j. differential geometry 51 (1999) 97-147

THE WEYL UPPER BOUND ON THE DISCRETE SPECTRUM OF LOCALLY SYMMETRIC SPACES

LIZHEN JI

1. Introduction

1.1. Let G be a reductive Lie group with finitely many connected components, and Γ a cofinite volume discrete subgroup of G. Let K ⊂ G be a maximal compact subgroup, and X = G/K be the associated symmetric space, which is the product of a symmetric space of noncompact type and a possible Euclidean space. Then Γ\X is a locally symmetric space of finite volume. For simplicity, we assume, unless otherwise specified, that there exists a reductive algebraic group G defined over ℚ satisfying the conditions in [18, p. 1] such that G = G(ℝ), and Γ ⊂ G(ℚ) is an arithmetic subgroup.

Any finite dimensional unitary representation σ of K defines a homogeneous bundle Ẽ_{σ} on X and hence a locally homogeneous bundle E_{σ} on Γ\X. The bundle E_{σ} admits a locally invariant connection ▽ which is the push forward of the invariant connection on the homogeneous bundle Ẽ_{σ}. The connection ▽ defines a quadratic form D on sections of E_{σ}: For any f ∈ C_{0}^{∞}(Γ\X, σ),

D(f) = ∫_{Γ\X} | ▽ f(x)|^{2}dx.

This quadratic form D defines an elliptic operator Δ on L^{2}(Γ\X, σ), called the Laplace operator, where L^{2}(Γ\X, σ) denotes the space of L^{2}- sections of E_{σ}. If σ is irreducible, Δ is equal to a shift of the restriction of the negative of the Casimir element of the Lie algebra of G by a constant determined by σ (cf. [5, Theorem 2.5, p. 49] and [32, Equation 6.10, p. 386]).

Received October 8, 1996, and, in revised form, August 7, 1998. Author was partially supported by NSF postdoctoral fellowship 9407427 and NSF grant DMS-9704434.

If Γ\X is compact, Δ has a discrete spectrum Spec(Δ) : λ_{1} ≤ λ_{2} ≤ ⋯ (repeated with multiplicity) on L^{2}(Γ\X, σ), and the spectral counting function N(λ) = |{λ_{i} ∈ Spec(Δ) | λ_{i} ≤ λ}| satisfies the famous Weyl law:

lim_{λ→+∞} \frac{N(λ)}{λ^{n/2}} = (4π)^{−n/2} \frac{vol(Γ\X)}{Γ(\frac{n}{2} + 1)} dim σ,

where n = dim Γ\X, dim σ is the dimension of the representation space of σ, i.e., the rank of the bundle E_{σ}, and Γ(·) is the Gamma function.

From now on we assume that Γ\X is noncompact. Then Δ has both a continuous spectrum Spec_{con}(Δ) and a discrete spectrum Spec_{dis}(Δ) : λ_{1} ≤ λ_{2} ≤ ⋯ (repeated with multiplicity), which could be finite.

Denote the counting function of the discrete spectrum of Δ by N_{d}(λ) = |{λ_{i} ∈ Spec_{dis}(Δ) | λ_{i} ≤ λ}|. In [4, Theorem 1], Borel and Garland showed that N_{d}(λ) is finite for all λ > 0. Inspired by the above Weyl law for compact quotients and a result of Donnelly [11, Theorem 1.1] (see Lemma 2.3.2 below) on the cuspidal discrete spectrum, they raised the following question [4, §4.7]:

Question 1.1.1. Decide whether N_{d}(λ) satisfies the following Weyl upper bound

lim_{λ→+∞} sup \frac{N_{d}(λ)}{λ^{n/2}} ≤ (4π)^{−n/2} \frac{vol(Γ\X)}{Γ(\frac{n}{2} + 1)} dim σ.

One of the main results of this paper is to answer this question affirmatively in several cases. The results in this paper strongly suggest that the answer to the above question is always positive.

Theorem 1.1.2 (§5). If the ℚ-rank of G is equal to 1 or the ℝ-rank of G, i.e., the rank of X, is less than or equal to 2, then the counting function N_{d}(λ) of the discrete spectrum of Δ satisfies the Weyl upper bound, i.e.,

lim_{λ→+∞} sup \frac{N_{d}(λ)}{λ^{n/2}} ≤ (4π)^{−n/2} \frac{vol(Γ\X)}{Γ(\frac{n}{2} + 1)} dim σ,

where n = dim Γ\X, and dim σ is the dimension of the representation space of σ.

Siegel modular varieties Γ\Sp(2, ℝ)/U(2) and the Hilbert modular varieties satisfy the condition in the above theorem, and hence the Weyl upper bound holds for them, but the Siegel modular varieties Γ\Sp(n, ℝ)/U(n) for n ≥ 3 do not satisfy this condition. On the other hand, we show below that for Siegel modular varieties Γ\Sp(3, ℝ)/U(3), N_{d}(λ) satisfies a bound of the sharp order, though not the sharp constant.

The sharp upper bound on N_{d}(λ) was known only in the following cases. If G = SL(2, ℝ), the upper bound in Theorem 1.1.2 is due to Selberg and follows from the Selberg trace formula [46, p. 668]. For the general rank-one X, this upper bound is due to Donnelly [9, Theorem 1.1]. The only higher rank case where this upper bound was known to hold is when G = SL(n, ℝ) and Γ ⊂ SL(n, ℤ) is a congruence subgroup, n ≥ 3; in fact, the upper bound then follows from a result of Moeglin and Waldspurger [33] on an explicit description of the residual discrete spectrum that implies that every Eisenstein series induce at most one residual eigenfunction, and the result of Donnelly mentioned earlier on the Weyl upper bound on the cuspidal discrete spectrum in [11, Theorem 1.1] (see Lemma 2.3.2 below). Donnelly also proved in [10] a (nonsharp) polynomial upper bound on N_{d}(λ) in the Γ-rank-1, in particular the ℚrank-1, case (see below for the precise bound).

To state general result, we need more notation. For any proper rational parabolic subgroup P of G, there is a boundary locally symmetric space Γ_{M_{P}} \X_{P}, which is a boundary component of the reductive Borel–Serre compactification \overline{Γ\X}^{RBS}. The dimension of the split center of the Levi quotient of P is called the ℚ-rank (or the split rank, or the parabolic rank) of P, denoted by rank_{ℚ}(P). Then the ℚ-rank of G, denoted by rank_{ℚ}(G), is equal to the maximum of rank_{ℚ}(P) for all proper rational parabolic subgroups P. Since Γ\X is noncompact, rank_{ℚ}(G) ≥ 1. (See §2.2 for more details.)

Theorem 1.1.3 (§7). The counting function N_{d}(λ) of the discrete spectrum of Δ satisfies the following upper bound:

N_{d}(λ) ≤ (1 + o(1))(4π)^{−n/2} \frac{vol(Γ\X)}{Γ(\frac{n}{2} + 1)} dim σ λ^{n/2} + O(1)λ^{m/2},

where n = dim Γ\X, m is equal to the maximum of

(rank_{ℚ}(P) + 1) dim Γ_{M_{P}} \X_{P}

for all proper rational parabolic subgroups P of G such that

rank_{ℚ}(P) ≤ rank_{ℚ}(G) − 1,

where o(1) goes to zero, and O(1) is a bounded quantity as λ → +∞. In particular, there exists a positive constant C such that for all λ > 1,

N_{d}(λ) ≤ C(1 + λ^{\frac{1}{2} max\{n,m\}}) ≤ C(1 + λ^{\frac{n}{2} rank_{ℚ}(G)}).

When m < n, this theorem implies that N_{d}(λ) satisfies the Weyl upper bound. The condition m < n is clearly satisfied when rank_{ℚ}(G) is equal to 1 since there is no proper rational parabolic subgroup P satisfying rank_{ℚ}(P) ≤ rank_{ℚ}(G) − 1. When the ℝ-rank of G is equal to 2, then m = 0 and hence m < n. Therefore, this theorem implies Theorem 1.1.2.

For Siegel modular varieties Γ\Sp(3, ℝ)/U(3), n = 12 and m = 12. Theorem 1.1.3 implies the following bound of sharp order.

Corollary 1.1.4. When Γ\X = Γ\Sp(3)/U(3), there exists a constant c such that for all λ > 0

N_{d}(λ) ≤ c(1 + λ^{\frac{n}{2}} ),

where n = dim Γ\X = 12.

Theorem 1.1.3 improves the following bound on N_{d}(λ) due to Müller [35, Theorem 0.1] ^{1} that

N_{d}(λ) ≤ C(1 + λ^{\frac{n}{2} + \frac{3n}{2} rank_{ℚ}(G)})

for a positive constant C. (See also Remark 7.2.3 below for explanations.) When rank_{ℚ}(G) = 1, Donnelly proved earlier in [10, Theorem 4.11] that N_{d}(λ) ≤ C(1 + λ^{\frac{1}{2} (n+p)}) where p = max\{dim Γ_{M_{P}} \X_{P} | P is a proper rational parabolic subgroup of G}, and Langlands proved independently in [29] that N_{d}(λ) has a polynomial upper bound.

Such a polynomial upper bound on N_{d}(λ) is closely related to the trace class conjecture in the theory of the Selberg trace formula (see [35] and [48]). In fact, the method of proof of Theorem 1.1.3 is used in [21] to prove a polynomial upper bound on the discrete spectrum of L^{2}(Γ\G) and hence the trace class conjecture in full generality, i.e., the K–finiteness assumption on the convolution function in [35] will be removed.

1It is stated in [35, Theorem 0.1] that N_{d}(λ) ≤ C(1 + λ^{2n}). Actually, the proof only gives the above weaker bound.

Remark 1.1.5. The results of this paper along with [21] have been announced in [22]. It is claimed there that the Weyl upper bound is satisfied by the counting function N_{d}(λ) in all higher rank cases also. Unfortunately, in an early version of this paper, there is a gap, as in [35, p. 523], in the last step bounding the residual discrete spectrum associated with rational parabolic subgroups of rank greater than or equal to 2 (see Remark 7.2.3 below for details). Because of this, we can prove the Weyl upper bound only in the cases listed in Theorem 1.1.2 and get the weaker upper bound in Theorem 1.1.3 above for other cases. This weakening of the upper bound on N_{d}(λ), however, does not affect the solution of the trace class conjecture announced in [22].

1.2. On the other hand, relatively little is known about lower bounds for N_{d}(λ). Inspired by his upper bound on N_{d}(λ) and results for G = SL(2, ℝ) (see below), Müller made the following closely related but extremely difficult conjecture [36, Remarks 2, p. 180]. (This question is also raised in [4, §4.7].)

Conjecture 1.2.1. If Γ is an arithmetic subgroup of G, then the Weyl law holds:

lim_{λ→+∞} \frac{N_{d}(λ)}{λ^{n/2}} = (4π)^{−n/2} \frac{vol(Γ\X)}{Γ(\frac{n}{2} + 1)} dim σ.

An interesting corollary of Conjecture 1.2.1 is that the discrete subspace L_{d}^{2}(Γ\X, σ) is of infinite dimension, which is not known in general.

In [45, Conjecture 2], Sarnak made an even stronger conjecture that under the same assumption, the counting function for the cuspidal discrete spectrum alone satisfies the Weyl law. It seems that it is implicitly conjectured in [45] that the counting function of the residual discrete spectrum is of smaller order than the Weyl law (see Conjecture 1.3.1 below).

If Γ is an irreducible cofinite volume discrete subgroup and the rank of X is greater than or equal to two, then Margulis’ superrigidity theorem [31] implies that Γ is arithmetic, and hence the assumption that Γ is arithmetic is always satisfied in this case. On the other hand, if G = SL(2, ℝ) and Γ\X is a Riemann surface, the theory of disappearance of the cuspidal discrete spectrum under a generic deformation in the Teichmüller space of Γ\X developed by Phillips and Sarnak [45],

[41] and others shows that the arithmetic assumption on Γ is necessary. In fact, in [51], Wolpert showed that under the multiplicity one assumption on new forms, the Weyl law does not hold for a generic Riemann surface which is the sphere with even number of punctures and certain symmetries.

In view of the results of Efrat for function fields (see [15] for a summary), it may be necessary to require Γ to be a congruence subgroup in Conjecture 1.2.1.

Conjecture 1.2.1 has been proved only for the following cases:

1. G = SL(2), Γ a congruence subgroup, σ the trivial representation, a combination of results of Selberg [46, pp. 668 and 670], Hejhal [19, Chapter 11], and Huxley [20].

2. G = SO(n, 1) or SU(1, n), Γ a congruence subgroup, σ the trivial representation, by Reznikov [43].

3. G = R_{k/ℚ}SL(2)^{2}, Γ a congruence subgroup of the Hilbert modular group (any arithmetic subgroup is automatically congruent in this case), σ the trivial representation, by Efrat [13, p. 6] (the details are in [14]).

So Conjecture 1.2.1 is still open for a general arithmetic but noncongruence subgroup Γ of SL(2, ℤ).

In the above cases, the Weyl law is proved in two steps:

1. Obtain the Weyl–Selberg law from the Selberg trace formula:

N_{d}(λ) + N_{c}(λ) ∼ (4π)^{−n/2} \frac{vol(Γ\X)}{Γ(\frac{n}{2} + 1)}λ^{n/2},

where N_{c}(λ) counts the continuous spectrum and is an integral of the determinant of the scattering (or intertwining) matrix.

2. Study the scattering matrix and show that the term N_{c}(λ) is of smaller order than λ^{n/2} as λ → +∞.

In the general case, the trace formula in the original sense of Selberg is not as well developed as in the case of SL(2), though Arthur has developed a trace formula which expresses the trace of a convolution operator on the cuspidal subspace and is very powerful for applications to number theory and automorphic forms (see [1] and the references there). In particular, the Weyl–Selberg law is not known in general.

2In R_{k/ℚ}SL(2), k is a totally real number field, and R_{k/ℚ} is the functor of the restriction of the scalar.

Another main result of this paper gives bounds for the counting function of eigenvalues of the pseudo-Laplace operator (Theorem 3.3.2). This result can be considered as an analogue of the Weyl–Selberg law, since the spectrum of the pseudo-Laplacian involves both the discrete and the continuous spectra of the Laplacian. But the second step above is extremely difficult and seems to be out of reach now.

1.3. To sketch the proof of Theorem 1.1.2, we need to introduce more notation.

As mentioned above, we assume in the following that Γ\X is noncompact. Then the space L^{2}(Γ\X, σ) can be decomposed into two nontrivial subspaces:

L^{2}(Γ\X, σ) = L_{d}^{2}(Γ\X, σ) ⊕ L_{c}^{2}(Γ\X, σ),

where Δ has a discrete spectrum on L_{d}^{2}(Γ\X, σ) and a continuous spectrum on L_{c}^{2}(Γ\X, σ).

The discrete subspace L_{d}^{2}(Γ\X, σ) has a further decomposition:

L_{d}^{2}(Γ\X, σ) = L_{cus}^{2}(Γ\X, σ) ⊕ L_{res}^{2}(Γ\X, σ),

where L_{cus}^{2}(Γ\X, σ) is spanned by the cuspidal eigenfunctions of Δ, i.e., those eigenfunctions whose constant terms along all proper rational parabolic subgroups are zero (§2.3), and L_{res}^{2}(Γ\X, σ) is the orthogonal complement of L_{cus}^{2}(Γ\X, σ) in L_{d}^{2}(Γ\X, σ), called the residual discrete subspace.

In [11, Theorem 1.1] (see Lemma 2.3.2 below), Donnelly proved that the counting function N_{cus}(λ) of the cuspidal discrete spectrum, i.e., the spectrum of Δ on the cuspidal subspace L_{cus}^{2}(Γ\X, σ), satisfies the following Weyl upper bound:

lim_{λ→+∞} sup \frac{N_{cus}(λ)}{λ^{n/2}} ≤ (4π)^{−n/2} \frac{vol(Γ\X)}{Γ(\frac{n}{2} + 1)} dim σ.

Let N_{res}(λ) be the counting function of the residual discrete spectrum, i.e., the spectrum of Δ restricted to L_{res}^{2}(Γ\X, σ). Then

N_{d}(λ) = N_{cus}(λ) + N_{res}(λ).

Therefore, the problem is to bound N_{res}(λ) and to get the right constant for N_{d}(λ). Examples, whose residual discrete spectra are understood, and our work below strongly suggest the following conjecture.

Conjecture 1.3.1. The counting function of the residual discrete spectrum N_{res}(λ) is of smaller order than λ^{n/2} as λ → +∞, where n = dim Γ\X.

Remark 1.3.2. From the point of view of representation theory of G, the residual discrete subspace is contained in the subspace spanned by the complementary series subrepresentations^{3} and hence the residual discrete spectrum is contained in the complementary spectrum defined in [12]. If Γ\X is compact, it is proved in [12, Theorem 8.3] that the counting function of the complementary spectrum is indeed of smaller order than the Weyl law, i.e., the analogue of Conjecture 1.3.1 holds for compact spaces Γ\X.

To bound the residual discrete spectrum, we introduce a pseudo-Laplacian (or cut-off Laplacian) Δ_{T}, where T is the truncation parameter. The domain of Δ_{T} roughly consists of those functions whose constant terms along all proper rational parabolic subgroups above height T vanish (see §3 below for the precise definition). Then the spectrum of Δ_{T} is discrete. By modifying the arguments of Donnelly in [11], we prove that the spectral counting function of Δ_{T} satisfies the Weyl upper bound and a lower bound of the same order (Theorem 3.3.2), which is an analogue of the Weyl–Selberg law as mentioned earlier.

Then the problem is to understand the relation between the eigenvalues of Δ and Δ_{T}. All the cuspidal discrete spectrum of Δ is contained in the spectrum of Δ_{T} since truncating the constant terms above T does not affect the cuspidal functions (3.7.2), but the relation between the residual discrete spectrum of Δ and the spectrum of Δ_{T} is not entirely clear. Our guiding philosophy below is that Δ_{T} is a good perturbation of Δ and hence that the residual discrete spectrum can be approximated uniformly by a part of the spectrum of Δ_{T}.

To explain our approach to the residual discrete spectrum, we need to introduce further notation. As above, we assume that G is the real locus of a reductive algebraic group, and Γ ⊂ G(ℚ) is an arithmetic subgroup.

The space L^{2}(Γ\X, σ) can be decomposed according to the association classes C of rational parabolic subgroups P and the cuspidal spectra {μ} of the boundary locally symmetric spaces Γ_{M_{P}} \X_{P} associated with P (see Lemma 2.5.3):

3It is shown in [49] that tempered subrepresentations of L^{2}(Γ\G) is cuspidal, and hence residual discrete subspace is contained in the space spanned by nontempered subrepresentations.

\begin{matrix}L^{2}(Γ\X, σ) = ∑_{C} ⊕L_{C}^{2}(Γ\X, σ), \\ L_{C}^{2}(Γ\X, σ) = ∑_{μ} ⊕L_{C,μ}^{2}(Γ\X, σ). \end{matrix} (1)

When C = {G}, L_{C}^{2}(Γ\X, σ) is the cuspidal subspace L_{cus}^{2}(Γ\X, σ). For any 1 ≤ r ≤ rank_{ℚ}(G), the subspace

∑_{C} L_{C}^{2}(Γ\X, σ) ∩ L_{dis}^{2}(Γ\X, σ),

where the sum over C is over all the association classes of rational parabolic subgroups of rank r, is called the rank r residual discrete spectrum of Δ. The above decomposition of L^{2}(Γ\X, σ) also induces a decomposition of the pseudo-Laplacian Δ_{T} (§3.8). One significance of this decomposition of L^{2}(Γ\X, σ) is that after this decomposition, the discrete spectrum of Δ can be separated away from the continuous spectrum and the regular perturbation theory can be applied (see the remarks after Lemma 2.5.3 and Remark 4.2.4).

Using these decompositions and some positivity of the scattering matrices, we can show that the majority of the rank-one residual discrete spectrum can be approximated uniformly by the corresponding part of the spectrum of Δ_{T} (Corollary 5.2.7); in particular, Theorem 1.1.2 holds,

If such a uniform approximation also holds for the higher rank residual discrete spectrum, then the Weyl upper bound Δ_{T} (Theorem 3.3.2) implies immediately that the counting function N_{d}(λ) of the discrete spectrum of Δ satisfies the Weyl upper bound. We believe that such a uniform approximation holds, but we can not prove it. Instead, we follow Müller’s approach in [35] of reducing the bound on the higher rank residual discrete spectrum to bounds on poles of the rank-one scattering matrices. Using the above result on the rank-one residual discrete spectrum, the bound on the eigenvalues of Δ_{T}, and the decomposition of Δ and Δ_{T} induced from Equation (1) above, we can improve the estimates of Müller and prove Theorem 1.1.3.

Acknowledgements

I would like to thank Professor S.T.Yau for suggesting me to study the paper [35] many years ago. This paper depends essentially on [35].

The main difference is the emphasis on the relation between the Laplacian Δ and the pseudo-Laplacian (or cut-off Laplacian) Δ_{T}. I would also like to thank Professor W.Müller for his interests in early partial results. Especially, I would like to thank the referees for their extremely careful reading of the manuscript and pointing out several mistakes in an earlier version, and for their kind suggestions. I would also like to thank Dr. W.Hoffmann for meticulous reading of the manuscript, for pointing out several mistakes and helpful suggestions, and for sending me the preprint [8].

2. Spectral decomposition of L^{2}(Γ\X, σ)

2.1. In this section, we describe Langlands’ theory of Eisenstein series (2.4), various decompositions of L^{2}(Γ\X, σ) (2.5), the residual discrete spectrum as iterated residues of the cuspidal Eisenstein series (2.6), and some properties of the scattering matrices needed for the reduction to the rank-one case (2.7).

For convenience, throughout this paper, a function on Γ\X is identified with a Γ-invariant function on X, and the same convention applies to other quotient spaces.

2.2. Recall from §1.1 and §1.3 that G is the real locus G(ℝ) of a reductive algebraic group G defined over ℚ and satisfying the conditions in [18, p. 1], and Γ ⊂ G(ℚ) an arithmetic subgroup. Fix a maximal compact subgroup K of G and hence a basepoint x_{0} of X = G/K.

For any rational parabolic subgroup P, let N_{P} be the unipotent radical of P. Denote the real locus P(ℝ) of P by P, and the real locus N_{P}(ℝ) of N_{P} by N_{P}. Then P has a (rational) Langlands decomposition P = N_{P} M_{P} A_{P} such that M_{P} A_{P} is stabilized by the Cartan involution determined by K, where A_{P} is a lift of the connected component of the real locus of the split center of the Levi quotient N_{P}\P of P, and M_{P} is a lift of the real locus of the anisotropic part M_{P} of the Levi quotient. Denote the Lie algebras of A_{P} and N_{P} by 𝔞_{P} and 𝔫_{P} respectively. Then 𝔞_{P} acts on 𝔫_{P}, and the set of roots is denoted by Σ(P, A_{P}). The half sum of the roots in Σ(P, A_{P}) with multiplicity is denoted by ρ_{P}.

For any g ∈ P, write

g = n_{P} (g)m_{P} (g)a_{P} (g) = n_{P} (g)m_{P} (g) exp(H_{P} (g)), (1)

where n_{P} (g) ∈ N_{P} , a_{P} (g) ∈ A_{P}, H_{P} (g) ∈ 𝔞_{P}, m_{P} (g) ∈ M_{P} are uniquely determined by g.

Let K_{M_{P}} = K ∩ M_{P}. Then K_{M_{P}} is a maximal compact subgroup of M_{P}, and X_{P} = M_{P} /K_{M_{P}} is the boundary symmetric space associated with P. ^{4}

Since P acts transitively on X, the above decomposition induces the following decomposition of X:

X = N_{P} × X_{P} × A_{P} , (2)

i.e., any point x ∈ X = G/K can be uniquely written as

x = n_{P} (x)a_{P} (x)z_{P} (x)K,

where

n_{P} (x) ∈ N_{P} , z_{P} (x) ∈ X_{P} = M_{P} /K_{M_{P}} ,

a_{P} (x) ∈ A_{P}.

The arithmetic subgroup Γ induces several discrete subgroups associated with the Langlands decomposition of P. Define Γ_{P} = Γ ∩ P, Γ_{N_{P}} = Γ_{P} ∩ N_{P}, and Γ_{M_{P}} to be the image of Γ_{P} under the projection P = N_{P} A_{P} M_{P} → M_{P}. Then Γ_{N_{P}} is a cocompact subgroup of N_{P}, and Γ_{M_{P}} is an arithmetic subgroup of M_{P}. The quotient Γ_{M_{P}} \X_{P} is called the boundary locally symmetric space associated with P. As mentioned in §1.1, these boundary locally symmetric spaces form the boundary components of the reductive Borel–Serre compactification \overline{Γ\X}^{RBS} (see [24, §7]).

2.3. Let E_{σ} be the locally homogeneous bundle over Γ\X defined by a finite dimensional unitary representation σ of K. Let f be an E_{σ}-valued, locally bounded measurable function on Γ\X. The constant term f_{P} of f along a rational parabolic subgroup P is defined by

f_{P} (x) = ∫_{Γ_{N_{P}} \N_{P}} f(nx)dn, (1)

where dn is the Haar measure on N_{P} normalized by the condition that vol(Γ_{N_{P}} \N_{P}) = 1.

The subspace of L^{2}(Γ\X, σ) consisting of all functions whose constant terms along all proper rational parabolic subgroups vanish is called the cuspidal subspace and denoted by L_{cus}^{2}(Γ\X, σ).

4In general, X_{P} is not of noncompact type; instead, it is the product of a symmetric space of noncompact type and a Euclidean space, i.e., M_{P} is only reductive, not necessarily semisimple. This is the reason that we start with a reductive, instead of semisimple, algebraic group G.

Lemma 2.3.1. The spectrum of the Laplace operator Δ on L_{cus}^{2}(Γ\X, σ) is discrete, i.e., is of finite multiplicity and has no finite accumulation point. This spectrum is called the cuspidal spectrum and denoted by Spec_{cus}(Γ\X, σ).

Proof. It essentially follows from a result of Gelfand and Piatetski-Shapiro [17] [18, Theorem 3]. See [4] for a complete proof of this result.

q.e.d.

Let N_{cus}(λ) be the counting function of the cuspidal discrete spectrum in L_{cus}^{2}(Γ\X, σ). Then Lemma 2.3.1 implies that N_{cus}(λ) is finite for every λ > 0. As mentioned earlier, Donnelly proved the following Weyl upper bound on N_{cus}(λ) [11, Theorem 1.1].

Lemma 2.3.2. The counting function of the cuspidal spectrum N_{cus}(λ) satisfies the Weyl upper bound:

lim_{λ→+∞} sup \frac{N_{cus}(λ)}{λ^{n/2}} ≤ (4π)^{−n/2} \frac{vol(Γ\X)}{Γ(\frac{n}{2} + 1)} dim σ.

Remark 2.3.3. Donnelly proved this result for locally symmetric spaces of noncompact type. But the same method works in the slightly more general situation here. (See also Remark 3.4.3.)

2.4. We now introduce cuspidal Eisenstein series associated with a rational parabolic subgroup P. Let σ_{M} : K_{M_{P}} → GL(V) be the restriction of the representation σ : K → GL(V). Then σ_{M} defines a bundle E_{σ_{M}} on Γ_{M_{P}} \X_{P} and the space L^{2}(Γ_{M_{P}} \X_{P} , σ_{M}) of E_{σ_{M}} - valued functions on Γ_{M_{P}} \X_{P}. For any μ ∈ Spec_{cus}(Γ_{M_{P}} \X_{P} , σ_{M}), and a cuspidal eigenfunction Φ on Γ_{M_{P}} \X_{P}, ΔΦ = μΦ, where Δ is the Laplacian on L^{2}(Γ_{M_{P}} \X_{P} , σ_{M}) defined as in §1.1, define an Eisenstein series

E(P, Φ, Λ, x) = ∑_{γ∈Γ_{P} \Γ} e^{(ρ_{P} +Λ)(H_{P} (γx))}Φ(z_{P} (γx)),

where H_{P} (γx) ∈ 𝔞_{P} and z_{P} (γx) ∈ X_{P} are the components of γx in the decomposition of X in Equation 2.2.(2)^{5}, and Λ ∈ 𝔞_{P}^{∗} ⊗ ℂ, Re(Λ) ∈ ρ_{P} + 𝔞_{P}^{∗+}, 𝔞_{P}^{∗+} = {Λ ∈ 𝔞_{P}^{∗} | ⟨Λ, α⟩ > 0 for all α ∈ Σ(P, A_{P})}. This series converges uniformly over compact subsets of (ρ_{P} + 𝔞_{P}^{∗+} + i𝔞_{P}^{∗}) × X and can be continued as a meromorphic function in Λ to the whole complex space 𝔞_{P}^{∗} ⊗ ℂ (see Proposition 2.4.2 below).

5Here Equation 2.2.(2) means equation (2) in §2.2. The same convention applies to the rest of the paper.

The theory of constant terms of the Eisenstein series plays an important role in the spectral theory of Γ\X. To state it, we need the following definition.

Definition 2.4.1. Two rational parabolic subgroups P_{1}, P_{2} are called associated if there exists g ∈ G such that Ad(g)𝔞_{P_{1}} = 𝔞_{P_{2}}. The set of such isomorphisms between 𝔞_{P_{1}} and 𝔞_{P_{2}} is denoted by W (𝔞_{P_{1}}, 𝔞_{P_{2}}). For simplicity, W (𝔞_{P_{1}}, 𝔞_{P_{1}}) is denoted by W (𝔞_{P_{1}}).

Let P_{1}, P_{2} be two associated rational parabolic subgroups. For any cuspidal eigenvalue μ ∈ ∪_{i=1}^{2}Spec_{cus}(Γ_{M_{i}}\X_{P_{i}}, σ_{M_{i}}) denote the eigenspace of μ in L_{cus}^{2}(Γ_{M_{i}}\X_{P_{i}}, σ_{M_{i}}) by E_{cus}(Γ_{M_{i}}\X_{P_{i}}, σ_{M_{i}}, μ). Then for any s ∈ W (𝔞_{P_{1}}, 𝔞_{P_{2}}) and Λ ∈ 𝔞_{P_{1}}^{∗} ⊗ ℂ, there is an intertwining operator (or scattering matrix)

C_{P_{2},P_{1}}(s, Λ) : E_{cus}(Γ_{M_{1}}\X_{P_{1}}, σ_{M_{1}}, μ) → E_{cus}(Γ_{M_{2}}\X_{P_{2}}, σ_{M_{2}}, μ) (1)

such that for any Φ ∈ E_{cus}(Γ_{M_{1}}\X_{P_{1}}, σ_{M_{1}}, μ), the constant term of E(P_{1}, Φ, Λ) along P_{2} is given by

E_{P_{2}}(P_{1}, Φ, Λ, x) = ∑_{s∈W (𝔞_{P_{1}},𝔞_{P_{2}})}e^{(sΛ+ρ_{P_{2}})(H_{P_{2}}(x))}C_{P_{2},P_{1}}(s, Λ)Φ(x), (2)

where the s action on 𝔞_{P_{1}}^{∗} is defined as follows: For any Λ ∈ 𝔞_{P_{1}}^{∗}, H ∈ 𝔞_{P_{2}}, (sΛ)(H) = Λ(s^{−1}H) (see [18, Theorem 5]).

These intertwining operators satisfy functional equations. Let C be an association class of proper rational parabolic subgroups, and C_{1}, ⋯, C_{r} be the G(ℚ)-conjugacy classes in C. Let P_{1}, ⋯, P_{r} be representatives of the G(ℚ)-conjugacy classes. For any 1 ≤ i ≤ r, let P_{i1}, ⋯, P_{ir_{i}} be representatives of the Γ-conjugacy classes in C_{i}. Then P_{il}, 1 ≤ i ≤ r, 1 ≤ l ≤ r_{i}, are representatives of the Γ-conjugacy classes in C. Denote the split component 𝔞_{P_{i}} by 𝔞_{i}, and 𝔞_{P_{il}} by 𝔞_{il}. Choose y_{il} ∈ K such that P_{il} = y_{il}P_{i}y_{il}^{−1}. Then 𝔞_{il} = Ad(y_{il})𝔞_{i}. By duality, Ad(y_{il}) defines a map Ad(y_{il}) : 𝔞_{i}^{∗} → 𝔞_{il}^{∗}.

For the association class C, define

Spec_{cus}(C) = ∪_{i=1}^{r} ∪_{l=1}^{r_{i}} Spec_{cus}(Γ_{M_{il}}\X_{P_{il}}, σ_{M_{il}}),

called the cuspidal spectrum of the class C. For every μ ∈ Spec_{cus}(C), define

E_{cus}(C_{i}, μ) = ⊕_{l=1}^{r_{i}}E_{cus}(Γ_{M_{il}}\X_{P_{il}}, σ_{M_{il}}, μ). (3)

For any Φ = (Φ_{1}, ⋯, Φ_{r_{i}}) ∈ E_{cus}(C_{i}, μ), and Λ ∈ 𝔞_{i}^{∗} ⊗ ℂ, define

E(P_{i}, Φ, Λ) = ∑_{l=1}^{r_{i}} E(P_{il}, Φ_{l}, Ad(y_{il})Λ).

Then for 1 ≤ i, j ≤ r, s ∈ W (𝔞_{i}, 𝔞_{j}), Λ ∈ 𝔞_{i}^{∗} ⊗ ℂ, the scattering matrix

C_{ji}(s, Λ) : E_{cus}(C_{i}, μ) → E_{cus}(C_{j}, μ)

has entries C_{P_{jk},P_{il}}(Ad(y_{jk})s Ad(y_{il})^{−1}, Ad(y_{il})Λ), 1 ≤ k ≤ r_{j}, 1 ≤ l ≤ r_{i}, and the constant term of E(P_{i}, Φ, Λ) along P_{jk} is given as follows:

E_{P_{jk}}(P_{i}, Φ, Λ) = ∑_{s∈W (𝔞_{i},𝔞_{j})}e^{(sΛ+ρ_{P_{jk}})(H_{P_{jk}} (x))}(C_{ij}(s, Λ)Φ)_{jk}, (4)

where (C_{ij}(s, Λ)Φ)_{jk} is the jk-component in the decomposition E_{cus}(C_{j}, μ) in Equation (3) above.

A basic result in the theory of Eisenstein series is the following (see [18, Theorems 7, 8, 9] and [34, Proposition IV.1.11]).

Proposition 2.4.2.

1. For any 1 ≤ i, j ≤ r, s ∈ W (𝔞_{i}, 𝔞_{j}), the scattering matrix C_{ji}(s, Λ) is meromorphic in Λ ∈ 𝔞_{i}^{∗} ⊗ ℂ, whose all singularities lie along hyperplanes of the form ⟨α, Λ⟩ = c, where α ∈ Σ(P_{i}, A_{i}) and c is a constant. Moreover, the singular hyperplanes in the tube domain over the positive chamber 𝔞_{i}^{∗+} + \sqrt{−1}𝔞_{i}^{∗} are real and simple.

2. For any Φ ∈ E_{cus}(C_{i}, μ), the Eisenstein series E(P_{i}, Φ, Λ) satisfies the functional equation

E(P_{i}, Φ, Λ) = E(P_{j}, C_{ji}(s, Λ)Φ, sΛ).

3. The Eisenstein series E(P_{i}, Φ, Λ) is meromorphic in Λ ∈ 𝔞_{i}^{∗} ⊗ ℂ whose poles (with multiplicity) are contained in the union of the poles of C_{ij}(s, Λ), s ∈ W (𝔞_{i}, 𝔞_{j}), 1 ≤ j ≤ r.

Remark 2.4.3. To get the spectral decomposition of L^{2}(Γ\G) with respect to all invariant differential operators, one usually requires the cuspidal function Φ in E(P, Φ, Λ, x) to be an eigenfunction of all invariant differential operators as in [27] [34] [40]. Since we mainly deal with the spectral decomposition of the Laplacian on L^{2}(Γ\X, σ), we only require Φ to be an eigenfunction of the Laplacian on L^{2}(Γ_{M_{P}} \X_{P} , σ_{M}) to save notation.

2.5. In general,

E(P, Φ, Λ, x) ∉ L^{2}(Γ\X, σ),

though

E(P, Φ, Λ, x) ∈ L_{loc}^{2}(Γ\X, σ).

To get square integrable functions, we need to introduce pseudo-Eisenstein series. For any function f ∈ C_{0}^{∞}(𝔞_{P}), define

E(P, Φ, f, x) = ∑_{γ∈Γ_{P} \Γ} f(H_{P} (γx))Φ(z_{P} (γx)).

Lemma 2.5.1 ([18, Corollary to Lemma 26]). For any such f, E(P, Φ, f, x) ∈ L^{2}(Γ\X, σ).

The relation between the Eisenstein series and the pseudo-Eisenstein series is as follows. Let f̂(Λ) be the Fourier transform of f:

f̂(Λ) = ∫_{𝔞} f(H)e^{−(Λ+ρ_{P})(H)}dH.

Then

(1) E(P, Φ, f, x) = ∫_{Re(Λ)=λ} E(P, f̂(Λ)Φ, Λ, x)dΛ,

where λ ∈ ρ_{P} + 𝔞_{P}^{∗+} [18, Lemma 28].

Lemma 2.5.2 ([18, Lemma 39]). Let P_{1}, P_{2} be two rational parabolic subgroups, and Φ_{1}, Φ_{2} are two cuspidal eigenfunctions on Γ_{M_{1}}\X_{P_{1}} and Γ_{M_{2}}\X_{P_{2}} with eigenvalues μ_{1}, μ_{2} respectively. Then

(E(P_{1}, Φ_{1}, f_{1}), E(P_{2}, Φ_{2}, f_{2}))_{L^{2}} = 0

if either P_{1} and P_{2} are not associated, or μ_{1} ≠ μ_{2}.

For any association class C of rational parabolic subgroups and a cuspidal eigenvalue μ ∈ Spec_{cus}(C), which is the union of the cuspidal eigenvalues of L^{2}(Γ_{M_{P}} \X_{P} , σ_{M}) for all the parabolic subgroups P ∈ C, the pseudo-Eisenstein series E(P, Φ, f), where Φ ∈ E_{cus}(Γ_{M}\X_{P} , σ_{M}, μ), P ∈ C, f ∈ C_{0}^{∞}(𝔞_{P}), span a closed subspace of L^{2}(Γ\X, σ), denoted by L_{C,μ}^{2}(Γ\X, σ). For convenience, the pair (C, μ) is called a cuspidal pair.

For each association class C, define

L_{C}(Γ\X, σ) = ∑_{μ∈Spec_{cus}(C)} L_{C,μ}^{2}(Γ\X, σ).

Then we have the following decomposition of L^{2}(Γ\X, σ).

Lemma 2.5.3 ([18, Lemma 45]). With the above notation,

L^{2}(Γ\X, σ) = ∑_{C} ⊕L_{C}(Γ\X, σ) = ∑_{(C,μ)} ⊕L_{C,μ}^{2}(Γ\X, σ),

where the second sum is over all the cuspidal pairs (C, μ). The Laplace operator Δ preserves this decomposition and hence is the direct sum of the restrictions to the subspaces.

In general, the discrete spectrum of Δ on L^{2}(Γ\X, σ) is embedded in the continuous spectrum. But after this decomposition of L^{2}(Γ\X, σ) in Lemma 2.5.3, the discrete spectrum is separated out. For example, if C = {G}, then Δ has only a discrete spectrum on L_{C}^{2}(Γ\X, σ), which is equal to the cuspidal subspace L_{cus}^{2}(Γ\X, σ). For an association class C of rank-one parabolic subgroups, the discrete spectrum of Δ on L_{C,μ}^{2}(Γ\X, σ) is below its continuous spectrum on L_{C,μ}^{2}(Γ\X, σ) (see Lemma 4.2.2 below). For other association classes, it is more complicated (see Remark 4.2.4 below).

2.6. Next we recall the description of the residual discrete spectrum as iterated residues of cuspidal Eisenstein series. Besides the cuspidal eigenfunctions and cuspidal Eisenstein series, the residual discrete spectrum and the associated noncuspidal Eisenstein series are also needed to give the spectral decomposition of L^{2}(Γ\X, σ).

For any rational parabolic subgroup P, an affine subspace in 𝔞_{P}^{∗} ⊗ ℂ is called admissible if it is the intersection of affine hyperplanes of the form {Λ ∈ 𝔞_{P}^{∗} ⊗ ℂ | ⟨α, Λ⟩ = c}, where α ∈ Σ(P, A_{P}), and ⟨·, ·⟩ is induced from the Killing form. In particular, the affine hyperplanes {Λ ∈ 𝔞_{P}^{∗} ⊗ ℂ | ⟨α, Λ⟩ = c} are called admissible hyperplanes.

A complete flag of admissible subspaces of 𝔞_{P}^{∗}:

F : 𝔞_{P}^{∗} ⊗ ℂ = V_{r} ⊃ V_{r−1} ⊃ ⋯ ⊃ V_{0}, r = dim 𝔞_{P}^{∗} ,

is called an admissible flag. For any admissible flag, choose real unit vectors Λ_{i} ∈ V_{i}, i = 1, ⋯, r such that Λ_{i} ⊥ V_{i−1}. Then for any meromorphic function E(Λ) on 𝔞_{P}^{∗} ⊗ ℂ whose singularities lie on admissible hyperplanes, we can define the iterated residue Res_{F}E along the admissible flag F as follows: E_{r}(Λ) = E(Λ), and for i = 0, ⋯, r − 1, Λ ∈ V_{i},

E_{i}(Λ) = δ ∫_{0}^{1} E_{i+1}(Λ + δe^{2π\sqrt{−1}θ}Λ_{i+1})dθ,

where δ is a small positive number so that E_{i+1}(Λ + zΛ_{i+1}) has no singularities for 0 < |z| < δ, and

Res_{F}E(Λ) = E_{0}(Λ_{0}),

where Λ_{0} is the unique element in V_{0}.

By Lemma 2.5.3, to describe the residual discrete subspace L_{res}^{2}(Γ\X, σ), it suffices to describe the subspaces

L_{res}^{2}(Γ\X, σ) ∩ L_{C,μ}^{2}(Γ\X, σ)

for all cuspidal pairs (C, μ). If C = {G}, then

L_{C,μ}^{2}(Γ\X, σ) ⊂ L_{cus}^{2}(Γ\X, σ),

and this intersection

L_{res}^{2}(Γ\X, σ) ∩ L_{C,μ}^{2}(Γ\X, σ)

is empty. We assume that C is an association class of proper rational parabolic subgroups. Then

L_{res}^{2}(Γ\X, σ) ∩ L_{C,μ}^{2}(Γ\X, σ)

is given as follows.

For any P ∈ C, Φ ∈ E_{cus}(Γ_{M}\X_{P} , σ_{M}, μ), by Proposition 2.4.2, the singularities of E(P, Φ, Λ) lie on admissible hyperplanes of 𝔞_{P}^{∗} ⊗ ℂ. For any r singular hyperplanes H_{1}, ⋯, H_{r} with ∩_{i=1}^{r}H_{i} = {Λ_{0}}, where r = dim 𝔞_{P}^{∗}, set V_{r} = 𝔞_{P}^{∗} ⊗ ℂ, and for j = 1, ⋯, r, V_{r−j} = H_{1} ∩ ⋯ ∩ H_{j}. Then F : V_{r} ⊃ ⋯ V_{0} = {Λ_{0}} is an admissible flag. For every s ∈ W (𝔞_{P}), s · F is also an admissible flag.

For any f ∈ C_{0}^{∞}(𝔞_{P}), let f^{∧}(Λ) be its Fourier transformation which

is a holomorphic function on 𝔞_{P}^{∗} ⊗ ℂ. Then Res_{F}(E(P, Φ, Λ)f^{∧}(Λ)) is a function on Γ\X.

Proposition 2.6.1 ([27, Theorem 7.1, p. 222] [34, Theorem V.3.13, p. 221]). Let ^{+}𝔞_{P}^{∗} be the closed cone in 𝔞_{P}^{∗} dual to the positive chamber 𝔞_{P}^{+}, i.e., the span of the roots in Σ(P, A_{P}). If Λ_{0} ∈ ^{+}𝔞_{P}^{∗}, then

the following sum of iterated residues ∑_{s∈W (𝔞_{P})} Res_{s·F}(E(P, Φ, Λ)f^{∧}(Λ)) is square integrable, i.e., belongs to L^{2}(Γ\X, σ), and is an eigenfunction of Δ with eigenvalue |ρ_{P} |^{2} − |Λ_{0}|^{2} + μ. Furthermore, the residual discrete subspace L_{res}^{2}(Γ\X, σ) ∩ L_{C,μ}^{2}(Γ\X, σ) is spanned by these iterated residues ∑_{s∈W (𝔞_{P})} Res_{s·F}(E(P, Φ, Λ)f^{∧}(Λ)) at singular points Λ_{0} in the bounded, closed domain {Λ ∈ ^{+}𝔞_{P}^{∗} | |Λ| ≤ |ρ_{P} |}, where P ∈ C, Φ ∈ E_{cus}(Γ_{M}\X_{P} , σ_{M}, μ), and f ∈ C_{0}^{∞}(𝔞_{P}).

Remark 2.6.2. The sum over W (𝔞_{P}) is necessary for the square integrability of the residues. In fact, each individual iterated residue

Res_{F}(E(P, Φ, Λ)f^{∧}(Λ)) may not be square integrable. See [16, Remark 1 at the end of §6] for an explicit example. On the other hand, if

rank_{ℚ}(P) = 1 and Φ is cuspidal, then Res_{F}(E(P, Φ, Λ)f^{∧}(Λ)) is square integrable and the residual discrete subspace of L_{C,μ}^{2}(Γ\X, σ) is spanned by such residues.

Remark 2.6.3. The factor f^{∧}(Λ) is inserted because E(P, Φ, Λ) could have nonsimple poles outside the positive chamber 𝔞_{P}^{∗+}. This phenomenon can be seen in the example of G_{2} in [27, Appendix III].

Remark 2.6.4. The fact that the singular point Λ_{0} is restricted to the bounded domain {Λ ∈ ^{+}𝔞_{P}^{∗} | |Λ| ≤ |ρ_{P} |} could be briefly explained as follows: The Eisenstein series E(P, Φ, Λ) is holomorphic in Λ when Re(Λ) ∈ ρ_{P} + 𝔞_{P}^{∗+}. The residues are picked up when the contour of integration in Equation 2.5.(1) is moved from λ ∈ ρ_{P} + 𝔞_{P}^{∗+} to λ = 0. In the process, contours of smaller dimension based at one point λ_{1} are moved to another basepoint λ_{2}, where λ_{1} belongs to an admissible real affine subspace V which is the intersection of singular hyperplanes of E(P, Φ, Λ), and λ_{2} is the intersection of this admissible subspace V and the subspace spanned by the roots which define V. (See [28] and [27, Appendix III] for pictures and illustrations of the deformation of the contours of integration.) So after the whole deformation process, only iterated residues at points in the above bounded domain are picked up.

Remark 2.6.5. Once we get the residual discrete spectrum, we can describe the complete spectral decomposition of L_{C,μ}^{2}(Γ\X, σ) for C ≠ {G}. The complement of the residual discrete spectrum in L_{C,μ}^{2}(Γ\X, σ) is a continuous spectrum. This continuous subspace is spanned by wave packets of Eisenstein series E(Q, Ψ, Λ), Re(Λ) = 0, where Q is a rational parabolic subgroup containing a group P ∈ C, and Ψ is a residual eigenfunction on Γ_{M_{Q}}\X_{Q}, which is the residue of an Eisenstein series on Γ_{M_{Q}}\X_{Q} associated with a cuspidal eigenfunction Φ on Γ_{M_{P}} \X_{P}. If Q = P, then E(Q, Ψ, Λ) = E(P, Φ, Λ) is a cuspidal Eisenstein series. Otherwise, such Eisenstein series E(Q, Ψ, Λ) arise as sums of the iterated residues of the cuspidal Eisenstein series E(P, Φ, Λ) along an affine subspace of positive dimension.

2.7. The above description of the residual discrete spectrum as iterated residues and Proposition 2.4.2 show that to bound the counting function of the residual discrete spectrum, it suffices to bound the number of complete flags of singular hyperplanes of the scattering matrices C(s, Λ) in the bounded domain in Proposition 2.6.1.

In the rest of this section, we recall from [18, Chap. V] and [35, pp. 523–525] that the scattering matrices of higher rank parabolic subgroups can be expressed as products of the scattering matrices of rank-one parabolic subgroups of some Levi subgroups of G. This reduces the problem to bounding the poles of the scattering matrices of rank-one.

Let C be an association class of rational parabolic subgroups. Fix a parabolic subgroup P ∈ C with split component A = A_{P}. Then there is a canonical correspondence between the set of rational parabolic subgroups whose split components are equal to A and the set of chambers in 𝔞. Denote the chambers of 𝔞 by C_{1}, ⋯, C_{q}, and the corresponding parabolic subgroups by P_{1}, ⋯, P_{q}. Since every parabolic subgroup in C is G(ℚ)-conjugate to a parabolic subgroup whose split component is equal to A, the set of P_{1}, ⋯, P_{q} contains a set of representatives of the G(ℚ)-conjugacy classes in C in §2.4. For any 1 ≤ i ≤ q, denote the G(ℚ)-conjugacy class in C containing P_{i} by C_{i}. Then for any s ∈ W (𝔞) and 1 ≤ i, j ≤ q, we can define a scattering matrix C_{ji}(s, Λ) as in §2.4.

Lemma 2.7.1 ([18, Theorem 8]). For any 1 ≤ i, j, k ≤ q, and s, t ∈ W (𝔞),

C_{ji}(ts, Λ) = C_{jk}(t, sΛ)C_{ki}(s, Λ).

Lemma 2.7.2 ([18, Lemmas 80 and 115]). For any 1 ≤ i ≤ q, C_{ii}(1, Λ) = Id. For any 1 ≤ i, j ≤ q, s ∈ W (𝔞), if sC_{i} = C_{j}, then C_{ji}(s, Λ) is holomorphic in Λ ∈ 𝔞^{∗} ⊗ ℂ.

Lemma 2.7.3 ([18, p. 123]). For any two chambers C_{i} and C_{j}, there exists a chain of chambers C_{i_{1}}, ⋯, C_{i_{p}} with p ≤ |Σ(P, A_{P})| such that C_{i_{1}} = C_{i}, C_{i_{p}} = C_{j}, and C_{i_{l}} and C_{i_{l+1}} are adjacent for l = 1, ⋯, p − 1.

From Lemma 2.7.1, we get the following factorization of the scattering matrices.

Lemma 2.7.4. For any 1 ≤ i, j ≤ q and s ∈ W (𝔞), let C_{k} = sC_{i}, and C_{j_{1}}, ⋯, C_{j_{p}} a chain connecting C_{j} and C_{k} as in Lemma 2.7.3.

Then

(1)  C_{ji}(s, Λ) = C_{jk}(1, sΛ)C_{ki}(s, Λ)

= C_{j_{1},j_{2}}(1, sΛ) ⋯ C_{j_{p−1},j_{p}}(1, sΛ)C_{ki}(s, Λ). (2)

By Lemma 2.7.2, C_{ki}(s, Λ) is holomorphic. Since the chambers C_{j_{l}}, C_{j_{l+1}} are adjacent, to bound the singularities of C_{ij}(s, Λ), it suffices to bound the singularities of C_{ij}(1, Λ) when C_{i}, C_{j} are adjacent.

Assume that C_{i}, C_{j} are adjacent. Let Q be the unique rational parabolic subgroup containing both P_{i}, P_{j} with \overline{𝔞_{Q}^{+}} = \overline{C_{i}} ∩ \overline{C_{j}}, in particular, rank_{ℚ}(Q) = rank_{ℚ}(P_{i}) − 1. Then P_{i}, P_{j} determine two rankone parabolic subgroups ′P_{i},′P_{j} of M_{Q}. Denote the Lie algebra of the split component of ′P_{i} by ′𝔞_{i}. Then dim′𝔞_{i} = 1. Denote the M_{Q}(ℚ)- conjugacy classes of rational parabolic subgroups of M_{Q} containing ′P_{i},′P_{j} by ′C_{i},′C_{j}, and the scattering matrix by C_{′j,′i}(s,′Λ), s ∈ W (′𝔞_{i}), ′Λ ∈ (′𝔞_{i})^{∗} ⊗ ℂ, where (′𝔞_{i})^{∗} is the dual of ′𝔞_{i}.

Lemma 2.7.5 ([18, p. 124-125], [35, p. 524-525]). An entry of the scattering matrix C_{ji}(1, Λ) is either zero or equal to an entry of the rank-one scattering matrix C_{′j,′i}(s,′Λ), where s ∈ W (′𝔞_{i}), and ′Λ is the restriction of Λ to ′𝔞_{i} ⊗ ℂ.

3. The pseudo-Laplacian Δ_{T}

3.1. In this section, we recall the pseudo-Laplacian (or cut-off Laplacian) Δ_{T} introduced by Müller [35, §3] and show that its counting function satisfies the Weyl upper bound and a lower bound of the same order (§3.3). This result could be interpreted as an analogue of the Weyl–Selberg law as mentioned in §1.2. Then we characterize the eigenfunctions of Δ_{T} in order to establish connections between Δ and Δ_{T} (§3.7).

The pseudo-Laplacian Δ_{T} is closely connected with Arthur’s truncation operator Λ^{T} in [2], [39]. In fact, Δ_{T} can be defined through Λ^{T} (see §3.3).

3.2. Before defining the pseudo-Laplacian, we recall the precise reduction theory. Let P_{1}, ⋯, P_{m} be a set of representatives of Γ-conjugacy classes of rank-one, i.e., maximal rational parabolic subgroups of G. Let 𝔞_{j} = 𝔞_{P_{j}}, j = 1, ⋯, m. Define

𝔞_{0} = ⊕_{j=1}^{m}𝔞_{j}.

Then for any rational parabolic subgroup P, there is a well defined map

I_{P} : 𝔞_{0} → 𝔞_{P} (1)

such that if P = P_{j}, 1 ≤ j ≤ m, then I_{P_{j}} is the projection from 𝔞_{0} to the factor 𝔞_{P_{j}}, [39, p. 330], [44].

Let ρ_{j} be the half sum of the roots in Σ(P_{j}, A_{j}) with multiplicity. Then ρ_{j} defines a vector H_{ρ_{j}} in 𝔞_{j} under the duality defined by the Killing form. These vectors H_{ρ_{j}}, j = 1, ⋯, m, define a unique vector H_{ρ} in 𝔞_{0} such that I_{P_{j}}(H_{ρ}) = H_{ρ_{j}}.

Fix a large positive number t so that T = tH_{ρ} ∈ 𝔞_{0}. For any rational parabolic subgroup P, define

A_{P,T} ={e^{H} ∈ A_{P} | α(H) > α(I_{P}(T)), α ∈ Σ(P, A_{P})},
A_{P}^{T} ={e^{H} | ⟨I_{P} (T) − H, V ⟩ ≥ 0 for all V ∈ 𝔞_{P}^{+}},

i.e., A_{P}^{T} is a shift of the negative of the obtuse cone dual to the dominant cone A_{P,0} = exp 𝔞^{+}. Using the decomposition X = N_{P} × A_{P} × X_{P} in Equation 2.2.(2), we get a subset N_{P} × A_{P}^{T} × X_{P} in X.

Denote by X_{T} the intersection ∩_{P}N_{P} × A_{P}^{T} × X_{P} over all proper rational parabolic subgroups P. Then X_{T} is a Γ-invariant noncompact submanifold with corners of X of codimension 0 and is the central tile in [44, Equation (5.1)] associated with the parameter T.

Proposition 3.2.1 ([44, Proposition 2.2, Theorem 5.7]). For t ≫ 0 and T = tH_{ρ}, the quotient Γ\X_{T} is a compact submanifold with corners of Γ\X, denoted by (Γ\X)_{T}.

Intuitively, (Γ\X)_{T} is obtained by chopping off all the cusps at infinity at the height T and hence is a compact core of Γ\X. In the following, T will be referred to as the truncation parameter.

Let P_{1}, ⋯, P_{p} be a set of representatives of Γ-conjugacy classes of proper rational parabolic subgroups. Then the precise reduction theory in [44, Proposition 2.2 and Theorem 5.7] can be stated as follows.

Proposition 3.2.2. For every 1 ≤ i ≤ p, there exists a compact submanifold with corners ω_{i} ⊂ Γ_{P_{i}}\N_{P_{i}} ×X_{P_{i}} which is left N_{P_{i}}-invariant such that when t is sufficiently large as above and T = tH_{ρ}, the subset ω_{i} × A_{P_{i},T} ⊂ Γ_{P_{i}}\X is mapped injectively into Γ\X whose image is still denoted by ω_{i} × A_{P_{i},T}, and Γ\X admits the following disjoint decomposition:

Γ\X = (Γ\X)_{T} ∪ ∐_{i=1}^{p} ω_{i} × A_{P_{i},T} .

3.3. Denote by H^{1}(Γ\X, σ) the Sobolev space of sections f of the vector bundle E_{σ} such that

∫_{Γ\X} |f|^{2} + | ▽ f|^{2} < +∞.

For the truncation parameter T ∈ 𝔞_{0} as above, define a subspace H_{T}^{1} (Γ\X, σ) of H^{1}(Γ\X, σ) as follows:

H_{T}^{1} (Γ\X, σ) = {f ∈ H^{1}(Γ\X, σ) | f_{P} (nza) = 0
for n ∈ N_{P} , z ∈ X_{P} , a ∈ A_{P,T} },

where P runs over all proper maximal parabolic subgroups of G. For a non-maximal parabolic subgroup P, the constant term f_{P} (nza) vanishes when a ∈ A_{P} \ A_{P}^{T}, n ∈ N_{P}, z ∈ X_{P}. Roughly, H_{T}^{1} (Γ\X, σ) is the subspace of functions all whose constant terms vanish outside the compact core (Γ\X)_{T}. (See Remark 3.3.3 for another description of H_{T}^{1} (Γ\X, σ).)

Since H_{T}^{1} (Γ\X, σ) is a closed subspace of H^{1}(Γ\X, σ), the Dirichlet quadratic form

D(f) = ∫_{Γ\X} | ▽ f|^{2}

restricts to H_{T}^{1} (Γ\X, σ) and defines a self-adjoint operator Δ_{T} on the closure of H_{T}^{1} (Γ\X, σ) in L^{2}(Γ\X, σ) [35, p. 489]. This operator Δ_{T} is called the pseudo-Laplace (or cut-off Laplace) operator at the height T. If G = SL(2, ℝ), this operator was first defined by Lax and Phillips [30], and used by Colin de Verdiere [7] to study the meromorphic continuation of the Eisenstein series and the discrete spectrum.

Proposition 3.3.1 ([35, Theorem 3.23]). The spectrum of Δ_{T} is discrete and the counting function N_{T} (λ) = |{λ_{i} ∈ Spec(Δ_{T}) | λ_{i} ≤ λ}| satisfies N_{T} (λ) ≤ c(1 + λ^{n/2}), λ ≥ 0, where n = dim Γ\X and c is a positive constant.

The main result of this section is the following bounds for the pseudo-Laplacian Δ_{T}.

Theorem 3.3.2. Let N_{T} (λ) be the counting function of the eigenvalues of Δ_{T}. Then

(4π)^{−n/2} \frac{vol((Γ\X)_{T})}{Γ(\frac{n}{2} + 1)} dim σ ≤ lim_{λ→+∞} inf \frac{N_{T} (λ)}{λ^{n/2}}
≤ lim_{λ→+∞} sup \frac{N_{T} (λ)}{λ^{n/2}}
≤(4π)^{−n/2} \frac{vol(Γ\X)}{Γ(\frac{n}{2} + 1)} dim σ.

Only the upper bound for N_{T} (λ) is used in the proof of Theorems 1.1.2 and 1.1.3, but the lower bound is of independent interest. It will be shown below that N_{T} (λ) involves counting of both the discrete spectrum and the continuous spectrum of Δ, and hence this theorem can be looked upon as an analogue of the Weyl–Selberg law as mentioned in §1.2 (see Remark 3.5.2 for more details).

In the next two subsections §3.4 and §3.5, we prove Theorem 3.3.2. In §3.6, we introduce other pseudo-Laplace operators. In the ℚ-rank-one case, they are the same as Δ_{T} but different in the higher rank cases. These other operators seem to have nicer geometric properties.

Remark 3.3.3. Let L_{T}^{2} (Γ\X, σ) be the closure of H_{T}^{1} (Γ\X, σ) in L^{2}(Γ\X, σ). This subspace can be described by Arthur’s truncation operator Λ^{T}, [2, p. 89], [39], [35, pp. 487–489]. Let P_{1}, ⋯, P_{p} be the set of representatives of Γ-conjugacy classes of proper rational parabolic subgroups as in §3.2. For each P_{i}, let χ_{i} be the characteristic function of +𝔞_{P_{i}} = {H ∈ 𝔞_{P_{i}} | ⟨H, V ⟩ ≥ 0 for all V ∈ 𝔞_{P_{i}}^{+}}, which is the obtuse cone dual to the dominant cone 𝔞_{P_{i}}^{+}. Then for a large truncation parameter T as in §3.2, if f is continuous and belongs to L^{2}(Γ\X, σ), define

Λ_{P_{i}}^{T}f(x) = ∑_{γ∈Γ_{P_{i}}\Γ} χ_{i}(H_{P_{i}}(γx) − I_{P_{i}}(T))f_{P_{i}}(γx),

where I_{P_{i}}(T) is the image of T in 𝔞_{i} (see Equation 1 in §3.2), and

Λ^{T} f(x) = f(x) + ∑_{i=1}^{p}(−1)^{rank_{ℚ}(P_{i})}Λ_{P_{i}}^{T}f(x).

A basic property of the truncation operator Λ^{T} is that for every maximal proper rational parabolic subgroup P, (Λ^{T} f(x))_{P} (nza) = 0 for all n ∈ N_{P}, z ∈ X_{P}, a ∈ A_{P,T}, i.e., Λ^{T} f(x) satisfies precisely the vanishing conditions in the definition of H_{T}^{1} (Γ\X, σ), hence Λ^{T} f ∈ L_{T}^{2} (Γ\X, σ); furthermore, Λ^{T} extends to an orthogonal projection on L^{2}(Γ\X, σ) [2, Remark, p. 92], and L_{T}^{2} (Γ\X, σ) = Λ^{T} L^{2}(Γ\X, σ). Consequently, we can define H_{T}^{1} (Γ\X, σ) and hence the pseudo-Laplacian Δ_{T} by setting H_{T}^{1} (Γ\X, σ) = H^{1}(Γ\X, σ) ∩ Λ^{T} L^{2}(Γ\X, σ). This gives a more direct relation between the pseudo-Laplacian Δ_{T} and Arthur’s truncation operator Λ^{T}.

3.4. First we prove the upper bound for N_{T} (λ) in Theorem 3.3.2:

Proposition 3.4.1.

lim_{λ→+∞} sup \frac{N_{T} (λ)}{λ^{n/2}} ≤ (4π)^{−n/2} \frac{vol(Γ\X)}{Γ(\frac{n}{2} + 1)} dim σ.

Proof. The proof is a combination of the precise reduction theory and the proof of [11, Theorem 1.1]. Let

Γ\X = (Γ\X)_{T} ∪ ∐_{i=1}^{p} ω_{i} × A_{P_{i},T}

be the decomposition in Proposition 3.2.2. Let Y_{0,k}, k ≥ 1, be a family of smooth compact submanifolds with boundary such that (1) Y_{0,k} ⊂ Y_{0,k+1}, and ∪_{k=1}^{∞}Y_{0,k} = Γ\X, (2) the interior of Y_{0,k} contains

(Γ\X)_{T} ∪ ∐ ω_{i} × (A_{P_{i},T} ∩ A_{P_{i}}^{T_{k}}),

and Y_{0,k} is contained in the interior of

(Γ\X)_{T} ∪ ∐ ω_{i} × (A_{P_{i},T} ∩ A_{P_{i}}^{T_{k+1}}),

where T_{k} = kH_{ρ} + T = (k + t)H_{ρ}.

For every pair of i and k, let Y_{i,k} be a smooth manifold with boundary in Γ_{P_{i}}\X such that (1) Y_{i,k} is invariant under N_{P_{i}}, (2) the image of Y_{i,k} in Γ\X contains ω_{i} × A_{P_{i},T} − Y_{0,k}, and the images of Y_{1,k}, ⋯, Y_{P,k} in Γ\X together with Y_{0,k} cover Γ\X, (3) the image of Y_{i,k} in Γ\X is contained in the complement of Y_{0,k−1}, and the image in the split component A_{P_{i}} of points in Y_{i,k} is contained in the complement of A_{P_{i}}^{T_{k−1}} in A_{P_{i}} and hence shrinks to infinity, and the image in Γ_{M_{P_{i}}} \X_{P_{i}} of points in Y_{i,k} is bounded uniformly for k (in fact contained in a fixed neighborhood of (Γ_{M_{P_{i}}} \X_{P_{i}})_{T}).

For every i = 1, ⋯, p, the homogeneous bundle Ẽ_{σ} on X induces a bundle E_{σ} on Γ_{P_{i}}\X. Denote the space of L^{2}-sections of E_{σ} by L^{2}(Γ_{P_{i}}\X, σ), and the subspace for Y_{i,k} by L^{2}(Y_{i,k}, σ). Define the cuspidal subspace H_{cus}^{1}(Y_{i,k}, σ) by

H_{cus}^{1}(Y_{i,k}, σ) = {f ∈ H^{1}(Y_{i,k}, σ) | f_{P} = 0 for all proper P ⊃ P_{i}}.

Since Y_{i,k} is invariant under N_{P_{i}} and hence all N_{P} for P ⊃ P_{i}, the cuspidal subspace is well-defined. The Dirichlet quadratic form defines a self-adjoint operator Δ_{i,N} on the closure of H_{cus}^{1}(Y_{i,k}, σ) in L^{2}(Y_{i,k}, σ), where the subscript N stands for the Neumann boundary condition.

Lemma 3.4.2. The spectrum of Δ_{i,N} is discrete and its spectral counting function N_{i,k}(λ) satisfies the following bound:

N_{i,k}(λ) ≤ ε_{i}(k)λ^{\frac{n}{2}} , λ ≥ 1,

where ε_{i}(k) → 0 as k → +∞.

Proof. This follows from the same proof of [11, Corollary 7.6]. One crucial point here is that the component of Y_{i,k} is A_{P_{i}} shrinks to infinity, while the component of Y_{i,k} in Γ_{M_{P_{i}}} \X_{P_{i}} stays bounded as k → +∞.

Remark 3.4.3. As pointed out recently in [8], there is a possible gap in the construction of the cuspidal Neumann heat kernel in [11]. Briefly, the order of the three steps in constructing the cuspidal heat kernel of X_{i,k} with the Neumann boundary condition needs to be changed to the following: (1) Average the heat kernel of X over Γ_{P_{i}} to get the heat kernel of Γ_{P_{i}}\X. (2) Use the method of single layer potentials to adjust the heat kernel to satisfy the Neumann boundary condition. (3) Remove the constant term of the heat kernel in Step (2) to make it cuspidal.

In [11], Step (3) was performed before Step (2). This causes a problem since after the constant terms are removed, the heat kernel does not satisfy the bounds needed to carry out the method of single layer potentials. See [8] for details and also [37, pp. 332–334] for some related comments.

Proof of Proposition 3.4.1. Denote the counting function of the Neumann eigenvalues of Y_{0,k} by N_{0,k}(λ). Since Y_{0,k}, Y_{1,k}, ⋯, Y_{p,k} form a covering of Γ\X and the image of Y_{i,k} is contained in the complement of (Γ\X)_{T}, we have an inclusion

H_{T}^{1} (Γ\X, σ) ⊂ H^{1}(Y_{0,k}, σ) ⊕ H_{cus}^{1}(Y_{1,k}, σ) ⊕ ⋯ H_{cus}^{1}(Y_{p,k}, σ).

The principle of Neumann bracketing implies that

N_{T} (λ) ≤ N_{0,k}(λ) + ∑_{i=1}^{p} N_{i,k}(λ).

Since Y_{0,k} is a compact submanifold with boundary,

lim_{λ→+∞} \frac{N_{0,k}(λ)}{λ^{n/2}} = (4π)^{−n/2} \frac{vol(Y_{0,k})}{Γ(\frac{n}{2} + 1)} ≤ (4π)^{−n/2} \frac{vol(Γ\X)}{Γ(\frac{n}{2} + 1)},

and we get from Proposition 3.4.2 that

lim_{λ→+∞} sup \frac{N_{T} (λ)}{λ^{n/2}} ≤ (4π)^{−n/2} \frac{vol(Γ\X)}{Γ(\frac{n}{2} + 1)} + ∑_{i=1}^{p} ε_{i}(k).

Since ε_{i}(k) → 0 as k → +∞, the upper bound for N_{T} (λ) in Proposition 3.4.1 follows immediately.

Remark 3.4.4. In an early vision of this paper, the following slightly different proof of Proposition 3.4.1 was given, which turned out to be problematic, as explained below.

By the precise reduction theory, for each T′ ≥ T, we get a covering of Γ\X and hence a bound on N_{T} (λ):

(1)  lim_{λ→+∞} sup \frac{N_{T} (λ)}{λ^{n/2}} ≤ (4π)^{−n/2} \frac{vol(Γ\X)_{T′}}{Γ(\frac{n}{2} + 1)} dim σ + c(T′),

where c(T′) comes from bounds on the counting functions of the Neumann cuspidal eigenvalues of the covering of the infinity: ω_{i} × A_{P_{i},T′}, i = 1, ⋯, p (or their slight enlargements). We claimed that since A_{P_{i},T′} shrinks towards infinity as T′ → ∞, lim_{T′→∞} c(T′) = 0, which would imply the Weyl upper bound for N_{T} (λ). Hoffmann pointed out recently that this is not obvious. Actually, c(T′) is a product of two factors, one goes to zero and another to infinity. This can be seen through the example Γ\X = Γ_{1}\H^{2} × Γ_{2}\H^{2}, where H^{2} is the upper half plane, and Γ_{1}\H^{2}, Γ_{2}\H^{2} are non-compact and have finite volume. The reason is that when P_{i} is not a minimal parabolic subgroup, as T′ → ∞, ω_{i} becomes unbounded, i.e., not enough constant terms are removed on ω_{i} × A_{P_{i},T′} in this upper bound. Briefly, ω_{i} is a Γ_{N_{P_{i}}} \N_{P_{i}} bundle over (Γ_{M_{P_{i}}} \X_{P_{i}})_{T′}, and (Γ_{M_{P_{i}}} \X_{P_{i}})_{T′} develops cusps as T′ → ∞, and hence these new constant terms associated with the cusps of Γ_{M_{P_{i}}} \X_{P_{i}} (or rational parabolic subgroups of M_{P_{i}}) need to be removed also. If we remove these new constant terms, then the argument is equivalent to the above proof of Proposition 3.4.1.

Unfortunately, this approach outlined here was also used in [21] to prove the Weyl upper bound for the pseudo-Laplacian of Γ\G. The above discussions show that the arguments in [21, Proposition 4.3.2] need to be modified as indicated above or replaced by an analogue of the proof of Proposition 3.4.1 here. On the other hand, for a fixed T′ ≥ T, Equation (1) gives a bound N_{T} (λ) ≤ c(1 + λ^{\frac{n}{2}} ). Such a bound (in fact, any polynomial bound) is sufficient for the proof of the trace class conjecture in [21, Theorem 1.1.2].

The proof of Proposition 3.4.1 combines the covering in [11, p. 243] and the decomposition in Proposition 3.2.2. Since the constant terms of functions in H_{T}^{1} (Γ\X, σ) do not vanish along all horospheres, unlike cuspidal functions, this combination seems necessary.

3.5. Next we prove the lower bound for N_{T} (λ) in Theorem 3.3.2:

Proposition 3.5.1.

lim_{λ→+∞} inf \frac{N_{T} (λ)}{λ^{n/2}} ≥ (4π)^{−n/2} \frac{vol((Γ\X)_{T})}{Γ(\frac{n}{2} + 1)} dim σ.

Proof. Let H_{0}^{1}((Γ\X)_{T} , σ) be the Sobolev space with vanishing boundary values. Then it defines through the Dirichlet quadratic form a Laplacian Δ_{D} on L^{2}((Γ\X)_{T} , σ) satisfying the Dirichlet boundary condition. Let N_{D}(λ) be the counting function of the eigenvalues of Δ_{D}. Since (Γ\X)_{T} is a manifold with corners, it clearly satisfies the segment property (see [42, p. 256] for its definition and applications) and hence N_{D}(λ) satisfies the Weyl law (see [47, Corollary 2.5]);

lim_{λ→+∞} \frac{N_{D}(λ)}{λ^{n/2}} = (4π)^{−n/2} \frac{vol((Γ\X)_{T})}{Γ(\frac{n}{2} + 1)} dim σ.

For every function f in H_{0}^{1}((Γ\X)_{T} , σ), its extension by zero outside (Γ\X)_{T} gives a function f̃ in H^{1}(Γ\X, σ). We claim that f̃ ∈ H_{T}^{1} (Γ\X, σ). In fact, for any point x ∈ X which is mapped into the complement of (Γ\X)_{T} in Γ\X and any maximal rational parabolic subgroup P, by the definition of (Γ\X)_{T} in §3.2, the orbit N_{P} x of x under N_{P} is also projected into the complement of (Γ\X)_{T}. This implies that f̃ satisfies the vanishing conditions for the definition of H_{T}^{1} (Γ\X, σ) in §3.3. This proves the claim. Then the mini-max principle implies that

N_{T} (λ) ≥ N_{D}(λ),

and the lower bound for N_{T} (λ) in the proposition follows from the Weyl law for N_{D}(λ) above.

Remark 3.5.2. It was announced in [22, Theorem 2.4.1] that N_{T} (λ) satisfies the lower Weyl bound also, i.e., vol((Γ\X)_{T}) in the above proposition could be replaced by vol(Γ\X). Actually, the arguments only worked in the ℚ-rank-1 case, and for the higher rank case, they worked for a different pseudo–Laplacian (see §3.6 below for various pseudo-Laplacians in the higher rank case), though we believe that this sharp lower bound is true. One potential application of this lower bound for N_{T} (λ) in Proposition 3.5.1 is to serve as an analogue of the Weyl–Selberg law as mentioned near the end of §1.2. Specifically, suppose we could show that all eigenfunctions of Δ_{T} are either cuspidal eigenfunctions of Δ or truncation by Λ^{T} of some Eisenstein series, and show that the contribution coming from the continuous spectrum of Δ is of smaller order than λ^{n/2}, the analogue of Step 2 in §1.2, and that the residual discrete spectrum of Δ can be approximated well by the corresponding part of the spectrum of Δ_{T} (see Corollary 5.2.7 below for the rank-1 case). Then the Weyl upper bound of N_{T} (λ) would imply the Weyl upper bound of N_{d}(λ), and the lower bound for N_{T} (λ) would give a bound

lim_{λ→+∞} inf \frac{N_{d}(λ)}{λ^{\frac{n}{2}}} ≥ (4π)^{− \frac{n}{2}} \frac{vol((Γ\X)_{T})}{Γ(\frac{n}{2} + 1)} dim σ.

Since lim_{T →∞} vol((Γ\X)_{T}) = vol(Γ\X), by letting T → ∞, we would get the desired Weyl lower bound and hence the Weyl law for N_{d}(λ).

Remark 3.5.3. As mentioned in the previous remark, in the ℚrank-1 case, the pseudo-Laplacian Δ_{T} satisfies the Weyl law. In fact, this Weyl law can be used to derive the Weyl–Selberg law. These results, together with discussions of other pseudo-Laplacians in the higher rank, will be treated elsewhere.

Remark 3.5.4. In [21, Theorem 1.2.1], it is claimed as in an earlier version of this paper that the pseudo-Laplacian Δ_{T} for Γ\G satisfies the Weyl lower bound also. As pointed out in Remark 3.5.2, the arguments there actually only work for the ℚ-rank-1 case, and in the higher rank case, the lower bound in [21, Proposition 4.4.1] should be replaced by a weaker one that is obtained by substituting vol((Γ\G)_{T}) for vol(Γ\G). Since this lower bound for the pseudo-Laplacian of Γ\G is not used in the rest of that paper, in particular, the proof of [21, Theorem 1.1.2], this change does not affect the paper.

3.6. In this section, we introduce another pseudo-Laplace operator Δ̂_{T}. This operator is more geometric and can be generalized to other spaces, and hence is of independent interests, though not used in this paper.

By the precise reduction theory in Proposition 3.2.2, Γ\X admits the following disjoint decomposition Γ\X = (Γ\X)_{T} ∪ ∐_{i=1}^{p} ω_{i} × A_{P_{i},T}, where ω_{i} is a compact manifold with corners invariant under N_{P_{i}}. Using this decomposition, we define a subspace Ĥ_{T}^{1} (Γ\X, σ) of H^{1}(Γ\X, σ) as follows: Ĥ_{T}^{1} (Γ\X, σ) = {f ∈ H^{1}(Γ\X, σ) | for every i, points w ∈ ω_{i}, a ∈ A_{P_{i},T}, and all P containing P_{i}, f_{P} (wa) = 0.} As above, the Dirichlet quadratic form defines a self-adjoint operator Δ̂_{T} in the closure of H̃_{T}^{1} (Γ\X, σ) in L^{2}(Γ\X, σ).

It is conceivable that Δ̂_{T} has a discrete spectrum, and the counting function N̂_{T} (λ) of its spectrum satisfies the Weyl law. Though each ω_{i} ×A_{P_{i},T} is not a smooth manifold with boundary, it is a manifold with corners, and it seems that the method of single layers and the procedure of [11] plus the modification in [8] can still be applied to show that the spectrum of Δ̂_{T} is discrete and satisfies the Weyl upper bound. On the other hand, since the removal of the constant terms for functions in Ĥ_{T}^{1} (Γ\X, σ) is simpler, it can be shown that Δ̂_{T} also satisfies the Weyl lower bound, and also that Δ̂_{T} depends real analytically on t, where T = tH_{ρ}.

The pseudo-Laplacian Δ̂_{T} can be generalized to other spaces, for example, compact perturbations of Γ\X and manifolds with corners whose metrics respect the corner structure. In other words, Δ̂_{T} depends only locally on the geometry at infinity. On the other hand, it is not obvious whether the pseudo-Laplacian Δ_{T} can be generalized to other spaces.

To avoid the problems with the corners of ω_{i}×A_{P_{i},T}, there is another way to define a pseudo-Laplacian. By slightly enlarging ω_{i} × A_{P_{i},T} and rounding off the corners, we can get a covering of Γ\X by smooth manifolds with boundary. Then we can require the suitable constant terms to vanish near infinity to define a cut-off Sobolev space and hence a pseudo-Laplacian. Such a pseudo-Laplacian can also be shown to satisfy the Weyl law.

In the ℚ-rank-1 case, the three pseudo-Laplacians coincide. But they are different in the higher rank case. One problem with the latter two pseudo-Laplacians is that they are not connected to Arthur’s truncation operator Λ^{T}. These pseudo-Laplacians and their properties will be discussed elsewhere.

Remark 3.6.1. This operator Δ̂_{T} was not introduced in the early version of this paper, but Δ_{T} was confused with Δ̂_{T} in several places.

3.7. In order to apply the Weyl upper bound for Δ_{T} in Theorem 3.3.2 to study the discrete spectrum of Δ, we need to understand the relation between Δ and Δ_{T}.

First we recall the domain of Δ_{T} [35, p. 490]. Let H^{−1}(Γ\X, σ) denote the space of distributions in D′(Γ\X, σ) that extend to continuous linear functionals on H^{1}(Γ\X, σ).

Lemma 3.7.1. The domain of Δ_{T} consists of all φ ∈ H_{T}^{1} (Γ\X, σ) such that there exists a distribution D ∈ H^{−1}(Γ\X, σ) which vanishes on H_{T}^{1} (Γ\X, σ) and Δφ − D belongs to the closure of H_{T}^{1} (Γ\X, σ) in L^{2}(Γ\X, σ). Then Δ_{T} φ = Δφ − D.

Lemma 3.7.2. Assume that λ ∈ Spec_{cus}(Δ) and φ ∈ L_{cus}^{2}(Γ\X, σ) with Δφ = λφ. Then φ ∈ Dom(Δ_{T}) and Δ_{T} φ = λφ. In particular, Spec_{cus}(Δ) ⊂ Spec(Δ_{T}).

Proof. Since Δφ = λφ ∈ H_{T}^{1} (Γ\X, σ), the distribution D in Lemma 3.7.1 is equal to zero, and hence Δ_{T} φ = λφ.

3.8. In order to study other eigenfunctions of Δ_{T} and their connection with the spectrum of Δ, we decompose Δ_{T} according to the decomposition of L^{2}(Γ\X, σ) in Lemma 2.5.3.

Lemma 3.8.1. For every cuspidal pair (C, μ) as in Lemma 2.5.3, define H_{T,C,μ}^{1}(Γ\X, σ) = H_{T}^{1} (Γ\X, σ)∩L_{C,μ}^{2}(Γ\X, σ). Then H_{T}^{1} (Γ\X, σ) is equal to the completed sum in H^{1}(Γ\X, σ) of H_{T,C,μ}^{1}(Γ\X, σ) over all cuspidal pairs (C, μ).

Proof. By Lemma 2.5.3, any f ∈ H_{T}^{1} (Γ\X, σ) can be decomposed into f = ∑_{C,μ} f_{C,μ}, where f_{C,μ} ∈ L_{C,μ}^{2}(Γ\X, σ). Since

∫_{Γ\X} | ▽ f|^{2} = ∑_{C,μ} ∫_{Γ\X} | ▽ f_{C,μ}|^{2},

we have f_{C,μ} ∈ H^{1}(Γ\X, σ).

For any cuspidal pair (C, μ) and any rational parabolic subgroup P, either the constant term (f_{C,μ})_{P} of f_{C,μ} along P vanishes, or for any fixed a ∈ A_{P}, (f_{C,μ})_{P} (za) is a cuspidal eigenfunction on Γ_{M_{P}} \X_{P} of eigenvalue μ. This implies that if (C_{1}, μ_{1}) ≠ (C_{2}, μ_{2}), (f_{C_{1},μ_{1}})_{P} ≠ 0, and (f_{C_{2},μ_{2}})_{P} ≠ 0, then (f_{C_{1},μ_{1}})_{P} and (f_{C_{2},μ_{2}})_{P} are linearly independent as functions on Γ_{M_{P}} \X_{P}. Therefore, for a ∈ A_{P,T}, f_{P} (za) = 0 implies that (f_{C,μ})_{P} (za) = 0, and hence f_{C,μ} ∈ H_{T,C,μ}^{1}(Γ\X, σ) for all (C, μ). This completes the proof. q.e.d.

For every cuspidal pair (C, μ), the subspace H_{T,C,μ}^{1}(Γ\X, σ) defines a self-adjoint operator denoted by Δ_{T,C,μ} on the closure of H_{T,C,μ}^{1}(Γ\X, σ) in L_{C,μ}^{2}(Γ\X, σ). We note that when the parabolic subgroups in C are of rank-one, this operator Δ_{T,C,μ} is the operator Δ_{t} defined in [35, p. 500]. The above lemma shows that Δ_{T} is the direct sum of these operators Δ_{T,C,μ}.

3.9. When C = {G}, Lemma 3.7.2 shows that Δ_{T,C,μ} = Δ on L_{C,μ}^{2}(Γ\X, σ). In this subsection, we study the eigenfunction of Δ_{T,C,μ} when the parabolic subgroups in C are of rank-one. Fix an association class C of rank-one parabolic subgroups and a cuspidal eigenvalue μ ∈ Spec_{cus}(C).

To characterize the eigenfunctions of Δ_{T}, we need the truncation operator Λ^{T} [2, p. 89], [39] as recalled in Remark 3.3.3. For functions in L_{C,μ}^{2}(Γ\X, σ), Λ^{T} can be simplified as follows. Let P_{1}, ⋯, P_{m} be a set of representatives of Γ-conjugacy classes of maximal rational parabolic subgroups. Then

Λ^{T} f(x) = f(x) − ∑_{i=1}^{m} Λ_{P_{i}}^{T}f(x).

Recall from [39, p. 370] that a function f ∈ L_{loc}^{∞}(Γ\X, σ) is of moderate growth if for any invariant differential operator D, any rational parabolic subgroup P and any compact subset ω ⊂ N_{P} ×X_{P}, there exist Λ ∈ 𝔞_{P}^{∗} independent of D and a positive constant c = c(D) such that for any w ∈ ω, a ∈ A_{P,T},

|Df(wa)| ≤ ca^{Λ}. (1)

A basic property of the truncation operator Λ^{T} is that if a function f on Γ\X is of moderate growth, then Λ^{T} f decays rapidly at infinity, and in particular belongs to L^{2}(Γ\X, σ) (see [39, Theorem 5.2], [2, Lemma 1.4]).^{6} Denote the unique unit vector in 𝔞_{P_{i}}^{∗+} by Λ_{i}. For any

6If f satisfies an equation Δf = λf on Γ\X and has an upper bound |f(wa)| ≤ ca^{Λ} on every Siegel set ωA_{P,T}, then the elliptic theory implies that Inequality (1) above also holds for all other invariant differential operators D, and hence f is of moderate growth.

Φ = (Φ_{1}, ⋯, Φ_{m}),

where ΔΦ_{i} = μΦ_{i}, Φ_{i} ∈ L_{cus}^{2}(Γ_{M_{i}}\X_{P_{i}}, σ_{M_{P_{i}}}), s ∈ ℂ, define

E(Φ, s) = ∑_{i=1}^{m} E(P_{i}, Φ_{i}, sΛ_{i}), s ∈ ℂ.

Lemma 3.9.1 ([35, Lemma 3.14]). Let E(Φ, s) be an Eisenstein series defined above for Φ ≠ 0. If for a fixed s and all i = 1, ⋯, m, E_{P_{i}}(Φ, s, az) = 0 when a = exp I_{P_{i}}(T) and z ∈ X_{P}, then Λ^{T} E(Φ, s) is a nonzero eigenfunction of Δ_{T,C,μ} with eigenvalue |ρ_{P} |^{2} − s^{2} + μ, where |ρ_{P} | is equal to the norm of the half sum of the roots in Σ(P, A_{P}) with multiplicity for any P in C.

Remark 3.9.2. For a smooth function f of moderate growth, Λ^{T} f does not belong to H_{T}^{1} (Γ\X, σ) in general. In the above lemma, the vanishing condition on the constant terms implies that

Λ^{T} E(Φ, s) ∈ H_{T}^{1} (Γ\X, σ).

4. Perturbation of Δ_{T}.

4.1. In this section, we show that Δ_{T} is a continuous family of selfadjoint operators in T. This allows us to study branches of eigenvalues of Δ_{T}. The results in this section are not used in the proofs of Theorem 1.1.2 and 1.1.3, but the behavior of the eigenvalues of Δ_{T} as T varies is interesting in itself.

Lemma 4.1.1. For T = tH_{ρ}, t ≫ 0, Δ_{T} depends continuously on t, and hence the eigenvalues of Δ_{T} depend continuously on t.

Proof. Since Δ_{T} is defined on the closure of H_{T}^{1} (Γ\X, σ) through the Dirichlet quadratic form, the eigenvalues of Δ_{T} are given by the Rayleigh quotient and the mini-max principle. Since the vanishing conditions defining H_{T}^{1} (Γ\X, σ) depend continuously on T, the eigenvalues of Δ_{T} also change continuously.

Remark 4.1.2. In view of the close connection between the pseudo-Laplace operator Δ_{T} and Arthur’s truncation operator Λ^{T}, and the fact that for an invariant integral operator of compact support K(x, y) the trace of the truncated kernel Λ^{T} Λ^{T} K(x, y) is a polynomial in T for sufficiently large t [2, p. 88], it is conceivable that Δ_{T} depends real analytically on t, where T = tH_{ρ}.

4.2. We now the study convergence behavior of the continuous branches of eigenvalues of Δ_{T} as t → +∞. Since H_{T}^{1} (Γ\X, σ) converges to H^{1}(Γ\X, σ) as t → +∞ and the eigenvalues of Δ_{T} are given by the mini-max principle applied to the Rayleigh quotient associated with the Dirichlet quadratic form D on H_{T}^{1} (Γ\X, σ), it is conceivable that Spec(Δ_{T}) converges to Spec(Δ) as t → +∞.

Remark 4.2.1. There are different ways to define the convergence of Spec(Δ_{T}) to Spec(Δ). A weak convergence is to treat them as subsets of ℝ. Then Spec(Δ) is a union of finitely points and a half line going to the positive infinity (the most interesting eigenvalues are embedded in this half line, the continuous spectrum), while Spec(Δ_{T}) is a union of discrete points. It can be shown that below the continuous spectrum in Spec(Δ), the eigenvalues of Spec(Δ_{T}) converge to the corresponding eigenvalues in Spec(Δ), and the points in Spec(Δ_{T}) become dense in the half line in Spec(Δ). (See [23] for some related discussions about the spectral degeneration.) A more interesting question concerns the behavior of eigenfunctions, in particular those that might converge to the eigenfunctions whose eigenvalues are embedded in the continuous spectrum.

By Lemma 3.7.2, Spec_{cus}(Δ) ⊂ Spec(Δ_{T}). This implies that for any λ_{j} ∈ Spec_{cus}(Δ), there is a constant branch λ_{j}(T) = λ_{j} ∈ Spec(Δ_{T}). On the other hand, the relation between Spec_{res}(Δ) and Spec(Δ_{T}) is not entirely clear.

Let C be an association class of rational parabolic subgroups of rank-1, and μ ∈ Spec_{cus}(C). Consider the restriction Δ_{C,μ} of Δ to L_{C,μ}^{2}(Γ\X, σ) and the corresponding component Δ_{T,C,μ} of Δ_{T} in §3.8.

Lemma 4.2.2. The continuous spectrum of Δ_{C,μ} on L_{C,μ}^{2}(Γ\X, σ) is [μ + |ρ_{P} |^{2}, +∞), and the residual discrete spectrum is contained in [μ, μ + |ρ_{P} |^{2}), in particular, is outside the continuous spectrum, where |ρ_{P} | is the norm of the half sum of the roots in Σ(P, A_{P}) with multiplicity for any P ∈ C.

Proof. This follows from the spectral decomposition of L_{C,μ}^{2}(Γ\X, σ) in Remark 2.6.5. More precisely, E(Φ, Λ) satisfies

ΔE(Φ, Λ) = (μ + |ρ_{P} |^{2} − Λ^{2})E(Φ, Λ).

Since the continuous spectrum is spanned by E(Φ, iΛ), where Λ is real, and the residual discrete spectrum by residues of E(Φ, Λ) at points in (0, |ρ_{P} |], the lemma follows. q.e.d.

Since the residual discrete spectrum of Δ_{C,μ} is outside the continuous spectrum and the set of eigenvalues of Δ_{T,C,μ} converges to the spectrum of Δ_{C,μ} as t → +∞, we get the following result.

Proposition 4.2.3. Let λ_{1} ≤ ⋯ ≤ λ_{r} be the residual eigenvalues in L_{C,μ}^{2}(Γ\X, σ) repeated with multiplicities, and λ_{1}(t) ≤ ⋯ ≤ λ_{r}(t) the first r eigenvalues of Δ_{T,C,μ}. Then for i = 1, ⋯, r, lim_{t→+∞} λ_{i}(t) = λ_{i}.

It will be shown in §5 that except for at most one of them,

|λ_{i} − λ_{i}(t)| ≤ |ρ_{P} |^{2}.

Remark 4.2.4. If C is an association class of rational parabolic subgroups of rank greater than one, then the residual discrete spectrum of Δ_{C,μ} on L_{C,μ}^{2}(Γ\X, σ) could be embedded in the continuous spectrum of Δ_{C,μ}. On the other hand, if all invariant differential operators are considered, then the residual discrete spectrum is not embedded in the continuous spectrum anymore.

5. Bounds on the rank-1 residual discrete spectrum

5.1. In this section, we show that the majority of the rank-one residual discrete spectrum can be approximated uniformly by the corresponding eigenvalues of Δ_{T} and use it to prove Theorem 1.1.2.

5.2. It follows from §3.9 that in studying eigenfunctions of Δ_{T}, the key point is to understand the constant terms of Eisenstein series.

Let P be a maximal, i.e., ℚ-rank-1, rational parabolic subgroup of G, and C an association class containing P. Denote the opposite group N_{P}^{−}A_{P}M_{P} of P by P^{−}, where the Lie algebra 𝔫_{P}^{−} of N_{P}^{−} is spanned by the root spaces of the negative of the roots in Σ(P, A_{P}). In the notation of §2.7, P^{−} corresponds to the negative of the positive chamber 𝔞_{P}^{+}. Then P^{−} ∈ C, and any group in C is G(ℚ)-conjugate to either P or P^{−}. And P^{−} is G(ℚ)-conjugate to P if and only if −1 ∈ W (𝔞_{P}), i.e., W (𝔞_{P}) contains two elements.

Let P_{1}, ⋯, P_{r} be a set of representatives of the G(ℚ)-conjugacy classes in C. Then r = 1 or 2. For any 1 ≤ i ≤ r, let P_{i1}, ⋯, P_{ir_{i}} be a set of representatives of the Γ-conjugacy classes in the G(ℚ)-conjugacy

the weyl upper bound 131

class containing P_{i} as in §2.4. Then P_{il}, 1 ≤ i ≤ r, 1 ≤ l ≤ r_{i}, are representatives of all the Γ-conjugacy classes in the association class C.

Let Λ_{i} be the unique unit vector in 𝔞_{i}^{∗+}, 1 ≤ i ≤ r. If r = 1, then W (𝔞_{1}) = {±1}. Define C(z) = C_{11}(−1, zΛ_{1}), z ∈ ℂ. (1) If r = 2, then W (𝔞_{i}) = {1}, i = 1, 2, and W (𝔞_{1}, 𝔞_{2}) = {s}. Define

C(z) = (\begin{matrix}0 & C_{12}(s^{−1}, zΛ_{2}) \\ C_{21}(s, zΛ_{1}) & 0 \end{matrix}) . (2)

For μ ∈ Spec_{cus}(C), define E_{cus}(C, μ) = ⊕_{i=1}^{r}E_{cus}(C_{i}, μ). Then C(z) : E_{cus}(C, μ) → E_{cus}(C, μ), z ∈ ℂ. (3) By the functional equation in Lemmas 2.7.1 and 2.7.2, C(z) satisfies the following equations: C(z)C(−z) = Id, C(z)^{∗} = C(\overline{z}), z ∈ ℂ. (4) For any Φ = (Φ_{i})_{i=1}^{r} ∈ E_{cus}(C, μ), define

E(Φ, z) = ∑_{i=1}^{r} E(P_{i}, Φ_{i}, zΛ_{i}).

By Lemma 2.7.2, C_{ii}(1, Λ) = Id. Then it follows from Equation 2.4.(4) that for any P_{il} ∈ C_{i}, the constant term of E(Φ, z) along P_{il} is given by

E_{P_{il}}(Φ, z, x) =e^{(zAd(y_{il})Λ_{i}+ρ_{il})(H_{il}(x))}Φ_{il}
+ e^{(−zAd(y_{il})Λ_{i}+ρ_{il})(H_{il}(x))}(C(z)Φ)_{il}(x),

where Φ_{i} = (Φ_{il})_{l=1}^{r_{i}}, C(z)Φ = ((C(z)Φ)_{il}), i = 1, ⋯, r, l = 1, ⋯, r_{i}. For simplicity, define t_{il}(x) = (Ad(y_{il})Λ)(H_{il}(x)). Since ρ_{il}(H_{il}(x)) = |ρ_{il}|t_{il}(x) and |ρ_{il}| are the same for different i and l, we denote |ρ_{il}| by |ρ| and get E_{P_{il}}(Φ, z, x) = e^{(z+|ρ|)t_{il}(x)}Φ_{il} + e^{(−z+|ρ|)t_{il}(x)}(C(z)Φ)_{il}. (5)

Lemma 5.2.1.

1. The poles of E(Φ, z) are contained in the poles of C(z)Φ.

2. The poles of C(z) in the right half plane Re(z) ≥ 0 are simple, real and contained in the interval (0, |ρ|].

3. The residual discrete spectrum in L_{C,μ}^{2}(Γ\X, σ) is spanned by Res_{z_{0}}(E(Φ, z)), where Φ ∈ E_{cus}(C, μ), and z_{0} ∈ (0, |ρ|] is a pole of C(z).

Proof. Parts (1) and (2) follow from Proposition 2.4.2, and Part (3) from Proposition 2.6.1 and Remark 2.6.2. q.e.d.

For u ∈ (0, |ρ|], C(u)^{∗} = C(u), i.e., C(u) is Hermitian symmetric and hence can be diagonalized. Let u_{1}, ⋯, u_{m} ∈ (0, |ρ|] be the poles of C(z) in the half plane Re(z) ≥ 0 and d = dim E_{cus}(C, μ). Then we have the following [35, p. 485 and Proposition 3.6].

Lemma 5.2.2. For u ∈ (0, |ρ|], there exists an analytic family of bases Φ_{1}(u), ⋯, Φ_{d}(u) of E_{cus}(C, μ) such that the following hold:

1. Each Φ_{k}(u) is an eigenfunction of C(u), C(u)Φ_{k}(u) = λ_{k}(u)Φ_{k}(u).

2. Every eigenvalue λ_{k}(u) is a real valued analytic function in u ∈ (0, |ρ|] \ {u_{1}, ⋯, u_{m}}.

3. Near each pole u_{j} of C(u), j = 1, ⋯, m, λ_{k}(u) has the following expansion

λ_{k}(u) = \frac{μ_{kj}}{u − u_{j}} + ∑_{i=0}^{∞} a_{ki}(u − u_{j})^{i}

with μ_{kj} ≥ 0. In particular, λ_{k}(u) is singular at u_{j} if and only if μ_{kj} > 0.

Proof. The eigenfunctions Φ_{k}(u) are not mentioned explicitly in [35, Proposition 3.6]. But the existence of such analytic families of eigenfunctions is a part of Rellich’s theorem. (See [26, Chap. 2, §4.5 and 6.2].) q.e.d.

For every 1 ≤ k ≤ d, 1 ≤ j ≤ m with μ_{kj} > 0, E(Φ_{k}(u_{j}), u) has a nonzero residue Res_{u_{j}}E(Φ_{k}(u_{j}), u) at u = u_{j} which is an eigenfunction of eigenvalue λ_{k,j} = μ + |ρ|^{2} − u_{j}^{2}, and these eigenfunctions generate the residual discrete subspace in L_{C,μ}^{2}(Γ\X, σ).

For each 1 ≤ k ≤ d, denote the poles of λ_{k}(u) in (0, |ρ|] by u_{k,1} < ⋯ < u_{k,m_{k}}, which are the subsets of the poles u_{1}, ⋯, u_{m} with μ_{kj} > 0. Then for any j = 1, ⋯, m_{k},

lim_{u→u_{k,j}±0} λ_{k}(u) = ±∞.

In particular, if m_{k} ≥ 2, then in each interval (u_{k,j−1}, u_{k,j}), 2 ≤ j ≤ m_{k}, the value of λ_{k}(u) changes from +∞ to −∞. Therefore, for any truncation parameter T = tH_{ρ}, there exists a point u = u_{k,j}(t) ∈ (u_{k,j−1}, u_{k,j}) such that

e^{(u+|ρ|)t_{il}(exp I_{il}(T))} + λ_{k}(u)e^{(−u+|ρ|)t_{il}(exp I_{il}(T))} = 0. (1)

By Lemma 3.9.1, we get the following lemma.

Lemma 5.2.3. For any 1 ≤ k ≤ d with m_{k} ≥ 2, 2 ≤ j ≤ m_{k} and u_{k,j}(t) above, Λ^{T} E(Φ_{k}(u_{k,j}(t)), u_{k,j}(t)) is a nonzero eigenfunction of the pseudo-Laplacian Δ_{T} with eigenvalue λ_{k,j}(t) = μ + |ρ|^{2} − u_{k,j}(t)^{2}.

The above discussions show that between every pair of poles u_{k,j−1}, u_{k,j}, there is an eigenfunction of Δ_{T}. In fact, such an eigenfunction is unique.

Proposition 5.2.4. For 2 ≤ j ≤ m_{k}, there exists a unique solution u_{k,j}(t) in (u_{k,j−1}, u_{k,j}) to the above Equation (1) when t ≫ 0.

Proof. Since H_{T,C,μ}^{1}(Γ\X, σ) increases and converges to

H^{1}(Γ\X, σ) ∩ L_{C,μ}^{2}(Γ\X, σ)

as t → +∞, the mini-max principle implies that the eigenvalues of Δ_{T,C,μ} are decreasing functions of t and converge to the spectrum of Δ_{C,μ}. Let N_{T,C,μ}(λ) be the counting function of Δ_{T,C,μ}, and N_{C,μ}(λ) the counting function of Δ_{C,μ}. Then, for any λ ≤ μ + |ρ|^{2}, which is the bottom of the continuous spectrum of Δ_{C,μ} (see Lemma 4.4.2),

N_{T,C,μ}(λ) ≤ N_{C,μ}(λ). (2)

Since lim_{u→u_{k,1}−0} λ_{k,1}(u) = −∞, we can find a point u_{k,1}(t) ∈ (0, u_{k,1}) when t ≫ 0 such that Λ^{T} E(Φ_{k}(u_{k,1}(t)), u_{k,1}(t)) is an eigenfunction of Δ_{T}. As shown above, Equation (1) has at least one solution in each interval (u_{i,j−1}, u_{i,j}), and each such solution produces an eigenfunction of Δ_{T,C,μ}. If there are two solutions to Equation (1) in some interval (u_{i,j−1}, u_{i,j}), the above inequality (2) will not hold for λ slightly smaller than μ + |ρ|^{2}. This proves the proposition. q.e.d.

As pointed out in the above proof, when t ≫ 0, we can get u_{k,1}(t) ∈ (0, u_{k,1}) such that Λ^{T} E(Φ_{k}(u_{k,1}(t)), u_{k,1}(t)) is an eigenfunction of Δ_{T}. But it is not clear that such a point u_{k,1}(t) should exist for all t ≥ t_{0}.

Lemma 5.2.5. For 1 ≤ k ≤ d and 2 ≤ j ≤ m_{k}, the eigenfunction (u_{k,j}(t) − u_{k,j})Λ^{T} E(Φ_{k}(u_{k,j}(t)), u_{k,j}(t)) of Δ_{T} converges to the residual eigenfunction Res_{u_{k,j}}E(Φ_{k}(u_{k,j}), u) as t → +∞.

Proof. Clearly, when t → +∞, u_{k,j}(t) → u_{k,j}, and hence λ_{k,j}(t) → λ_{k,j} and

(u_{j,k}(t) − u_{k,j})Λ^{T} E(Φ_{k}(u_{k,j}(t)), u_{k,j}(t)) → Res_{u_{k,j}}E(Φ_{k}(u_{k,j}), u).

q.e.d.

Lemma 5.2.6. For every 1 ≤ k ≤ d and 2 ≤ j ≤ m_{k}, the branch λ_{k,j}(t) in Lemma 5.2.5 converging to λ_{k,j} satisfies the following bound:

λ_{k,j} ≤ λ_{k,j}(t) ≤ λ_{k,j} + |ρ|^{2}.

Proof. It follows from the fact that u_{k,j}(t) ∈ (u_{k,j−1}, u_{k,j}) ⊂ [0, |ρ|).

Corollary 5.2.7. For the residual eigenvalues {λ_{k}} in L_{C,λ}^{2}(Γ\X, σ), except for at most dim E_{cus}(C, λ) of them, the branches λ_{k}(t) above converging to λ_{k} satisfy the following inequality:

λ_{k} ≤ λ_{k}(t) ≤ λ_{k} + |ρ|^{2}.

Recall that C is an association class of rank-one rational parabolic subgroups of G. Let N_{res,C}(λ) be the counting function of the discrete spectrum in L_{C}^{2}(Γ\X, σ), N_{il,cus}(λ) the counting function of

L_{cus}^{2}(Γ_{M_{il}}\X_{P_{il}}, σ_{M_{il}}),

and N_{T,C}(λ) the counting function of Δ_{T} restricted to the closure of

H_{T}^{1} (Γ\X, σ) ∩ L_{C}^{2}(Γ\X, σ)

in L_{C}^{2}(Γ\X, σ) (see §3.8). Then we have the following bound on N_{res,C}(λ).

Proposition 5.2.8. For an association class C of rank-one,

N_{res,C}(λ) ≤ N_{T,C}(λ + |ρ|^{2}) + ∑_{i=1}^{r} ∑_{l=1}^{r_{i}} N_{il,cus}(λ).

Proof. It follows from Corollary 5.2.7 and the fact that for any Φ ∈ E_{cus}(C, μ), any residual eigenfunction induced from E(Φ, Λ) has an eigenvalue greater than μ.

Proposition 5.2.9. Let N′_{d}(λ) be the counting function of the discrete spectrum in the subspace L_{cus}^{2}(Γ\X, σ) ⊕ ∑_{C} L_{C}^{2}(Γ\X, σ), where C runs over all association classes of rank-1. Then

lim_{λ→+∞} sup \frac{N′_{d}(λ)}{λ^{n/2}} ≤ (4π)^{−n/2} \frac{vol(Γ\X)}{Γ(\frac{n}{2} + 1)} dim σ.

Proof. Let C_{1}, ⋯, C_{m} be all the association classes of rank-one rational parabolic subgroups. Then

N′_{d}(λ) = N_{cus}(λ) + ∑_{i=1}^{m} N_{res,C_{i}}(λ),

N_{cus}(λ) + ∑_{i=1}^{m} N_{T,C_{i}}(λ) ≤ N_{T} (λ).

Here we have used the fact that L_{cus}^{2}(Γ\X, σ) is contained in the domain of Δ_{T} (Lemma 3.7.2) and orthogonal to L_{C_{i}}^{2}(Γ\X, σ). For each C_{i}, let P_{i1}, ⋯, P_{ir_{i}} be representatives of the Γ-conjugacy classes in C_{i}. Then Proposition 5.2.8 implies that

N_{res,C_{i}}(λ) ≤ N_{T,C_{i}}(λ + |ρ|^{2}) + ∑_{l=1}^{r_{i}} N_{il,cus}(λ),

and hence

N′_{d}(λ) ≤ N_{T} (λ + |ρ|^{2}) + ∑_{i=1}^{m} ∑_{l=1}^{r_{i}} N_{il,cus}(λ).

Since dim X_{P_{il}} < dim X = n, Lemma 2.3.2 gives that for every pair i, l above,

N_{il,cus}(λ) = O(λ^{dim X_{P_{il}}/2}) = o(λ^{n/2})

as λ → +∞. Then the bound for N′_{d}(λ) follows from the Weyl upper bound in Theorem 3.3.2. q.e.d.

Proof of Theorem 1.1.2. If the ℚ-rank of G is equal to 1, then all proper rational parabolic subgroups are of rank-1, and hence all the residual discrete spectrum is of rank-1. Therefore, N_{d}(λ) = N′_{d}(λ), and the Weyl upper bound on N_{d}(λ) is given by Proposition 5.2.9.

If the ℝ-rank of G is equal to 2, there are two cases to consider. If ℚ-rank of G is equal to 1, then it is a special case of the above paragraph. If the ℚ-rank of G is equal to 2, then only the minimal rational parabolic subgroups are of rank 2 and hence greater than one. For any minimal rational parabolic subgroup P, dim X_{P} = 0, since the rank of X is equal to 2 and dim A_{P} = 2. Let C be the association class containing P. Then the residual discrete spectrum in L_{C}^{2}(Γ\X, σ) is of finite dimension. This implies N_{d}(λ) = N′_{d}(λ) + O(1), and the Weyl upper bound on N_{d}(λ) also follows from Proposition 5.2.9.

6. Bounds on the poles of the rank-one scattering matrices

6.1. In this section, we bound the number of poles of the rank-one scattering matrix on the negative real line, which is needed to bound the poles of the higher rank scattering matrices in the bounded domain {Λ ∈ ^{+}𝔞_{P}^{∗} | |Λ| ≤ |ρ_{P} |} in Proposition 2.6.1, i.e., the higher rank residual discrete spectrum.

6.2. Let C be an association class of rank-one rational parabolic subgroups, μ ∈ Spec_{cus}(C), and C(z) : E_{cus}(C, μ) → E_{cus}(C, μ) the scattering matrix defined in §5.2. Then det C(z) is a meromorphic function of z.

Lemma 6.2.1 ([35, Theorem 5.10]). The determinant det C(z) is the quotient of two holomorphic functions of order less than n + 2.

Let u_{1}, ⋯, u_{l} ∈ (0, |ρ|] be the poles of det C(z) in the right halfplane Re(z) ≥ 0, and let η_{k}, k ≥ 1, be the poles of det C(z) in the left half-plane Re(z) ≤ 0, both repeated with multiplicities. Denote dim E_{cus}(C, μ) by d as above. Fix a truncation parameter T = tH_{ρ} with t ≫ 0 as in §3.2. Using the functional equation for C(z) and Hadamard’s factorization theorem, Müller proved the following result for det C(z).

Proposition 6.2.2 ([35, Theorem 6.9]). There exists a negative number a such that

det C(z) = q^{z} ∏_{j=1}^{l} \frac{z + u_{j}}{z − u_{j}} ∏_{k≥1} \frac{z + \overline{η_{k}}}{z − η_{k}} , z ∈ ℂ,

where q = e^{2d(t+1)+1+a}, and hence for v ∈ ℝ,

\frac{d}{dz} log(det C(iv)) = log q + ∑_{j=1}^{l} \frac{2u_{j}}{v^{2} + u_{j}^{2}} + ∑_{k≥1} \frac{2Re(η_{k})}{Re(η_{k})^{2} + (v − Im(η_{k}))^{2}} .

6.3. We use the above factorization of det C(z) to bound the number of its poles on the negative half of the real line.

Lemma 6.3.1. Let N_{T,C,μ}(x) denote the counting function of the eigenvalues of Δ_{T,C,μ} (see §3.8). Then for any V > 0,

| ∫_{−V}^{V} \frac{d}{dz} log det C(iv)dv| ≤2πN_{T,C,λ}(μ + |ρ|^{2} + V^{2})
+ (4t|ρ|V + 4π) dim E_{cus}(C, μ).

Proof. Recall that T = tH_{ρ} is the truncation parameter. Define C_{t}(z) = e^{−2zt|ρ|}C(z). By the functional equation of C(z) in Equation 5.2.(4), C(iv), v ∈ ℝ, is unitary and hence C_{t}(iv) is also unitary. So C_{t}(iv) can be diagonalized. Since C_{t}(z) is holomorphic in a neighborhood of Re(z) = 0, the regular perturbation theory in [26, Chap. 2, §4.5 and 6.2] implies that there exist real valued analytic functions β_{1}(v), ⋯, β_{d}(v) such that e^{iβ_{1}(v)}, ⋯, e^{iβ_{d}(v)} are the eigenvalues of C(iv), v ∈ ℝ. Denote the corresponding analytic branches of eigenfunctions by Φ_{1}(v), ⋯, Φ_{d}(v). Then

\frac{d}{dz} log det C(iv) = 2t|ρ|d + \frac{d}{dz} log det C_{t}(iv) = 2t|ρ|d + ∑_{j=1}^{d} β′_{j}(v),

and

| ∫_{−V}^{V} \frac{d}{dz} log det C(iv)dv| ≤ 4t|ρ|V d + ∑_{j=1}^{d} | ∫_{−V}^{V} β′_{j}(v)dv|. (1)

Each β_{j}(v) is only determined up to 2πℤ. Since C_{t}(0)^{2} = Id, we can fix β_{j}(v) by picking either β_{j}(0) = 0 or π. For every j = 1, ⋯, d, let v_{j,1} < ⋯ < v_{j,n_{j}} be the all the points in [−V, V] such that e^{iβ_{j}(v)} = −1. Then

| ∫_{−V}^{v_{j,1}} β′_{j}(v)dv| ≤2π,
| ∫_{v_{j,n_{j}}}^{V} β′_{j}(v)dv| ≤2π,
| ∫_{v_{j,1}}^{v_{j,n_{j}}} β′_{j}(v)dv| ≤2πn_{j},

and hence

| ∫_{−V}^{V} β′_{j}(v)dv| ≤ 4π + 2πn_{j}. (2)

For each v_{j,k}, 1 ≤ k ≤ n_{j}, Λ^{T} E(Φ_{k}(v_{j,k}), iv_{j,k}) is a nonzero eigenfunction of Δ_{T,C,μ} with eigenvalue μ+|ρ|^{2}+v_{j,k}^{2} as in Lemma 5.2.3. Since different points v_{j,k}, 1 ≤ j ≤ d, 1 ≤ k ≤ n_{j}, produce linearly independent eigenfunctions Λ^{T} E(Φ_{k}(v_{j,k}), iv_{j,k}) of Δ_{T,C,μ}, we obtain

∑_{j=1}^{d} 2πn_{j} ≤ 2πN_{T,C,μ}(μ + |ρ|^{2} + V^{2}).

Combined with the above inequalities (1) and (2), this completes the proof of the lemma.

Proposition 6.3.2. For any V > 0, there exists a constant b depending on V such that the number of poles, counted with multiplicity, of det C(z) in [−V, 0] is bounded by b(N_{T,C,μ}(μ + |ρ|^{2} + 1) + (t + t|ρ|) dim E_{cus}(C, μ) + 1).

Proof. For any positive pole u_{j} of det C(z),

∫_{−1}^{1} \frac{u_{j}}{u_{j}^{2} + v^{2}} dv ≤ ∫_{−∞}^{∞} \frac{1}{1 + v^{2}} dv = π.

By Proposition 6.2.2, we get

| ∫_{−1}^{1} log q + ∑_{k≥1} \frac{2Re(η_{k})}{Re(η_{k})^{2} + (v − Im(η))^{2}} dv|

≤ 2lπ + | ∫_{−1}^{1} \frac{d}{dz} log det C(iv)dv|,

where l is the number of the poles of det C(u) in (0, |ρ|] counted with multiplicity and is bounded by the dimension of the residual discrete spectrum in L_{C,μ}^{2}(Γ\X, σ) (see the discussion after Lemma 5.2.2). By Corollary 5.2.7, we get

l ≤ N_{T,C,μ}(μ + |ρ|^{2}) + d,

where d = dim E_{cus}(C, μ). Combined with Lemma 6.3.1, this implies

| ∫_{−1}^{1} log q + ∑_{k≥1} \frac{2Re(η_{k})}{Re(η_{k})^{2} + (v − Im(η))^{2}} dv|

≤ 4πN_{T,C,μ}(μ + |ρ|^{2} + 1) + (4t|ρ| + 5π)d.

If q ≥ 1, then 0 ≤ log q ≤ 2(t + 1)d + 1, where we have used the fact that a ≤ 0, and hence

\begin{matrix}| ∫_{−1}^{1} ∑_{k≥1} \frac{2Re(η_{k})}{Re(η_{k})^{2} + (v − Im(η))^{2}} dv| \\ ≤ 4πN_{T,C,μ}(μ + |ρ|^{2} + 1) \\ + (4t|ρ| + 4t + 4 + 5π)d + 2. \end{matrix} (1)

If 0 < q < 1, then log q < 0. Since every term

∫_{−1}^{1} \frac{2Re(η_{k})}{Re(η_{k})^{2} + (v − Im(η))^{2}} dv

is negative, the same inequality (3) holds.

Let m be the number of poles of det C(z) in the interval [−V, 0]. Then

m ∫_{−1/V}^{1/V} \frac{1}{1 + v^{2}} dv ≤ ∑_{η_{k}∈ℝ,−V ≤η_{k}≤0}∫_{−1}^{1} \frac{|η_{k}|}{η_{k}^{2} + v^{2}} dv

≤ ∑_{k≥1} ∫_{−1}^{1} \frac{2|Re(η_{k})|}{Re(η_{k})^{2} + (v − Im(η))^{2}} dv.

Together with the above inequality (3), we get

m ≤ b(N_{T,C,μ}(μ + |ρ|^{2} + 1) + (t + t|ρ|) dim E_{cus}(C, μ) + 1),

where b is a constant depending on V. This completes the proof.

6.4. Next we estimate the poles of the scattering matrix

C(z) : E_{cus}(C, μ) → E_{cus}(C, μ)

in [−V, V] counted with multiplicity and the rank of the residue.

The functional equation in Equation 5.2.(4) shows that for v ∈ [−V, 0], C(v) is Hermitian, and hence there exist meromorphic functions λ_{1}(v), ⋯, λ_{d}(v), which are eigenvalues of C(v), where d = dim E_{cus}(C, μ).

Proposition 6.4.1. For any V > 0, the number of the poles of λ_{1}(v), ⋯, λ_{d}(v) in [−V, 0], counted with multiplicity, is bounded by b′(N_{T,C,μ}(μ + |ρ|^{2} + 1) + (t + t|ρ|)d + 1), where b′ is a constant depending on V, and hence the number of poles of the scattering matrix C(s) in [−V, V] is bounded by

b′′(N_{T,C,μ}(μ + |ρ|^{2} + 1) + (t + t|ρ|)d + 1),

where b′′ is a constant depending on V.

Proof. By definition, det C(v) = ∏_{j=1}^{d} λ_{j}(v). For any pole η_{k} of C(v) in [−V, 0], if none of λ_{j}(v) vanishes at η_{k}, then the order of C(v) at η_{k} is equal to the sum of the order of poles of λ_{i}(v) at η_{k}. On the other hand, if some λ_{j}(η_{k}) = 0, then the functional equation of C(z) implies that λ_{j}(v) has a pole at −η_{k} > 0. The pole −η_{k} of λ_{j}(u) is simple and gives rise to a residual eigenfunction Res_{−η_{k}}E(Φ_{j}(η_{k}), v) in L_{C,μ}^{2}(Γ\X, σ). This implies that λ_{j}(v) has a simple zero at η_{k}. Since different such poles −η_{k} produce linearly independent residual eigenfunctions, the sum of the number of the poles of λ_{j}(v), j = 1, ⋯, d, in [−V, 0] is bounded by the number of the poles of the determinant detC(v) in [−V, 0] and the dimension of the residual discrete spectrum in L_{C,μ}^{2}(Γ\X, σ). By Corollary 5.2.7, the latter is bounded by

N_{T,C,μ}(μ + |ρ|^{2}) + dim E_{cus}(C, μ). (1)

Combined with Proposition 6.3.2, this gives the desired bound on the number of poles of C(v) in [−V, 0].

On the other hand, the number of the poles of C(v) in [0, V] is bounded by the dimension of the residual discrete spectrum, which is bounded by the number in Equation (1) as mentioned above. Combined with the previous paragraph, this completes the proof of the proposition.

7. Bounds on the higher rank residual discrete spectrum and proof of Theorem 1.1.3

7.1. In this section, we shall complete the proof of Theorem 1.1.3 by using the bounds on the number of poles of rank-1 scattering matrices in Proposition 6.4.1 and the factorization of the scattering matrices in §2.7 to bound the number of singular hyperplanes of the higher rank scattering matrices and hence the counting function of the higher rank residual discrete spectrum.

7.2. For any rational parabolic subgroup P of G, denote the counting function of the cuspidal spectrum of L^{2}(Γ_{M_{P}} \X_{P} , σ_{M_{P}}) by N_{cus,P} (λ), and the counting function of the pseudo-Laplacian Δ_{T} for L^{2}(Γ_{M_{P}} \X_{P} , σ_{M_{P}}) by N_{T,P} (λ). For an association class C, the counting function of the cuspidal spectrum of L_{C}^{2}(Γ\X, σ) is denoted by N_{cus,C}(λ), and the corresponding counting function of the pseudo-laplacian Δ_{T} by N_{T,C}(λ).

Let C be an association class of rational parabolic subgroups of rank greater than or equal to 2. As in §2.4, let C_{1}, ⋯, C_{r} be the G(ℚ)- conjugacy classes in C. Then for every μ ∈ Spec_{cus}(C), i, j ∈ {1, ⋯, r}, s ∈ W (𝔞_{i}, 𝔞_{j}), there is a scattering matrix

C_{ij}(s, Λ) : E_{cus}(C_{i}, μ) → E_{cus}(C_{j}, μ), Λ ∈ 𝔞_{i}^{∗} ⊗ ℂ.

Proposition 7.2.1. For every λ > 0, and an association C of rank greater than or equal to 2 the total number of singular hyperplanes of all the scattering matrices C_{ij}(s, Λ) acting on E_{cus}(C_{i}, μ) with μ ≤ λ which meet the bounded region {Λ ∈ 𝔞_{i}^{∗} ⊗ ℂ | |Λ| ≤ |ρ_{P} |}, is bounded by

c(∑_{Q} N_{T,Q}(λ + |ρ_{P} |^{2} + 1) + ∑_{Q} ∑_{′P} N_{cus,′P} (λ) + 1),

where Q runs over a set of representatives of Γ-conjugacy classes of all the rational parabolic subgroups containing a group P in C such that rank_{ℚ}(Q) = rank_{ℚ}(P) − 1, and for every Q in the first sum ′P runs over a set of representatives of Γ_{M_{Q}}-conjugacy classes of rank-1 rational parabolic subgroups of M_{Q}, and c is a constant depending only on G and T.

Proof. For any μ ∈ Spec_{cus}(C) with μ ≤ λ, we first bound the singular hyperplanes of the scattering matrices on E_{cus}(C, μ). We use the notation in §2.7. Let A be the split component of a parabolic subgroup P ∈ C. Then Lemma 2.7.4 shows that to bound the number of poles of the scattering matrices for the parabolic subgroups in C, it suffices to bound the poles of C_{ji}(1, Λ) in the region {Λ ∈ 𝔞^{∗} ⊗ ℂ | |Λ| ≤ |ρ_{P} |}, where C_{i}, C_{j} are adjacent chambers of 𝔞.

Let P_{i}, P_{j} be the rational parabolic subgroups corresponding to C_{i}, C_{j}, and Q the parabolic subgroup containing both P_{i}, P_{j} with

rank_{ℚ}(Q) = rank_{ℚ}(P_{i}) − 1.

Then P_{i}, P_{j} determine parabolic subgroups ′P_{i},′P_{j} of M_{Q} as in §2.7. Lemma 2.7.5 implies that the poles of C_{ji}(1, Λ) are contained in the poles of C_{′j,′i}(s,′Λ).

Since the association class ′C is of rank-one, from Proposition 6.4.1 it follows that the number of poles of C_{′j,′i}(s,′Λ) in the region {′Λ ∈ ′𝔞^{∗} ⊗ ℂ | |′Λ| ≤ |ρ_{P} |}, counted with the multiplicity and the rank of the residue, is bounded by

b(N_{T,′C,μ}(μ + |ρ_{′P} |^{2} + 1) + dim E_{cus}(′C, μ) + 1),

where b is a constant depending only on G and the truncation parameter T, and N_{T,′C,μ}(λ) is the counting function of Δ_{T,′C,λ} (§3.8), ′P ∈ ′C . Since

∑_{μ≤λ} N_{T,′C,μ}(μ + |ρ_{′P} |^{2} + 1) ≤ ∑_{μ≤λ} N_{T,′C,μ}(λ + |ρ_{′P} |^{2} + 1)
=N_{T,′C}(λ + |ρ_{′P} |^{2} + 1)
≤N_{T,Q}(λ + |ρ_{P} |^{2} + 1),
∑_{μ≤λ} dim E_{cus}(′C, μ) =N_{cus,′C}(λ),

we get that for the two adjacent chambers C_{i}, C_{j} above, the total number of poles in the region {′Λ ∈ ′𝔞^{∗} ⊗ ℂ | |′Λ| ≤ |ρ_{P} |} of the rank-one scattering matrices C_{′i,′j}(1,′Λ) for all μ ∈ Spec_{cus}(′C) with μ ≤ λ is bounded by

b(N_{T,′C}(λ + |ρ_{P} |^{2} + 1) + N_{cus,′C}(λ) + 1) ≤b(∑_{Q} N_{T,Q}(λ + |ρ_{P} |^{2} + 1)

+ ∑_{Q} ∑_{′P} N_{cus,′P} (λ) + 1),

where Q,′P run over the same set of parabolic subgroups as in the proposition.

By Lemma 2.7.3, for any C_{ji}(s, Λ), the number of the above rankone scattering matrices C_{′i,′j}(1,′Λ) which appear in the factorization of C_{ji}(s, Λ) in Lemma 2.7.4 is bounded by a constant which only depends on G. Thus Proposition 7.2.1 follows from the above bound for the number of the poles of C_{′i,′j}(1,′Λ).

Proposition 7.2.2. For an association class C of rational parabolic subgroups P with rank_{ℚ}(P) ≥ 2, the counting function N_{res,C}(λ) of the residual discrete spectrum of Δ in L_{C}^{2}(Γ\X, σ) is bounded as follows:

N_{res,C}(λ) ≤ c(∑_{Q} N_{T,Q}(λ + |ρ_{P} |^{2} + 1) + ∑_{Q} ∑_{′P} N_{cus,′P} (λ) + 1)^{rank_{ℚ}(P)},

where Q runs over a set of representatives of Γ-equivalence classes of all the rational parabolic subgroups containing a group P in C such that rank_{ℚ}(Q) = rank_{ℚ}(P) − 1, and for every Q in the first sum ′P runs over a set of representatives of Γ_{M_{Q}}-conjugacy classes of rank-1 rational parabolic subgroups of M_{Q}, and c is a constant depending only on G and T. In particular, as λ → +∞,

N_{res,C}(λ) = O(1)λ^{m/2},

where m is the maximum of (rank_{ℚ}(Q)+1) dim Γ_{M_{Q}}\X_{Q} for all rational parabolic subgroups Q ⊂ G with rank_{ℚ}(Q) ≤ rank_{ℚ}(G) − 1.

Proof. First we note that if μ > λ, the residual eigenvalues in L_{C,μ}^{2}(Γ\X, σ) are greater than λ and hence do not contribute to N_{res,C}(λ). Proposition 2.6.1 shows that the number of residual eigenvalues in L_{C,μ}^{2}(Γ\X, σ) is bounded by the number of complete flags of singular hyperplanes of the corresponding scattering matrices meeting the bounded region {Λ ∈ 𝔞^{∗} ⊗ ℂ | |Λ| ≤ |ρ_{P} |}. Then the bound on the number of the singular hyperplanes of the scattering matrices in Proposition 7.2.1 gives a bound on the number of complete flags by raising it to the power rank_{ℚ}(P) and hence proves the first bound on N_{res,C}(λ) in Proposition 7.2.2.

To prove the second bound on N_{res,C}(λ), we notice that

N_{T,Q}(λ + |ρ_{P} |^{2} + 1) = O(1)λ^{\frac{1}{2} dim Γ_{M_{Q}}\X_{Q}}

by Theorem 3.3.2, and

N_{cus,′P} (λ) = O(1)λ^{\frac{1}{2} dim Γ_{M_{′P}} \X_{′P}}

by Lemma 2.3.2. Since dim Γ_{M_{′P}} \X_{′P} ≤ dim Γ_{M_{Q}}\X_{Q} and

rank_{ℚ}(Q) = rank_{ℚ}(P) − 1 ≤ rank_{ℚ}(G) − 1,

we get that as λ → +∞

N_{res,C}(λ) = O(1)λ^{m/2}.

q.e.d.

Proof of Theorem 1.1.3. Let C_{1}, ⋯, C_{k} be representatives of association classes of rational parabolic subgroups of rank greater than or equal to 2. Let N′_{d}(λ) be the counting function of combination of the cuspidal spectrum and the rank-one residual spectrum as in Proposition 5.2.9. Then

N_{d}(λ) = N′_{d}(λ) + ∑_{i=1}^{k} N_{res,C_{i}}(λ),

and the bounds in Propositions 5.2.9 and 7.2.2 give the bound on N_{d}(λ) stated in Theorem 1.1.3.

Remark 7.2.3. In an early version of this manuscript, it was mistakenly claimed that N_{res,C}(λ) is bounded by the total number of singular hyperplanes of the scattering matrices which meet the bounded region {Λ ∈ 𝔞^{∗} ⊗ ℂ | |Λ| ≤ |ρ_{P} |} above for all the cuspidal pairs (C, μ), μ ≤ λ. Since the bound for the latter in Proposition 7.2.1 is of smaller order than λ^{n/2}, if the rank of C is strictly greater than 1, it would imply that N_{res,C}(λ) is of smaller order than λ^{n/2}, and hence the Weyl upper bound is satisfied by N_{d}(λ). The same problem occurred in [35, p. 523]. Because of this, the bound in [35, Theorem 0.1] should be replaced by the weaker one:

N_{d}(λ) ≤ c(1 + λ^{\frac{n}{2} + \frac{3n}{2} rank_{ℚ}(G)})

as mentioned in §1.1.

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