On the Existence and Temperedness of Cusp Forms for SL_{3}(ℤ)

Stephen D. Miller^{∗} Department of Mathematics Yale University P.O. Box 208283 New Haven, CT 06520-8283

           stephen.miller@yale.edu

Abstract

We develop a partial trace formula which circumvents some technical difficulties in computing the Selberg trace formula for the quotient SL_{3}(ℤ)\SL_{3}(ℝ)/SO_{3}(ℝ). As applications, we establish the Weyl asymptotic law for the discrete Laplace spectrum and prove that almost all of its cusp forms are tempered at infinity. The technique shows there are non-lifted cusp forms on SL_{3}(ℤ)\SL_{3}(ℝ)/SO_{3}(ℝ) as well as non-self-dual ones. A self-contained description of our proof for SL_{2}(ℤ)\ℍ is included to convey the main new ideas. Heavy use is made of truncation and the Maass-Selberg relations.

1 Introduction

In the 1950s A. Selberg ([Sel1]) developed his trace formula to prove the existence of non-holomorphic, everywhere-unramified, cuspidal “Maass” forms. These are real-valued functions on the upper-half plane ℍ = {x + iy | y > 0} which are invariant under the action of SL_{2}(ℤ) by fractional linear transformations. Unlike the holomorphic cusp forms, which can all be explicitly described, no Maass form for SL_{2}(ℤ) has ever been constructed and they are believed to be intrinsically transcendental.

∗The author was supported by an NSF Postdoctoral Fellowship during this work.

The non-constant Laplace eigenfunctions in L^{2}(SL_{2}(ℤ)\ℍ) are all Maass forms. Since SL_{2}(ℤ)\ℍ is noncompact, their existence is no triviality; they very likely do not exist on the generic finite-volume quotient of ℍ (see [Sarnak]). However, Selberg showed that for any congruence subgroup Γ ⊂ SL_{2}(ℤ), the discrete spectrum is as large as one can expect – namely, it obeys the same asymptotics as the spectrum of a compact surface of the same size:

Theorem 1.1. (Selberg) Let 0 = λ_{0} < λ_{1} ≤ λ_{2} ≤ ⋯ be the discrete eigenvalues (with multiplicity) of the non-Euclidean laplacian Δ on Γ\ℍ. If Γ ⊂ SL_{2}(ℤ) is a congruence subgroup, then

N(T) = #\{λ_{j} ≤ T \} ∼ \frac{area(Γ\ℍ)}{4π}T  (1.1)

as T → ∞.

Similar asymptotics hold for any compact manifold by theorems of H. Weyl and others; we will refer to such an asymptotic for N(T) as the Weyl law for the space in question. For noncompact manifolds one can count the discrete and continuous spectra together, but it is extremely difficult to decouple the two. The main contribution of this paper is a technical novelty for separating them and proving the Weyl law for certain noncompact quotients.

One can generalize Maass forms to groups other than SL_{2}(ℝ) in a variety of ways. We are most interested in the linear group and so will focus our attention there, to G = SL_{3}(ℝ) in particular. Let K = SO_{3}(ℝ) be a maximal compact subgroup of G, H = G/K, Γ = SL_{3}(ℤ), and X = Γ\H. The ring D of (G-)invariant differential operators on the symmetric space H is a polynomial ring in two generators; it is explicitly described in [Bump], p. 32 for example. We will concentrate first on a particular element of D, the laplacian Δ, which is normalized so that its continuous spectrum on L^{2}(H) is the interval [1, ∞). Here again, the non-constant discrete eigenfunctions of Δ in L^{2}(X) are cusp forms (see Section 3 for definitions), which are the appropriate generalizations of Maass forms and our basic objects of study.

On the Existence and Temperedness of Cusp Forms for SL_{3}(Z) 3

By symmetry considerations, it is not difficult to prove the existence of odd cusp forms on SL_{2}(ℤ)\ℍ and SL_{3}(ℤ)\H. These comprise only half of the expected spectrum; the deeper issue is the existence of even cusp forms. The only other known cusp forms on SL_{3}(ℤ)\H are Gelbart-Jacquet lifts ([GelJac]) of the forms on SL_{2}(ℤ)\ℍ that Selberg discovered (many of which have been numerically identified–see [Hejhal]). However, no version of the trace formula has been used for the higher rank SL_{3}(ℤ)\H to prove the Weyl law for cusp forms, ala Selberg’s Theorem 1.1.^{1} For example, Arthur’s trace formula ([Art2]) computes the traces of certain integral operators over the discrete spectrum, but in it the discrete spectrum is paired with parabolic orbital integrals. Though it appears difficult, it would be very interesting to separate the two and give estimates on the size of the spectrum – especially because Arthur’s formula has been developed for general quotients.

We will shortcut the trace formula to prove the Weyl law for SL_{3}(ℤ)\H as well as some qualitative results about the spectrum.

Theorem 1.2. Let 0 = λ_{0} < λ_{1} ≤ λ_{2} ≤ ⋯ denote the eigenvalues of Δ on SL_{3}(ℤ)\H with multiplicity. Then

N(T) := #\{λ_{j} ≤ T \} ∼ \frac{vol(SL_{3}(ℤ)\H)}{Γ(7/2)} (\frac{T}{4π} )^{5/2}  (1.2)

as T → ∞.

These are the same asymptotics that the spectra of closed five-dimensional manifolds obey. This result confirms a conjecture of Sarnak ([Sarnak]), who asserted that the Weyl law holds for the cuspidal spectrum of the laplacian on any congruence quotient of SL_{n}(ℝ)/SO_{n}(ℝ). Theorem 1.2 is actually a special case of the more-general Theorem 5.3, a kind of equidistribution theorem which counts the two-dimensional joint spectrum of D lying in various sets. This generalization is used to establish many of the other findings below, so first we shall briefly describe the joint spectrum before stating the applications.

1In [StaWal] a proof of Theorem 1.2 is claimed. This proof appears to be incorrect because it is based on taking a specifically-chosen sequence of functions in the trace formula of [Wallace]. Unfortunately that trace formula is incomplete, because in its estimation of the continuous spectrum, it ignores the poles of the intertwining operators which arise from contour shifts. This seems to be a complicated problem to fix, though doing so would lead to much more specific information about the discrete spectrum (e.g. perhaps an error term).

The ring of invariant differential operators D on H is commutative and we may take an orthonormal set of common eigenfunctions

\frac{1}{\sqrt{area(SL_{3}(ℤ)\H)}} = ϕ_{0}, ϕ_{1}, ϕ_{2}, …  (1.3)

such that

Δϕ_{j} = λ_{j}ϕ_{j}.

Each ϕ_{j} in turn induces a homomorphism

λ : D → ℂ, where Dϕ = λ(D)ϕ, D ∈ D.

This map is best described by “principal series” or “Langlands” parameters

{ℓ_{1}, ℓ_{2}, ℓ_{3}}, ℓ_{1} + ℓ_{2} + ℓ_{3} = 0

(see Section 4 for more details). It is possible to compute λ(D) in terms of these spectral parameters, e.g.

λ(Δ) = 1 − \frac{ℓ_{1}^{2} + ℓ_{2}^{2} + ℓ_{3}^{2}}{2} .  (1.4)

The real parts of the ℓ_{i} are bounded, for example by a trivial bound from representation theory:

|Re ℓ_{i}| < \frac{1}{2} ([JacSha]).  (1.5)

Hence the Weyl law essentially counts the number of spectral parameters (ℓ_{1}, ℓ_{2}, ℓ_{3}) lying in a large ball. In Theorem 5.3 we compute the asymptotics of how many points of the joint spectrum lie in certain sets of various shapes and sizes. Along with some representation theory, we can count how many cusp forms obey given properties. The statement of Theorem 5.3 is technical so we will only describe its corollaries here.

A cusp form is called tempered if each Re ℓ_{i} = 0. The archimedean Ramanujan-Selberg conjecture asserts that all cusp forms on SL_{3}(ℤ)\H are tempered. The only nontrivial improvement of (1.5) on individual forms is the result of [LRS].

Theorem 1.3. Almost all of the cusp forms on SL_{3}(ℤ)\SL_{3}(ℝ)/SO_{3}(ℝ) obey the archimedean Ramanujan-Selberg conjectures in the sense that

lim_{T →∞} \frac{#\{λ_{j} ≤ T | Δϕ_{j} = λ_{j}ϕ_{j}, ϕ_{j} tempered\}}{#\{λ_{j} ≤ T \}} = 1.

For SL_{2}(ℤ)\ℍ the nontempered cusp forms are those with Laplace eigenvalue less than 1/4, and the Weyl law shows there are only a finite number of these. In higher rank, though, there may be nontempered forms with large Laplace eigenvalues. The archimedean Ramanujan-Selberg conjecture is an analog of Selberg’s famous “1/4 conjecture,” as it automatically implies that every cuspidal Laplace eigenvalue satisfies

λ^{cusp}(SL_{3}(ℤ)\H) ≥ λ_{1}(H) = 1.

In fact, this has already been proved in [Miller], but without establishing temperedness itself. The technique here is different and further-reaching in that it works whenever the Weyl law can be proved by our method.

Corollary 1.4. If

N^{cusp}(T) := #{λ_{j} ≤ T | Δϕ_{j} = λ_{j}ϕ_{j} for a nonzero cusp form ϕ_{j}},

then

N^{cusp}(T) ∼ N(T) ∼ \frac{vol(SL_{3}(ℤ)\H)}{Γ(7/2)} (\frac{T}{4π} )^{5/2}

as T → ∞.

Proof: The only other eigenfunctions are residues of Eisenstein series, which are not tempered, and Theorem 1.3 shows these have measure zero.

Another and more-precise reason uses the classification of the discrete spectrum established by Mœglin and Waldspurger ([MœWal], confirming the conjecture of [Jacquet]). A consequence is that the only non-cuspidal discrete eigenfunctions of Δ on SL_{p}(ℤ)\SL_{p}(ℝ)/SO_{p}(ℝ) are constants when p is prime. Thus for T ≥ 0, N(T) = N^{cusp}(T) + 1, as alluded to earlier. □

Corollary 1.5. There are “native” cusp forms on SL_{3}(ℤ)\H which are not Gelbart-Jacquet lifts of Maass forms from SL_{2}(ℤ)\ℍ. Moreover, these native forms comprise 100% of the spectrum in the sense that

lim_{T →∞} \frac{#\{λ_{j} ≤ T | Δϕ_{j} = λ_{j}ϕ_{j}, ϕ_{j} a Gelbart-Jacquet lift\}}{#\{λ_{j} ≤ T \}} = 0.

A cusp form is called self-dual if ϕ(g) = ϕ((g^{t})^{−1}) for all g ∈ G. Alternatively, its joint spectral parameters are

{ℓ_{1}, ℓ_{2}, ℓ_{3}} = {μ, 0, −μ} for some μ ∈ ℂ.

Theorem 1.6. There exists a non-self-dual cusp form on SL_{3}(ℤ)\SL_{3}(ℝ)/SO_{3}(ℝ), and in fact these are also of full measure:

lim_{T →∞} \frac{#\{λ_{j} ≤ T | Δϕ_{j} = λ_{j}ϕ_{j}, ϕ_{j} self-dual\}}{#\{λ_{j} ≤ T \}} = 0.

All Gelbart-Jacquet lifts are self-dual and it is conjectured that the converse is true.

In the next section we present a self-contained account of the main ideas of this paper, but specialized towards giving a new proof of Selberg’s Theorem 1.1 for Γ = SL_{2}(ℤ). The main improvement is a circumvention of the usual derivation of the trace formula. Usually one integrates the automorphic kernel only over a truncated fundamental domain, in order to prevent divergence from the parabolic orbital integrals and the Eisenstein series. After matching the growth rates of these two terms, a trace formula is obtained in the limit as the truncation parameter moves to infinity through the cusp. Here we merely truncate, and by positivity considerations arrive at an inequality giving a lower bound on the size of the spectrum. This gives the Weyl lower bound, but with the wrong constant – it involves the area of the truncated fundamental domain instead. By pushing the truncation parameter to infinity only at the end, we recover the correct lower bound; the upper bound has already been established by [Donnelly].

Acknowledgements

I wish to thank Peter Sarnak, under whose direction some of this material first appeared in my Princeton University dissertation. Also I am indebted to Donna Belli, Don Blasius, Sol Friedberg, Serge Lang, Alex Lubotzky, Jonathan Rogawski, Ze’ev Rudnick, Nolan Wallach, Gregg Zuckerman, and the referee for helpful comments. This work was supported by NSF graduate and post-doctoral fellowships, the Yale Hellman fund, and NSA grant MDA904-99-1-0046.

2 The Weyl Law for SL_{2}(ℤ)\ℍ

This is a self-contained section illustrating our method on SL_{2}(ℤ)\ℍ. We will reprove the Weyl Law (Theorem 1.1) using the same “partial trace” technique we use in higher rank for Theorem 1.2. We stress that the reason the partial trace is successful has little to do with the complications of higher rank per se, but rather because it serves as a substitute for the trace formula in counting the spectrum. As of yet, it appears difficult to give a direct proof of the Weyl law in higher rank using a trace formula, though this is how and why Selberg proved it for SL_{2}(ℤ)\ℍ to begin with. Our technique is also applicable to congruence covers of SL_{2}(ℤ)\ℍ, but we will focus on the full-level situation here in order to demonstrate how the main difficulty – non-compactness – is addressed. Basic references for this section are [Sel1], [Sel2], [Hejhal], and [Terras].

2.1 Definitions

The hyperbolic plane can be modeled by the complex upper-half plane

ℍ = {u + iv | v > 0}

with area element

dA = \frac{dudv}{v^{2}} ,

line element

dσ^{2} = \frac{du^{2} + dv^{2}}{v^{2}} ,

and laplacian

Δ = −v^{2} (\frac{∂^{2}}{∂u^{2}} + \frac{∂^{2}}{∂v^{2}} ) .

The group G = SL_{2}(ℝ) acts by isometries on ℍ by fractional linear transformations

( \begin{matrix}a & b \\ c & d \end{matrix} ) : z ↦ \frac{az + b}{cz + d}.

The stabilizer of any point in ℍ is a maximal compact subgroup isomorphic to K = SO_{2}(ℝ) = Stab(i), and so ℍ is isomorphic to the symmetric space G/K.

Let Γ = SL_{2}(ℤ), Γ̄ = P SL_{2}(ℤ) and X = Γ\ℍ = Γ̄\ℍ. The space X is noncompact but has finite volume, \frac{π}{3}. Thus the laplacian Δ has a continuous spectrum on L^{2}(X) furnished by Eisenstein series

E_{s}(z) = \frac{1}{2} ∑_{\{( \begin{matrix}1 & n \\ 0 & 1 \end{matrix} )|n∈ℤ\}\Γ} Im(γz)^{s}.

(This infinite series is only defined for Re(s) > 1, where it converges absolutely, but it has a meromorphic continuation to s ∈ ℂ.) The continuous spectrum comes from

E_{\frac{1}{2} +it}(z) , t ∈ ℝ.

Also, since Δ annihilates constant functions, it has a discrete spectrum. Our aim here is to count it. Let

0 = λ_{0} < λ_{1} ≤ λ_{2} ≤ ⋯

be the discrete spectrum of Δ on L^{2}(X) and

\sqrt{\frac{3}{π}} = ϕ_{0}, ϕ_{1}, ϕ_{2}, …

be an orthonormal set of eigenfunctions

ϕ_{j} ∈ L^{2}(X), Δϕ_{j} = λ_{j}ϕ_{j}.

We will prove

Theorem 2.1. (=Theorem 1.1)(Selberg) As T → ∞

N(T) = #\{λ_{j} ≤ T \} ∼ \frac{T}{12}.

2.2 Convolution Operators

Every g ∈ C_{c}^{∞}(K\ℍ) = C_{c}^{∞}(K\G/K) acts on f ∈ L^{2}(G/K) by convolution:

(L_{g}f)(x) = (f ∗ g)(x) = ∫_{G/K} f(y)g(y^{−1}x)dy.  (2.1)

Bi-K-invariant functions are functions of the distance of a point in ℍ to i. Their convolution operators form a commutative algebra which also commutes with the Laplace operator. These operators share common eigenfunctions as well, so Laplace eigenfunctions on ℍ = G/K play a special role. The simplest examples are furnished by power functions

Δv^{s} = s(1 − s)v^{s}  (2.2)

and by the bi-K-invariant “spherical” functions

ϕ̃_{s}(z) = ∫_{K} Im (kz)^{s}dk, ∫_{K} dk = 1,

Δϕ̃_{s}(z) = s(1 − s)ϕ̃_{s}.  (2.3)

The integral operators L_{g} also act on f ∈ L^{2}(Γ\G/K):

(L_{g}f)(x) =  ∫_{G/K} f(y)g(y^{−1}x)dy
= ∑_{γ∈Γ̄} ∫_{Γ\G/K} f(γ^{−1}y)g(y^{−1}γx)dy
=  ∫_{Γ\G/K} f(y)K(x, y)dy,  (2.4)

where

K(x, y) = ∑_{γ∈Γ̄} g(x^{−1}γy)  (2.5)

is called the automorphic kernel. Again, the L_{g} commute with each other and with Δ, so we may assume that the ϕ_{j} are taken to be an orthonormal set of eigenfunctions of all L_{g}, g ∈ C_{c}^{∞}(K\G/K) as well:

(L_{g}ϕ)(x) = ∫_{G/K} ϕ(y)g(y^{−1}x)dy = ĝ(ϕ) · ϕ(x).  (2.6)

Selberg showed that

∫_{K} ϕ_{j}(kg)dk = ϕ_{j}(I)ϕ̃_{s_{j}}(g), Δϕ_{j} = s_{j}(1 − s_{j})ϕ_{j}

and hence by averaging (2.6) over K he found a formula for ĝ(ϕ):

Proposition 2.2. (Selberg’s Uniqueness Principle) The L_{g}-eigenvalue ĝ(ϕ_{j}) of any ϕ_{j} as above depends only on its Laplace eigenvalue. Namely, if

Δϕ_{j} = (\frac{1}{4} − ν_{j}^{2})ϕ_{j},  (2.7)

then

L_{g}ϕ_{j} = ĝ(ν_{j})ϕ_{j},

where the Selberg transform

ĝ(ν) = ∫_{ℍ} g(u + iv)v^{1/2+ν} \frac{dudv}{v^{2}} .  (2.8)

Remark 2.3. Formula (2.8) is in fact (2.6) applied to the eigenfunction v^{1/2+ν}, and evaluated at the identity.

The kernel K(x, y) thus has two expansions:

Proposition 2.4. (Spectral Expansion) For g ∈ C_{c}^{∞}(K\G/K),

K(x, y) = ∑_{γ∈Γ̄} g(y^{−1}γx) =

∑ ; disc. L^{2} spec.^{j≥0} ĝ(ν_{j})ϕ_{j}(x)ϕ_{j}(y) + \frac{1}{4π} ∫_{ℝ} ĝ(it)E_{1/2+it}(x)\overline{E_{1/2+it}(y)}dt. (2.9)

We will now recall some analytic properties of the Selberg transform that will be needed in constructing our choices of g later. First is the inversion formula

g(x) = ∫_{ℝ} ĝ(it)ϕ̃_{1/2−it}(x)\frac{t tanh πt}{4π}dt.  (2.10)

Secondly, we need to clarify the relationship between the Selberg transform and the Mellin transform M. If

ḡ(v) = ∫_{ℝ} g(u + iv)du  (2.11)

denotes the “Harish transform,” then (2.8) expresses

ĝ(ν) = M (\frac{ḡ(x)}{\sqrt{x}}) (ν).  (2.12)

Through the Mellin inversion formula, formula (2.10) essentially amounts to a statement about the Harish transform. A more-precise fact was proven by Ehrenpreis and Mautner (see also [Gangolli]).

Theorem 2.5. ([EhrMau]). The Harish transform g ↦ ḡ is a bijection between

C_{σ}^{∞}(K\ℍ) = {g ∈ C_{c}^{∞}(K\ℍ) | supp(g) ⊂ K · i(σ^{−1}, σ)}

and

C_{σ}^{∞}(0, ∞) = {h ∈ C_{c}^{∞}(0, ∞) | h(v) = h(v^{−1}), supp(h) ⊂ (σ^{−1}, σ)}.

2.3 Construction of Functions

Recall that there exist smooth, non-negative functions on ℝ with arbitrarilysmall support whose Fourier transforms are also non-negative. (The easiest way to make one is to convolve a non-negative, compactly supported function with itself and rescale.) Using (2.12) and Theorem 2.5, we may fix some g ∈ C_{σ}^{∞}(K\ℍ) such that g ≥ 0, ∫_{ℝ} ĝ(it)dt = 1, and ĝ(it) ≥ 0 for t ∈ ℝ. We would like to scale ĝ so that it resembles the characteristic function of a large interval. However, in this non-euclidean setting it is difficult to control g and ĝ simultaneously. Instead, we will convolve ĝ(it) with χ_{[−T,T]}:

ĝ_{T} (it) = ∫_{−T}^{T} ĝ(it + ir)dr ≥ 0,  (2.13)

which is the Mellin transform of

\frac{ḡ(x)}{\sqrt{x}} ∫_{−T}^{T} x^{ir}dr ∈ C_{σ}^{∞}(0, ∞).

Hence Theorem 2.5 also guarantees the existence of some function g_{T} ∈ C_{σ}^{∞}(K\ℍ) whose Selberg transform

ĝ_{T} = ĝ_{T} .

We will use the g_{T} ’s in the automorphic kernel. Our construction allows us to conclude two important analytic properties:

Proposition 2.6.

max_{x∈ℍ}|g_{T} (x)| = g_{T} (i).

Proof: This follows from the inversion formula (2.10), the positivity of ĝ_{T} (2.13), and the inequality

|ϕ̃_{1/2+it}(x)| ≤ 1 = ϕ̃_{1/2+it}(i)

(see [DKV], (2.13)). □

Proposition 2.7. For any m ≥ 0,

ĝ_{T} (it) = \begin{cases}1 + O_{m} ((|T | − |t| + 1)^{−m}) & t < T, \\ O_{m} ((|t| − |T | + 1)^{−m}) & t ≥ T. \end{cases}

Proof: Since g is smooth, ĝ(it) = O ((1 + |t|)^{−m}) for any m ≥ 0. Hence

∫_{T}^{∞} ĝ(it)dt = O ((1 + T)^{−m}) , T > 0

and the proposition follows because of our normalization ∫_{ℝ} ĝ(it)dt = 1. □

2.4 Partial Trace

Let C ≥ 1 be a fixed truncation parameter. We will adjust it only in the final step of our analysis. Denote the usual fundamental domain for X as

F = {z ∈ ℍ | |z + z̄| ≤ 1 ≤ |z|}

and its truncation as

F_{C} = {z ∈ F | Im(z) ≤ C}.

Of course, F_{C} is compact and has area \frac{π}{3} − \frac{1}{C}; as C → ∞, it exhausts F. If X were compact, we would take the trace of the integral operator L_{g} by integrating the two expressions for K(x, x) in (2.9) over F. However, these integrals diverge and we instead take a partial trace over F_{C}:

∫_{F} K(x, x)dx = ∑_{γ∈Γ̄} ∫_{F_{C}} g(x^{−1}γx)dx

= ∑_{j≥0} ĝ(ν_{j}) ∫_{F_{C}} |ϕ_{j}(x)|^{2}dx + \frac{1}{4π} ∫_{ℝ} ĝ(it) ∫_{F_{C}} |E_{1/2+it}(x)|^{2}dxdt. (2.14)

Up until now we have essentially followed Selberg’s derivation of the trace formula. His next step is the understand the divergence of each side of (2.14) as a function of C. He is able to cancel it from each side and derive a trace formula for ∑ ĝ(ν_{j}) by letting C → ∞. As a final step, he uses a family of functions such as the g_{T} to deduce the spectral asymptotics.

This cancellation can be found in great generality (e.g. [Art2]), but the resulting formula is quite complicated. We will now deviate from this path by keeping Cfixed for now, and performing an analysis of (2.14) with the g_{T}. By noting

∫_{F_{C}} |ϕ_{j}(x)|^{2}dx ≤ ∫_{F} |ϕ_{j}(x)|^{2}dx = 1,  (2.15)

(2.14) becomes an inequality, giving a lower bound on ∑ ĝ(ν_{j}) (which, as we will make precise later, roughly equals N(T^{2})).

Since only a finite number of Γ-translates of F_{C} neighbor it, we may take σ to be small enough so that

g(x^{−1}γx) ≠ 0, x ∈ F_{C}

only for the finitely-many γ ∈ Γ̄ which have a fixed-point on the boundary of F_{C}.

Lemma 2.8. For σ small,

|∑_{γ≠I^{γ∈Γ̄}} ∫_{F_{C}} g(x^{−1}γx)dx| = O (σ^{2} max g) .

Proof: By the above remark, it suffices to show

∫_{ℍ} |g(x^{−1}γx)|dx = O (σ^{2} max g)

for γ a rotation. Since rotations sweep distant points a proportional amount, only x of distance O(σ) from the fixed point can have g(x^{−1}γx) ≠ 0. □

Applying (2.14), (2.15), and Proposition 2.6 we have that

∫_{F_{C}} g_{T} (x^{−1}Ix) ≤  ∑_{j≥0} ĝ_{T} (ν_{j})
+\frac{1}{4π} ∫_{ℝ} ĝ_{T} (it) ∫_{F_{C}} |E_{1/2+it}(x)|^{2}dxdt
+O(σ^{2}g_{T} (I)).  (2.16)

The next subsection is devoted to analyzing the Eisenstein series term. In Lemma 2.10 we show that it is O(T log T). Thus for some c_{0} > 0

(area(F_{C}) − c_{0}σ^{2}) g_{T} (I) = (area(F_{C}) − c_{0}σ^{2}) \frac{1}{4π} ∫_{ℝ} ĝ_{T} (it)t tanh(πt)dt

≤ ∑_{j≥0} ĝ_{T} (ν_{j}) + O(T log T).

Proof of Theorem 2.1:

We need to show

lim_{T →∞} \frac{N(T)}{T} = \frac{1}{12}.

The general theorem of [Donnelly], for example, shows that

lim sup_{T →∞} \frac{N(T)}{T} ≤ \frac{1}{12}.  (2.17)

By Proposition 2.7

∫_{ℝ} ĝ_{T} (it)t tanh(πt)dt = ∫_{−T}^{T} t tanh(πt)dt + O(T)
=  T^{2} + o(T^{2}).  (2.18)

Again using (2.17),

∑_{j≥0} ĝ_{T} (ν_{j}) ≤ N((T + \sqrt{T})^{2}) + O(1).

This proves

lim inf_{T →∞} \frac{N(T)}{T} ≥ \frac{\frac{π}{3} − \frac{1}{C} − c_{0}σ^{2}}{4π} .

Taking σ → 0 and C → ∞, we conclude the Weyl law. □

2.5 The Eisenstein Series Bounds

Recall that the Eisenstein series E_{s}(z) has the constant term

c(v, s) = ∫_{0}^{1} E_{s}(u + iv)du = v^{s} + ϕ(s)v^{1−s},

where

Z(s) = π^{−s/2}Γ(\frac{s}{2})ζ(s),

and

ϕ(s) = \frac{Z(2s − 1)}{Z(2s)} = \sqrt{π} \frac{Γ(s − \frac{1}{2})ζ(2s − 1)}{Γ(s)ζ(2s)}.

Define the truncated Eisenstein series for u + iv ∈ F as

Λ^{C}E_{s}(u + iv) = \{ \begin{matrix}E_{s}(u + iv) − c(v, s), & v > C \\ E_{s}(u + iv), & v ≤ C. \end{matrix}  (2.19)

The truncated Λ^{C}E_{s}(z) can be extended from F to Γ\ℍ; it decays rapidly as v → ∞, and so is in L^{2}(Γ\ℍ). Thus for any s_{1}, s_{2} ∈ ℂ which are not poles of the Eisenstein series E_{s}(z),

∫_{Γ\ℍ} Λ^{C}E_{s_{1}}(z)Λ^{C}E_{s_{2}}(z)\frac{dudv}{v^{2}} < ∞.

In fact the same is true if we only truncate E_{s_{1}}(z):

Lemma 2.9. With s_{1} and s_{2} as above,

∫_{Γ\ℍ} Λ^{C}E_{s_{1}}(z)Λ^{C}E_{s_{2}}(z)\frac{dudv}{v^{2}} = ∫_{Γ\ℍ} Λ^{C}E_{s_{1}}(z)E_{s_{2}}(z)\frac{dudv}{v^{2}} .

Proof: Unfolding, their difference

∫_{Γ\ℍ} Λ^{C}E_{s_{1}}(z) (E_{s_{2}}(z) − Λ^{C}E_{s_{2}}(z)) \frac{dudv}{v^{2}} = ∫_{C}^{∞} \frac{dv}{v^{2}} c(y, s_{2}) ∫_{0}^{1} (E_{s_{1}}(z) − c(v, s_{1})) du

= ∫_{C}^{∞} \frac{dv}{v^{2}} c(v, s_{2}) (c(v, s_{1}) − c(v, s_{1})) = 0.

□

It will simplify notation to introduce δ_{C}(z), the characteristic function of {z ∈ ℂ | Im z > C}. Then for z ∈ F

Λ^{C}E_{s}(z) = E_{s}(z) − δ_{C}(z)c(v, s) = ∑_{γ∈Γ_{∞}\Γ} [Im (γz)^{s_{1}} − δ_{C}(γz)c(Im (γz), s_{1})] .

Because the rightmost expression is automorphic, it agrees with Λ^{C}E_{s}(z) for all z ∈ ℍ; at most one of the terms indicated by the δ_{C} functions is actually subtracted. We can thus unfold this series representation of Λ^{C}E_{s_{1}}(z) in the integral:

∫_{Γ\ℍ} Λ^{C}E_{s_{1}}(z)E_{s_{2}}(z)\frac{dudv}{v^{2}} = ∫_{Γ_{∞}\ℍ} E_{s_{2}}(z) (v^{s_{1}} − δ_{C}(z)c(v, s_{1})) \frac{dudv}{v^{2}}

= ∫_{0}^{∞} \frac{dv}{v^{2}} (v^{s_{1}} − δ_{C}(z)c(v, s_{1})) ∫_{0}^{1} duE_{s_{2}}(z)

= ∫_{0}^{∞} (v^{s_{2}} + ϕ(s_{2})v^{1−s_{2}}) (v^{s_{1}} − δ_{C}(v) (v^{s_{1}} + ϕ(s_{1})v^{1−s_{1}})) \frac{dv}{v^{2}}

= ∫_{0}^{C} v^{s_{2}+s_{1}−2}dv + ϕ(s_{2}) ∫_{0}^{C} v^{s_{1}−s_{2}−1}dv
−ϕ(s_{1}) ∫_{C}^{∞} v^{s_{2}−s_{1}+1}dv − ϕ(s_{1})ϕ(s_{2}) ∫_{C}^{∞} v^{−s_{1}−s_{2}}dv

= \frac{C^{s_{2}+s_{1}−1}}{s_{2} + s_{1} − 1} + \frac{ϕ(s_{2})C^{s_{1}−s_{2}}}{s_{1} − s_{2}} + \frac{ϕ(s_{1})C^{s_{2}−s_{1}}}{s_{2} − s_{1}} + ϕ(s_{1})ϕ(s_{2})\frac{C^{1−s_{2}−s_{1}}}{1 − s_{2} − s_{1}} , (2.20)

provided Re s_{1} > 1 + Re s_{2}. Each side has a meromorphic continuation in s_{1}, s_{2}, and together are the Maass-Selberg relations.

Because the Eisenstein series and hence their truncations are holomorphic for Re(s_{1}) = Re(s_{2}) = \frac{1}{2}, the Maass-Selberg relations must have a removable singularity there. The limiting value is

∫_{Γ\ℍ} |Λ^{C}E_{1/2+it}(z)|^{2} \frac{dudv}{v^{2}} = 2 log C − \frac{ϕ′}{ϕ} (\frac{1}{2} + it) + \frac{C^{2it}ϕ(\frac{1}{2} − it) − C^{−2it}ϕ(\frac{1}{2} + it)}{2it}. (2.21)

The functional equations

Z(s) = Z(1 − s) , Z(s̄) = \overline{Z(s)}

show that

|ϕ(\frac{1}{2} + it)| = |\frac{Z(it)}{Z(it + 1)}| = 1.  (2.22)

Also,

\frac{ϕ′}{ϕ} (\frac{1}{2} + it) = 2 (\frac{Z′}{Z} (2it) + \frac{Z′}{Z} (−2it))  (2.23)

and

\frac{Z′}{Z} (s) = ∑_{ρ|Z(ρ)=0} \frac{1}{s − ρ} − \frac{1}{s} − \frac{1}{s − 1},  (2.24)

a sum over the poles and critical zeroes of ζ(s).

Lemma 2.10.

∫_{ℝ} ĝ_{T} (it) ∫_{F_{C}} |E_{1/2+it}(x)|^{2}dxdt = O(T log T).

Proof: By the definition of the truncation,

∫_{F_{C}} |E_{1/2+it}(x)|^{2}dx ≤ ∫_{F} |Λ^{C}E_{1/2+it}(x)|^{2}dx,

which is calculated in (2.21). In view of (2.22) and (2.23), it suffices to show

Re ∫_{ℝ} ĝ_{T} (it)\frac{Z′}{Z} (2it)dt = O(T log T).

This follows from (2.24) and the zero counting estimate (see Proposition 7.1 or [Titchmarsh])

Re ∫_{i(H−1)}^{i(H+1)} ∑_{ρ|Z(ρ)=0} \frac{ds}{s − ρ} = O(#\{ρ | H − 1 ≤ Im (ρ) ≤ H + 1\}) = O(log H).

□

3 Coordinates on SL_{3}(ℝ) and H = SL_{3}(ℝ)/SO_{3}(ℝ)

The Iwasawa decomposition G = SL_{3}(ℝ) = NAK states that every element g ∈ G can be uniquely factored as g = nak, where

n ∈ N_{0} = \{( \begin{matrix}1 & ∗ & ∗ \\ 0 & 1 & ∗ \\ 0 & 0 & 1 \end{matrix} ) ∈ G\} ,

a ∈ A_{0} = \{( \begin{matrix}a_{1} & 0 & 0 \\ 0 & a_{2} & 0 \\ 0 & 0 & a_{3} \end{matrix} ) ∈ G , a_{i} > 0\} ,

and

k ∈ K = SO_{3}(ℝ).

The minimal standard parabolic is

P_{0} = N_{0}A_{0} = A_{0}N_{0} = \{( \begin{matrix}∗ & ∗ & ∗ \\ 0 & ∗ & ∗ \\ 0 & 0 & ∗ \end{matrix} ) ∈ G\} .

There are two associate maximal standard parabolics

P_{1} = \{( \begin{matrix}∗ & ∗ & ∗ \\ ∗ & ∗ & ∗ \\ 0 & 0 & ∗ \end{matrix} ) ∈ G\} , P_{2} = \{( \begin{matrix}∗ & ∗ & ∗ \\ 0 & ∗ & ∗ \\ 0 & ∗ & ∗ \end{matrix} ) ∈ G\} .

Each can be decomposed uniquely as P_{i} = M′_{i}N_{i}A_{i} = N_{i}M′_{i}A_{i}, where

N_{1} = \{( \begin{matrix}1 & 0 & ∗ \\ 0 & 1 & ∗ \\ 0 & 0 & 1 \end{matrix} ) ∈ G\} ,

M′_{1} = \{( \begin{matrix}∗ & ∗ & 0 \\ ∗ & ∗ & 0 \\ 0 & 0 & ±1 \end{matrix} ) ∈ G\} ,

A_{1} = \{( \begin{matrix}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a^{−2} \end{matrix} ) ∈ G , a > 0\} ,

N_{2} = \{( \begin{matrix}1 & ∗ & ∗ \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) ∈ G\} ,

M′_{2} = \{( \begin{matrix}±1 & 0 & 0 \\ 0 & ∗ & ∗ \\ 0 & ∗ & ∗ \end{matrix} ) ∈ G\} ,

and

A_{2} = \{( \begin{matrix}a^{2} & 0 & 0 \\ 0 & a^{−1} & 0 \\ 0 & 0 & a^{−1} \end{matrix} ) ∈ G , a > 0\} .

The period of an automorphic function ψ ∈ L^{2}(Γ\G/K) along the parabolic P is defined as the integral

ψ_{P} (g) = ∫_{Γ∩N\N} ψ(ng)dn.  (3.1)

If all periods along each of the parabolics P_{0}, P_{1}, and P_{2} vanish, then ψ is called a cusp form.

We shall use the following sets of coordinates for the Lie algebras. Let

a_{0} = \{( \begin{matrix}h_{1} & 0 & 0 \\ 0 & h_{2} & 0 \\ 0 & 0 & h_{3} \end{matrix} ) | h_{1}, h_{2}, h_{3} ∈ ℝ , h_{1} + h_{2} + h_{3} = 0\}

≃ {H = (h_{1}, h_{2}, h_{3}) ∈ ℝ^{3} | h_{1} + h_{2} + h_{3} = 0}.

It has simple roots α_{1}, α_{2} ∈ a_{0}^{∗}, which are the following linear functions on a_{0}:

α_{1}(H) = h_{1} − h_{2} , α_{2}(H) = h_{2} − h_{3}.

There is also a third root α_{3} = α_{1} + α_{2} which acts by α_{3}(H) = h_{1} − h_{3}. We can similarly coordinatize

a_{0}^{∗} = {λ = (ℓ_{1}, ℓ_{2}, ℓ_{3}) ∈ ℝ^{3} | ℓ_{1} + ℓ_{2} + ℓ_{3} = 0}

and its complexification

a_{0ℂ}^{∗} = {λ = (ℓ_{1}, ℓ_{2}, ℓ_{3}) ∈ ℂ^{3} | ℓ_{1} + ℓ_{2} + ℓ_{3} = 0} .

Since these are subsets of ℂ^{3}, we will use the standard norm there to define norms in these spaces. The simple roots are bases of a_{0}^{∗} and a_{0ℂ}^{∗} as vector spaces over ℝ and ℂ, respectively. Also, they have corresponding co-roots

α_{1}^{∨} = (1, −1, 0) ∈ a_{0} , α_{2}^{∨} = (0, 1, −1) ∈ a_{0}.

The Weyl groups Ω(a_{0}) and Ω(a_{0}^{∗}) are isomorphic to the symmetric group S_{3}, and act in a compatible way by permuting the standard basis vectors in ℝ^{3} or ℂ^{3} (see the chart in the appendix for more details). The Cartan subgroups A_{1}, A_{2} of the maximal parabolics have one-dimensional Lie algebras, coordinatized by

a_{1} = {H = (h, h, −2h) | h ∈ ℝ} ≃ {h ∈ ℝ}

and

a_{2} = {H = (2h, −h, −h) | h ∈ ℝ} ≃ {h ∈ ℝ}.

These each have one-dimensional dual spaces, a_{1}^{∗} and a_{2}^{∗} respectively. Also, each dual has a special vector ρ, half the sum of the positive roots:

ρ_{0} = \frac{α_{1} + α_{2} + α_{3}}{2} : (h_{1}, h_{2}, h_{3}) ↦ h_{1} − h_{3},

ρ_{1} : (h, h, −2h) ↦ 3h,

and

ρ_{2} : (2h, −h, −h) ↦ 3h.

Under these coordinates ρ_{1} and ρ_{2} are naturally identified with α_{2} and α_{1}, respectively.

Each parabolic P_{i} has the Langlands decomposition

P_{i} = N_{i}A_{i}M′_{i}

and logarithm maps H_{i} : G → a_{i} such that

a = e^{H_{i}(a)} , a ∈ A_{i}  (3.2)

and

g ∈ N_{i}e^{H_{i}(g)}M′_{i}K.

These maps are well-defined despite the fact that the decomposition G = P_{i}K is in general not unique. Similarly for the maximal parabolics, there are maps m_{1} and m_{2} mapping G onto M′_{1}/(K ∩ M′_{1}) and M′_{2}/(K ∩ M′_{2}), respectively.

Truncation

Let Δ_{P} denote the set of simple roots which do not vanish identically on P:

Δ_{P_{0}} = {α_{1}, α_{2}} , Δ_{P_{1}} = {α_{2}} , Δ_{P_{2}} = {α_{1}}.

The group G can also be viewed as a parabolic, with Δ_{G} = {}.

We can now define Langlands’ truncation ([Lan1], see [Art1]). For any parabolic P = P_{0}, P_{1}, P_{2}, or G, define τ̂_{P} (x) to be the characteristic function of

{x = c_{1}α_{1}^{∨} + c_{2}α_{2}^{∨} ∈ a_{0} | c_{i} > 0 ∀α_{i} ∈ Δ_{P} }.

Thus, τ̂_{P_{0}} is the characteristic function of

{x = c_{1}α_{1}^{∨} + c_{2}α_{2}^{∨} ∈ a_{0} | c_{1}, c_{2} > 0},
τ̂_{P_{1}} of {x = c_{1}α_{1}^{∨} + c_{2}α_{2}^{∨} ∈ a_{0} | c_{2} > 0},
τ̂_{P_{2}} of {x = c_{1}α_{1}^{∨} + c_{2}α_{2}^{∨} ∈ a_{0} | c_{1} > 0},

and

τ̂_{G} of a_{0}.

Let C ∈ a_{0} be a fixed parameter. The truncation of an automorphic form ψ is a sum over all standard parabolic subgroups

(Λ^{C}ψ)(x) := ∑_{P}(−1)^{dim A} ∑_{γ∈Γ∩P \Γ} τ̂_{P} (H_{0}(γx) − C) ∫_{Γ∩N\N} ψ(nγx)dn

= ∑_{P}(−1)^{dim A} ∑_{γ∈Γ∩P \Γ} τ̂_{P} (H_{0}(γx) − C)ψ_{P} (γx),  (3.3)

which itself is clearly an automorphic form. It can be proven that it also decays rapidly in the cusp because of the way its constant terms have been removed (see [Art1]). Note that if ψ is a cusp form to begin with, by definition all its constant terms ψ_{P} vanish identically in proper parabolics, and thus Λ^{C}ψ = ψ.

Spectral Density

We will later need to use the spectral density function β(λ) on ia_{0}^{∗}, which is a constant multiple of

|\frac{Γ(\frac{1}{2} + λ(α_{1}^{∨}))}{Γ(λ(α_{1}^{∨}))} \frac{Γ(\frac{1}{2} + λ(α_{2}^{∨}))}{Γ(λ(α_{2}^{∨}))} \frac{Γ(\frac{1}{2} + λ(α_{3}^{∨}))}{Γ(λ(α_{3}^{∨}))}|^{2} .  (3.4)

This function will appear below in (4.2); see [DKV] for more information. For λ such that λ(α_{1}^{∨}), λ(α_{2}^{∨}), and λ(α_{3}^{∨}) are all large, Stirling’s formula shows that β(λ) behaves as a constant times |λ(α_{1}^{∨})λ(α_{2}^{∨})λ(α_{3}^{∨})|.

The spectral density function is normalized so that

∫_{∥λ∥≤T} β(λ)dλ ∼ \frac{T^{5/2}}{Γ(7/2)(4π)^{5/2}} .

One can also view this as normalizing the measure dλ.

4 Convolution operators

Every g ∈ C_{c}^{∞}(K\G/K) acts by convolution on L^{2}(G/K):

(L_{g}f)(x) = (f ∗ g)(x) = ∫_{G} f(y)g(y^{−1}x)dy = ∫_{G/K} f(y)g(y^{−1}x)dy.

The convolution operator L_{g} also acts on f ∈ L^{2}(Γ\G/K) by

(L_{g}f)(x) = ∫_{G/K} f(y)g(y^{−1}x)dy

= ∑_{γ∈Γ} ∫_{Γ\G/K} f(γ^{−1}y)g(y^{−1}γx)dy = ∫_{Γ\G/K} f(y)K(x, y)dy,

where

K(x, y) = ∑_{γ∈Γ} g(y^{−1}γx)

is the automorphic kernel.

Suppose furthermore that g(x) is real; then L_{g} is an operator on L^{2}(Γ\G/K) which commutes with the laplacian Δ and all other invariant differential operators. We may thus choose an orthonormal set

\frac{1}{\sqrt{vol(Γ\G/K)}} = ϕ_{0}, ϕ_{1}, ϕ_{2}, …

of common eigenfunctions of L_{g} and the ring of invariant differential operators D. Continuing, we may resolve any multiplicities that remain by using the Hecke operators, which commute with these other operators.

Proposition 4.1. (Selberg’s Uniqueness Principle)([Sel1]) If ϕ is a common eigenfunction of the ring of invariant differential operators, then

(L_{g}ϕ)(x) = ĝ(ϕ)ϕ(x),

where ĝ(ϕ) only depends on ϕ’s eigenvalues under D. In fact, one can always find some λ ∈ a_{ℂ}^{∗} such that the function ϕ_{λ} = e^{(λ+ρ)(H_{0}(x))} on G/K has the same eigenvalues as ϕ under any D ∈ D. This provides the formula

ĝ(ϕ) = ĝ(λ) = (L_{g}ϕ_{λ})(I) = ∫_{G/K} g(x)e^{(λ+ρ)(H_{0}(x))}dx.  (4.1)

If s is any permutation in the Weyl group Ω(a_{0}^{∗}), then in fact ϕ_{λ}(x) = e^{(λ+ρ)(H_{0}(x))} has the same eigenvalues as ϕ_{sλ}. The uniqueness principle thus implies that ĝ is invariant under Ω(a_{0}^{∗}). Formula (4.1) shows that the transform ĝ(λ) is the composition of an average over N_{0}, and a Fourier transform on A_{0}. More precisely, if g is a function on G/K, denote

ḡ(a) = ∫_{N_{0}} g(na)dn.

For a function f on A_{0}, define the Fourier transform as

(Ff)(λ) = ∫_{A_{0}} f(a)e^{λ(H_{0}(a))}da , λ ∈ ia_{0ℂ}^{∗}.

Then ĝ = F (ḡ(·)e^{−ρ(H_{0}(·))}).

We will make use of the following theorem in constructing our choices of functions g. Before stating it, let us introduce the notation

A(σ) = {a ∈ A_{0} | ∥H_{0}(a)∥ ≤ σ}.

Theorem 4.2. ([Gangolli]) The map g ↦ ḡ provides a bijection between the sets

{g ∈ C^{∞}(K\G/K) | supp g ⊂ KA(σ)K}

and

{h ∈ C^{∞}(A) | supp h ⊂ A(σ) and h(sa) = h(a) for all s ∈ Ω(a_{0})}.

Construction of functions

Let σ > 0 be fixed for the remainder of this section. It is possible to find a non-negative function g ∈ C^{∞}(K\G/K) which is supported in KA(σ)K, and whose transform ĝ is non-negative on the joint spectrum. Having such a function without the positivity requirement on ĝ is straightforward; one then rescales and convolves it with itself to achieve positivity (see lemma 6.2 of [DKV] for example). We shall again normalize g so that

∫∫_{ia_{0}^{∗}} ĝ(λ)dλ = 1.

Let us state a property of g which is implied by the positivity of ĝ on ia_{0}^{∗}:

Proposition 4.3. If g ∈ C_{C}^{∞}(K\G/K) is such that ĝ(λ) ≥ 0 for all λ ∈ ia_{0}^{∗}, then

max_{x∈G} |g(x)| = g(I).

Remark 4.4. As the proof will demonstrate, the analogous fact is true for the usual Euclidean Fourier transform.

Proof of Proposition 4.3: The proof uses the inversion formula

g(x) = ∫_{ia_{0}^{∗}} ĝ(λ)\overline{ϕ̃_{λ}(x)}β(λ)dλ,  (4.2)

where

|ϕ̃_{λ}(x)| ≤ ϕ̃_{λ}(I) = 1

is a spherical function (e.g. (2.3)) and β(λ) is the spectral density. Trivially

|g(x)| ≤ ∫_{ia_{0}^{∗}} ĝ(λ)β(λ)dλ = g(I).

□

Now if Σ is a measurable, bounded subset of ia_{0}^{∗} which is invariant under the Weyl group, define

ĝ_{Σ}(λ) = ĝ ∗ χ_{Σ}(λ) = ∫_{Σ} ĝ(λ − μ)dμ.  (4.3)

The function ĝ_{Σ}(λ) is roughly concentrated on Σ, especially for large Σ; we shall use it to estimate the spectrum in Σ. Also ĝ_{Σ}(λ) decays rapidly, since g_{Σ} is a smooth function of compact support. If Σ is open and its boundary has a finite Hausdorff length, then

ĝ_{tΣ}(λ) ≪ (1 + dist(λ, tΣ))^{−m}  (4.4)

(cf. p. 85 of [DKV]). Of course ĝ_{Σ}(λ) is the Fourier transform of

ḡ(a) ∫_{Σ} e^{(−μ−ρ)(H_{0}(a))}dμ,

which is a smooth function on A_{0} whose support is contained in A(σ). It is furthermore invariant under the action of the Weyl group Ω(a_{0}) because Σ is. Thus by Theorem 4.2 there exists a smooth, bi-K-invariant function g_{Σ} supported in KA(σ)K such that

\overline{g_{Σ}}(a) = ḡ(a) ∫_{Σ} e^{−μ(H_{0}(a))}dμ  (4.5)

and

ĝ_{Σ} = ĝ_{Σ}.

In summary, while not changing the support of our function g, we can still smear its transform ĝ over Σ. Of course g_{Σ} becomes more concentrated near the identity as Σ gets larger; in the classical Euclidean case this is analogous to multiplying a function by Fejer’s kernel. In (5.5) we will use a result analogous to Proposition 2.7 comparing ĝ_{Σ} to χ_{Σ}.

5 The Partial Trace

In addition to the discrete spectrum there is also a continuous spectrum, furnished by Eisenstein series. Because it is complicated we will just refer to its terms in the spectral expansion as “Eis_{g}(x, y)” until we need to be more explicit:

K(x, y) = ∑_{j≥0} ĝ(λ_{j})ϕ_{j}(x)ϕ_{j}(y) + Eis_{g}(x, y) = ∑_{γ∈Γ} g(x^{−1}γy).  (5.1)

Let F be a fundamental domain for Γ\G/K and F_{C} be a compact subset of F. Then

∫_{F_{C}} K(x, x)dx =  ∑_{γ∈Γ} ∫_{F_{C}} g(x^{−1}γx)dx
= ∑_{j≥0} ĝ(λ_{j}) ∫_{F_{C}} ϕ_{j}(x)^{2}dx + ∫_{F_{C}} Eis_{g}(x, x)dx (5.2)

Now,

∫_{F_{C}} ϕ_{j}(x)^{2}dx ≤ ∫_{F} ϕ_{j}(x)^{2}dx = 1

so

∑_{γ∈Γ} ∫_{F_{C}} g(x^{−1}γx)dx ≤ ∑_{j≥0} ĝ(λ_{j}) + ∫_{F_{C}} Eis_{g}(x, x)dx.  (5.3)

Now we will analyze the integrals

∫_{F_{C}} g(x^{−1}γx)dx

more systematically. Note that we do not group these into conjugacy classes as is done in the trace formula.

Proposition 5.1. For σ sufficiently small and supp g ⊂ A(σ), the integral

∫_{F_{C}} g(x^{−1}γx)dx = 0

for all but finitely many γ – those which have fixed points in the closure of F_{C}.

Proposition 5.2. If γ ≠ I has a fixed-point in the closure of F_{C} and supp g ⊂ A(σ), then

∫_{F_{C}} g(x^{−1}γx)dx ≪ | max g|vol(A(σ)).

Although the implied constant above depends on F_{C}, it may be taken to be independent of γ ∈ SL_{3}(ℤ) since only finitely many elements produce a non-zero integral.

Clearly

∫_{F_{C}} g(x^{−1}Ix)dx = vol(F_{C}) · g(I).

Now take g = g_{Σ} as in (4.5). Using Propositions 4.3 and 5.2, the inequality (5.3) becomes

vol(F_{C})g_{Σ}(I) + O (g_{Σ}(I)vol(A(σ))) ≤ ∑_{j=0}^{∞} ĝ_{Σ}(λ_{j}) + ∫_{F_{C}} Eis_{g_{Σ}}(x, x)dx. (5.4)

The proof of Theorem 8.5 in [DKV] shows that both

|∑_{j=0}^{∞} ĝ_{Σ}(λ_{j}) − #{Im λ_{j} ∈ Σ}| ≪ ∫_{BΣ} β(λ)dλ

and

|g_{Σ}(I) − ∫_{Σ} β(λ)dλ| ≪ ∫_{BΣ} β(λ)dλ,

BΣ = {λ ∈ ia_{0}^{∗} | dist(λ, ∂Σ) ≤ 1},  (5.5)

assuming Σ is a bounded, open, and Weyl-group invariant subset of ia_{0}^{∗}. Now to control the error term we shall assume Σ’s boundary is piece-wise smooth (cf. Lemma 8.7 of [DKV]). Then (3.4) and (5.5) show that for large t

g_{tΣ}(I) ∼ ∫_{tΣ} β(λ)dλ,

and by using (5.4),

(∫_{tΣ} β(λ)dλ) (vol(F_{C}) + O(vol(A(σ)))) (1 + o(1))

≤ #{Im λ_{j} ∈ tΣ} + ∫_{F_{C}} Eis_{g_{tΣ}}(x, x)dx.

We will see at the end of Section 7 that

∫_{F_{C}} Eis_{g_{tΣ}}(x, x)dx = o (∫_{tΣ} β(λ)dλ) .  (5.6)

Proof of Theorem 1.2: Taking t → ∞

lim inf_{t→∞} \frac{#\{Im λ_{j} ∈ tΣ\}}{∫_{tΣ} β(λ)dλ} ≥ vol(F_{C}) + O(vol(A(σ))).

We had insisted that supp g ⊂ A(σ), so taking σ → 0 and exhausting F through compact sets F_{C} we conclude that

vol(F) ≤ lim inf_{t→∞} \frac{#\{Im λ_{j} ∈ tΣ\}}{∫_{tΣ} β(λ)dλ}.  (5.7)

If

Σ = {λ = (ℓ_{1}, ℓ_{2}, ℓ_{3}) ∈ ia_{0}^{∗} | ∥λ∥^{2} ≤ 1},

then this indicates

vol(F) ≤ lim inf_{T →∞} \frac{N(T)}{(\frac{T}{4π})5/2 \frac{1}{Γ(7/2)}}

because the Laplace eigenvalue of e^{(λ+ρ)(H(g))} is 1 − \frac{ℓ_{1}^{2}+ℓ_{2}^{2}+ℓ_{3}^{2}}{2} .^{2} (Note that |Re ℓ_{j}| < \frac{1}{2} by unitary – [JacSha].)

The upper bound

lim sup_{T →∞} \frac{N(T)}{(\frac{T}{4π})5/2 \frac{1}{Γ(7/2)}} ≤ vol(F)  (5.8)

due to [Donnelly] shows in fact that

N(T) ∼ (\frac{T}{4π} )^{5/2} \frac{vol(F)}{Γ(7/2)}.  (5.9)
□

Theorem 5.3. (Spectral Equidistribution) If Σ is a bounded, open, Weylgroup invariant subset of ia_{0}^{∗} with a piece-wise smooth boundary, then

lim_{t→∞} \frac{#\{Im λ_{j} ∈ tΣ\}}{∫_{tΣ} β(λ)dλ} = vol(F).  (5.10)

Proof: The lower bound is in (5.7). To prove the upper bound, we will use this lower bound along with the asymptotics of the Weyl law for counting in balls. Let r be the length of the longest vector in Σ (which is bounded by assumption). By defining Σ_{c} as the complement of Σ inside the r-ball B_{r}, we have that

#{Im λ_{j} ∈ tΣ} + #{Im λ_{j} ∈ tΣ_{c}} = #{Im λ_{j} ∈ t(Σ ∪ Σ_{c})}.

2A check of this normalization is provided by the compact case, where there are no Eisenstein series and the Weyl law is already known.

Given any ϵ > 0 we can find t large enough so that

#{Im λ_{j} ∈ tΣ} ≤ (1 + ϵ) (∫_{t(Σ∪Σ_{c})} β(λ)dλ − ∫_{tΣ_{c}} β(λ)dλ) ,  (5.11)

which implies the upper bound when ϵ → 0. □

Proofs of Theorems 1.6 and 1.3: If the cusp form ϕ_{j} is not tempered, then its spectral parameter λ = (ℓ_{1}, ℓ_{2}, ℓ_{3}) is not a purely-imaginary vector. By the classification of the unitary dual, we have equality of the sets

{ℓ_{1}, ℓ_{2}, ℓ_{3}} = {−ℓ̄_{1}, −ℓ̄_{2}, −ℓ̄_{3}}.

We know that |Re(ℓ_{i})| < \frac{1}{2} by unitary ([JacSha]), so the vectors λ all lie near the hyperplanes defined by

λ(α_{1}^{∨}) = 0 , λ(α_{2}^{∨}) = 0 , λ(α_{1}^{∨} + α_{2}^{∨}) = 0.

Similarly, if ϕ is self dual then

{ℓ_{1}, ℓ_{2}, ℓ_{3}} = {μ, 0, −μ} for some μ.

This again constrains λ to lie along a hyperplane.

However, by taking the shape Σ in Theorem 5.3 to be further and further away from any fixed hyperplane, we conclude that no hyperplane has a positive percentage of the spectrum near it. Thus each of the exceptional sets we are considering is of measure zero compared to the rest of the spectrum. □

A related argument was used in [DKV] for cocompact subgroups Γ.

Proof of Theorem 1.5: All Gelbart-Jacquet lifts are self-dual forms on SL_{3}.

Another proof would be simply by counting: the lift quadruples the Laplace eigenvalue of a cusp form on SL_{2}(ℤ)\ℍ, and the number of these with SL_{3}(ℝ)-Laplace eigenvalue ≤ T is O(T) by Selberg’s theorem 1.1. Though forms ψ on congruence covers of SL_{2}(ℤ)\ℍ may also lift to SL_{3}(ℤ)\SL_{3}(ℝ)/SO_{3}(ℝ), such ψ are actually twists of forms on SL_{2}(ℤ)\ℍ, so one need only count the lifts from SL_{2}(ℤ)\ℍ itself, and not from other congruence covers. □

6 Eisenstein Series

These periodized functions on Γ\G/K are constructed using the Langlands decompositions of G’s parabolics. Recall that any element p ∈ P = NAM′ factors uniquely into a product p = nam of elements from their respective subgroups. Writing the diagonal matrices a = e^{H_{P} (p)}, we get a map which extends to G:

g ↦ H_{P} (g) , g ∈ Ne^{H_{P} (g)}M′K.

For the parabolics P_{1} and P_{2}, where M′ ≃ GL_{2}(ℝ), there are corresponding maps m : G → M′/(K ∩ M′) (see (3.2)).

If λ ∈ a_{0ℂ}^{∗} and g ∈ G, the minimal parabolic Eisenstein series is defined as

E(P_{0}, g, λ) = E(P_{0}, g, λ, 1) = ∑_{Γ∩P_{0}\Γ} e^{(λ+ρ_{0})(H_{0}(γg))}.  (6.1)

This sum only converges when α_{1}(λ) and α_{2}(λ) have large real parts, but it has a meromorphic continuation to all of a_{0ℂ}^{∗}.

Since

Γ ∩ M′_{1}\M′_{1}/(K ∩ M′_{1}) ≃ GL_{2}(ℤ)\ℍ,

the discrete eigenfunctions on the former are just the even^{3} discrete eigenfunctions for SL_{2}(ℤ). Take such a cusp form and λ ∈ a_{1ℂ}^{∗}, and define the maximal parabolic Eisenstein series

E(P_{1}, g, λ, ϕ) = ∑_{Γ∩P_{1}\Γ} e^{(λ+ρ_{1})(H_{1}(γg))}ϕ(m_{1}(γg)).  (6.2)

There is a similar Eisenstein series E(P_{2}, g, λ_{1}, ϕ) for the other maximal parabolic P_{2}, related through a functional equation. Each series again is initially only defined for certain values of λ but extends via a meromorphic continuation to a_{ℂ}^{∗} ([Lan1],[Lan3]).

We are now in a position to define what “Eis_{g}(x, y)” is. The spectral expansion of the automorphic kernel is

K(x, y) = ∑_{on SL_{3}(ℤ)\SL_{3}(ℝ)/SO_{3}(ℝ)^{ϕ_{j} an L^{2} discrete eigenfunction}} ĝ(λ_{j})ϕ_{j}(x)ϕ_{j}(y) + Eis_{g}(x, y),

3i.e. f(x + iy) = f(−x + iy).

and up to a normalizing constant for the measure, Eis_{g}(x, y) is ([Art2])

\frac{1}{3(2πi)^{2}} ∫∫_{ia_{0}^{∗}} ĝ(λ)E(P_{0}, x, λ, 1)\overline{E(P_{0}, y, λ, 1)}dλ+

\frac{1}{2πi} ∑_{on SL_{2}(ℤ)\ℍ , Δϕ_{j}=( \frac{1}{4} −ν_{j}^{2})ϕ_{j}^{ϕ_{j} an even L^{2} discrete eigenfunction}}∫_{ia_{1}^{∗}} ĝ(λ + (ν_{j}, −ν_{j}, 0))E(P_{1}, x, λ, ϕ_{j})\overline{E(P_{1}, y, λ, ϕ_{j})}dλ.

(6.3)

We also may assume that each ϕ_{j} on ℍ is a Hecke eigenform. There is a beautiful formula for the inner-products of truncated Eisenstein series, due to Langlands. It generalizes the Maass-Selberg formula (2.20) for SL_{2}(ℝ). See [Art1] for details.

Theorem 6.1. Langlands’ inner product formula ([Lan1], [Art1])

∫_{Γ\G/K} [(Λ^{C}E)(P, x, ϕ, λ_{1})] [(Λ^{C}E)(P, x, ϕ, λ_{2})] dx

= ∑_{P ∼P′ associate} vol(a′/ < α^{∨} | α ∈ Δ_{P′} >)×

∑_{s_{1},s_{2}∈Ω(a,a′)} \frac{e^{(s_{1}λ_{1}+s_{2}λ_{2})(C)}}{∏_{α∈Δ_{P′}}(s_{1}λ_{1} + s_{2}λ_{2})(α^{∨})} ⟨M(s_{1}, λ_{1})ϕ, M(s_{2}, λ_{2})ϕ⟩ .

Here M(s, λ) is an intertwining operator, which sends ϕ to a cusp form on the potentially-different parabolic P′. We will discuss it in the instances it arises for us, where it essentially acts as scalar multiplication. The last expression ⟨ψ, ψ′⟩ is an inner product over Γ ∩ M′\M′, and the Weyl group Ω(a, a′) is the set of isomorphisms of a to a′ coming from restrictions of elements in Ω(a_{0}).

Interlude: SL_{2}

Write C = (c, −c), c > 0, and let λ_{1} = (it + ϵ, −it − ϵ), λ_{2} = (−it, it) so that

∫_{F} Λ^{C}E(P, g, λ_{1}, ϕ)E(P, g, λ_{2}, ϕ)dg

is a constant times

\frac{e^{(ϵ,−ϵ)C}}{2ϵ} +\frac{e^{(2it+ϵ,−2it−ϵ)C}}{4it + 2ϵ} R(−2it)+\frac{e^{(−2it−ϵ,2it+ϵ)C}}{−4it − 2ϵ} R(2it+2ϵ)+\frac{e^{(−ϵ,ϵ)C}}{−2ϵ} R(−2it)R(2it+2ϵ).

Here the intertwining operator is

R(s) = \sqrt{π} \frac{Γ(\frac{s}{2})}{Γ(\frac{s+1}{2})} \frac{ζ(s)}{ζ(s + 1)} = \frac{Z(s)}{Z(s + 1)}  (6.4)

with

Z(s) = π^{−s/2}Γ(\frac{s}{2})ζ(s) = Z(1 − s).

Take ϵ → 0 so that λ_{1} → λ̄_{2}. Then the last expression approaches

∫_{F} |Λ^{C}E(P, g, λ)|^{2}dg = (const) [4c − \frac{R′}{R} (2it) + \frac{e^{4itc}}{4it} R(−2it) − \frac{e^{−4itc}}{4it} R(2it)] .

In (2.20) we derived this from a direct calculation. It is a key step in Selberg’s trace formula for SL_{2}(ℤ).

7 Bounding the Eisenstein Contribution

In this final section we will complete the proofs by establishing the estimate in (5.6). From (6.3) it is sufficient to show that both

∫_{F_{C}} ∫∫_{ia_{0}^{∗}} ĝ_{tΣ}(λ)|E(P_{0}, x, λ, 1)|^{2}dλdx = o (∫_{tΣ} β(λ)dλ) ,  (7.1)

and

∫_{F_{C}} ∫_{ia_{1}^{∗}} ∑ ; eigenfunction^{ϕ_{j} an even L^{2} discrete}_{Δϕ_{j}=( \frac{1}{4} −ν_{j}^{2})ϕ_{j}^{on SL_{2}(ℤ)\ℍ}} ĝ_{tΣ}(λ + (ν_{j}, −ν_{j}, 0))|E(P_{1}, x, λ, ϕ_{j})|^{2}dλdx = o (∫_{tΣ} β(λ)dλ) .

(7.2)

To do this we shall use the rapid decay of ĝ_{tΣ} in (4.4). Among other benefits, this allows us to interchange the order of integration. Keep in mind that ĝ_{tΣ} is roughly the characteristic function of tΣ.

Some background on L-functions

We will require some information about the density of zeroes of certain Lfunctions: the Riemann ζ function and the standard L-function of an even cusp form ϕ on SL_{2}(ℤ)\ℍ, L(s, ϕ). Each is defined as an Euler product^{4} over the primes

ζ(s) = ∏_{p}(1 − p^{−s})^{−1} , L(s, ϕ) = ∏_{p}(1 − α_{p}p^{−s})^{−1}(1 − α_{p}^{−1}p^{−s})^{−1}, (7.3)

where the α_{p} satisfy the bound |α_{p}| ≤ p^{5/28} ([BDHI]). Each L-function can be completed:

Z(s) = Γ_{ℝ}(s)ζ(s) , Λ(s, ϕ) = Γ_{ℝ}(s + ν)Γ_{ℝ}(s − ν)L(s, ϕ),

where

Γ_{ℝ}(s) = π^{−s/2}Γ(s/2)

and ν is related to ϕ’s Laplace eigenvalue by

Δϕ = (\frac{1}{4} − ν^{2})ϕ.

With this convention Z(s) = Z(1 − s), Λ(s, ϕ) = Λ(1 − s, ϕ), and each is entire except for the simple poles of Z(s) at s = 0, 1.

The following estimate will be used to bound Eisenstein series integrals later. The analogous statement for Z(s) is classical (e.g. see [Titchmarsh]) and the proof for Λ(s, ϕ) is essentially identical. However we do not know of a reference in the literature and include it for completeness.

Proposition 7.1. For T ∈ ℝ

Re ∫_{T −1}^{T +1} \frac{Λ′}{Λ} (1 + it, ϕ)dt ≪ log(|T | + |ν|).  (7.4)

Proof: Using entirety of Λ(s, ϕ) and its Mittag-Leffler expansion,

\frac{Λ′}{Λ} (s) = \frac{Γ′_{ℝ}}{Γ_{ℝ}}(s + ν) + \frac{Γ′_{ℝ}}{Γ_{ℝ}}(s − ν) + \frac{L′}{L} (s, ϕ) = ∑_{\{ρ|Λ(ρ)=0\}} \frac{1}{s − ρ}

4Recall that ϕ is assumed to be both a Hecke and Laplace eigenfunction.

(the sum of the zeroes is actually only conditionally convergent, so the term with ρ should always be summed with the term containing the zero at 1−ρ). From the Euler product,

\frac{L′}{L} (s, ϕ) = − ∑_{p} ∑_{n=1}^{∞}(α_{p}^{n} + α_{p}^{−n})p^{−ns} log p,

so the bound |α_{p}| ≤ p^{5/28} implies that |\frac{L′}{L} (2 + it, ϕ)| is uniformly bounded in both ν and t. By Stirling’s formula,

\frac{Γ′_{ℝ}}{Γ_{ℝ}}(s) = −\frac{1}{2} log π + \frac{1}{2} log s + O(1/|s|)

and

\frac{Γ′_{ℝ}}{Γ_{ℝ}}(2 + it + ν) + \frac{Γ′_{ℝ}}{Γ_{ℝ}}(2 + it − ν) ≤ log(|t| + |ν|) + O(1).

Thus,

∑_{ρ} \frac{1}{1 + |t − γ|^{2}} ≪ ∑_{ρ} \frac{1}{2 + it − ρ} ≤ log(|t| + |ν|) + O(1).  (7.5)

It follows that there are no more than O(log(|T | + |ν|)) zeroes between with imaginary part between T − 1 and T + 1. By writing

Re ∫_{T −1}^{T +1} \frac{Λ′}{Λ} (1 + it, ϕ)dt = Re ∫_{T −1}^{T +1} ∑_{ρ} \frac{1}{1 + it − ρ}dt

= Re ∫_{T −1}^{T +1} ∑_{|ρ−iT |≤2} \frac{1}{1 + it − ρ}dt + Re ∫_{T −1}^{T +1} ∑_{|ρ−iT |>2} \frac{1}{1 + it − ρ}dt,

invoking (7.5), and using the fact that

Re ∫_{1+i(T −1)}^{1+i(T +1)} \frac{ds}{s − ρ} = O(1),

we bound each of the terms above by O(log(|T | + |ν|)). □

Minimal Parabolic Eisenstein Series

We have appended a table of the 36 terms in the Langlands inner-product formula (also referred to as the Maass-Selberg relations) for the minimal parabolic.

The calculation is aided by the identity

M(s, (ℓ_{1}, ℓ_{2}, ℓ_{3})) = ∏_{s(i)>s(j)^{1≤i<j≤3}} R(ℓ_{i} − ℓ_{j}).  (7.6)

This is a special case of a general result of Langlands ([Lan2],pp. 36-47;[Lan4], p. 134;[Art2],p. 854); the function R here is the same as the one used above in (6.4).

Using a limiting procedure as in the interlude we can compute:

Proposition 7.2. (Diagonal terms) Let C = (c, 0, −c), λ_{1} = (it_{1} + ϵ_{1}, it_{2} + ϵ_{2}, it_{3} + ϵ_{3}), λ_{2} = (−it_{1}, −it_{2}, −it_{3}). Then

lim_{ϵ_{1},ϵ_{2},ϵ_{3}→0} ∑_{s∈Ω(a_{0})} \frac{e^{s(λ_{1}+λ̄_{2})(C)}}{[s(λ_{1} + λ̄_{2})(α_{1}^{∨})][s(λ_{1} + λ̄_{2})(α_{2}^{∨})]} ⟨M(s, λ_{1}), M(s, λ_{2})⟩

= 3c^{2} − 2c\frac{R′}{R} (it_{1} − it_{2}) − 2c\frac{R′}{R} (it_{2} − it_{3}) − 2c\frac{R′}{R} (it_{1} − it_{3})

+\frac{R′}{R} (it_{1} − it_{2})\frac{R′}{R} (it_{2} − it_{3})

+\frac{R′}{R} (it_{1} − it_{3})\frac{R′}{R} (it_{2} − it_{3}) + \frac{R′}{R} (it_{1} − it_{2})\frac{R′}{R} (it_{1} − it_{3}).

Proposition 7.3. For any ϵ > 0 and λ ∈ ia_{0}^{∗} with ∥λ∥ large, we have that

∫_{F} |Λ^{C}E(P_{0}, x, λ)|^{2}dx = O_{ϵ}(∥λ∥^{ϵ}).

Proof: We start with some estimates on ζ(s). The following may be found in Chapter 3 of [Titchmarsh]: There is an absolute constant κ > 0 such that

\frac{1}{log(|t| + 2)} ≪ |ζ(σ + it)| ≪ log(|t| + 2)  (7.7)

and

|\frac{ζ′}{ζ} (σ + it)| ≪ log(|t| + 2)

in the region σ ≥ 1 − \frac{κ}{log(|t|+2)}. Thus

R(s) = \frac{Z(1 − s)}{Z(1 + s)} = \frac{π^{−(1−s)/2}Γ (\frac{1−s}{2}) ζ(1 − s)}{π^{−(1+s)/2}Γ (\frac{1+s}{2}) ζ(1 + s)}

≪ (1 + |Im s|)^{−Re s} log(|Im s| + 2)^{2}

and

\frac{R′}{R} (s) ≪ log(|Im s| + 2)

for

|Re s| ≤ \frac{κ}{log(|Im s| + 2)}.

Of course both R(s) and \frac{R′}{R} (s) are analytic in this region because of the nonvanishing.

Fix λ = (it_{1}, it_{2}, it_{3}). Then each of the six terms from Proposition 7.2 is trivially O_{ϵ}(∥λ∥^{ϵ}). Of the remaining thirty terms (see the chart in the appendix), some may have singularities when either of the denominators

s(λ_{1} + λ̄_{2})(α_{1}^{∨}) or s(λ_{1} + λ̄_{2})(α_{2}^{∨})

vanish. Nevertheless, the sum of all of the terms represents a holomorphic function for all λ ∈ ia_{0}^{∗}, so the poles cancel with other terms. If λ is such that each

|s(λ_{1} + λ̄_{2})(α_{j}^{∨})| ≥ \frac{κ}{2 log(∥λ∥ + 2)},

then each term is trivially O_{ϵ}(∥λ∥^{ϵ}) as well (the numerators have modulus one when Re λ = 0).

Otherwise, if some denominator is small, we will take estimates further away and appeal to the maximum principle. Take a small neighborhood in

s_{1} = s(λ_{1} + λ̄_{2})(α_{1}^{∨}) , s_{2} = s(λ_{1} + λ̄_{2})(α_{2}^{∨}).

Now, suppose that

|s_{1}| ≤ \frac{κ}{2 log(∥λ∥ + 2)} ≤ |s_{2}|

(the other cases have almost-identical proofs). Then for |s_{1}| = \frac{κ}{2 log(∥λ∥+2)}, (7.7) shows that each term is O_{ϵ}(∥λ∥^{ϵ}). The maximum modulus principle shows that this bound holds uniformly for |s_{1}| ≪ log ∥λ∥. □

Maximal Parabolic P_{1}

Now we turn to the sum in (7.2). We will first consider the simpler case that ϕ is a cusp form. If s is the lone permutation in Ω(a_{1}, a_{2}), then the intertwining operator acts as

M(s, λ)ϕ = R(λ(α^{∨}), ϕ)ϕ′,

where ϕ and ϕ′ are cusp forms on M_{1} and M_{2} coming from the same even cusp form on SL_{2}(ℤ)\ℍ, and

R(s, ϕ) = \frac{Λ(s, ϕ)}{Λ(s + 1, ϕ)}

is a ratio formed from ϕ’s completed standard L-function (7.3). As before with (7.6), since ϕ is everywhere unramified, this can be derived from Langlands’ formula ([Lan2],[Lan4]). Then the Maass-Selberg relations take the following form:

Proposition 7.4. The inner product

∫_{F} |Λ^{C}E(P, g, λ, ϕ)|^{2} dg

is a constant multiple of 3/2c − \frac{R′}{R} (λ(α^{∨}), ϕ).

Proof: The inner product formula is a constant times

lim_{ϵ→0} \frac{e^{(2it+2ϵ)c}}{2it + 2ϵ} − \frac{e^{−(2it+2ϵ)c}}{2it + 2ϵ} \frac{R(2it + 2ϵ)}{R(2it)}.

□

Proposition 7.5. The integral

∫_{t=T −1}^{T +1} ∫_{F} |Λ^{C}E(P, g, λ, ϕ_{j})|^{2} dgdt = O(log(T + ∥λ_{j}∥)).

Proof: Using the previous proposition, this requires only the estimate on ∫_{T −1}^{T +1} \frac{Λ′}{Λ} (1 + it, ϕ)dt in Proposition 7.1 (cf. (2.23)). □

Remark 7.6. (On summing over the eigenvalues) The formula (6.3) includes a sum over the SL_{2}(ℤ)\ℍ spectrum as well. Through a more-precise statement of Selberg’s Theorem 1.1 such as

N(T) = \frac{1}{12}T^{2} + O(T log T),

we can bound the spectral points ν_{j} in the interval [ν−1, ν+1] by O(\sqrt{|ν|} log |ν|).

The Constant Function on SL_{2}

There is only one Eisenstein series left to estimate,

E(P_{1}, g, λ, 1),

the Eisenstein series induced from the constant function on the maximal parabolic P_{1}.

The constant functions on SL_{2}(ℤ)\ℍ may be viewed as multiples of

Res_{s=1}E_{s}(z) = \frac{1}{2}Res_{s=1} ∑_{(c,d)=1} \frac{y^{s}}{|cz + d|^{2s}} .

We may ignore the actual value of the constant since we are only trying to get an order-of-magnitude estimate. Thus, E(P_{1}, g, λ, 1)is a constant multiple of the residue

Res_{δ=0}E(P_{0}, g, (\frac{1}{2} + it + iδ, −\frac{1}{2} + it − iδ, −2it), 1).  (7.8)

Taking the residue of the inner product

∫_{F} |Λ^{C}E(P_{1}, g, λ, 1)|^{2}dg

is more complicated, though a limiting value must exist since the Eisenstein series is meromorphic there. We will explicitly see this cancellation occurring. Slicker arguments are possible, but we will present a detailed proof for the sake of clarity.

Let

λ_{1} = (\frac{1}{2} + it + iδ_{1} + ϵ, −\frac{1}{2} + it − iδ_{1} + ϵ, −2it − 2ϵ),

and

λ_{2} = (\frac{1}{2} − it + iδ_{2} , − \frac{1}{2} − it − iδ_{2} , 2it).

Then we are interested in

lim_{δ_{1},δ_{2}→0} δ_{1}δ_{2} ∑_{s_{1},s_{2}∈Ω(a_{0})} \frac{e^{(s_{1}λ_{1}+s_{2}λ_{2})(C)}}{(s_{1}λ_{1} + s_{2}λ_{2})(α_{1}^{∨})(s_{1}λ_{1} + s_{2}λ_{2})(α_{2}^{∨})} ⟨M(s_{1}, λ_{1})M(s_{2}, λ_{2})⟩ .

This expression is holomorphic in δ_{1} and δ_{2} near zero, so it does not matter how we take the limits δ_{1}, δ_{2} → 0. We will first take the residue in δ_{1}. Of the 36 terms, we will of course ignore those which do not have a pole at δ_{1} = 0. These can occur only in M(s_{1}, λ_{1}), for the denominators do not yet vanish (note ϵ, δ_{2} ≠ 0 at this stage). Since

M(s_{1}, λ_{1}) = ∏_{s_{1}(i)>s_{1}(j)^{i<j}} R(λ_{1_{i}} − λ_{1_{j}}),

these permutations s_{1} must interchange 1 and 2, which forces

s_{1} ∈ {(12), (13), (321)}.

Next, when the residue at δ_{2} = 0 is taken, poles can occur in two different ways: M(s_{2}, λ_{2}) might have a pole, or a denominator might vanish. The former occurs for s_{2} ∈ {(12), (13), (321)} as well.

Proposition 7.7. Let s_{1} ∈ {(12), (13), (321)}. If one of the denominator terms

(s_{1}λ_{1} + s_{2}λ_{2})(α_{1}^{∨}) = 0

On the Existence and Temperedness of Cusp Forms for SL_{3}(Z) 41

or

(s_{1}λ_{1} + s_{2}λ_{2})(α_{2}^{∨}) = 0,

then

s_{2} ∈ {e, (23), (123)} = S_{3} − {(12), (13), (321)},

i.e. M(s_{2}, λ_{2}) has no pole at δ_{2} = 0.

Proof: Now that

λ_{1} = (\frac{1}{2} + it + ϵ, −\frac{1}{2} + it + ϵ, −2it − 2ϵ)

and

λ_{2} = (\frac{1}{2} − it, −\frac{1}{2} − it, 2it),

the only possible way to get consecutive entries of s_{1}λ_{1} + s_{2}λ_{2} to equal is if

s_{1} = s_{2}(12).

□

We see that only the following terms have poles:

two from M ′s : s_{1}, s_{2} ∈ {(12), (13), (321)}

or one from M, one from a denominator : s_{1} = (12), s_{2} = e , s_{1} = (13), s_{2} = (123). Note that (321) × (23) fails to have poles in the denominators. The 11 terms (see the chart in the appendix) are:

(12) × (12) : \frac{e^{(−1+ϵ,1+ϵ,−2ϵ)(c,0,−c)}}{(−2)(1 + 3ϵ)}

(12) × (13) : \frac{e^{(− \frac{1}{2} +3it+ϵ,ϵ, \frac{1}{2} −3it−2ϵ)(c,0,−c)}}{(−\frac{1}{2} + 3it)(−\frac{1}{2} + 3it + 3ϵ)}R(\frac{1}{2} − 3it)R(−\frac{1}{2} − 3it)

(12) × (321) : \frac{e^{(−1+ϵ, \frac{1}{2} +3it+ϵ, \frac{1}{2} −3it−2ϵ)(c,0,−c)}}{(−\frac{3}{2} − 3it)(6it + 3ϵ)} R(\frac{1}{2} − 3it)

(13) × (12) : \frac{e^{(− \frac{1}{2} −3it−2ϵ,ϵ, \frac{1}{2} +3it+ϵ)(c,0,−c)}}{(−\frac{1}{2} − 3it − 3ϵ)(−\frac{1}{2} − 3it)}R(\frac{1}{2} + 3it + 3ϵ)R(−\frac{1}{2} + 3it + 3ϵ)

(13)×(13) : \frac{e^{(−2ϵ,−1+ϵ,1+ϵ)(c,0,−c)}}{(1 − 3ϵ)(−2)} R(\frac{1}{2}+3it+3ϵ)R(−\frac{1}{2}+3it+3ϵ)R(\frac{1}{2}−3it)R(−\frac{1}{2}−3it)

(13)×(321) : \frac{e^{(− \frac{1}{2} −3it−2ϵ,− \frac{1}{2} +3it+ϵ,1+ϵ)(c,0,−c)}}{(−6it − 3ϵ)(−\frac{3}{2} + 3it)} R(\frac{1}{2}+3it+3ϵ)R(−\frac{1}{2}+3it+3ϵ)R(−\frac{1}{2}−3it)

(321) × (12) : \frac{e^{(−1+ϵ, \frac{1}{2} −3it−2ϵ, \frac{1}{2} +3it+ϵ)(c,0,−c)}}{(−\frac{3}{2} + 3it + 3ϵ)(−6it − 3ϵ)} R(\frac{1}{2} + 3it + 3ϵ)

(321)×(13) : \frac{e^{(− \frac{1}{2} +3it+ϵ,− \frac{1}{2} −3it−2ϵ,1+ϵ)(c,0,−c)}}{(6it + 3ϵ)(−\frac{3}{2} − 3it − 3ϵ)} R(\frac{1}{2}+3it+3ϵ)R(\frac{1}{2}−3it)R(−\frac{1}{2}−3it)

(321) × (321) : \frac{e^{(−1+ϵ,−2ϵ,−1+ϵ)(c,0,−c)}}{(−1 + 3ϵ)(−1 − 3ϵ)} R(\frac{1}{2} + 3it + 3ϵ)R(\frac{1}{2} − 3it).

The next two terms had limits taken in δ_{1}, δ_{2} → 0 and, up to constants from the residues, are

(12) × e : \frac{e^{(ϵ,ϵ,−2ϵ)(c,0,−c)}}{3ϵ}

(13)×(123) : \frac{e^{(−2ϵ,ϵ,ϵ)(c,0,−c)}}{−3ϵ} R(\frac{1}{2}+3it+3ϵ)R(−\frac{1}{2}+3it+3ϵ)R(\frac{1}{2}−3it)R(−\frac{1}{2}−3it).

Note that

R(−\frac{1}{2} + x)R(\frac{1}{2} + x) = \frac{Z(−\frac{1}{2} + x)}{Z(\frac{3}{2} + x)} = \frac{Z(\frac{3}{2} − x)}{Z(\frac{3}{2} + x)}

by the functional equation. Accordingly, we may set ϵ = 0 in the first 9 terms; the last two involve a limit at ϵ = 0 producing derivatives.

On the Existence and Temperedness of Cusp Forms for SL_{3}(Z) 43 Proposition 7.8.

(i) R(\frac{1}{2} + it) ≪ 1  (7.9)
(ii) |R(−\frac{1}{2} + it)R(\frac{1}{2} + it)| = 1  (7.10)
(iii) \frac{d}{dϵ} \frac{Z(\frac{3}{2} + it + ϵ)}{Z(\frac{3}{2} + it − ϵ)} |_{ϵ=0}= O(log t).  (7.11)

Proof: (ii) follows from the facts that Z(s̄) = \overline{Z(s)} and R(s)R(−s) = 1. For (i), recall

R(s) = \sqrt{π} \frac{Γ(\frac{s}{2})}{Γ(\frac{s+1}{2})} \frac{ζ(s)}{ζ(s + 1)}.

By Stirling’s formula

|\frac{Γ(\frac{1}{4} + \frac{it}{2})}{Γ(\frac{3}{4} + \frac{it}{2})}| ∼ \sqrt{\frac{2}{t}} as t → ∞,

and by the convexity bound,

ζ(\frac{1}{2} + it) = O_{ϵ}(t^{1/4+ϵ})  (7.12)

(better bounds can be obtained – see [Titchmarsh]). Also, taking the logarithm of the Euler product of ζ(s) we find

− log ζ(\frac{3}{2} + it) = ∑_{p prime} log(1 − p^{−3/2−it}) = O(1),

which proves (i). Differentiating,

\frac{ζ′}{ζ} (\frac{3}{2} + it) = − ∑_{p} \frac{p^{−3/2−it} log p}{1 − p^{−3/2−it}} = O(1)

also. So

\frac{Z′}{Z} (\frac{3}{2} + it) = −\frac{1}{2} log π + \frac{1}{2} \frac{Γ′}{Γ} (\frac{3}{4} + \frac{it}{2}) + \frac{ζ′}{ζ} (\frac{3}{2} + it) = O(log t),

proving (iii). □

Summarizing these pointwise bounds,

Proposition 7.9. For T large,

∫_{T −1}^{T +1} ∫_{F} |Λ^{C}E(P_{1}, g, (it, it, −2it), 1)|^{2}dgdt = O_{ϵ}(T^{ϵ}).

Assembling the Bounds

Proof of (5.6): Choose c large enough so that F_{C} is contained in {x | τ̂_{P_{1}}(H_{0}(x − C)), τ̂_{P_{2}}(H_{0}(x − C)) ≤ 0}. Then the truncation does not affect F_{C} and

∫_{F_{C}} |E(g, λ)|^{2}dg ≤ ∫_{F} |Λ^{C}E(g, λ)|^{2}dg

for each Eisenstein series E(g, λ) on H. In the propositions we bounded local integrals of all of the Eisenstein series as growing slower than any polynomial, with the exception of the maximal parabolic Eisenstein series (Remark 7.6). For this we must sum over the SL_{2}(ℤ)\ℍ spectrum as well, so the contribution of the left-hand side of (7.2) near the point λ is bounded by O_{ϵ}(∥λ∥^{1+ϵ}).

We note in comparison with Proposition 2.7 that (4.4) shows

ĝ_{tΣ}(λ) = \begin{cases}1 + O_{m}(dist(λ, ∂(tΣ))^{−m}), & λ ∈ tΣ, \\ O_{m}(dist(λ, ∂(tΣ))^{−m}), & λ ∈ tΣ \end{cases} , m ≥ 0.

Since we have shown that all the Eisenstein series grow polynomially, we may switch the order of integration, ignore the tail, and conclude (see (5.5))

∫_{F_{C}} Eis_{tΣ}(x, x)dx ≪ ∫_{tΣ}(1 + ∥λ∥)^{1+ϵ}dλ.

Since β(λ) grows at the rate of ∥λ∥^{3} in almost all directions, we have completed the proof that

∫_{F_{C}} Eis_{g_{tΣ}}(x, x)dx = o (∫_{tΣ} β(λ)dλ) .
□

References

[Art1] James Arthur, A trace formula for reductive groups. II. Applications of a truncation operator, Compositio Math. 40 (1980), no. 1, 87– 121.

[Art2] James Arthur, The trace formula for noncompact quotient in Proceedings of the International Congress of Mathematicians, vol. II, pp. 849–859 (1984).

[Bump] Daniel Bump, Automorphic Forms on GL(3, ℝ), Lecture Notes in Math. 1083, Springer, Berlin, 1984.

[BDHI] Daniel Bump, W. Duke, Jeffrey Hoffstein, and Henryk Iwaniec, An estimate for the Hecke eigenvalues of Maass forms, Internat. Math. Res. Notices, 1992, No. 4, 75–81.

[Donnelly] H. Donnelly, On the cuspidal spectrum for finite volume symmetric spaces, J. Diff. Geom. 17 (1982), 239-253.

[DKV] J.J Duistermaat, J.A.C. Kolk, and V.S. Varadarajan, Spectra of Compact Locally Symmetric Manifolds of Negative Curvature, Inventiones 52 (1979), 27-93.

[EhrMau] L. Ehrenpreis and F. I. Mautner, Some properties of the Fourier transform on semi-simple Lie groups. I., Ann. of Math. (2) 61, (1955). 406–439.

[Gangolli] R. Gangolli, On the Plancherel formula and the Paley-Wiener theorem for spherical functions on semisimple Lie groups, Ann. of Math. 93, 150-165 (1971).

[GelJac] Stephen Gelbart and Hervé Jacquet, A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. École Norm. Sup. 4^{e}, Serie 11 (1978), 4, 471–542.

[Hejhal] Dennis A. Hejhal, The Selberg trace formula for P SL(2, ℝ), volume 2, Lecture Notes in Math. 1001, Springer, 1983.

[Jacquet] Hervé Jacquet, On the residual spectrum of GL(n), in Lie group representations, II, 185–208, Lecture Notes in Math., 1041, Springer, 1984.

[JacSha] H. Jacquet and J.A. Shalika Rankin-Selberg Convolutions: Archimdean Theory, Israel Math. Conf. Proceedings 2 (1990), 125-208.

[Lan1] Robert Langlands, Eisenstein series, in Albegraic groups and discontinuous subgroups, A. Borel and G. Mostow, ed., Proc. Symp. Pure Math. IX, 1966, 235–252.

[Lan2] Robert Langlands, Euler products, Yale Mathematical Monographs 1, Yale University Press, 1971.

[Lan3] Robert Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Math. 544, Springer, 1976.

[Lan4] Robert Langlands, Eisenstein series, the trace formula, and the modern theory of automorphic forms, in Number theory, trace formulas, and discrete groups, K. E. Aubert, E. Bombieri, and Dorian Goldfeld, ed. Academic Press, 1989, pp. 125-155.

[LRS] W. Luo, Z. Rudnick, and P. Sarnak, On Selberg’s Eigenvalue Conjecture, Geom. Funct. Anal. 5 (1995), no. 2, 387–401.

[Miller] Stephen D. Miller, Spectral and Cohomological Applications of the Rankin-Selberg Method, Internat. Math. Res. Notices, 1996, No. 1, 15-26.

[MœWal] C. Mœglin and J.-L. Waldspurger, Spectral decomposition and Eisenstein series, Cambridge Univ. Press, 1995.

[Sarnak] Peter Sarnak, On cusp forms, in The Selberg trace formula and related topics, Dennis Hejhal, Peter Sarnak, and Audrey Terras, ed., Contemporary Math. 53, Amer. Math. Soc., 1984, 393-407.

[Sel1] Atle Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87. (Also in Collected Works, Vol. I, Springer-Verlag, New York, 1989, pp. 423-463.)

[Sel2] Atle Selberg, Harmonic Analysis, in Collected Papers, Vol. I, Springer-Verlag, New York , 1989, pp. 626-674.

[Sel3] Atle Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Sympos. Pure Math., Amer. Math. Soc., Vol. VIII (1965), pp. 1-15.

[Shahidi] Freydoon Shahidi, On nonvanishing of L-functions, Bull. Amer. Math. Soc. 2 (N.S.), No. 3 (1980), pp. 462-464.

[StaWal] Eric  Stade and D.  I.  Wallace,  Weyl’s

law for SL(3, ℤ)\SL(3, ℝ)/SO(3, ℝ), Pacific J. Math 173 (1996), no. 1, 241-261.

[Terras] Audrey Terras, Harmonic Analysis on Symmetric Spaces and Applications. I., Springer-Verlag, New York, 1985.

[Titchmarsh] E.C.Titchmarsh, The Theory of the Riemann Zeta-function, second edition revised by D.R. Heath-Brown, Clarendon Press, Oxford, 1986.

[Wallace] Dorothy I. Wallace The Selberg trace formula for SL(3, ℤ)\SL(3, ℝ)/SO(3, ℝ), Trans. Amer. Math. Soc. 345 (1994), no. 1, 1-36.