} = bψ_{1} and H_{ℓ}^{∗}ψ_{1} = b′ψ_{1}, we have ||H_{ℓ}|| ≤ \sqrt{bb′} by Schur’s criterion [?, p. 102]. Let p = ((d − 1)ℓ, −ℓ, …, −ℓ).

From [?, (3.5)] and [?, (3.3)], and using the limit

(2.8)  lim_{(x_{1},...,x_{d})→(1,...,1)} ∑_{k=1}^{d} \frac{x_{k}^{m}}{∏_{i≠k} (x_{k} − x_{i})} = \binom{m}{d − 1},

we obtain bb′ = (1 − q^{−1})^{2(d−1)}\binom{dℓ}{d−1}\binom{d(ℓ+1)−2}{d−1}q^{d(d−1)ℓ} < (dℓ)^{2d}q^{d(d−1)ℓ}, so ||H_{ℓ}|| < (dℓ)^{d}q^{d(d−1)ℓ/2}. (Note that the action of H(G, K) on the spherical functions in [?] is via the multiplication of the Hecke algebra, unlike ours; see Equation (??)).

In a similar manner, ω(H_{ℓ}) is equal to ĉ_{−p}(ω_{s}) of [?, (3.3)], and has the form q^{d(d−1)ℓ/2} ∑_{r=1}^{d} α_{r}z_{r}^{−dℓ} where α_{r} = ∏_{i≠r} \frac{z_{i}−q^{−1}z_{r}}{z_{i}−z_{r}} if all the z_{i} are different (see [?, III.(2.2)]). From the continuity of ω we proved

∑_{r=1}^{d} α_{r}z_{r}^{−dℓ} ≤ Cℓ^{d}

for some constant C and every ℓ.

Order the z_{i} by absolute value, so that |z_{1}| ≤ ⋯ ≤ |z_{d}|. Then α_{1} ≠ 0 and from the last bound it follows that |z_{1}| ≥ 1, but z_{1} … z_{d} = 1 so (z_{1}, …, z_{d}) ∈ S. If the z_{i} are not assumed to be different, one computes the coefficients of the z_{i}^{−ℓd} by Equation (??), and the same arguments apply. □

The sets 𝔖_{d,k} are explicitly described in [?]: 𝔖_{d,k} is the simply connected domain with boundary the complex curve

{q^{k(d−k)/2}σ_{k}(e^{iθ}, …, e^{iθ}, e^{−(d−1)iθ}) : θ ∈ [0, 2π]}

where i = \sqrt{−1}.

Notice that the equations λ_{k} = q^{k(d−k)/2}σ_{k}(z_{1}, …, z_{d}) always have a solution, but unless (λ_{1}, …, λ_{d}) ∈ 𝔖_{d}, the z_{i} do not have to be unitary—even if each λ_{k} ∈ 𝔖_{d,k}.

In terms of characters, Theorem ?? implies that the eigenvalues corresponding to f_{χ} (see Proposition ??) are in the simultaneous spectrum of (A_{1}, …, A_{d−1}) acting on L^{2}(B) iff χ is unitary. This can be used to give a representation theoretic definition of being Ramanujan, as in Proposition ??.

Proof of Proposition ??. Assume every irreducible spherical infinitedimensional sub-representation of L^{2}(Γ\G(F)) is tempered. As Γ is cocompact, L^{2}(Γ\G(F)) is a direct sum of irreducible representations. Let f ∈ L^{2}(Γ\G(F)/K) be a non-trivial simultaneous eigenvector of the A_{k}, with A_{k}f = λ_{k}f. By Proposition ??, the λ_{k} are determined by some unramified character χ . Consider f as a K-fixed vector in L^{2}(Γ\G(F)). Since the only finite-dimensional representations of G(F) are the trivial ones, the representation ⟨fG(F)⟩ is infinite-dimensional. Let V be an irreducible quotient of ⟨fG(F)⟩ in which f ≠ 0, then V is an irreducible infinite-dimensional spherical sub-representation of L^{2}(Γ\G(F)) so by assumption V is tempered. It then follows from Proposition ?? that χ is unitary, and so (λ_{1}, …, λ_{d−1}) ∈ 𝔖_{d}.

In the other direction, let V be an irreducible spherical infinitedimensional sub-representation of L^{2}(Γ\G(F)); then its unique K-fixed vector f is a simultaneous eigenvector of the A_{k}, where A_{k}f = λ_{k}f. By assumption (λ_{1}, …, λ_{d}) ∈ 𝔖_{d}. The eigenvalues induce a homomorphism ω = ω_{χ} for some unitary character χ, and by Proposition ??, V is isomorphic to a subquotient of I_{χ}. Consequently, V is tempered. □

2.4. Bounds on the spectrum of A_{k} on L^{2}(Γ\B). In the previous subsection we computed the spectrum of (A_{1}, …, A_{d−1}) in their action on L^{2}(B). For a discrete subgroup Γ ≤ G, let spec_{Γ\B}(A_{1}, …, A_{d−1}) denote the spectrum of these operators in their action on L^{2}(Γ\B) (which is a finite set).

In this subsection we apply the classification of unitary representations of GL_{d}(F) to give an upper bound on spec_{Γ\B}(A_{1}, …, A_{d−1}) (which is independent of Γ). In addition we state an Alon-Boppana type theorem, due to W. Li, that for suitable families of quotients \{Γ_{i}\B\} of B, \overline{∪ spec_{Γ_{i}\B}(A_{1}, …, A_{d−1})} ⊇ 𝔖_{d}.

Let f ∈ L^{2}(Γ\G/K) be a simultaneous eigenvector of the A_{k}. Lift f to L^{2}(Γ\G), and recall that the representation ⟨fG⟩ is unitary (since the action of G on L^{2}(Γ\G) is unitary) and spherical (since f is Kfixed).

The unitary spherical representations were described by Tadic [?], as part of the classification of all the unitary representations of GL_{d}(F). Such a spherical representation is induced by a character χ = χ_{1}⊕⋯⊕ χ_{d}, where the χ_{i} are combined into blocks. For the Satake parameters (z_{i_{1}}, …, z_{i_{s}}) of each block χ_{i_{1}}, …, χ_{i_{s}}, one of the following three options holds: either s = 1 and z_{i_{1}} ∈ S^{1} = {z ∈ ℂ: |z| = 1}; (z_{i_{1}}, …, z_{i_{s}}) is of the form

(q^{(s−1)/2}z, …, q^{(1−s)/2}z)

for z ∈ S^{1}; or (if s = 2s′ is even) it is of the form

(q^{(s′−1)/2+α}z, …, q^{(1−s′)/2+α}z, q^{(s′−1)/2−α}z, …, q^{(1−s′)/2−α}z)

for z ∈ S^{1} and 0 < α < 1/2.

This set of possible parameters (z_{1}, …, z_{d}) determines the eigenvalues (λ_{1}, …, λ_{d−1}) via Proposition ??. In particular if d ≥ 3, we obtain for the non-trivial eigenvalues

|λ_{k}| ≤ q^{k(d−k)/2} · σ_{k}(q^{(d−2)/2}, …, q^{(2−d)/2}, 1) ≈ q^{k(d−k− \frac{1}{2} )}

for every k ≤ d/2 (and λ_{d−k} = λ̄_{k}).

One can see that if d ≥ 3 then |λ_{k}| < \left[{d \atop k}\right]_{q} for every non-trivial unitary representation (where \left[{d \atop k}\right]_{q} denotes the number of subspaces of dimension k in 𝔽_{q}^{d}, which is the number of neighbors of color k of each vertex). In particular the non-trivial eigenvalues of A = A_{1} +⋯+A_{d−1} are bounded away from the trivial one. This demonstrates the fact that PGL_{d}(F) has Kazhdan property (T) and the quotient graphs Γ\B^{1} are expanders for every Γ [?].

On the other hand, for d = 2 the eigenvalues q^{1/2}σ_{1}(q^{α}, q^{−α}) approach the degree q+1 when α→1/2, in accordance with the fact that PGL_{2}(F) does not have property (T).

For the lower bound, we quote

Theorem 2.12 ([?, Thm. H]). Let X_{i} be a family of finite quotients of B with unbounded injective radius. Then \overline{∪ spec_{X_{i}}(A_{1}, …, A_{d−1})} ⊇ 𝔖_{d}.

This also follows from a multi-dimensional version of [?].

2.5. Super-cuspidal and square-integrable representations. Let G denote the group GL_{d}(F) or PGL_{d}(F), and Z = Cent(G) its center. Let ρ: G→ End(V) be a unitary representation. Recall that the matrix coefficients of ρ are the functions ρ_{v,w}: g ↦ ⟨ρ(g)v, w⟩ where v, w ∈ V. A unitary representation of G is called super-cuspidal, if its matrix coefficients are compactly supported modulo the center. Notice that the irreducible representations of GL_{1}(F) are all super-cuspidal (as the group equals its center).

We say that a unitary representation ρ is square-integrable, if ρ_{v,w} ∈ L^{2}(G/Z) for every v, w ∈ V. A representation is square-integrable iff it is isomorphic to a sub-representation of L^{2}(G) [?, Prop. 9.6]. Note that super-cuspidal representations are square-integrable, and squareintegrable representations are tempered.

Let s | d be any divisor, and let P_{s}(F) denote the parabolic subgroup corresponding to the partition of d into s equal parts. For a representation ψ of GL_{d/s}(F), we denote

(2.9) M_{s}(ψ) = Ind_{P_{s}(F)}^{GL_{d}(F)}(|det|F^{\frac{1−s}{2}} ψ ⊕ |det|F^{\frac{3−s}{2}} ψ ⊕ ⋯ ⊕ |det|F^{\frac{s−1}{2}} ψ).

The unique irreducible sub-representation of M_{s}(|det|_{F}^{(s−1)/2} ψ) will be denoted by C_{s}(ψ). It is known that if ψ is irreducible and supercuspidal, then the induced representation M_{s}(|det|_{F}^{(s−1)/2} ψ) has precisely 2^{s−1} irreducible subquotients, two of which (if s > 1) are unitary [?, p. 32] (notice that M_{1}(ψ) = ψ). These subquotients are C_{s}(ψ), and a certain irreducible quotient, called the generalized Steinberg representation (or sometimes ”special representation”) and denoted by Sp_{s}(ψ).

Proposition 2.13 ([?, p. 32], [?]). For s > 1 and ψ an irreducible super-cuspidal representation of GL_{d/s}(F), Sp_{s}(ψ) is square-integrable, and C_{s}(ψ) is not tempered.

Every square-integrable representation of GL_{d}(F) is either supercuspidal, or of the form Sp_{s}(ψ) for a unique divisor s of d and a unique super-cuspidal representation ψ of GL_{d/s}(F).

Remark 2.14. If s > 1, C_{s}(ψ) is not tempered for any unitary representation ψ.

Example 2.15. Let ϕ: F^{×}→ℂ^{×} be a character, and ψ = |·|_{F}^{(1−d)/2} ϕ. Then C_{d}(ψ) = ϕ ∘ det, which is one-dimensional.

Proof. Let B(F) denote the standard Borel subgroup of GL_{d}(F). By definition, C_{d}(ψ) is the unique irreducible sub-representation of

M_{d}(ϕ) = Ind_{B(F)}^{GL_{d}(F)}(|·|_{F}^{(1−d)/2} ϕ ⊕ ⋯ ⊕ |·|_{F}^{(d−1)/2}ϕ),

which is the unitary induction of Δ^{−1/2} · (ϕ ∘ det) to GL_{d}(F). In particular, this representation, when restricted to B(F), contains the representation ϕ ∘ det, which is thus a sub-representation of M_{d}(ϕ), so by definition C_{d}(ψ) = ϕ ∘ det. □

3. From local to global

3.1. The global field. Let k be a global field, V = {ν} its nonarchimedean discrete valuations, and V_{∞} the Archimedean valuations. For ν ∈ V, k_{ν} is the completion, O_{ν} = {x: ν(x) ≥ 0} the valuation ring of k_{ν} (which is the closed unit ball of k_{ν} and thus compact), and P_{ν} = {x: ν(x) > 0} the valuation ideal.

Note that the ring of ν-adic integers k ∩ O_{ν} of k is a local ring, with maximal ideal k ∩ P_{ν}. Fix a valuation ν_{0} ∈ V, and set F = k_{ν_{0}}. Consider the intersection

R_{0} = {x ∈ k: ∀(ν ∈ V − {ν_{0}}) ν(x) ≥ 0} = ⋂_{ν∈V−{ν_{0}}}(k ∩ O_{ν}).

Recall that the valuations of k = 𝔽_{q}(y) are all nonarchimedean. They are indexed by the prime polynomials of 𝔽_{q}[y] and 1/y. For a prime p the valuation is ν_{p}(p^{i}f/g) = i when f and g are prime to p, and the valuation corresponding to 1/y is the minus degree valuation, defined by ν_{1/y}(f/g) = deg(g) − deg(f). If ν_{0} = ν_{1/y} then R_{0} = 𝔽_{q}[y].

For every x ∈ k^{×} we have that

(3.1)  ν_{1/y}(x) + ∑_{p} deg(p)ν_{p}(x) = 0,

so ν_{0}(x) ≤ 0 for every x ∈ R_{0}. As a result R_{0} is discrete in F. It also follows that R_{0} ∩O_{ν_{0}} = 𝔽_{q}. For every ν, choose a uniformizer ϖ_{ν} ∈ R_{0}, so that ν(ϖ_{ν}) = 1. Then, the completion of k at ν is 𝔽_{q}((ϖ_{ν})), and the local ring of integers is 𝔽_{q}[[ϖ_{ν}]]. Note that if ν_{0} = ν_{1/y}, we can choose the uniformizers, ϖ_{ν}, to be the prime polynomials of 𝔽_{q}[y], and ϖ_{ν_{0}} = 1/y.

Let I denote the set of functions i⃗ : V→ℕ ∪ {0}, such that i_{ν} = 0 for almost all ν and i_{ν_{0}} = 0. The ideals of R_{0} are indexed by functions i⃗ ∈ I, in the following way: For i⃗ ∈ I, we define

(3.2)  Ii⃗ = {x ∈ R_{0} : ν(x) ≥ i_{ν}} = ⋂_{ν∈V−{ν_{0}}}(k ∩ P_{ν}^{i_{ν}}),

where we make the notational convention that P_{ν}^{0} = O_{ν}. From our choice of the uniformizers, it follows that if ν_{0} = ν_{1/y}, then Ii⃗ is the (principal) ideal generated by ∏_{ν≠ν_{0}} ϖ_{ν}^{i_{ν}}. Notice that for the zero vector i⃗ = 0 (i_{ν} = 0 for all ν), we obtain the trivial ideal I_{0} = R_{0}.

Let ×k_{ν} be the direct product of the fields k_{ν} over all the valuations ν ∈ V ∪ V_{∞} of k, and recall that the ring 𝔸 of adèles over k is defined to be the restricted product

(3.3)  𝔸 = {x = (x_{ν}) ∈ ×k_{ν} : ν(x_{ν}) ≥ 0 for almost all ν}.

The field k embeds in 𝔸 diagonally. In a similar manner to the construction of R_{0}, we define

(3.4)  R̃_{0} = {(x_{ν}) ∈ 𝔸: ∀(ν ∈ V − {ν_{0}}) ν(x_{ν}) ≥ 0}
= F × ∏_{ν∈V−{ν_{0}}} O_{ν} × ∏_{ν∈V_{∞}} k_{ν}.

The ideals of finite index of R̃_{0} are again indexed by I, and are of the form

(3.5)  Ĩi⃗ = {(x_{ν}) ∈ 𝔸: ∀(ν ∈ V − {ν_{0}}) ν(x_{ν}) ≥ i_{ν}},

and with respect to the diagonal embedding, we have R_{0} = k ∩ R̃_{0} and Ii⃗ = k ∩ Ĩi⃗ for every i⃗ ∈ I. In fact, R̃_{0} and Ĩi⃗ are the topological closures of R_{0} and Ii⃗, respectively.

3.2. Strong Approximation. Let G be a connected, simply connected, almost simple linear algebraic group, defined over k (e.g. SL_{d}), with a fixed embedding into GL_{r} for some r. For a subring R of a k-algebra A, we denote G(R) = G(A) ∩ GL_{r}(R). For simplicity of notation (and as our applications are mainly for positive characteristic), we assume G(k_{ν}) is compact for all Archimedean places ν.

The diagonal embedding k ↪ 𝔸, which is obviously discrete, induces a discrete embedding G(k) ↪ G(𝔸). Let T be the set of valuations θ such that G(k_{θ}) is compact; this is a finite set [?]. Fix a valuation ν_{0} ∈ V − T, and let F = k_{ν_{0}} denote the completion with respect to this special valuation. G(k) is a lattice of finite co-volume in G(𝔸), and moreover if T ≠ ∅, G(k) is a cocompact lattice [?, Thm. 5.5].

Theorem 3.1 (Strong Approximation [?], [?]). The product G(k)G(F) is dense in G(𝔸).

So for every open subgroup U of G(𝔸),

(3.6)  G(k)G(F)U = G(𝔸).

Corollary 3.2. Let U ⊆ G(𝔸) be a compact subgroup such that G(F)U is open, and G(F)∩U = 1. Set Γ_{U} = G(k)∩G(F)U. Then its projection to G(F) (which we will also denote by Γ_{U}) is discrete, and

(3.7)  G(k)\G(𝔸)/U ≅ Γ_{U}\G(F).

For example, if U = ∏_{ν∈V−{ν_{0}}} G(O_{ν}) × ∏_{ν∈V_{∞}} G(k_{ν}), then Γ = Γ_{U} is the arithmetic subgroup G(R_{0}). More generally, let i⃗ = (i_{ν}) ∈ I be a function corresponding to an ideal Ĩi⃗, and let

Ui⃗ = ∏_{ν∈V−{ν_{0}}} G(O_{ν}, P_{ν}^{i_{ν}}) × ∏_{ν∈V_{∞}} G(k_{ν})

where G(O_{ν}, P_{ν}^{i_{ν}}) = Ker(G(O_{ν})→G(O_{ν}/P_{ν}^{i_{ν}})) is a congruence subgroup. Then G(F)Ui⃗ = G(R̃_{0}, Ĩi⃗) = Ker(G(R̃_{0}))→G(R̃_{0}/Ĩi⃗) is an open subgroup of G(𝔸), and we set

(3.8)  Γi⃗ = G(R_{0}, Ii⃗) = G(k) ∩ G(F)Ui⃗ ,

called the principal congruence subgroup mod Ii⃗ of G(R_{0}). Again, when T ≠ ∅, this is a cocompact lattice in G(F).

3.3. Automorphic representations. The group G(𝔸) acts on the space L^{2}(G(k)\G(𝔸)) by multiplication from the right. The sub-modules are called automorphic representations of G(𝔸). The closed irreducible sub-modules are said to be discrete, or to belong to the discrete spectrum. Its complement is called the continuous spectrum. If T ≠ ∅ then there is no continuous spectrum.

Let K_{ν} = G(O_{ν}) and recall that for every (g_{ν}) ∈ G(𝔸), g_{ν} ∈ K_{ν} for almost all ν. Given irreducible representations π_{ν} : G(k_{ν})→ End(V_{ν}), with all but finitely many being K_{ν}-spherical, one defines the restricted tensor product π = ⊗π_{ν} : G(𝔸)→ End(⊗′V_{ν}) [?].

A fundamental theorem [?, Thm. 3.3.3] states that any irreducible automorphic representation of G(𝔸) is isomorphic to such a restricted tensor product. The representations π_{ν} in π = ⊗π_{ν} are called the (local) components of π, and since π is irreducible, they are also irreducible. Moreover, π is admissible iff all its components are.

Proposition 3.3. Assume that G(k_{ν}) is non-compact. If the component π_{ν} of an irreducible automorphic representation π of G(𝔸) is trivial, then π is trivial.

Proof. Let π = ⊗π_{ν} be an automorphic representation acting on V ≤ L^{2}(G(k)\G(𝔸)), where π_{ν} is trivial. For f ∈ V, (assumed to be Kfinite, see [?, Thm. 3.3.4]), f is G(k_{ν})-invariant from the right and G(k)- invariant from the left. However by Strong Approximation, G(k_{ν})G(k) is dense, so f must be constant everywhere, making π trivial. □

Recall that an irreducible representation is Ui⃗ -spherical if it has a Ui⃗ -fixed vector. We assume G(F) is non-compact where F = k_{ν_{0}}.

Proposition 3.4. Let π be an irreducible, Ui⃗ –spherical automorphic representation of G(𝔸). Then π_{ν_{0}} is a sub-representation of L^{2}(Γi⃗ \G(F)).

Conversely, if ρ ≤ L^{2}(Γi⃗ \G(F)) is irreducible, then there exists an irreducible Ui⃗ –spherical automorphic representation π of G(𝔸) such that π_{ν_{0}} is isomorphic to ρ.

The second assertion is seen by lifting a function f ∈ V_{ρ} (where V_{ρ} is the representation space) from Γi⃗ \G(F) to G(k)\G(𝔸) using Corollary ??, and taking π to be an irreducible quotient of the (right) G(𝔸)- module generated by f.

3.4. The conductor. For a representation ρ of G(k_{ν}), the conductor of ρ, cond(ρ) = i, is defined to be the minimal i ≥ 0, for which there is a G(O_{ν}, P_{ν}^{i})-fixed vector in V (such an i exists since the representation is admissible). In particular, cond(ρ) = 0 iff ρ is spherical.

Now let π be an irreducible automorphic representation of G(𝔸). Since almost all the local components are spherical, cond(π_{ν}) = 0 for almost every ν. We thus let cond(π) be the function i⃗ : V→ℕ ∪ {0} defined by i_{ν} = cond(π_{ν}) (note that i⃗ is not in I in general, as we do not assume i_{ν_{0}} = 0).

Remark 3.5. Let i⃗ = cond(π). Then H = G(O_{ν_{0}}, P_{ν_{0}}^{i_{ν_{0}}})Ui⃗ is the maximal principal congruence subgroup for which π has an H-fixed vector.

The results of this section will be used later for non-simply connected cases, which requires some minor modifications. Let G be a connected, almost simple algebraic group over k. Let Π : G→G be its simply connected cover. Then Π(G(𝔸))⊴G(𝔸) and the quotient is abelian (of finite exponent). In this situation, Proposition ?? becomes

Proposition 3.6. Assume that G(k_{ν}) is non-compact. If the component π_{ν} of an irreducible automorphic representation π of G(𝔸) is one dimensional, then π is one dimensional.

4. Global automorphic representations

Let G be an almost simple, connected algebraic group defined over k, where k is a global field of arbitrary characteristic. The discrete spectrum of automorphic representations is composed of cuspidal and residual representations. The cuspidal representation space is comprised of functions f ∈ L^{2}(G(k)\G(𝔸)) which satisfy ∫_{N(k)\N(𝔸)} f(ng)dn = 0 for every g ∈ G(𝔸) and for every N, where N is a unipotent radical of a parabolic subgroup of G. Since the cuspidal condition involves integration from the left, and the action is by right translation, this is a sub-representation space. The other discrete irreducible representations are called residual.

Recently, L. Lafforgue has proved the following version of the Ramanujan conjecture:

Theorem 4.1 ([?], [?]). Assume k is of positive characteristic and G = GL_{d}. Let π = ⊗π_{ν} be an irreducible, cuspidal representation with finite central character. For all ν, if π_{ν} is spherical then π_{ν} is tempered.

4.1. The residual spectrum. All the one-dimensional representations are residual, and when G = GL_{d} and d is prime these are the only residual representations. If d is not a prime, the other residual representations can be described in terms of the cuspidal representations of smaller rank, as follows:

An element (a_{ν}) ∈ 𝔸 is invertible only if for almost all ν, a_{ν} ∈ O_{ν}^{×}. We can thus define an absolute value on 𝔸^{×} by |(a_{ν})|_{𝔸} = ∏ |a_{ν}|_{k_{ν}}, which is a finite product. The modular function for parabolic subgroups of GL_{d}(𝔸) is defined as in the local case (see Equation (??), with |det (a)|_{𝔸} for each block), and likewise we have a unitary induction from parabolic subgroups, with similar properties to the local case.

Let s > 1 be a divisor of d, and let π be any cuspidal automorphic representation of GL_{d/s}(𝔸). The representation

(4.1) T_{s}(π) = Ind_{P_{s}(𝔸)}^{GL_{d}(𝔸)}(|det|𝔸^{\frac{1−s}{2}} π ⊕ |det|𝔸^{\frac{3−s}{2}} π ⊕ ⋯ ⊕ |det|𝔸^{\frac{s−1}{2}} π)

has a unique irreducible sub-representation J(T_{s}(π)) (here P_{s}(𝔸) is the parabolic subgroup of GL_{d}(𝔸) associated to the decomposition into s blocks of size d/s).

Theorem 4.2 ([?]). The residual spectrum of L^{2}(GL_{d}(k)\GL_{d}(𝔸)) consists of the representations J(T_{s}(π)) for proper divisors s | d and π a cuspidal representation of GL_{d/s}(𝔸).

Comparing Equations (??) and (??), the local ν-component of J(T_{s}(π)) is seen to be the (unique) irreducible sub-representation of M_{s}(π_{ν}), namely C(π_{ν}) which was defined in Subsection ??. From Remark ?? we then obtain

Corollary 4.3. (a) Every local component of a residual representation is non-tempered.

(b) If π is an irreducible automorphic representation of GL_{d} where one of its local components is tempered, then π is cuspidal (and in positive characteristic, all of its spherical components are tempered by Theorem ??).

4.2. The Jacquet-Langlands correspondence. Let D be a division algebra of degree d over k, and let D_{ν} = D⊗_{k}k_{ν}. Then by the Albert-Brauer-Hasse-Noether theorem, D_{ν} ≅ M_{d}(k_{ν}) for almost every completion k_{ν}. Let G′ = D^{×}, which is a form of inner type of G = GL_{d}. Let T denote the (finite) set of valuations θ such that D⊗k_{θ} is not split. We assume that for every θ ∈ T, D⊗k_{θ} is a division algebra.

There is an injective correspondence, called the local Jacquet-Lang-lands correspondence, which maps every irreducible, unitary representation ρ′ of G′(k_{θ}) (θ ∈ T) to an irreducible, unitary square-integrable (modulo the center) representation ρ = JL_{θ}(ρ′) of G(k_{θ}) (see [?] or [?, p. 29] for details).

If ϕ is a character of k_{θ}^{×}, then [?, p. 32]

(4.2)  JL_{θ}(ϕ ∘ det) = Sp_{d}(|·|_{k_{θ}}^{(1−d)/2} ϕ)

where Sp_{d} is defined in Subsection ??. Recall by Example ??, that C_{d}(|·|_{k_{θ}}^{(1−d)/2} ϕ) is a one-dimensional representation.

The global Jacquet-Langlands correspondence maps an irreducible automorphic representation π′ of G′(𝔸) to an irreducible automorphic representation π = JL(π′) of G(𝔸) which occurs in the discrete spectrum (see [?, p. 195]). If ν ∉ T, then

JL(π′)_{ν} ≅ π′_{ν}.

Note that the restrictions of cond(π) and cond(π′) to V − T are equal.

The situation in the other local components is as follows: let θ ∈ T, and consider the component π′_{θ} of π′. The local Jacquet-Langlands correspondence maps π′_{θ} to an irreducible square-integrable representation JL_{θ}(π′_{θ}) of G(k_{θ}), which is by Proposition ?? a generalized Steinberg representation, of the form Sp_{s}(ψ) for some divisor s | d and supercuspidal representation ψ of GL_{d/s}(k_{θ}). Then JL(π′)_{θ} is isomorphic to either Sp_{s}(ψ) or C_{s}(ψ).

Theorem 4.4 ([?, p. 196]). The image of JL (for a fixed D) is the set of irreducible automorphic representations π of GL_{d}(𝔸) such that π occurs in the discrete spectrum and for every θ ∈ T there is a positive integer s_{θ}|d and an irreducible super-cuspidal representation ψ_{θ} of GL_{d/s_{θ}}(k_{θ}) such that π_{θ} is isomorphic to either Sp_{s_{θ}}(ψ_{θ}) or C_{s_{θ}}(ψ_{θ}).

Throughout the book [?], the authors assume characteristic zero. However, see Remark ??.

5. Proofs of Theorems ?? and ??

Let k be a global field of prime characteristic, D a division algebra of degree d over k, G′ the algebraic group D^{×}/Z^{×} where Z is the center, and G = PGL_{d}.

Let T denote the set of ramified primes, namely valuations θ for which D_{θ} = D⊗k_{θ} is non-split. We again assume that for such primes, D_{θ} is a division algebra. It follows that G′(k_{θ}) is compact for θ ∈ T. The valuation θ extends uniquely to a valuation of D_{θ}, and we let O_{D_{θ}} denote the ring of integers there.

The group G′(O_{θ}) depends on the specific embedding G′(k) ↪ GL_{r}(k), namely, G′(k_{θ}) is the subgroup of GL_{r}(k_{θ}) defined by the equations defining G′(k), and G′(O_{θ}) = G′(k_{θ}) ∩ GL_{r}(O_{θ}). For most of our applications the precise embedding is irrelevant (G′(O_{θ}) is well defined up to commensurability anyway). However, for Theorem ??(b), we need the embedding to satisfy

(5.1)  G′(O_{θ}) ⊇ k_{θ}^{×}O_{D_{θ}}^{×}/k_{θ}^{×},

where both groups are viewed as subgroups of G′(k_{θ}) = (D⊗_{k}k_{θ})^{×}/k_{θ}^{×}, which is embedded in GL_{r}(k_{θ}) for some r.

This condition is in fact satisfied by a natural embedding. Let E be a cyclic extension of dimension d over k, which is unramified at every θ ∈ T (the existence of E is guaranteed by Grunwald’s theorem for function fields [?, Chap. 10]). From Albert-Brauer-Hasse-Noether theorem it follows that E is a splitting field of D, making D a cyclic division algebra. Moreover there is an element z ∈ D such that D = E[z] and conjugation by z is an automorphism of E, generating Gal(E/k).

Let e_{1}, …, e_{d} be an integral basis of E/k (with respect to every θ ∈ T). Then, for every valuation θ ∈ T, O_{E_{θ}} = ∑ O_{θ}e_{i}, where E_{θ} = E⊗_{k}k_{θ} and O_{E_{θ}} is its ring of integers. Now, z can be chosen so that O_{D_{θ}} = O_{E_{θ}}[z] = ∑_{i,j} O_{θ}e_{i}z^{j}. The left regular representation of D via the basis {e_{i}z^{j}} defines an embedding D^{×}→GL_{d^{2}}(k) which sends O_{D_{θ}}^{×} into GL_{d^{2}}(O_{θ}) (and central elements to scalar matrices). Composing this with the adjoint representation of PGL_{d^{2}}(k) (into GL_{d^{4}}(k)), we obtain an embedding of G′(k) = D^{×}/k^{×} which satisfies Equation (??).

Lemma 5.1. For θ ∈ T, O_{D_{θ}}^{×} is normal in D_{θ}^{×}, and D_{θ}^{×}/k_{θ}^{×}O_{D_{θ}}^{×} ≅ ℤ/d.

Proof. The uniformizer ϖ of k_{θ} is a uniformizer for E_{θ} as well, and (by choosing the generator σ ∈ Gal(E_{θ}/k_{θ}) appropriately) we may assume D_{θ} = E_{θ}[z] where z^{d} = ϖ and conjugation by z induces σ. Since z normalizes O_{D_{θ}}^{×}, this is a normal subgroup of D_{θ}^{×}.

The elements of value zero in O_{D_{θ}} are invertible there, so every element of D_{θ}^{×} is of the form cz^{i} for some c ∈ O_{D_{θ}}^{×} and an integer i. Such an element is equivalent to z^{i} in D_{θ}^{×}/k_{θ}^{×}O_{D_{θ}}^{×}, and z^{d} = ϖ ∈ k_{θ}^{×}. Finally, z^{i} induces a non-trivial automorphism on E for for every 0 < i < d, so the order of z modulo the center is equal to d. □

By our assumption (??), the lemma implies that G′(k_{θ})/G′(O_{θ}) is a quotient of ℤ/d.

For i⃗ ∈ I set Γi⃗ = G′(R_{0}, Ii⃗), as in Equation (??). For i⃗, j⃗ ∈ I, we say that i⃗ ≤_{T} j⃗ if i_{ν} ≤ j⃗_{ν} for every ν ∈ V−T.

Proposition 5.2. Let i⃗ ∈ I. The complex Γi⃗ \B is Ramanujan iff every spherical infinite-dimensional ν_{0}-component of an irreducible automorphic discrete representation π′ of G′(𝔸) with cond(π′) ≤_{T} i⃗, is tempered.

Proof. This follows immediately from Propositions ?? and ?? (and Remark ??). □

We can now prove the theorems stated in the Introduction.

Proof of Theorems ?? and ??(a). Write the given ideal of R_{0} as I = Ii⃗ for i⃗ ∈ I (see Equation (??)). Let π′ be an irreducible discrete automorphic representation of G′(𝔸) with cond(π′) ≤_{T} i⃗, and assume ρ = π′_{ν_{0}} is spherical and infinite-dimensional. By Proposition ??, Γ(I)\B is Ramanujan iff in all such cases π′_{ν_{0}} is tempered.

By the Jacquet-Langlands correspondence, there is an irreducible automorphic sub-representation π of L^{2}(G(k)\G(𝔸)) such that π_{ν} = π′_{ν} for every ν ∉ T. In particular, π_{ν_{0}} = π′_{ν_{0}}.

Assume d is prime, then all the infinite dimensional automorphic representations of G(𝔸) are cuspidal, so π is cuspidal. By Lafforgue’s Theorem ??, the components of a cuspidal representation are tempered. Therefore, ρ = π_{v_{0}} is tempered, and Theorem ?? is proved.

Now let d be arbitrary, and assume i_{θ} = 0 for some θ ∈ T (namely I is prime to θ). Thus, π′_{θ} has a G′(O_{θ}) - fixed vector. By Lemma ??, G′(O_{θ}) is normal in G′(k_{θ}), and π′_{θ} is an irreducible representation of the cyclic quotient, so it is one-dimensional. Write π′_{θ} = ϕ ∘ det for a suitable character ϕ: k_{θ}^{×}→ℂ (of order d), where here det stands for the reduced norm of G′(k_{θ}).

By Equation (??) we have that JL_{θ}(π′_{θ}) = JL_{θ}(ϕ ∘ det) = Sp_{d}(ψ) for the character ψ = |·|^{(1−d)/2} ϕ of k_{θ}^{×}. By Example ??, C_{d}(ψ) is one-dimensional.

As mentioned in Subsection ??, π_{θ} is isomorphic to either Sp_{s}(ψ) or C_{s}(ψ), but π_{θ} cannot be one-dimensional (by Remark ??). Therefore, π_{θ} = Sp_{s}(ψ), which is square-integrable (Proposition ??) and, in particular, tempered. Now, Corollary ??(a) implies that π is cuspidal, and by Theorem ??, π_{ν_{0}} = ρ is tempered too. □

Proof of Theorem ??(b). By Proposition ??, we need to find an irreducible sub-representation π′ of L^{2}(G′(k)\G′(𝔸)) such that π′_{ν_{0}} is spherical and non-tempered. Then π′_{ν_{0}} would be a sub-representation of L^{2}(G′(R_{0}, I)\G(F)) for some I⊲R_{0}, and G′(R_{0}, I)\B would not be Ramanujan.

We use the following result, which is a variant of a special case of [?, Thm. 2.2].

Proposition 5.3. Let T = {θ_{1}, …, θ_{t}} and ν_{1} ∉ T be valuations of k. For i = 1, …, t, let ψ_{i} be a super-cuspidal representation of PGL_{m}(k_{θ_{i}}), where m > 1 is fixed.

Then, there exists an automorphic cuspidal representation π of PGL_{m}(𝔸), such that π_{θ_{i}} = ψ_{i} for i = 1, …, t, and π_{ν′} is spherical for every valuation ν′ ∉ T ∪ {ν_{1}}.

Proof. Here we let G denote the group PGL_{m}. Let f_{θ_{i}} be matrix coefficients of ψ_{i}, and let U_{θ_{i}} denote the (compact and open) support. For ν ∉ T ∪ {ν_{1}} let U_{ν} = G(O_{ν}), and choose an open compact subgroup U_{ν_{1}} of G(k_{ν_{1}}) such that U = ∏ U_{ν} ⊆ G(𝔸) intersects G(k) only in the identity. For ν ≠ θ_{1}, …, θ_{t}, let f_{ν} be the characteristic function of U_{ν}. Let f = ⊗f_{ν} ∈ L^{2}(G(𝔸)).

Define an operator R_{f} : L^{2}(G(k)\G(𝔸))→L^{2}(G(k)\G(𝔸)) by

R_{f}φ(g) = ∫_{G(𝔸)} f(g^{−1}x)φ(x)dx.

The image of R_{f} is in the discrete spectrum. Let π be an irreducible representation in the image, then π_{θ_{i}} = ψ_{i} and in particular π is cuspidal. Moreover, f_{ν′} is a fixed vector of π_{ν′} so these are spherical for every ν′ ∉ T ∪ {ν_{1}}. It remains to show that R_{f} ≠ 0:

R_{f}φ(g) = ∫_{G(k)\G(𝔸)} K_{f}(g, x)φ(x)dx

where K_{f}(g, x) = ∑_{γ∈G(k)} f(g^{−1}γx), which is a finite sum since f is compactly supported. But K_{f}(1, 1) = f(1) + ∑_{1≠γ∈G(k)} f(γ) = 1, showing that K_{f} ≠ 0 and R_{f} ≠ 0. □

For T we take the usual set of places in which D remains a division algebra, and we choose an arbitrary ν_{1} ∉ T ∪ {ν_{0}}.

Now pick any proper divisor s of d. For every i = 1, …, t choose a super-cuspidal representation ψ_{i} of PGL_{d/s}(k_{θ_{i}}), and let π be the representation of PGL_{d/s}(𝔸) given by Proposition ??; in particular π_{ν_{0}} is spherical. Then let π̃ = T_{s}(π), as in Equation (??), and let \overline{π} = J(π̃) be its unique irreducible sub-representation. By Proposition ??, \overline{π} is in the residual spectrum, and in particular \overline{π}_{ν_{0}} is spherical (since ν_{0} ≠ ν_{1}) and non-tempered (Corollary ??(a)).

Now, for every i = 1, …, t, \overline{π}_{θ_{i}} = C_{s}(|det|_{F}^{(1−s)/2} ψ_{i}) (see the remark preceding Corollary ??), so by Theorem ??, \overline{π} is in the image of the Jacquet-Langlands correspondence, corresponding to a representation \overline{π}′ of G′(𝔸) where G′ = D^{×}/Z^{×}. But \overline{π}_{ν_{0}} = \overline{π}′_{ν_{0}}, so this component is spherical and not tempered. □

6. Outer forms

Theorem ?? (especially when compared to Theorem ??(b)) may suggest that if d is an odd prime, then every finite quotient of the Bruhat-Tits building B = B_{d}(F) is Ramanujan, where F is a local field.

Indeed, if Y is such a finite quotient of B, then the fundamental group Γ_{1} = π_{1}(Y) acts on B, the universal cover of Y, and Y = Γ_{1}\B. By a well known result of Tits, Aut(B) is G = PGL_{d}(F), up to compact extension. It seems likely that Γ_{1} has a subgroup of finite index Γ which is contained in G, and the corresponding finite cover of Y can be obtained as Γ\G/K. Now, by Margulis’ arithmeticity theorem [?], Γ is an arithmetic lattice of G.

A well known conjecture of Serre [?] asserts that arithmetic lattices of G (where d ≥ 3) satisfy the congruence subgroup property. This essentially means that every finite index subgroup is a congruence subgroup. If Γ is of inner type, our Theorem ?? applies to it, and shows that the quotients are really Ramanujan. However, there are other arithmetic subgroups (see for example the classification of the k-forms of GL_{d} in [?, III.1.4]).

The outer forms of PGL_{d} all come from the following general construction: let k be a global field, k′/k a quadratic separable extension, and A a k′-central simple algebra with an involution u ↦ u^{∗} which induces the non-trivial automorphism of k′/k on the center of A. Let N_{k′/k} denote the norm map. The algebraic group G′ = {u ∈ A: uu^{∗} = 1}/Z (where Z = Ker(N_{k′/k}) is the center) gives a form of PGL_{d}. Now, if d is a prime, A may be either a division algebra, or the matrix algebra M_{d}(k′). The second case corresponds to Hermitian forms [?], i.e. G′ is PU_{d}(q, k′) = {a ∈ M_{d}(k′) : q(a(v)) = q(v)}/Z of operators preserving the Hermitian form q: (k′)^{d}→k′. In this situation, the involution on A is a ↦ b^{−1}ā^{t}b, where b is a skew-symmetric matrix representing q.

But, if char k = p > 0, every Hermitian form over k represents 0 if d ≥ 3. Indeed this is known to be true for local fields [?, Sec. 4.2] and by Hasse Principal [?, Sec. 4.5], this is also true for k. Now in order to form a cocompact arithmetic lattice Γ in PGL_{d}(F), the form G′ should be anisotropic (i.e. have k-rank zero), but if q represents 0 over k, the k-rank is greater than zero. Thus, there are no arithmetic lattices of Hermitian form type if d ≥ 3 and F is of positive characteristic (the situation is different for characteristic zero, see below).

On the other hand, the case when A is a division algebra is possible (e.g. the cyclic algebra A = 𝔽_{q^{d}}(t)[z | z^{d} = t] where z induces the Frobenius automorphism on 𝔽_{q^{d}}, is a division algebra with center k′ = 𝔽_{q}(t), and has an involution defined by z^{∗} = z^{−1} and α^{∗} = α for α ∈ 𝔽_{q^{d}}, which is non-trivial on k′). We do not know if Theorem ?? is valid in this case, but if it is true then together with Serre’s conjecture this would imply the remarkable possibility that if char F > 0 and d ≥ 3 is a prime, then all the finite quotients of B_{d}(F) are Ramanujan. We leave it, however, as an open problem.

For d = 2, i.e. PGL_{2}(F), all arithmetic lattices are of inner type, as the Dynkin diagram of A_{1} does not have graph automorphisms, so Theorem ?? applies for all lattices (a result which has been proved before by Morgenstern [?]). Still we have

Proposition 6.1. If d = 2, for every non-archimedean local field F, of any characteristic PGL_{2}(F) has cocompact (arithmetic) lattices, such that the quotient Γ\B_{2}(F) of the tree B_{2}(F) is not Ramanujan.

Proof. The group PGL_{2}(F) has cocompact (arithmetic) lattices, and these are virtually free (cf. [?]). Let Γ be a free cocompact lattice in PGL_{2}(F), so Γ′ = [Γ, Γ] is of infinite index in Γ. Let Γ_{n} be a sequence of finite index subgroups of Γ, such that ⋂ Γ_{n} = Γ′. By [?, Sec. 4.3], the graphs Γ_{n}\B are not expanders, let alone Ramanujan graphs . Of course, in light of Theorem ?? (or [?]) for positive characteristic, and [?, Thm. 7.3.1] (see also [?]) for zero characteristic, almost all the Γ_{n} are non-congruence subgroups. □

Lemma 6.2. Let F be a local nonarchimedean field of characteristic zero. For every d ≥ 2, there exists a number field k with a quadratic extension k′ such that k ⊆ k′ ⊆ F, and an anisotropic Hermitian form q of dimension d over k′/k.

Proof. Let p be the prime such that ℚ_{p} ⊆ F. Choose a natural number δ > 0 such that −δ is a quadratic residue modulo p if p is odd (e.g. δ = p−1), and take δ = 7 if p = 2. Let k = ℚ and k′ = ℚ[\sqrt{−δ}], and let u ↦ ū denote the non-trivial automorphism of k′/k. Let q(u_{1}, …, u_{d}) = u_{1}ū_{1} + ⋯ + u_{d}ū_{d}. Writing u_{i} = x_{i} + \sqrt{−δ}y_{i} for x_{i}, y_{i} ∈ ℚ, we have that q(u_{1}, …, u_{d}) = x_{1}^{2} + ⋯ + x_{d}^{2} + δ(y_{1}^{2} + ⋯ + y_{d}^{2}), which does not represent zero even over ℝ. □

Proof of Theorem ??. Let k′/k be the quadratic extension and q the anisotropic Hermitian form as in the lemma, and let G′ = PU_{d}(q) and G = PGL_{d}. Then G′(F) ≅ G(F), because k′ ⊆ F, so k′⊗_{k}F = F × F and

G′(F) = {(a, b) ∈ GL_{d}(F) × GL_{d}(F) : b = a^{∗}}/Z^{×} = G(F).

Choose Γ = G′(R_{0}) (where R_{0} is as defined in Equation (??)) and ν_{0} is the valuation on F. This is a cocompact lattice of G′(F) since G′(k) = PU_{d}(q, k) has rank zero. Moreover if we let q_{1} denote the sum of the first d−1 terms in a diagonal form of q, then q_{1} does not represent zero, and setting H′ = PU_{d}(q_{1}), H′(F) = PU_{d}(q_{1}, F) embeds in G′(F) as (d − 1) × (d − 1) matrices (and is isomorphic to H(F) = PGL_{d−1}(F) for the same reasons as for q). We deduce that Λ = Γ ∩ H′(F) is a cocompact lattice in H′(F) = H(F).

By Proposition ?? it remains to find a spherical non-tempered subrepresentation ρ of L^{2}(Γ_{I}\G(F)) for a congruence subgroup Γ_{I}.

Now, since Λ\H(F) is compact,

L^{2}(GL_{d−1}(F)\GL_{d}(F)) = L^{2}(H(F)\G(F)) ⊆ L^{2}(Λ\G(F)).

The group GL_{d}(F) acts on V ⊕ V^{∗} where V = F^{d} and V^{∗} is the dual space. Fixing e_{1} ∈ V, the stabilizer of e_{1}⊕e_{1}^{∗} is isomorphic to GL_{d−1}(F), so L^{2}(PGL_{d−1}(F)\PGL_{d}(F)) ⊆ L^{2}(V ⊕ V^{∗}) ≅ L^{2}(V ⊗F^{2}). Now, using the action of GL_{d}×GL_{2} on V ⊗F^{2}, one can prove that L^{2}(V ⊗F^{2}) is the direct integral of ρ′⊗ρ over tempered ρ ∈ ĜL_{2} (the unitary dual), where ρ′ is the representation of GL_{d} obtained by inducing ρ⊗id_{d−2} from GL_{2}. In particular ρ′ is not tempered if d ≥ 4 (since id_{s} is non-tempered if s ≥ 2). Thus, L^{2}(PGL_{d−1}(F)\PGL_{d}(F)) has spherical non-tempered sub-representations. We thank R. Howe for this argument.

For an ideal I⊲R_{0}, let Λ_{I} = ΛΓ(I). The Λ_{I} have finite index in Γ and so are cocompact in G(F). Moreover, ∩_{I}Λ_{I} = Λ.

Now, L^{2}(Λ\G(F)) is weakly contained in ∪_{I}L^{2}(Λ_{I}\G(F)) [?],[?], so for some I⊲R_{0}, L^{2}(Λ_{I}\PGL_{d}(F)) contains a spherical non-tempered sub-representation (which is discrete since Λ_{I} is cocompact). It follows that Λ_{I}\B(F) as well as Γ(I)\B are non-Ramanujan. □

A final remark is in order: so far all the Ramanujan complexes constructed were quotients of Ã_{d−1}(F) where F is an arbitrary local field of positive characteristic. For characteristic zero the problem is still open, except for d = 2. Of course one hopes eventually to define and construct Ramanujan complexes as quotients of the Bruhat-Tits buildings of other simple groups as well.

References

[AT] E. Artin and J. Tate, Class Field Theory, W.A. Benjamin, 1967.

[B1] C.M. Ballantine, Ramanujan Type Buildings, Canad. J. Math. 52(6), 1121– 1148, (2000).

[B2] C.M. Ballantine, A Hypergraph with Commuting Partial Laplacians, Canad. Math. Bull. 44(4), 385–397, (2001).

[Bu] D. Bump, Automorphic Forms and Representations, Cambridge SAM 55, 1998.

[BLS1] M. Burger, J.-S. Li and P. Sarnak, Ramanujan duals and automorphic spectrum, unpublished, (1990).

[BLS2] M. Burger, J.-S. Li and P. Sarnak, Ramanujan duals and automorphic spectrum, Bull. AMS 26(2), 253–257, (1992).

[C] P. Cartier, Representations of p-adic Groups: A survey, Proc. Symposia Pure Math. 33(1), 111–155, (1979).

[Cw] D.I. Cartwright, Spherical Harmonic Analysis on Buildings of Type Ã_{n}, Monatsh. Math. 133, 93–109, (2001).

[CM] D.I. Cartwright and W. Młotkowski, Harmonic Analysis for groups acting on triangle buildings, J. Austral. Math. Soc. (A) 56(1), 345–383, (1994).

[CS] D.I. Cartwright and T. Steger, Elementary Symmetric Polynomials in Numbers of Modulus 1, Canad. J. Math. 54(2), 239–262, (2002).

[CSZ] D.I. Cartwright, P. Solé and A. Żuk, Ramanujan Geometries of type Ã_{n}, Discrete Math. 269, 35–43, (2003).

[Gr] Y. Greenberg, On the spectrum of graphs and their universal covering (Hebrew), Doctoral Dissertation, Hebrew University, Jerusalem, 1995.

[GZ] R.I. Grigorchuk and A. Żuk, On the asymptotic spectrum of random walks on infinite families of graphs, Random walks and discrete potential theory (Cortona, 1997), Sympos. Math. XXXIX, Cambridge Univ. Press, Cambridge, 188–204, (1999).

[HT] R. Harris and R. Taylor, The Geometry and Cohomology of Simple Shimura Varieties, Anals of Math. Studies. 151, Princeton Univ. Press, 2001.

[JL] B.W. Jordan and R. Livne, The Ramanujan property for regular cubical complexes, Duke Mathematical Journal 105(1), 85–103, (2000).

[Kn] A.W. Knapp, Representation theory of semisimple groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1986.

[L] L. Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands (French), Invent. Math. 147(1), 1–241, (2002).

[LRS] G. Laumon, M. Rapoport, and U. Stuhler, D-elliptic sheaves and the Langlands correspondence, Invent. Math. 113(2), 217–338, (1993)

[Li] W.-C. W. Li, Ramanujan Hypergraphs, Geom. Funct. Anal. 14, 380-399, (2004).

[Lu1] A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, Progress in Math. 125, Birkhäuser, 1994.

[Lu2] A. Lubotzky, Cayley graphs: eigenvalues, expanders and random walks, Surveys in combinatorics (Stirling), London Math. Soc. Lecture Notes Ser. 218, 155–189, (1995).

[LPS] A. Lubotzky, R. Philips and P. Sarnak, Ramanujan graphs, Combinatorica 8, 261–277, (1988).

[LSV] A. Lubotzky, B. Samuels and U. Vishne, Explicit constructions of Ramanujan complexes, European J. of Combinatorics, to appear.

[M] I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2^{nd} edition, Oxford Math. Monographs, 1995.

[Ma1] G. Margulis, Explicit group-theoretic construction of combinatoric schemes and their applications in the construction of expanders and concentrators (Russian), Problemy Peredachi Informatsii, 24(1), 51–60. Trans. Prob. Inform. Transmission, 24(1), 39–46, (1988).

[Ma2] G. Margulis, Discrete subgroups of semisimple Lie groups, Results in Mathematics and Related Areas (3), 17, Springer-Verlag, 1991.

[Mo] M. Morgenstern, Existence and explicit constructions of q+1 regular Ramanujan graphs for every prime power q, J. Combin. Theory Ser. B 62(1), 44–62, (1994).

[MW] C. Mœglin and J.-L. Waldspurger, Le spectre rèsidual de GL_{n}, Ann. Scient. Èc. Norm. Sup. 4^{e} sèrie, 22, 605–674, (1989).

[P] G.K. Pedersen, Analysis Now, GTM 118, Springer, New York, 1989.

[PR] V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Pure and Applied Mathematics 139, Academic Press, 1994.

[Pr] G. Prasad, Strong approximation for semi-simple groups over function fields, Ann. Math. 105, 553–572, (1977).

[R] M. Rapoport, The mathematical work of the 2002 Fields medalists: The work of Laurent Lafforgue, Notices of the AMS, 50(2), 212–214, (2003).

[Ro] J. Rogawski, Representations of GL_{n} and division algebras over a p-adic field, Duke Math. J. 50, 161–196, (1983).

[Sa] A. Sarveniazi, Ramunajan (n_{1}, n_{2}, ..., n_{d−1})-regular hypergraphs based on Bruhat-Tits Buildings of type Ã_{d−1}, arxiv.org/math.NT/0401181.

[Sc] W. Scharlau, Quadratic Forms, Queen’s papers in Pure and Applied Math. 22, Queen’s Univ., Kingston, Ontario, 1969.

[Se1] J.-P. Serre, Galois Cohomology, Springer, translated from the 1964 French text, 1996.

[Se2] J.-P. Serre, Le probléme des groupes de congruence pour SL_{2}, Ann. of Math. 92(2), 489–527, (1970).

[Se3] J.-P. Serre, Trees, 2^{nd} edition, Springer Mono. Math., 2003.

[T] M. Tadić, An External approach to unitary representations, Bull. Amer. Math. Soc. 28, 215–252, (1993).

[V] M.F. Vigneras, Correspondances entre representations automorphes de GL(2) sur une extension quadratique de GSp(4) sur Q, conjecture locale de Langlands pour GSp(4), Contemporary Math. 53, 463–527, (1986).

[Z] A.V. Zelevinsky, Induced Representations of Reductive p-adic groups II: on irreducible representations of GL_{n}, Ann. Sci. E.N.S. 13(4), 165–210, (1980).

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., Yale University, 10 Hillhouse Av., New-Haven CT 06520, USA

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

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