WEYL’S LAW FOR THE CUSPIDAL SPECTRUM OF SL_{n}

WERNER MÜLLER

Abstract. Let Γ be a principal congruence subgroup of SL_{n}(ℤ) and let σ be an irreducible unitary representation of SO(n). Let N_{cus}^{Γ}(λ, σ) be the counting function of the eigenvalues of the Casimir operator acting in the space of cusp forms for Γ which transform under SO(n) according to σ. In this paper we prove that the counting function N_{cus}^{Γ}(λ, σ) satisfies Weyl’s law. Especially, this implies that there exist infinitely many cusp forms for the full modular group SL_{n}(ℤ).

Contents

1. Preliminaries 7

2. Heat kernel estimates 9

3. Estimations of the discrete spectrum 13

4. Rankin-Selberg L-functions 20

5. Normalizing factors 28

6. The spectral side 39

7. Proof of the main theorem 49

References 54

Let G be a connected reductive algebraic group over ℚ and let Γ ⊂ G(ℚ) be an arithmetic subgroup. An important problem in the theory of automorphic forms is the question of the existence and the construction of cusp forms for Γ. By Langlands’ theory of Eisenstein series [La], cusp forms are the building blocks of the spectral resolution of the regular representation of G(ℝ) in L^{2}(Γ\G(ℝ)). Cusp forms are also fundamental in number theory. Despite of their importance, very little is known about the existence of cusp forms in general. In this paper we will address the question of existence of cusp forms for the group G = SL_{n}. The main purpose of this paper is to prove that cusp forms exist in abundance for congruence subgroups of SL_{n}(ℤ), n ≥ 2.

Date: March 11, 2005.

1991 Mathematics Subject Classification. Primary: 22E40; Secondary: 58G25.

Key words and phrases. trace formula, Automorphic forms, spectral decomposition.

To formulate our main result we need to introduce some notation. For simplicity assume that G is semisimple. Let K_{∞} be a maximal compact subgroup of G(ℝ) and let X = G(ℝ)/K_{∞} be the associated Riemannian symmetric space. Let Z(𝔤_{ℂ}) be the center of the unviersal enveloping algebra of the complexification of the Lie algebra 𝔤 of G(ℝ). Recall that a cusp form for Γ in the sense of [La] is a smooth and K_{∞}-finite function ϕ : Γ\G(ℝ) → ℂ which is a simultaneous eigenfunction of Z(𝔤_{ℂ}) and which satisfies

∫_{Γ∩N_{P} (ℝ)\N_{P} (ℝ)} ϕ(nx) dn = 0,

for all unipotent radicals N_{P} of proper rational parabolic subgroups P of G. We note that each cusp form f ∈ C^{∞}(Γ\G(ℝ)) is rapidly decreasing on Γ\G(ℝ) and hence square integrable. Let L2_{cus}(Γ\G(ℝ)) be the closure of the linear span of all cusp forms. Let (σ, V_{σ}) be an irreducible unitary representation of K_{∞}. Set

L^{2}(Γ\G(ℝ), σ) = (L^{2}(Γ\G(ℝ)) ⊗ V_{σ})^{K_{∞}}

and define L2_{cus}(Γ\G(ℝ), σ) similarly. Then L2_{cus}(Γ\G(ℝ), σ) is the space of cusp forms with fixed K_{∞}-type σ. Let Ω_{G(ℝ)} ∈ Z(𝔤_{ℂ}) be the Casimir element of G(ℝ). Then −Ω_{G(ℝ)} ⊗ Id induces a selfadjoint operator Δ_{σ} in the Hilbert space L^{2}(Γ\G(ℝ), σ) which is bounded from below. If Γ is torsion free, L^{2}(Γ\G(ℝ), σ) is isomorphic to the space L^{2}(Γ\X, E_{σ}) of square integrable sections of the locally homogeneous vector bundle E_{σ} associated to σ, and Δ_{σ} = (∇^{σ})^{∗}∇^{σ} − λ_{σ} Id, where ∇^{σ} is the canonical invariant connection and λ_{σ} the Casimir eigenvalue of σ. This shows that Δ_{σ} is a second order elliptic differential operator. Especially, if σ_{0} is the trivial representation, then L^{2}(Γ\G(ℝ), σ_{0}) ≅ L^{2}(Γ\X) and Δ_{σ_{0}} equals the Laplacian Δ of X.

The restriction of Δ_{σ} to the subspace L2_{cus}(Γ\G(ℝ), σ) has pure point spectrum consisting of eigenvalues λ_{0}(σ) < λ_{1}(σ) < ⋯ of finite multiplicity. We call it the cuspidal spectrum of Δ_{σ}. A convenient way of counting the number of cusp forms for Γ is to use their Casimir eigenvalues. For this purpose we introduce the counting function N Γ_{cus}(λ, σ), λ ≥ 0, for the cuspidal spectrum of type σ which is defined as follows. Let E(λ_{i}(σ)) be the eigenspace corresponding to the eigenvalue λ_{i}(σ). Then

N Γ_{cus}(λ, σ) = ∑_{λ_{i}(σ)≤λ} dim E(λ_{i}(σ)).

For non-uniform lattices Γ the selfadjoint operator Δ_{σ} has a large continuous spectrum so that almost all of the eigenvalues of Δ_{σ} will be embedded in the continous spectrum. This makes it very difficult to study the cuspidal spectrum of Δ_{σ}.

WEYL’S LAW FOR SL_{n} 3

The first results concerning the growth of the cuspidal spectrum are due to Selberg [Se]. Let H be the upper half-plane and let Δ be the hyperbolic Laplacian of H. Let N Γ_{cus}(λ) be the counting function of the cuspidal spectrum of Δ. In this case the cuspidal eigenfunctions of Δ are called Maass cusp forms. Using the trace formula, Selberg [Se, p.668] proved that for every congruence subgroup Γ ⊂ SL_{2}(ℤ), the counting function satisfies Weyl’s law, i.e.

(0.1)  N Γ_{cus}(λ) ∼ \frac{vol(Γ\H)}{4π}λ

as λ → ∞. In particular this implies that for congruence subgroups of SL_{2}(ℤ) there exist as many Maass cusp forms as one can expect. On the other hand, it is conjectured by Phillips and Sarnak [PS] that for a non-uniform lattice Γ of SL_{2}(ℝ) whose Teichmüller space T is non trivial and different from the Teichmüller space corresponding to the once punctured torus, a generic lattice Γ ∈ T has only finitely many Maass cusp forms. This indicates that the existence of cusp forms is very subtle and may be related to the arithmetic nature of Γ.

Let d = dim X. In has been conjectured in [Sa] that for rank(X) > 1 and Γ an irreducible lattice

(0.2)  lim sup_{λ→∞} \frac{N Γ_{cus}(λ)}{λ^{d/2}} = \frac{vol(Γ\X)}{(4π)^{d/2}Γ(d/2 + 1)},

where Γ(s) denotes the Gamma function. A lattice Γ for which (0.2) holds is called by Sarnak essentially cuspidal. An analogous conjecture was made in [Mu3, p.180] for the counting function N Γ_{dis}(λ, σ) of the discrete spectrum of any Casimir operator Δ_{σ}. This conjecture states that for any arithmetic subroup Γ and any K_{∞}-type σ

(0.3)  lim sup_{λ→∞} \frac{N Γ_{dis}(λ, σ)}{λ^{d/2}} = dim(σ)\frac{vol(Γ\X)}{(4π)^{d/2}Γ(d/2 + 1)}.

Up to now these conjectures have been verified only in a few cases. Besides of Selberg’s result, Weyl’s law (0.2) has been proved in the following cases: For congruence subgroups of G = SO(n, 1) by Reznikov [Rez], for congruence subgroups of G = R_{F/ℚ} SL_{2}, where F is a totally real number field, by Efrat [Ef, p.6], and for SL_{3}(ℤ) by St. Miller [Mil].

In this paper we will prove that each principal congruence subgroup Γ of SL_{n}(ℤ), n ≥ 2, is essentially cuspidal, i.e. Weyl’s law holds for Γ. Actually we prove the corresponding result for all K_{∞}-types σ. Our main result is the following theorem.

Theorem 0.1. For n ≥ 2 let X_{n} = SL_{n}(ℝ)/ SO(n). Let d_{n} = dim X_{n}. For every principal congruence subgroup Γ of SL_{n}(ℤ) and every irreducible unitary representation σ of SO(n) such that σ|_{Z_{Γ}} = Id we have

(0.4)  N Γ_{cus}(λ, σ) ∼ dim(σ)\frac{vol(Γ\X_{n})}{(4π)^{d_{n}/2}Γ(d_{n}/2 + 1)}λ^{d_{n}/2}

as λ → ∞.

The method that we use is similar to Selberg’s method [Se]. In particular, it does not give any estimation of the remainder term. For n = 2 a much better estimation of the remainder term exists. Using the full strength of the trace formula, it is possible to get a three-term asymptotic expansion of N Γ_{cus}(λ) with remainder term of order O(\sqrt{λ}/ log λ)

[He, Theorem 2.28], [Ve, Theorem 7.3]. The method is based on the study of the Selberg zeta function. It is quite conceivable that the Arthur trace formula can be used to obtain a good estimation of the remainder term for arbitrary n.

Next we reformulate Theorem 0.1 in the adèlic language. Let G = GL_{n} regarded as an algebraic group over ℚ. Let 𝔸 be the ring of adèles of ℚ. Denote by A_{G} the split component of the center of G and let A_{G}(ℝ)^{0} be the component of 1 in A_{G}(ℝ). Let ξ_{0} be the trivial character of A_{G}(ℝ)^{0} and denote by Π(G(𝔸), ξ_{0}) the set of equivalence classes of irreducible unitary representations of G(𝔸) whose central character is trivial on A_{G}(ℝ)^{0}. Let L2_{cus}(G(ℚ)A_{G}(ℝ)^{0}\G(𝔸)) be the subspace of cusp forms in L^{2}(G(ℚ)A_{G}(ℝ)^{0}\G(𝔸)). Denote by Π_{cus}(G(𝔸), ξ_{0}) the subspace of all π in Π(G(𝔸), ξ_{0}) which are equivalent to a subrepresentation of the regular representation in L2_{cus}(G(ℚ)A_{G}(ℝ)^{0}\G(𝔸)). By [Sk] the multiplicity of any π ∈ Π_{cus}(G(𝔸), ξ_{0}) in the space of cusp forms L2_{cus}(G(ℚ)A_{G}(ℝ)^{0}\G(𝔸)) is one. Let A_{f} be the ring of finite adèles. Any irreducible unitary representation π of G(𝔸) can be written as π = π_{∞} ⊗ π_{f}, where π_{∞} and π_{f} are irreducible unitary representations of G(ℝ) and G(𝔸_{f}), respectively. Let H_{π_{∞}} and H_{π_{f}} denote the Hilbert space of the representation π_{∞} and π_{f}, respectively. Let K_{f} be an open compact subgroup of G(𝔸_{f}). Denote by H_{π_{f}}^{K_{f}} the subspace of K_{f}-invariant vectors in H_{π_{f}}. Let G(ℝ)^{1} be the subgroup of all g ∈ G(ℝ) with | det(g)| = 1. Given π ∈ Π(G(𝔸), ξ_{0}), denote by λ_{π} the Casimir eigenvalue of the restriction of π_{∞} to G(ℝ)^{1}. For λ ≥ 0 let Π_{cus}(G(𝔸), ξ_{0})_{λ} be the space of all π ∈ Π_{cus}(G(𝔸), ξ_{0}) which satisfy |λ_{π}| ≤ λ. Set ε_{K_{f}} = 1, if −1 ∈ K_{f} and ε_{K_{f}} = 0 otherwise. Then we have

Theorem 0.2. Let G = GL_{n} and let d_{n} = dim SL_{n}(ℝ)/ SO(n). Let K_{f} be an open compact subgroup of G(𝔸_{f}) and let (σ, V_{σ}) be an irreducible unitary representation of O(n) such that σ(−1) = Id if −1 ∈ K_{f}. Then

∑_{π∈Π_{cus}(G(𝔸),ξ_{0})_{λ}} dim(H_{π_{f}}^{K_{f}} ) dim(H_{π_{∞}} ⊗ V_{σ})^{O(n)}

∼ dim(σ)\frac{vol(G(ℚ)A_{G}(ℝ)^{0}\G(𝔸)/K_{f})}{(4π)^{d_{n}/2}Γ(d_{n}/2 + 1)}(1 + ε_{K_{f}})λ^{d_{n}/2}

(0.5)

as λ → ∞.

Here we have used that the multiplicity of any π ∈ Π(G(𝔸), ξ_{0}) in the space of cusp forms is one.

The asymptotic formula (0.5) may be regarded as the adèlic version of Weyl’s law for GL_{n}. A similar result holds if we replace ξ_{0} by any unitary character of A_{G}(ℝ)^{0}. If we specialize Theorem 0.2 to the congruence subgroup K(N) which defines Γ(N), we obtain Theorem 0.1.

Theorem 0.2 will be derived from the Arthur trace formula combined with the heat equation method. The heat equation method is a very convenient way to derive Weyl’s law for the counting function of the eigenvalues of the Laplacian on a compact Riemannian manifold [Cha]. It is based on the study of the asymptotic behaviour of the trace of the heat operator.

Our approach is similar. We will use the Arthur trace formula to compute the trace of the heat operator on the discrete spectrum and to determine its asymptotic behaviour as t → 0.

We will now describe our method in more detail. Let G(𝔸)^{1} be the subgroup of all g ∈ G(𝔸) satisfying | det(g)| = 1. Then G(ℚ) is contained in G(𝔸)^{1} and the noninvariant trace formula of Arthur [A1] is an identity

(0.6)  ∑_{χ∈𝔛}J_{χ}(f) = ∑_{𝔬∈𝔒}J_{𝔬}(f), f ∈ C_{c}^{∞}(G(𝔸)^{1}),

between distributions on G(𝔸)^{1}. The left hand side is the spectral side J_{spec}(f) and the right hand side the geometric side J_{geo}(f) of the trace formula. The distributions J_{χ} are defined in terms of truncated Eisenstein series. They are parametrized by the set of cuspidal data 𝔛. The distributions J_{𝔬} are parametrized by semisimple conjugacy in G(ℚ) and are closely related to weighted orbital integrals on G(𝔸)^{1}.

For simplicity we consider only the case of the trivial K_{∞}-type. We choose a certain family of test functions ϕ̃_{t}^{1} ∈ C_{c}^{∞}(G(𝔸)^{1}), depending on t > 0, which at the infinite place are given by the heat kernel h_{t} ∈ C^{∞}(G(ℝ)^{1}) of the Laplacian on X, multiplied by a certain cutoff function φ_{t}, and which at the finite places is given by the normalized characteristic function of an open compact subgroup K_{f} of G(𝔸_{f}). Then we evaluate the spectral and the geometric side at ϕ̃_{t}^{1} and study their asymptotic bahaviour as t → 0. Let Π_{dis}(G(𝔸), ξ_{0}) be the set of irreducible unitary representations of G(𝔸) which occur discretely in the regular representation of G(𝔸) in L^{2}(G(ℚ)A_{G}(ℝ)^{0}\G(𝔸)). Given π ∈ Π_{dis}(G(𝔸), ξ_{0}), let m(π) denote the multiplicity with which π occurs in L^{2}(G(ℚ)A_{G}(ℝ)^{0}\G(𝔸)). Let H_{π_{∞}}^{K_{∞}} be the space of K_{∞}-invariant vectors in H_{π_{∞}}. Comparing the asymptotic behaviour of the two sides of the trace formula, we obtain

∑_{π∈Π_{dis}(G(𝔸),ξ_{0})}m(π)e^{tλ_{π}} dim(H_{π_{f}}^{K_{f}}) dim(H_{π_{∞}}^{K_{∞}})

∼ \frac{vol(G(ℚ)\G(𝔸)^{1}/K_{f})}{(4π)^{d_{n}/2}}(1 + ε_{K_{f}})t^{−d_{n}/2}

(0.7)

as t → 0, where the notation is as in Theorem 0.2. Applying Karamatas theorem [Fe, p.446], we obtain Weyl’s law for the discrete spectrum with respect to the trivial K_{∞}-type. A nontrivial K_{∞}-type can be treated in the same way. The discrete spectrum is the union of the cuspidal and the residual spectrum. It follows from [MW] combined with Donnelly’s estimation of the cuspidal spectrum [Do], that the order of growth of the counting function of the residual spectrum for GL_{n} is at most O(λ^{(d_{n}−1)/2}) as λ → ∞. This implies (0.5).

To study the asymptotic behaviour of the geometric side, we use the fine 𝔬-expansion [A10]

(0.8)  J_{geo}(f) = ∑_{M∈L} ∑_{γ∈(M(ℚ_{S}))_{M,S}} a^{M}(S, γ)J_{M}(f, γ),

which expresses the distribution J_{geo}(f) in terms of weighted orbital integrals J_{M}(γ, f). Here M runs over the set of Levi subgroups L containing the Levi component M_{0} of the standard minimal parabolic subgroup P_{0}, S is a finite set of places of ℚ, and (M(ℚ_{S}))_{M,S} is a certain set of equivalence classes in M(ℚ_{S}). This reduces our problem to the investigation of weighted orbital integrals. The key result is that

lim_{t→0} t^{d_{n}/2}J_{M}(ϕ̃_{t}^{1}, γ) = 0,

unless M = G and γ = ±1. The contributions to (0.8) of the terms where M = G and γ = ±1 are easy to determine. Using the behaviour of the heat kernel h_{t}(±1) as t → 0, it follows that

(0.9)  J_{geo}(ϕ̃_{t}^{1}) ∼ \frac{vol(G(ℚ)\G(𝔸)^{1}/K_{f})}{(4π)^{d/2}}(1 + ε_{K_{f}})t^{−d_{n}/2}

as t → 0.

To deal with the spectral side, we use the results of [MS]. Let C^{1}(G(𝔸)^{1}) denote the space of integrable rapidly decreasing functions on G(𝔸)^{1} (see [Mu2, §1.3] for its definition). By Theorem 0.1 of [MS], the spectral side is absolutely convergent for all f ∈ C^{1}(G(𝔸)^{1}). Furthermore, it can be written as a finite linear combination

J_{spec}(f) = ∑_{M∈L} ∑_{L∈L(M)} ∑_{P ∈P(M)} ∑_{s∈W^{L}(𝔞_{M})_{reg}} a_{M,s}J_{M,P}^{L} (f, s).

of distributions J_{M,P}^{L} (f, s), where L(M) is the set of Levi subgroups containing M, P(M) denotes the set of parabolic subgroups with Levi component M and W^{L}(𝔞_{M})_{reg} is a certain set of Weyl group elements. Given M ∈ L, the main ingredients of the distribution J_{M,P}^{L} (f, s) are generalized logarithmic derivatives of the intertwining operators

M_{Q|P} (λ) : A^{2}(P) → A^{2}(Q), P, Q ∈ P(M), λ ∈ 𝔞_{M,ℂ}^{∗},

acting between the spaces of automorphic forms attached to P and Q, respectively. First of all, Theorem 0.1 of [MS] allows us to replace ϕ̃_{t}^{1} by a similar function ϕ_{t}^{1} ∈ C^{1}(G(𝔸)^{1}) which is given as the product of the heat kernel at the infinite place and the normalized characteristic function of K_{f}. Consider the distribution where M = L = G. Then s = 1 and

(0.10)  J_{G,G}^{G}(ϕ_{t}^{1}) = ∑_{π∈Π_{dis}(G(𝔸),ξ_{0})}m(π)e^{tλ_{π}} dim(H_{π_{f}}^{K_{f}}) dim(H_{π_{∞}}^{K_{∞}}).

This is exactly the left hand side of (0.7). Thus in order to prove (0.7) we need to show that for all proper Levi subgroups M, all L ∈ L(M), P ∈ P(M) and s ∈ W^{L}(𝔞_{M})_{reg}, we have

(0.11)  J_{M,P}^{L} (ϕ_{t}^{1}, s) = O(t^{−(d_{n}−1)/2})

as t → 0. This is the key result where we really need that our group is GL_{n}. It relies on estimations of the logarithmic derivatives of intertwining operators for λ ∈ i𝔞_{M}^{∗}. Given π ∈ Π_{dis}(M(𝔸), ξ_{0}), let M_{Q|P} (π, λ) be the restriction of the intertwining operator M_{Q|P} (λ) to the subspace A_{π}^{2}(P) of automorphic forms of type π. The intertwining operators can be normalized by certain meromorphic functions r_{Q|P} (π, λ) [A7]. Thus

M_{Q|P} (π, λ) = r_{Q|P} (π, λ)^{−1}N_{Q|P} (π, λ),

where N_{Q|P} (π, λ) are the normalized intertwining operators. Using Arthur’s theory of (G, M)-families [A5], our problem can be reduced to the estimation of derivatives of N_{Q|P} (π, λ) and r_{Q|P} (π, λ) on i𝔞_{M}^{∗}. The derivatives of N_{Q|P} (π, λ) can be estimated using Proposition 0.2 of [MS]. Let M = GL_{n_{1}} × ⋯ × GL_{n_{r}}. Then π = ⊗_{i}π_{i} with π_{i} ∈ Π_{dis}(GL_{n_{i}}(𝔸)^{1}) and the normalizing factors r_{Q|P} (π, λ) are given in terms of the Rankin-Selberg L-functions L(s, π_{i} ×π̃_{j}) and the corresponding ϵ-factors ϵ(s, π_{i} ×π̃_{j}). So our problem is finally reduced to the estimation of the logarithmic derivative of Rankin-Selberg L-functions on the line Re(s) = 1. Using the available knowledge of the analytic properties of Rankin-Selberg L-functions together with standard methods of analytic number theory, the necessary estimates can be derived.

In the proof of Theorem 0.1 and Theorem 0.2 we have used the following key results which at present are only known for GL_{n}: 1) The nontrivial bounds of the Langlands parameters of local components of cuspidal automorphic representations [LRS] which are needed in [MS], 2) The description of the residual spectrum given in [MW], 3) The theory of the Rankin-Selberg L-functions [JPS].

The paper is organized as follows. In section 2 we prove some estimations for the heat kernel on a symmetric space. In section 3 we establish some estimates for the growth of the discrete spectrum in general. We are essentially using Donnelly’s result [Do] combined with the description of the residual spectrum [MW]. The main purpose of section 4 is to prove estimates for the growth of the number of poles of Rankin-Selberg L-functions in the critical strip. We use these results in section 5 to establish the key estimates for the logarithmic derivatives of normalizing factors. In section 6 we study the asymptotic behaviour of the spectral side J_{spec}(ϕ_{t}^{1}). Finally, in section 7 we study the asymptotic behaviour of the geometric side, compare it to the asymptotic behaviour of the spectral side and prove the main results.

Acknowledgment. The author would like to thank W. Hoffmann, D. Ramakrishnan and P. Sarnak for very helpful discussions on parts of this paper. Especially Lemma 7.1 is due to W. Hoffmann.

1. Preliminaries

1.1. Fix a positive integer n and let G be the group GL_{n} considered as algebraic group over ℚ. By a parabolic subgroup of G we will always mean a parabolic subgroup which is defined over ℚ. Let P_{0} be the subgroup of upper triangular matrices of G. The Levi subgroup M_{0} of P_{0} is the group of diagonal matrices in G. A parabolic subgroup P of G is called standard, if P ⊃ P_{0}. By a Levi subgroup we will mean a subgroup of G which contains M_{0} and is the Levi component of a parabolic subgroup of G defined over ℚ. If M ⊂ L are Levi subgroups, we denote the set of Levi subgroups of L which contain M by L^{L}(M). Furthermore, let F^{L}(M) denote the set of parabolic subgroups of L defined over ℚ which contain M, and let P^{L}(M) be the set of groups in F^{L}(M) for which M is a Levi component. If L = G, we shall denote these sets by L(M), F(M) and P(M). Write L = L(M_{0}). Suppose that P ∈ F^{L}(M). Then

P = N_{P} M_{P} ,

where N_{P} is the unipotent radical of P and M_{P} is the unique Levi component of P which contains M.

Let M ∈ L and denote by A_{M} the split component of the center of M. Then A_{M} is defined over ℚ. Let X(M)_{ℚ} be the group of characters of M defined over ℚ and set

𝔞_{M} = Hom(X(M)_{ℚ}, ℝ).

Then 𝔞_{M} is a real vector space whose dimension equals that of A_{M}. Its dual space is

𝔞_{M}^{∗} = X(M)_{ℚ} ⊗ ℝ.

Let P and Q be groups in F(M_{0}) with P ⊂ Q. Then there is a canonical surjection 𝔞_{P} → 𝔞_{Q} and a canonical injection 𝔞_{Q}^{∗} ↪ 𝔞_{P}^{∗}. The kernel of the first map will be denoted by 𝔞_{P}^{Q}. Then the dual vector space of 𝔞_{P}^{Q} is 𝔞_{P}^{∗} /𝔞_{Q}^{∗}.

Let P ∈ F(M_{0}). We shall denote the roots of (P, A_{P}) by Σ_{P}, and the simple roots by Δ_{P}. Note that for GL_{n} all roots are reduced. They are elements in X(A_{P})_{ℚ} and are canonically embedded in 𝔞_{P}^{∗}.

For any M ∈ L there exists a partition (n_{1}, ..., n_{r}) of n such that

M = GL_{n_{1}} × ⋯ × GL_{n_{r}} .

Then 𝔞_{M}^{∗} can be canonically identified with (ℝ^{r})^{∗} and the Weyl group W (𝔞_{M}) coincides with the group S_{r} of permutations of the set {1, ..., r}.

1.2. Let F be a local field of characteristic zero. If π is an admissible representation of GL_{m}(F), we shall denote by π̃ the contragredient representation to π. Let π_{i}, i = 1, ..., r, be irreducible admissible representations of the group GL_{n_{i}}(F). Then π = π_{1} ⊗ ⋯ ⊗ π_{r} is an irreducible admissible representation of

M(F) = GL_{n_{1}}(F) × ⋯ × GL_{n_{r}}(F).

For s ∈ ℂ^{r} let π_{i}[s_{i}] be the representation of GL_{n_{i}}(F) which is defined by

π_{i}[s_{i}](g) = | det(g)|^{s_{i}}π_{i}(g), g ∈ GL_{n_{i}}(F).

Let

I_{P}^{G}(π, s) = Ind_{P (F)}^{G(F)}(π_{1}[s_{1}] ⊗ ⋯ ⊗ π_{r}[s_{r}])

be the induced representation and denote by H_{P} (π) the Hilbert space of the representation I_{P}^{G}(π, s). We refer to s as the continuous parameter of I_{P}^{G}(π, s). Sometimes we will write I_{P}^{G}(π_{1}[s_{1}], ..., π_{r}[s_{r}]) in place of I_{P}^{G}(π, s).

1.3. Let G be a locally compact topological group. Then we denote by Π(G) the set of equivalence classes of irreducible unitary representations of G.

1.4. Let M ∈ L. Denote by A_{M}(ℝ)^{0} the component of 1 of A_{M}(ℝ). Set

M_{P} (𝔸)^{1} = ⋂_{χ∈X(M)_{ℚ}} ker(|χ|).

This is a closed subgroup of M(𝔸), and M(𝔸) is the direct product of M(𝔸)^{1} and A_{M}(ℝ)^{0}.

Given a unitary character ξ of A_{M}(ℝ)^{0}, denote by L^{2}(M(ℚ)\M(𝔸), ξ) the space of all measurable functions ϕ on M(ℚ)\M(𝔸) such that

ϕ(xm) = ξ(x)ϕ(m), x ∈ A_{M}(ℝ)^{0}, m ∈ M(𝔸),

and ϕ is square integrable on M(ℚ)\M(𝔸)^{1}. Let L2_{dis}(M(ℚ)\M(𝔸), ξ) denote the discrete subspace of L^{2}(M(ℚ)\M(𝔸), ξ) and let L2_{cus}(M(ℚ)\M(𝔸), ξ) be the subspace of cusp forms in L^{2}(M(ℚ)\M(𝔸), ξ). The orthogonal complement of L2_{cus}(M(ℚ)\M(𝔸), ξ) in the discrete subspace is the resiudal subspace L2_{res}(M(ℚ)\M(𝔸), ξ). Denote by Π_{dis}(M(𝔸), ξ), Π_{cus}(M(𝔸), ξ), and Π_{res}(M(𝔸), ξ) the subspace of all π ∈ Π(M(𝔸), ξ) which are equivalent to a subrepresentation of the regular representation of M(𝔸) in L^{2}(M(ℚ)\M(𝔸), ξ), L2_{cus}(M(ℚ)\M(𝔸), ξ), and L2_{res}(M(ℚ)\M(𝔸), ξ), respectively.

Let Π_{dis}(M(𝔸)^{1}) be the subspace of all π ∈ Π(M(𝔸)^{1}) which are equivalent to a subrepresentation of the regular representation of M(𝔸)^{1} in L^{2}(M(ℚ)\M(𝔸)^{1}). We denote by Π_{cus}(M(𝔸)^{1}) (resp. Π_{res}(M(𝔸)^{1})) the subspaces of all π ∈ Π_{dis}(M(𝔸)^{1}) occurring in the cuspidal (resp. residual) subspace L2_{cus}(M(ℚ)\M(𝔸)^{1}) (resp. L2_{res}(M(ℚ)\M(𝔸)^{1})).

1.5. Let P be a parabolic subgroup of G. We denote by A^{2}(P) the space of square integrable automorphic forms on N_{P} (𝔸)M_{P} (ℚ)A_{P} (ℝ)^{0}\G(𝔸) (see [Mu2, §1.7]).

Given π ∈ Π_{dis}(M_{P} (𝔸), ξ_{0}), let A_{π}^{2}(P) be the subspace of A^{2}(P) of automorphic forms of type π [A1, p.925]. Let π ∈ Π(M_{P} (𝔸)^{1}). We identify π with a representation of M_{P} (𝔸) which is trivial on A_{P} (ℝ)^{0}. Hence we can define A_{π}^{2}(P) for any π ∈ Π(M_{P} (𝔸)^{1}). It is a space of square integrable functions on N_{P} (𝔸)M_{P} (ℚ)A_{P} (ℝ)^{0}\G(𝔸) such that for every x ∈ G(𝔸), the function

ϕ_{x}(m) = ϕ(mx), m ∈ M_{P} (𝔸),

belongs to the π-isotypical subspace of the regular representation of M_{P} (𝔸) in the Hilbert space L^{2}(A_{P} (ℝ)^{0}M_{P} (ℚ)\M_{P} (𝔸)).

2. Heat kernel estimates

In this section we shall prove some estimates for the heat kernel of the Bochner-Laplace operator acting on sections of a homogeneous vector bundle over a symmetric space. Let G be a connected semisimple algebraic group defined over ℚ. Let K_{∞} be a maximal compact subgroup of G(ℝ) and let (σ, V_{σ}) be an irreducible unitary representation of K_{∞} on a complex vector space V_{σ}. Let Ẽ_{σ} = (G(ℝ) × V_{σ})/K_{∞} be the associated homogeneous vector bundle over X = G(ℝ)/K_{∞}. We equip Ẽ_{σ} with the G(ℝ)–invariant Hermitian fibre metric which is induced by the inner product in V_{σ}. Let C^{∞}(Ẽ_{σ}), C_{c}^{∞}(Ẽ_{σ}) and L^{2}(Ẽ_{σ}) denote the space of smooth sections, the space of compactly supported smooth sections and the Hilbert space of square integrable sections of Ẽ_{σ}, respectively. Then we have

(2.1)  C^{∞}(Ẽ_{σ}) = (C^{∞}(G(ℝ)) ⊗ V_{σ})^{K_{∞}}, L^{2}(Ẽ_{σ}) = (L^{2}(G(ℝ)) ⊗ V_{σ})^{K_{∞}}

and similarly for C_{c}^{∞}(Ẽ_{σ}). Let Ω ∈ Z(𝔤_{ℂ}) be the Casimir element of G(ℝ) and let R be the right regular representation of G(ℝ) on C^{∞}(G(ℝ)). Let Δ̃_{σ} be the second order elliptic operator which is induced by −R(Ω) ⊗ Id in C^{∞}(Ẽ_{σ}). Let ˜∇^{σ} be the canonical connection on Ẽ_{σ}, and let Ω_{K} be the Casimir element of K_{∞}. Let λ_{σ} = σ(Ω_{K}) be the Casimir eigenvalue of σ. Then with respect to the identification (2.1), we have

(2.2)  ( ˜∇^{σ})^{∗} ˜∇^{σ} = −R(Ω) ⊗ Id + λ_{σ} Id

[Mia, Proposition 1.1], and therefore

(2.3)  Δ̃_{σ} = ( ˜∇^{σ})^{∗} ˜∇^{σ} − λ_{σ}Id.

Hence Δ̃_{σ}: C_{c}^{∞}(Ẽ_{σ}) → L^{2}(Ẽ_{σ}) is essentially selfadjoint and bounded from below. We continue to denote its unique selfadjoint extension by Δ̃_{σ}. Let exp(−tΔ̃_{σ}) be the associated heat semigroup. The heat operator is a smoothing operator on L^{2}(Ẽ_{σ}) which commutes with the representation of G(ℝ) on L^{2}(Ẽ_{σ}). Therefore, it is of the form

(2.4)  (e^{−tΔ̃_{σ}}φ)(g) = ∫_{G(ℝ)} H_{t}^{σ}(g^{−1}g_{1})(φ(g_{1}))dg_{1}, g ∈ G(ℝ),

where φ ∈ (L^{2}(G(ℝ)) ⊗ V_{σ})^{K_{∞}} and H_{t}^{σ}: G(ℝ) → End(V_{σ}) is in L^{2} ∩ C^{∞} and satisfies the covariance property

(2.5)  H_{t}^{σ}(g) = σ(k)H_{t}^{σ}(k^{−1}gk′)σ(k′)^{−1}, for g ∈ G(ℝ), k, k′ ∈ K_{∞}.

In order to get estimates for H_{t}^{σ}, we proceed as in [BM] and relate H_{t}^{σ} to the heat kernel of the Laplace operator of G(ℝ) with respect to a left invariant metric on G(ℝ). Let 𝔤 and 𝔨 denote the Lie algebras of G(ℝ) and K_{∞}, respectively. Let 𝔤 = 𝔨 ⊕ 𝔭 be the Cartan decomposition and let θ be the corresponding Cartan involution. Let B(Y_{1}, Y_{2}) be the Killing form of 𝔤. Set ⟨Y_{1}, Y_{2}⟩ = −B(Y_{1}, θY_{2}), Y_{1}, Y_{2} ∈ 𝔤. By translation of ⟨·, ·⟩ we get a left invariant Riemannian metric on G(ℝ). Let X_{1}, ⋯, X_{p} be an orthonormal basis for 𝔭 with respect to B|𝔭 × 𝔭 and let Y_{1}, ⋯, Y_{k} be an orthonormal basis for 𝔨 with respect to −B|𝔨 × 𝔨. Then we have

Ω = ∑_{i=1}^{p} X_{i}^{2} − ∑_{i=1}^{k} Y_{i}^{2} and Ω_{K} = − ∑_{i=1}^{k} Y_{i}^{2}.

Let

(2.6)  P = −Ω + 2Ω_{K} = − ∑_{i=1}^{p} X_{i}^{2} − ∑_{i=1}^{k} Y_{i}^{2}.

Then R(P) is the Laplace operator Δ_{G} on G(ℝ) with respect to the left invariant metric defined above. The heat semigroup e^{−tΔ_{G}} is represented by a smooth kernel p_{t}, i.e.

(2.7)  (e^{−tΔ_{G}}f) (g) = ∫_{G(ℝ)} p_{t}(g^{−1}g′)f(g′)dg′, f ∈ L^{2}(G(ℝ)), g ∈ G(ℝ),

where p_{t} ∈ C^{∞}(G(ℝ)) ∩ L^{2}(G(ℝ)). In fact, p_{t} belongs to L^{1}(G(ℝ)) [N] so that (2.7) can be written as

e^{−tΔ_{G}} = R(p_{t}).

Let

Q = ∫_{K_{∞}} R(k) ⊗ σ(k) dk

be the orthogonal projection of L^{2}(G(ℝ))⊗V_{σ} onto its K_{∞}–invariant subspace (L^{2}(G(ℝ))⊗ V_{σ})^{K_{∞}}. By (2.6) we have

Δ̃_{σ} = −Q(R(Ω) ⊗ Id)Q
= Q(R(P) ⊗ Id)Q − 2Q(R(Ω_{K}) ⊗ Id)Q
= Q(Δ_{G} ⊗ Id)Q − 2λ_{σ} Id_{L^{2}(Ẽ_{σ})} .

Hence, we get

e^{−tΔ̃_{σ}} = Q(e^{−tΔ_{G}} ⊗ Id)Q · e^{t2λ_{σ}}

which implies that

(2.8)  H_{t}^{σ}(g) = e^{t2λ_{σ}} ∫_{K_{∞}} ∫_{K_{∞}} p_{t}(k^{−1}gk′)σ(kk′^{−1}) dk dk′.

Let C^{1}(G(ℝ)) be Harish-Chandra’s space of integrable rapidly decreasing functions on G(ℝ). Then (2.8) can be used to show that

(2.9)  H_{t}^{σ} ∈ (C^{1}(G(ℝ)) ⊗ End(V_{σ}))^{K_{∞}×K_{∞}}

[BM, Proposition 2.4].

Now we turn to the estimation of the derivatives of H_{t}^{σ}. By (2.8), this problem can be reduced to the estimation of the derivatives of p_{t}. Let ∇ denote the Levi-Civita connection and ρ(g, g′) the geodesic distance of g, g′ ∈ G(ℝ) with respect to the left invariant metric. Then all covariant derivatives of the curvature tensor are bounded and the injectivity radius has a positive lower bound. Let a = dim G(ℝ), l ∈ ℕ_{0} and T > 0. Then it follows from Corollary 8 in [CLY] that there exist C, c > 0 such that

(2.10)  ∥ ∇^{l}p_{t}(g) ∥≤ Ct^{−(a+l)/2} exp (−\frac{cρ^{2}(g, 1)}{t})

for all 0 < t ≤ T and g ∈ G(ℝ). By (2.8) and (2.10) we get

\begin{aligned}∥ ∇^{l}H_{t}^{σ}(g) ∥ & ≤ e^{2tλ_{σ}} ∫_{K_{∞}} ∫_{K_{∞}} ∥ (∇^{l}p_{t})(k^{−1}gk′) ∥ dkdk′ \\ & ≤ Ct^{−(a+l)/2} ∫_{K_{∞}} ∫_{K_{∞}} exp (−\frac{cρ^{2}(gk, k′)}{t}) dkdk′ \end{aligned} ; (2.11)

for all 0 < t ≤ T. Choose the invariant Riemannian metric on X which is defined by the restriction of the Killing form to T_{e}X ≅ 𝔭. Then the canonical projection map G(ℝ) → X is a Riemannian submersion. Let d(x, y) denote the geodesic distance on X. Then it follows that

ρ(g, e) ≥ d(gK_{∞}, K_{∞}), g ∈ G(ℝ).

Set r(g) = d(gK_{∞}, K_{∞}), g ∈ G(ℝ). Together with (2.11) we get the following result.

Proposition 2.1. Let a = dim G(ℝ), l ∈ ℕ_{0} and T > 0. There exist C, c > 0 such that

(2.12)  ∥ ∇^{l}H_{t}^{σ}(g) ∥≤ Ct^{−(a+l)/2} exp (−\frac{cr^{2}(g)}{t})

for all 0 < t ≤ T and g ∈ G(ℝ).

We note that the exponent of t on the right hand side of (2.12) is not optimal. Using the method of Donnelly [Do2], this estimate can be improved for l ≤ 1. Indeed by Theorem 3.1 of [Mu1] we have

Proposition 2.2. Let n = dim X and T > 0. There exist C, c > 0 such that

(2.13)  ∥ ∇^{l}H_{t}^{σ}(g) ∥≤ Ct^{−n/2−l} exp (−\frac{cr^{2}(g)}{t})

for all 0 < t ≤ T, 0 ≤ l ≤ 1, and g ∈ G(ℝ).

We also need the asymptotic behaviour of the heat kernel on the diagonal. It is described by the following lemma.

Lemma 2.3. Let n = dim X and let e ∈ G(ℝ) be the identity element. Then

tr H_{t}^{σ}(e) = \frac{dim(σ)}{(4π)^{n/2}}t^{−n/2} + O(t^{−(n−1)/2})

as t → 0.

Proof. Note that for each x ∈ X, the injectivity radius at x is infinite. Hence we can construct a parametrix for the fundamental solution of the heat equation for Δ_{σ} as in [Do2]. Let ϵ > 0 and set

U_{ϵ} = {(x, y) ∈ X × X | d(x, y) < ϵ}.

For any l ∈ ℕ we define an approximate fundamental solution P_{l}(x, y, t) on U_{ϵ} by the formula

P_{l}(x, y, t) = (4πt)^{−n/2} exp (\frac{−d^{2}(x, y)}{4t}) (∑_{i=0}^{l} Φ_{i}(x, y)t^{i}) ,

where the Φ_{i}(x, y) are smooth sections of E_{σ} ⊠ E_{σ}^{∗} over U_{ϵ} × U_{ϵ} which are constructed recursively as in Theorem 2.26 of [BGV]. In particular, we have

Φ_{0}(x, x) = Id_{V_{σ}}, x ∈ X.

Let ψ ∈ C^{∞}(X × X) be equal to 1 on U_{ϵ/4} and 0 on X × X − U_{ϵ/2}. Set

Q_{l}(x, y, t) = ψ(x, y)P_{l}(x, y, t).

If l > n/2, then the section Q_{l} of E_{σ} ⊠ E_{σ}^{∗} is a parametrix for the heat equation. Since X is a Riemannian symmetric space, we get

H_{t}^{σ}(e) = Id_{V_{σ}}(4πt)^{−n/2} + O(t^{−(n−1)/2})

as t → 0. This implies the lemma. □

3. Estimations of the discrete spectrum

In this section we shall establish a number of facts concerning the growth of the discrete spectrum. Let M = GL_{n_{1}} × ⋯ × GL_{n_{r}}, r ≥ 1, and let

M(ℝ)^{1} = M(ℝ) ∩ M(𝔸)^{1}.

Then M(ℝ) = M(ℝ)^{1} · A_{M}(ℝ)^{0}. Let K_{M,∞} ⊂ M(ℝ) be the standard maximal compact subgroup. Then K_{M,∞} is contained in M(ℝ)^{1}. Let

X_{M} = M(ℝ)^{1}/K_{M,∞}

be the associated Riemannian symmetric space. Let Γ_{M} ⊂ M(ℚ) be an arithmetic subgroup and let (τ, V_{τ}) be an irreducible unitary representation of K_{M,∞} on V_{τ}. Set

C^{∞}(Γ_{M}\M(ℝ)^{1}, τ) := (C^{∞}(Γ_{M}\M(ℝ)^{1}) ⊗ V_{τ})^{K_{M,∞}}.

If Γ_{M} is torsion free, then Γ_{M}\X_{M} is a Riemannian manifold and the homogeneous vector bundle Ẽ_{τ} over X_{M}, which is associated to τ, can be pushed down to a vector bundle E_{τ} → Γ_{M}\X_{M}. Then C^{∞}(Γ_{M}\M(ℝ)^{1}, τ) equals C^{∞}(Γ_{M}\X_{M}, E_{τ}), the space of smooth sections of E_{τ}. Define C_{c}^{∞}(Γ_{M}\M(ℝ)^{1}, τ) and L^{2}(Γ_{M}\M(ℝ)^{1}, τ) similarly. Let Ω_{M(ℝ)^{1}} be the Casimir element of M(ℝ)^{1} and let Δ_{τ} be the operator in C^{∞}(Γ_{M}\M(ℝ)^{1}, τ) which is induced by −Ω_{M(ℝ)^{1}} ⊗ Id. As unbounded operator in L^{2}(Γ_{M}\M(ℝ)^{1}, τ) with domain C_{c}^{∞}(Γ_{M}\M(ℝ)^{1}, τ), Δ_{τ} is essentially selfadjoint. Let L2_{cus}(Γ_{M}\M(ℝ)^{1}, τ) be the subspace of cusp forms of L^{2}(Γ_{M}\M(ℝ)^{1}, τ). Then L2_{cus}(Γ_{M}\M(ℝ)^{1}, τ) is an invariant subspace of Δ_{τ}, and Δ_{τ} has pure point spectrum in this subspace consisting of eigenvalues λ_{0} < λ_{1} < ⋯ of finite multiplicity. Let E(λ_{i}) be the eigenspace of λ_{i}. Set

N Γ_{M}_{cus} (λ, τ) = ∑_{λ_{i}≤λ} dim E(λ_{i}).

Let d = dim X_{M} and let

C_{d} = \frac{1}{(4π)^{d/2}Γ(\frac{d}{2} + 1)}

be Weyl’s constant, where Γ(s) denotes the Gamma function. Then Donnelly [Do, Theorem 9] has established the following basic estimation of the counting function of the cuspidal spectrum.

Theorem 3.1. For every τ ∈ Π(K_{M,∞}) we have

lim sup_{λ→∞} \frac{N Γ_{M}_{cus} (λ, τ)}{λ^{d/2}} ≤ C_{d} dim(τ) vol(Γ_{M}\X_{M}).

Actually, Donnelly proved this theorem only for the case of a torsion free discrete group. However, it is easy to extend his result to the general case.

We shall now reformulate this theorem in the representation theoretic context. Let ξ_{0} be the trivial character of A_{M}(ℝ)^{0} and let π ∈ Π(M(𝔸), ξ_{0}). Let m(π) be the multiplicity with which π occurs in the regular representation of M(𝔸) in L^{2}(A_{M}(ℝ)^{0}M(ℚ)\M(𝔸)). Then Π_{dis}(M(𝔸), ξ_{0}) consists of all π ∈ Π(M(𝔸), ξ_{0}) with m(π) > 0. Write

π = π_{∞} ⊗ π_{f},

where π_{∞} ∈ Π(M(ℝ)) and π_{f} ∈ Π(M(𝔸_{f})). Denote by H_{π_{∞}} (resp. H_{π_{f}}) the Hilbert space of the representation π_{∞} (resp. π_{f}). Let K_{M,f} be an open compact subgroup of M(𝔸_{f}) and let τ ∈ Π(K_{M,∞}). Denote by H_{π_{∞}}(τ) the τ-isotypical subspace of H_{π_{∞}} and let H_{π_{f}}^{K_{M,f}} be the subspace of K_{M,f}-invariant vectors in H_{π_{f}} . Denote by λ_{π} the Casimir eigenvalue of the restriction of π_{∞} to M(ℝ)^{1}. Given λ > 0, let

Π_{dis}(M(𝔸), ξ_{0})_{λ} = {π ∈ Π_{dis}(M(𝔸), ξ_{0}) | |λ_{π}| ≤ λ}.

Define Π_{cus}(M(𝔸), ξ_{0})_{λ} and Π_{res}(M(𝔸), ξ_{0})_{λ} similarly.

Lemma 3.2. Let d = dim X_{M}. For every open compact subgroup K_{M,f} of M(𝔸_{f}) and every τ ∈ Π(K_{M,∞}) there exists C > 0 such that

∑_{π∈Π_{cus}(M(𝔸),ξ_{0})_{λ}} m(π) dim(H_{π_{f}}^{K_{M,f}}) dim(H_{π_{∞}}(τ)) ≤ C(1 + λ^{d/2})

for λ ≥ 0.

Proof. Extending the notation of §1.4, we write Π(M(ℝ), ξ_{0}) for the set of representations in Π(M(ℝ)) whose central character is trivial on A_{M}(ℝ)^{0}. Given π_{∞} ∈ Π(M(ℝ), ξ_{0}), let m(π_{∞}) be the multiplicity with which π_{∞} occurs discretely in the regular representation of M(ℝ) in L^{2}(A_{M}(ℝ)^{0}M(ℚ)\M(𝔸))^{K_{M,f}}. Then

(3.1)  m(π_{∞}) = ∑′_{π′∈Π_{cus}(M(𝔸),ξ_{0})}m(π′) dim(H_{π′_{f}}^{K_{M,f}}),

where the sum is over all π′ ∈ Π_{dis}(M(𝔸), ξ_{0}) such that the Archimedean component π′_{∞} of π′ equals π_{∞}.

Let Π_{cus}(M(ℝ), ξ_{0}) be the subset of all π_{∞} ∈ Π(M(ℝ), ξ_{0}) which are equivalent to an irreducible subrepresentation of the regular representation of M(ℝ) in the Hilbert space L2_{cus}(A_{M}(ℝ)^{0}M(ℚ)\M(𝔸))^{K_{M,f}}. Given π_{∞} ∈ Π_{cus}(M(ℝ), ξ_{0}), denote by λ_{π_{∞}} the Casimir eigenvalue of the restriction of π_{∞} to M(ℝ)^{1}. For λ ≥ 0, let

Π_{cus}(M(ℝ), ξ_{0})_{λ} = {π_{∞} ∈ Π_{cus}(M(ℝ), ξ_{0}) | |λ_{π_{∞}}| ≤ λ}.

Then by (3.1), it suffices to show that for each τ ∈ Π(K_{M,∞}) there exists C > 0 such that

∑_{π_{∞}∈Π_{cus}(M(ℝ),ξ_{0})_{λ}} m(π_{∞}) dim(H_{π_{∞}}(τ)) ≤ C(1 + λ^{d/2}).

To deal with this problem recall that there exist arithmetic subgroups Γ_{M,i} ⊂ M(ℝ), i = 1, …, l, such that

M(ℚ)\M(𝔸)/K_{M,f} ≅ ⨆_{i=1}^{l}(Γ_{M,i}\M(ℝ))

(cf. [Mu1, section 9]). Hence

(3.2)  L^{2}(A_{M}(ℝ)^{0}M(ℚ)\M(𝔸))^{K_{M,f}} ≅ ⨁_{i=1}^{l} L^{2}(A_{M}(ℝ)^{0}Γ_{M,i}\M(ℝ))

as M(ℝ)-modules. For each i, i = 1, …, l, and π_{∞} ∈ Π(M(ℝ)) let m_{Γ_{M,i}}(π_{∞}) be the multiplicity with which π_{∞} occurs discretely in the regular representation of M(ℝ) in L^{2}(A_{M}(ℝ)^{0}Γ_{M,i}\M(ℝ)). Then m(π_{∞}) = ∑_{i=1}^{l} m_{Γ_{M,i}}(π_{∞}) and

∑_{π_{∞}∈Π_{cus}(M(ℝ),ξ_{0})_{λ}}m(π_{∞}) dim(H_{π_{∞}}(τ))

= ∑_{i=1}^{l} ∑_{π_{∞}∈Π_{cus}(M(ℝ),ξ_{0})_{λ}} m_{Γ_{M,i}}(π_{∞}) dim(H_{π_{∞}}(τ)).

The interior sum can be interpreted as follows. Fix i and set Γ_{M} := Γ_{M,i}. Let λ_{1} < λ_{2} < ⋯ be the eigenvalues of Δ_{τ} in the space of cusp forms L2_{cus}(Γ_{M}\M(ℝ)^{1}, τ) and let E(λ_{i}) be the eigenspace of λ_{i}. By Frobenius reciprocity it follows that

dim E(λ_{i}) = ∑_{−λ_{π∞}=λ_{i}} m_{Γ_{M}} (π_{∞}),

where the sum is over all π_{∞} ∈ Π_{cus}(M(ℝ), ξ_{0}) such that the Casimir eigenvalue λ_{π_{∞}} equals −λ_{i}. Hence we obtain

∑_{π_{∞}∈Π_{cus}(M(ℝ),ξ_{0})_{λ}} m_{Γ_{M}} (π_{∞}) dim(H_{π_{∞}}(τ)) = N Γ_{M}_{cus} (λ, τ).

Combined with Theorem 3.1 the desired estimation follows. □

Next we consider the residual spectrum.

Lemma 3.3. Let d = dim X_{M}. For every open compact subgroup K_{M,f} of M(𝔸_{f}) and every τ ∈ Π(K_{M,∞}) there exists C > 0 such that

∑_{π∈Π_{res}(M(𝔸),ξ_{0})_{λ}} m(π) dim(H_{π_{f}}^{K_{M,f}}) dim(H_{π_{∞}}(τ)) ≤ C(1 + λ^{(d−1)/2})

for λ ≥ 0.

Proof. We can assume that M = GL_{n_{1}} × ⋯ × GL_{n_{r}}. Let K_{M,f} be an open compact subgroup of M(𝔸_{f}). There exist open compact subgroups K_{i,f} of GL_{n_{i}}(𝔸_{f}) such that K_{1,f} × ⋯ × K_{r,f} ⊂ K_{M,f}. Thus we can replace K_{M,f} by K_{1,f} × ⋯ × K_{r,f}. Next observe that K_{M,∞} = O(n_{1})×⋯×O(n_{r}) and therefore, τ is given as τ = τ_{1}⊗⋯⊗τ_{r}, where each τ_{i} is an irreducible unitary representation of O(n_{i}). Finally note that every π ∈ Π(M(𝔸), ξ_{0}) is of the form π = π_{1} ⊗ ⋯ ⊗ π_{r}. Hence we get m(π) = ∏_{i=1}^{r} m(π_{i}) and

dim(H_{π_{f}}^{K_{M,f}} ) = ∏_{i=1}^{r} dim(H_{π_{i,f}}^{K_{i,f}} ), dim(H_{π_{∞}}(τ)) = ∏_{i=1}^{r} dim(H_{π_{i,∞}}(τ_{i})).

This implies immediately that it suffices to consider a single factor.

With the analogous notation the proof of the proposition is reduced to the following problem. For m ∈ ℕ set X_{m} = SL_{m}(ℝ)/ SO(m) and d_{m} = dim X_{m}. Then we need to show that for every open compact subgroup K_{m,f} of GL_{m}(𝔸_{f}) and every τ ∈ Π(O(m)) there exists C > 0 sucht that

∑_{π∈Π_{res}(GL_{m}(𝔸),ξ_{0})_{λ}} m(π) dim(H_{π_{f}}^{K_{m,f}}) dim(H_{π_{∞}}(τ)) ≤ C(1 + λ^{(d_{m}−1)/2})

for λ ≥ 0. To deal with this problem recall the description of the residual spectrum of GL_{m} by Mœglin and Waldspurger [MW]. Let π ∈ Π_{res}(GL_{m}(𝔸)) and suppose that π is trivial on A_{GL_{m}}(ℝ)^{0}. There exist k|m, a standard parabolic subgroup P of GL_{m} of type (l, …, l), l = m/k, and a cuspidal automorphic representation ρ of GL_{l} which is trivial on A_{GL_{l}}(ℝ)^{0}, such that π is equivalent to the unique irreducible quotient J(ρ) of the induced representation

I_{P (𝔸)}^{GL_{m}(𝔸)}(ρ[(k − 1)/2] ⊗ ⋯ ⊗ ρ[(1 − k)/2]).

Here ρ[s] denotes the representation g ⟼ ρ(g)| det g|^{s}, s ∈ ℂ. At the Archimedean place, the corresponding induced representation

I_{P}^{GL_{m}}(ρ_{∞}, k) := I_{P (ℝ)}^{GL_{m}(ℝ)}(ρ_{∞}[(k − 1)/2] ⊗ ⋯ ⊗ ρ_{∞}[(1 − k)/2])

has also a unique irreducible quotient J(ρ_{∞}). Comparing the definitions, we get J(ρ)_{∞} = J(ρ_{∞}). Hence the Casimir eigenvalue of π_{∞} = J(ρ)_{∞} equals the Casimir eigenvalue of J(ρ_{∞}) which in turn coincides with the Casimir eigenvalue of the induced representation I_{P}^{GL_{m}}(ρ_{∞}, k). Let λ_{ρ} be the Casimir eigenvalue of ρ_{∞}. Then it follows that there exists C > 0 such that |λ_{π} − kλ_{ρ}| ≤ C for all π ∈ Π_{res}(GL_{m}(𝔸), ξ_{0}). Using the main theorem of [MW, p.606] it follows that it suffices to fix l|m, l < m, and to estimate

(3.3)  ∑_{ρ∈Π_{cus}(GL_{l}(𝔸),ξ_{0})_{λ}} m(ρ) dim(H_{J(ρ)_{f}}^{K_{m,f}} ) dim(H_{J(ρ)_{∞}}(τ)).

First note that by [Sk], we have m(ρ) = 1 for all ρ ∈ Π_{cus}(GL_{l}(𝔸), ξ_{0}). So it remains to estimate the dimensions. We begin with the infinite place. Observe that dim(H_{J(ρ)_{∞}}(τ)) = dim(τ)[J(ρ_{∞})|_{O(m)} : τ]. Thus in order to estimate dim(H_{J(ρ)_{∞}}(τ)) it suffices to estimate the multiplicity [J(ρ_{∞})|_{O(m)} : τ]. Since J(ρ_{∞}) is an irreducible quotient of I_{P}^{GL_{m}}(ρ_{∞}, k), we have

[J(ρ_{∞})|_{O(m)} : τ] ≤ [I_{P}^{GL_{m}}(ρ_{∞}, k)|_{O(m)} : τ].

Let K_{l,∞} = O(l) × ⋯ × O(l). Using Frobenius reciprocity as in [Kn, p.208], we obtain

\begin{matrix}[I_{P}^{GL_{m}}(ρ_{∞}, k) & |_{O(m)} : τ] \\ = ∑_{ω∈Π(K_{l,∞})}[(ρ_{∞} ⊗ ⋯ ⊗ ρ_{∞})|_{K_{l,∞}} : ω] · [τ|_{K_{l,∞}} : ω]. \end{matrix}

Finally note that ω = ω_{1} ⊗ ⋯ ⊗ ω_{k} with ω_{i} ∈ Π(O(l)). Therefore we have

[(ρ_{∞} ⊗ ⋯ ⊗ ρ_{∞})|_{K_{l,∞}} : ω] = ∏_{i=1}^{k}[ρ_{∞}|_{O(l)} : ω_{i}].

At the finite places we proceed in an analogous way. This implies that there exist an open compact subgroup K_{l,f} of GL_{l}(𝔸_{f}) and ω_{1}, ..., ω_{p} ∈ Π(O(l)) such that (3.3) is bounded from above by a constant times

∑_{i=1}^{p} ( ∑_{ρ∈Π_{cus}(GL_{l}(𝔸),ξ_{0})_{λ}} m(ρ) dim(H_{ρ_{f}}^{K_{l,f}} ) dim(H_{ρ_{∞}}(ω_{i})))^{k} .

By Lemma 3.2 this term is bounded by a constant times (1+λ^{d_{l}/2})^{k}, where d_{l} = l(l+1)/2−1. Since m = k · l and k > 1, we have

d_{l}k = \frac{l(l + 1)k}{2} − k ≤ \frac{m(m + 1)}{2} − 2 = d_{m} − 1.

This proves the desired estimation in the case of M = GL_{m}, and as explained above, this suffices to prove the lemma. □

Combining Lemma 3.2 and Lemma 3.3, we obtain

Proposition 3.4. Let d = dim X_{M}. For every open compact subgroup K_{M,f} of M(𝔸_{f}) and every τ ∈ Π(K_{M,∞}) there exists C > 0 such that

∑_{π∈Π_{dis}(M(𝔸),ξ_{0})_{λ}} m(π) dim(H_{π_{f}}^{K_{M,f}}) · dim(H_{π_{∞}}(τ)) ≤ C(1 + λ^{d/2})

for λ ≥ 0.

Next we restate Proposition 3.4 in terms of dimensions of spaces of automorphic forms. Let P ∈ P(M) and let A^{2}(P) be the space of square integrable automorphic forms on N_{P} (𝔸)M_{P} (ℚ)A_{P} (ℝ)^{0}\G(𝔸). Given π ∈ Π_{dis}(M(𝔸), ξ_{0}) let A_{π}^{2}(P) be the subspace of A^{2}(P) of automorphic forms of type π [A1, p.925]. Let K_{∞} be the standard maximal compact subgroup of G(ℝ). Given an open compact subgroup K_{f} of G(𝔸_{f}) and σ ∈ Π(K_{∞}), let A_{π}(P)_{K_{f}} denote the subspace of K_{f}-invariant automorphic forms in A_{π}^{2}(P) and let A_{π}^{2}(P)_{K_{f} ,σ} be the σ-isotypical subspace of A_{π}^{2}(P)_{K_{f}}.

Proposition 3.5. Let d = dim X_{M}. For every open compact subgroup K_{f} of G(𝔸_{f}) and every σ ∈ Π(K_{∞}) there exists C > 0 such that

∑_{π∈Π_{dis}(M(𝔸),ξ_{0})_{λ}} dim A_{π}^{2}(P)_{K_{f} ,σ} ≤ C(1 + λ^{d/2})

for λ ≥ 0.

Proof. Let π ∈ Π_{dis}(M(𝔸), ξ_{0}). Let H_{P} (π) be the Hilbert space of the induced representation I_{P (𝔸)}^{G(𝔸)}(π). There is a canonical isomorphism

(3.4)  j_{P} : H_{P} (π) ⊗ Hom_{M(𝔸)}(π, I_{M(ℚ)A_{M} (ℝ)^{0}}^{M(𝔸)}(ξ_{0})) → \overline{A}_{π}^{2}(P),

which intertwines the induced representations. Let π = π_{∞} ⊗ π_{f}. Let H_{P} (π_{∞}) (resp. H_{P} (π_{f})) be the Hilbert space of the induced representation I_{P (ℝ)}^{G(ℝ)}(π_{∞})) (resp. I_{P (𝔸_{f})}^{G(𝔸_{f})}(π_{f})). Denote by H_{P} (π_{∞})_{σ} the σ-isotypical subspace of H_{P} (π_{∞}) and by H_{P} (π_{f})^{K_{f}} the subspace of K_{f}-invariant vectors of H_{P} (π_{f}). Then it follows from (3.4) that

(3.5)  dim A_{π}^{2}(P)_{K_{f} ,σ} = m(π) dim(H_{P} (π_{f})^{K_{f}}) dim(H_{P} (π_{∞})_{σ}).

Using Frobenius reciprocity as in [Kn, p.208] we get

[I_{P (ℝ)}^{G(ℝ)}(π_{∞})|_{K_{∞}} : σ] = ∑_{τ∈Π(K_{M,∞})}[π_{∞}|_{K_{M,∞}} : τ] · [σ|_{K_{M,∞}} : τ].

Hence we get

(3.6)  dim(H_{P} (π_{∞})_{σ}) ≤ dim(σ) ∑_{τ∈Π(K_{M,∞})} dim(H_{π_{∞}}(τ))[σ|_{K_{M,∞}} : τ].

Next we consider π_{f} = ⊗_{p<∞}π_{p}. Replacing K_{f} by a subgroup of finite index if necessary, we can assume that K_{f} = Π_{p<∞}K_{p}. For any p < ∞, denote by H_{P} (π_{p}) the Hilbert space of the induced representation I_{P (ℚ_{p})}^{G(ℚ_{p})}(π_{p}). Let H_{P} (π_{p})^{K_{p}} be the subspace of K_{p}-invariant vectors. Then dim H_{P} (π_{p})^{K_{p}} = 1 for alomost all p and

H_{P} (π_{f})^{K_{f}} ≅ ⨂_{p<∞} H_{P} (π_{p})^{K_{p}}.

Furthermore

\begin{aligned}I_{P (ℚ_{p})}^{G(ℚ_{p})}(π_{p})^{K_{p}} & = (I_{P (ℤ_{p})}^{G(ℤ_{p})}(π_{p}))^{K_{p}} \\ & ↪ ⨁_{G(ℤ_{p})/K_{p}} I_{K_{p}∩P}^{K_{p}} (π_{p})^{K_{p}} \\ & ≅ ⨁_{G(Z_{p})/K_{p}} π_{p}^{K_{p}∩P} . \end{aligned} ; (3.7)

Let K_{M,f} = K_{f} ∩M(𝔸_{f}). Using (3.5)–(3.7), it follows that in order to prove the proposition, it suffices to fix τ ∈ Π(K_{M,∞}) and to estimate

∑_{π∈Π_{dis}(M(𝔸),ξ_{0})_{λ}} m(π) dim(H_{π_{f}}^{K_{M,f}}) dim(H_{π_{∞}}(τ)).

The proof is now completed applying Proposition 3.4. □

Finally we consider the analogous statement of Lemma 3.3 at the Archimedean place. For simplicity we consider only the case M = G. Let K_{∞} be the standard maximal compact subgroup of G(ℝ). Let Γ ⊂ G(ℚ) be an arithmetic subgroup and σ ∈ Π(K_{∞}). Then the discrete subspace L2_{dis}(Γ\G(ℝ)^{1}, σ) of Δ_{σ} decomposes as

L2_{dis}(Γ\G(ℝ)^{1}, σ) = L2_{cus}(Γ\G(ℝ)^{1}, σ) ⊕ L2_{res}(Γ\G(ℝ), σ),

where L2_{res}(Γ\G(ℝ)^{1}, σ) is the subspace which corresponds to the residual spectrum of Δ_{σ}. Let

L2_{res}(Γ\G(ℝ)^{1}, σ) = ⨁_{i} E_{res}(λ_{i})

be the decomposition into eigenspaces of Δ_{σ}. For λ ≥ 0 set

N Γ_{res}(λ, σ) = ∑_{λ_{i}≤λ} dim E_{res}(λ_{i}).

Proposition 3.6. Let d = G(ℝ)^{1}/K_{∞}. Let Γ ⊂ G(ℚ) be an arithmetic subgroup. For every σ ∈ Π(K_{∞}) there exists C > 0 such that

N Γ_{res}(λ, σ) ≤ C(1 + λ^{(d−1)/2})

for λ ≥ 0.

Proof. First assume that Γ ⊂ SL_{n}(ℤ). Let Γ(N) ⊂ Γ be a congruence subgroup. Then

(3.8)  N Γ_{res}(λ, σ) ≤ N Γ(N)_{res}(λ, σ).

Let

N = Π_{p}p^{r_{p}}, r_{p} ≥ 0.

Set

K_{p}(N) = {k ∈ GL_{n}(ℤ_{p}) | k ≡ 1 mod p^{r_{p}}ℤ_{p}}

and

(3.9)  K(N) = Π_{p<∞}K_{p}(N).

Then K(N) is an open compact subgroup of G(𝔸_{f}) and

(3.10)  A_{G}(ℝ)^{0}G(ℚ)\G(𝔸)/K(N) ≅ ⨆_{(ℤ/Nℤ)^{∗}}(Γ(N)\ SL_{n}(ℝ))

(cf. [A9]). Hence

L2_{res}(A_{G}(ℝ)^{0}G(ℚ)\G(𝔸))^{K(N)} ≅ ⨁_{(ℤ/Nℤ)^{∗}} L2_{res}(Γ(N)\ SL_{n}(ℝ))

as SL_{n}(ℝ)-modules. Put M = G in Lemma 3.3. Then

∑_{π∈Π_{res}(G(𝔸),ξ_{0})_{λ}} m(π) dim(H_{π_{f}}^{K_{f}}) dim(H_{π_{∞}}(σ)) = φ(N)N Γ(N)_{res}(λ, σ),

where φ(N) = #[(ℤ/Nℤ)^{∗}]. Hence by Lemma 3.3 it follows that there exists C > 0 such that

N Γ(N)_{res}(λ, σ) ≤ C(1 + λ^{(d−1)/2}).

This proves the proposition for Γ ⊂ SL_{n}(ℤ). Since an arithmetic subgroup Γ ⊂ G(ℚ) is commensurable with G(ℤ), the general case can be easily reduced to this one.

□

4. Rankin-Selberg L-functions

The main purpose of this section is to prove estimates for the number of zeros of Rankin-Selberg L-functions. We shall consider the Rankin-Selberg L-functions over an arbitrary number field, although in the present paper we shall use them only in the case of ℚ. We begin with the description of the local L-facors.

Let F be a local field of characteristic zero. Recall that any irreducible admissible representation of GL_{m}(F) is given as a Langlands quotient: There exist a standard parabolic subgroup P of type (m_{1}, ..., m_{r}), discrete series representations δ_{i} of GL_{m_{i}}(F) and complex numbers s_{1}, ..., s_{r} satisfying Re(s_{1}) ≥ Re(s_{2}) ≥ ⋯ ≥ Re(s_{r}) such that

(4.1)  π = J_{P}^{GL_{m}}(δ_{1}[s_{1}] ⊗ ⋯ ⊗ δ_{r}[s_{r}]),

where the representation on the right is the unique irreducible quotient of the induced representation I_{P}^{GL_{m}}(δ_{1}[s_{1}] ⊗ ⋯ ⊗ δ_{r}[s_{r}]) [MW, I.2]. Furthermore any irreducible generic representation π of GL_{m}(F) is equivalent to a fully induced representation I_{P}^{GL_{m}}(δ_{1}[s_{1}] ⊗ ⋯ ⊗ δ_{r}[s_{r}]). If π is generic and unitary, it follows from the classification of the unitary dual of GL_{m}(F) that the parameters s_{i} satisfy

(4.2)  | Re(s_{i})| < 1/2, i = 1, ..., r.

Suppose that π is given as Langlands quotient of the form (4.1). Then the L-function satisfies

(4.3)  L(s, π) = ∏_{j} L(s + s_{j}, δ_{j})

[J]. Furthermore, suppose that π_{1} and π_{2} are irreducible admissible representations of G_{1} = GL_{m_{1}}(ℝ) and G_{2} = GL_{m_{2}}(ℝ), respectively. Let

π_{i} ≅ J_{P_{i}}^{GL_{n_{i}}}(τ_{i1}[s_{i1}], …, τ_{ir_{i}}[s_{ir_{i}}])

be the Langlands parametrizations of π_{i}, i = 1, 2. Then it follows from the multiplicativity of the local Rankin-Selberg L-factors [JPS, (9.4)], [Sh6] that

(4.4)  L(s, π_{1} × π_{2}) = ∏_{i=1}^{r_{1}} ∏_{j=1}^{r_{2}} L(s + s_{1i} + s_{2j}, τ_{1i} × τ_{2j}).

This reduces the description of the local L-factors to the square-integrable case. Now we distinguish three cases according to the type of the field.

1. F non-Archimedean

Let O_{F} denote the ring of integers of F and 𝔓 the maximal ideal of O_{F}. Set q = O_{F} /𝔓. The square-integrable case can be further reduced to the supercuspidal one. Finally for supercuspidal representations the L-factor is given by an elementary polynomial in q^{−s}. For details see [JPS] (see also [MS]). If we put together all steps of the reduction, we get the following result. Let π_{1} and π_{2} be irreducible admissible representations of GL_{n_{1}}(F) and GL_{n_{2}}(F), resprectively. Then there is a polynomial P_{π_{1},π_{2}}(x) of degree at most n_{1} · n_{2} with P_{π_{1},π_{2}}(0) = 1 such that

L(s, π_{1} × π_{2}) = P_{π_{1},π_{2}}(q^{−s})^{−1}.

In the special case where π_{1} and π_{2} are unitary and generic the L-factor has the following special form.

Lemma 4.1. Let π_{1} and π_{2} be irreducible unitary generic representations of GL_{n_{1}}(F) and GL_{n_{2}}(F), respectively. There exist complex numbers a_{i}, i = 1, ..., n_{1} · n_{2}, with |a_{i}| < q such that

(4.5)  L(s, π_{1} × π_{2}) = ∏_{i=1}^{n_{1}·n_{2}}(1 − a_{i}q^{−s})^{−1}.

Proof. Let δ_{1} and δ_{2} be square-integrable representations of GL_{d_{1}}(F) and GL_{d_{2}}(F), respectively. As explained above there is a polynomial P_{δ_{1},δ_{2}}(x) of degree at most d_{1} · d_{2} with P_{δ_{1},δ_{2}}(0) = 1 such that

L(s, δ_{1} × δ_{2}) = P_{δ_{1},δ_{2}}(q^{−s})^{−1}.

By (6) of [JPS], p. 445, L(s, δ_{1} × δ_{2}) is holomorphic in the half-plane Re(s) > 0. Hence P_{δ_{1},δ_{2}}(x) has no zeros in the unit disc. Thus there exist complex numbers b_{i} with |b_{i}| < 1 such that

(4.6)  L(s, δ_{1} × δ_{2}) = ∏_{i=1}^{d_{1}·d_{2}}(1 − b_{i}q^{−s})^{−1}.

Now let π_{1} and π_{2} be unitary and generic. Then L(s, π_{1}×π_{2}) can be written as a product of the form (4.4) and by (4.2) the parameters s_{ij} satisfy | Re(s_{ij})| < 1/2, i = 1, 2, j = 1, ..., r_{i}. Combined with (4.6) the lemma follows. □

If F is Archimedean the L-factors are defined in terms of the L-factors attached to semisimple representations of the Weyl group W_{F} by means of the Langlands correspondence [La1]. The structure of the L-factors are described, for example, in [MS, §3]. We briefly recall the result.

2. F = ℝ.

First note that GL_{m}(ℝ) does not have square-integrable representations if m ≥ 3. To describe the principal L-factors in the remaining cases d = 1 and d = 2, we define Gamma factors by

(4.7)  Γ_{ℝ}(s) = π^{−s/2}Γ (\frac{s}{2}) , Γ_{ℂ}(s) = 2(2π)^{−s}Γ(s).

In the case d = 1, the unitary representations of GL_{1}(ℝ) = ℝ^{×} are of the form ψ_{ϵ,t}(x) = sign^{ϵ}(x)|x|^{t} with ϵ ∈ {0, 1} and t ∈ iℝ. Then

L(s, ψ_{ϵ,t}) = Γ_{ℝ}(s + t + ϵ).

For k ∈ ℤ let D_{k} be the k-th discrete series representation of GL_{2}(ℝ) with the same infinitesimal character as the k-dimensional representation. Then the unitary square-integrable representations of GL_{2}(ℝ) are unitary twists of D_{k}, k ∈ ℤ, for which the L-factor is given by

L(s, D_{k}) = Γ_{ℂ}(s + |k|/2).

Let ψ_{ϵ} = sign^{ϵ}, ϵ ∈ {0, 1}. Then up to twists by unramified characters the following list describes the Rankin-Selberg L-factors in the square-integrable case:

(4.8)  \begin{aligned}L(s, D_{k_{1}} × D_{k_{2}}) & = Γ_{ℂ}(s + |k_{1} − k_{2}|/2) · Γ_{ℂ}(s + |k_{1} + k_{2}|/2) \\ L(s, D_{k} × ψ_{ϵ}) & = L(s, ψ_{ϵ} × D_{k}) = Γ_{ℂ}(s + |k|/2) \\ L(s, ψ_{ϵ_{1}} × ψ_{ϵ_{2}}) & = Γ_{ℝ}((s + ϵ_{1,2})), \end{aligned}

where 0 ≤ ϵ_{1,2} ≤ 1 with ϵ_{1,2} ≡ ϵ_{1} + ϵ_{2} mod 2.

3. F = ℂ.

There exist square-integrable representations of GL_{k}(ℂ) only if k = 1. For r ∈ ℤ let χ_{r} be the character of GL_{1}(ℂ) = ℂ^{×} which is given by χ(z) = (z/\overline{z})^{r}, z ∈ ℂ^{∗}. Then

(4.9)  L(s, χ_{r}) = Γ_{ℂ}(s + |r_{i}|/2).

If χ_{r_{1}} and χ_{r_{2}} are two characters as above, then we have

L(s, χ_{r_{1}} × χ_{r_{2}}) = Γ_{ℂ}(s + |r_{1} + r_{2}|/2).

Up to twists by unramified characters, these are all possibilities for the L-factors in the square-integrable case.

To summarize we obtain the following description of the local L-factors in the complex case. Let π be an irreducible unitary representation of GL_{m}(ℂ). It is given by a Langlands quotient of the form

π = J_{B}^{GL_{m}}(χ_{1}[s_{1}] ⊗ ⋯ ⊗ χ_{m}[s_{m}]),

where B is the standard Borel subgroup of GL_{m} and the χ_{i}’s are characters of GL_{1}(ℂ) = ℂ^{×} which are defined by χ(z) = (z/\overline{z})^{r_{i}}, r_{i} ∈ ℤ, i = 1, …, m. Then

(4.10)  L(s, π) = ∏_{i=1}^{m} Γ_{ℂ}(s + s_{i} + |r_{i}|/2).

Let π_{1} and π_{2} be irreducible unitary representations of GL_{m_{1}}(ℂ) and GL_{m_{2}}(ℂ), respectively. Let B_{i} ⊂ GL_{m_{i}} be the standard Borel subgroup. There exist characters χ_{ij} of ℂ^{×} of the form χ_{ij}(z) = (z/\overline{z})^{r_{ij}}, r_{ij} ∈ ℤ, and complex numbers s_{ij}, i = 1, …, m_{1}, j = 1, …, m_{2}, satisfying

Re(s_{i1}) ≥ ⋯ ≥ Re(s_{im_{i}}), | Re(s_{ij}| < 1/2,

such that

(4.11)  π_{i} = J_{B_{i}}^{GL_{m_{i}}}(χ_{i1}[s_{i1}] ⊗ ⋯ ⊗ χ_{im_{i}}[s_{im_{i}}]), i = 1, 2.

Then the Rankin-Selberg L-factor is given by

(4.12)  L(s, π_{1} × π_{2}) = ∏_{i=1}^{m_{1}} ∏_{j=1}^{m_{2}} Γ_{ℂ}(s + s_{1i} + s_{2j} + |r_{1i} + r_{2j}|/2).

If F = ℝ, the L-factors have a similar form.

The description of the L-factors in the Archimedean case can be unified in the following way. By the duplication formula of the Gamma function we have

(4.13)  Γ_{ℂ}(s) = Γ_{ℝ}(s)Γ_{ℝ}(s + 1).

Let F be Archimedean. Set e_{F} = 1, if F = ℝ, and e_{F} = 2, if F = ℂ. Let π ∈ Π(GL_{m}(F)). Then it follows from (4.13) and the definition of the L-factors, that there exist complex numbers μ_{j}(π), j = 1, …, me_{F} , such that

(4.14)  L(s, π) = ∏_{j=1}^{me_{F}} Γ_{ℝ}(s + μ_{j}(π)).

The numbers μ_{j}(π) are determined by the Langlands parameters of π. Similarly, if π_{i} ∈ Π(GL_{m_{i}}(F)), i = 1, 2, it follows from the description of the Rankin-Selberg L-factors that there exist complex numbers μ_{j,k}(π_{1} × π_{2}) such that

(4.15)  L(s, π_{1} × π_{2}) = ∏_{j,k} Γ_{ℝ}(s + μ_{j,k}(π_{1} × π_{2})).

Lemma 4.2. Let F be Archimedean. There exists C > 0 such that

∑_{j,k} |μ_{j,k}(π_{1} × π_{2})|^{2} ≤ C (∑_{i} |μ_{i}(π_{1})|^{2} + ∑_{j} |μ_{j}(π_{2})|^{2})

for all generic π_{i} ∈ Π(GL_{m_{i}}(F)), i = 1, 2.

Proof. First consider the case F = ℂ. Let π_{1} and π_{2} be irreducible unitary generic representations of GL_{m_{1}}(ℂ) and GL_{m_{2}}(ℂ), respectively. Write π_{i} as Langlands quotient of the form (4.11). Using (4.10) and (4.12) together with (4.13), it follows that it suffices to prove that there exist C > 0 such that

∑_{j,k} |s_{1j} + s_{2k} + |r_{1j} + r_{2k}|/2|^{2} ≤ C ∑_{i,j} |s_{ij} + |r_{ij}|/2|^{2}

for all generic π_{i} ∈ Π(GL_{m_{i}}(ℂ)), i = 1, 2. This follows immediately, if we use the fact that the parameters s_{ij} satisfy | Re(s_{ij})| < 1/2 and the r_{ij}’s are integers.

The proof in the case F = ℝ is essentially the same. We only have to check the different possible cases for the L-factors as listed above. □

Next we consider the global Rankin-Selberg L-functions. Let E be a number filed and let 𝔸_{E} be the ring of adèles of E. Given m ∈ ℕ, let Π_{dis}(GL_{m}(𝔸_{E})) and Π_{cus}(GL_{m}(𝔸_{E})) be defined in the same way as in the case of ℚ (see §1.4). Recall that the Rankin-Selberg L-function attached to a pair of automorphic representations π_{1} of GL_{m_{1}}(𝔸_{E}) and π_{2} of GL_{m_{2}}(𝔸_{E}) is defined by the Euler product

(4.16)  L(s, π_{1} × π_{2}) = ∏_{v} L(s, π_{1,v} × π_{2,v}),

where v runs over all places of E. The Euler product is known to converge in a certain halfplane Re(s) > c. Suppose that π_{1} and π_{2} are unitary cuspidal automorphic representations of GL_{m_{1}}(𝔸_{E}) and GL_{m_{2}}(𝔸_{E}), respectively. Then L(s, π_{1} × π_{2}) has the following basic properties:

i) The Euler product L(s, π_{1} × π_{2}) converges absolutely for all s in the half-plane Re(s) > 1.

ii) L(s, π_{1} × π_{2}) admits a meromorphic continuation to the entire complex plane with

at most simple poles at 0 and 1. iii) L(s, π_{1} × π_{2}) is of order one and is bounded in vertical strips outside of the poles.

iv) It satisfies a functional equation of the form

(4.17)  L(s, π_{1} × π_{2}) = ϵ(s, π_{1} × π_{2})L(1 − s, π̃_{1} × π̃_{2})

with

(4.18)  ϵ(s, π_{1} × π_{2}) = W (π_{1} × π_{2})(D_{E}^{m_{1}m_{2}}N(π_{1} × π_{2}))^{1/2−s},

where D_{E} is the discriminant of E, W (π_{1} × π_{2}) is a complex number of absolute value 1 and N(π_{1} × π_{2}) ∈ ℕ.

The absolute convergence of the Euler product (4.16) in the half-plane Re(s) > 1 was proved in [JS1]. The functional equation is established in [Sh1, Theorem 4.1] combined with [Sh3, Prop. 3.1] and [Sh3, Theorem 1]. See also [Sh5] for the general case. The location of the poles has been determined in the appendix of [MW]. Property iii) was proved in [RS, p.280].

Now let π_{1} ∈ Π_{dis}(GL_{m_{1}}(𝔸_{E})) and π_{2} ∈ Π_{dis}(GL_{m_{2}}(𝔸_{E})). Using the description of the residual spectrum for GL_{n} [MW], L(s, π_{1} × π_{2}) can be expressed in terms of Rankin-Selberg L-functions attached to cuspidal automorphic representations. Indeed, by [MW] there exist k_{i} ∈ ℕ with k_{i}|m_{i}, parabolic subgroups P_{i} of G_{i} = GL_{m_{i}} of type (d_{i}, ..., d_{i}), d_{i} = m_{i}/k_{i}, and unitary cuspidal automorphic representations δ_{i} of GL_{d_{i}}(𝔸_{E}) such that

(4.19)  π_{i} = J_{P_{i}}^{G_{i}}(δ_{i}[(k_{i} − 1)/2] ⊗ ⋯ ⊗ δ_{i}[(1 − k_{i})/2]),

where the right hand side denotes the unique irreducible quotient of the induced representation I_{P_{i}}^{G_{i}}(δ_{i}[(k_{i} − 1)/2] ⊗ ⋯ ⊗ δ_{i}[(1 − k_{i})/2]). Set k = k_{1} + k_{2} − 2. Then it follows from (4.4) that

(4.20)  L(s, π_{1} × π_{2}) = ∏_{i=0}^{k_{1}−1} ∏_{j=0}^{k_{2}−1} L(s + k/2 − i − j, δ_{1} × δ_{2}).

Using this equality and i)–iv) above, we deduce immediately the corresponding properties satisfied by L(s, π_{1} × π_{2}). Especially, L(s, π_{1} × π_{2}) satisfies a functional equation of the form (4.17) with an ϵ-factor similar to (4.18).

We shall now investigate the logarithmic derivatives of the Rankin-Selberg L-functions. First we need to introduce some notation. Let π_{i} ∈ Π_{dis}(GL_{m_{i}}(𝔸_{E})), i = 1, 2. For each Archimedean place w of E let μ_{j,k}(π_{1,w} × π_{2,w}) j = 1, …, r_{w}, k = 1, …, h_{w}, be the parameters attached to (π_{1,w}, π_{2,w}) by means of (4.15). Set

(4.21)  c(π_{1} × π_{2}) = ∑_{w|∞} ∑_{j,k} |μ_{j,k}(π_{1,w} × π_{2,w})|.

Let N(π_{1} × π_{2}) be the integer that is determined by the ϵ-factor as in (4.18). Set

(4.22)  ν(π_{1} × π_{2}) = D_{E}^{m_{1}m_{2}}N(π_{1} × π_{2})(2 + c(π_{1} × π_{2})).

We call ν(π_{1} × π_{2}) the level of (π_{1}, π_{2}). Given π ∈ Π(GL_{m}(𝔸_{E})), set

π_{∞} = ⊗_{v|∞}π_{v}, π_{f} = ⊗_{v<∞}π_{v}.

Lemma 4.3. For every ϵ > 0 there exists C > 0 such that

|\frac{L′(s, π_{1,f} × π_{2,f})}{L(s, π_{1,f} × π_{2,f})}| ≤ C

for all s in the half-plane Re(s) ≥ 2 + ϵ and all π_{i} ∈ Π_{cus}(GL_{m_{i}}(𝔸_{E})), i = 1, 2.

Proof. Let π_{i} ∈ Π_{cus}(GL_{m_{i}}(𝔸_{E})), i = 1, 2, and let v < ∞. By [Sk], π_{1,v} and π_{2,v} are unitary generic representations. Hence by Lemma 4.1 there exist complex numbers a_{i}(v), i = 1, …, m_{1} · m_{2}, with

(4.23)  |a_{i}(v)| < N(v)

such that

L(s, π_{1,v} × π_{2,v}) = ∏_{i=1}^{m_{1}·m_{2}}(1 − a_{i}(v)N(v)^{−s})^{−1}.

Suppose that Re(s) > 1. By (4.23) we have |a_{i}(v)/N(v)^{s}| < 1. Hence, taking the logarithmic derivative, we get

\begin{aligned}\frac{L′(s, π_{1,v} × π_{2,v})}{L(s, π_{1,v} × π_{2,v})} & = − ∑_{i} \frac{a_{i}(v) log N(v)}{N(v)^{s}(1 − a_{i}(v)N(v)^{−s})} \\ & = − log N(v) ∑_{i} ∑_{k=1}^{∞} \frac{a_{i}(v)^{k}}{N(v)^{sk}} . \end{aligned}

Suppose that σ = Re(s) > 1. Then by (4.23) we get

|\frac{L′(s, π_{1,v} × π_{2,v})}{L(s, π_{1,v} × π_{2,v})}| ≤ m_{1}m_{2} ∑_{k=1}^{∞} \frac{log N(v)}{N(v)^{(σ−1)k}} .

Let ζ_{E}(s) be the Dedekind zeta function of E. Let ϵ > 0 and set σ = 2 + ϵ. Then for Re(s) ≥ σ we get

|\frac{L′(s, π_{1,f} × π_{2,f})}{L(s, π_{1,f} × π_{2,f})}| ≤ m_{1}m_{2}|\frac{ζ′_{E}(σ − 1)}{ζ_{E}(σ − 1)}|.

□

Lemma 4.4. For every ϵ > 0 there exists C > 0 such that

|\frac{L′(s, π_{1,∞} × π_{2,∞})}{L(s, π_{1,∞} × π_{2,∞})}| ≤ C(1 + log(|s| + c(π_{1} × π_{2})))

for all s with Re(s) ≥ 1 + ϵ and all π_{i} ∈ Π_{cus}(GL_{m_{i}}(𝔸_{E})), i = 1, 2.

Proof. Let w|∞. By (4.15) we have

(4.24)  L(s, π_{1,w} × π_{2,w}) = ∏_{j,k} Γ_{ℝ}(s + μ_{j,k}(π_{1,w} × π_{2,w})).

Since π_{1,w} and π_{2,w} are unitary and generic, the complex numbers μ_{j,k}(π_{1,w} × π_{2,w}) satisfy

(4.25)  Re(μ_{j,k}(π_{1,w} × π_{2,w})) > −1.

Now recall that by Stirlings formula

\frac{Γ′}{Γ} (s) = log s + O(|s|^{−1})

is valid as |s| → ∞, in the angle −π + δ < arg s < π − δ, for any fixed δ > 0. Hence

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

holds in the same range of s. Let ϵ > 0. Using (4.24), (4.25) and (4.26), it follows that there exists C > 0 such that

|\frac{L′(s, π_{1,w} × π_{2,w})}{L(s, π_{1,w} × π_{2,w})}| ≤ C + ∑_{j,k} log(|s| + |μ_{j,k}(π_{1,w} × π_{2,w})|).

for all w|∞, all s with Re(s) ≥ 1 + ϵ and all π_{i} ∈ Π_{cus}(GL_{m}(𝔸_{E})), i = 1, 2. This implies the lemma. □

Let π_{i} ∈ Π_{dis}(GL_{m_{i}}(𝔸_{E})), i = 1, 2, and T > 0, be given. Denote by N(T; π_{1}, π_{2}) the number of zeros of L(s, π_{1} × π_{2}), counted with multiplicity, which are contained in the disc of radius T centered at 0.

Proposition 4.5. There exists C > 0 such that

N(T; π_{1}, π_{2}) ≤ CT log(T + ν(π_{1} × π_{2}))

for all T ≥ 1 and all π_{i} ∈ Π_{dis}(GL_{m}(𝔸_{E})), i = 1, 2.

Proof. By (4.20) we can assume that π_{1} and π_{2} are unitary cuspidal automorphic representations. Set

(4.27)  Λ(s) = s^{a}(1 − s)^{a} (D_{E}^{m_{1}m_{2}}N(π_{1} × π_{2}))^{s/2} L(s, π_{1} × π_{2}),

where a denotes the order of the pole of L(s, π_{1} × π_{2}) at s = 1. Note that a can be at most 1. Since π_{i} is unitary, we have π̃_{i} = \overline{π_{i}}, i = 1, 2. Hence by (4.17), it follows that Λ(s) satisfies the functional equation

(4.28)  Λ(s) = W (π_{1} × π_{2}) (D_{E}^{m_{1}m_{2}}N(π_{1} × π_{2}))^{1/2} \overline{Λ(1 − \overline{s})}.

Since L(s, π_{1} × π_{2}) is of order one, Λ(s) is an entire function of order one and hence, it admits a representation as a Weierstraß product of the form

Λ(s) = e^{A+Bs} ∏_{ρ}(1 − s/ρ)e^{s/ρ},

where A, B ∈ ℂ and the product runs over the set of zeros of Λ(s). We note that for s = σ + iT

(4.29)  Re ∑_{ρ} \frac{1}{s − ρ} = ∑_{ρ} \frac{σ − Re(ρ)}{(σ − Re(ρ))^{2} + (Im(ρ) − T)^{2}}

and this series is convergent since Λ(s) is of oder one. Taking the real part of the logarithmic derivative of Λ(s), and applying the functional equation (4.28) to the right hand side, we get

Re(B) + Re ∑ \frac{1}{ρ} + Re ∑_{ρ} \frac{1}{s − ρ} = − Re(\overline{B}) − Re ∑ \frac{1}{ρ}

+ Re ∑_{ρ} \frac{1}{s − (1 − \overline{ρ})}.

Now observe that by (4.28), ρ is a zero of Λ(s) if and only if 1 − \overline{ρ} is a zero of Λ(s). Hence the two sums involving s are equal, as they run over the same set of zeros. It follows that

(4.30)  Re(B) = − Re(∑_{ρ} \frac{1}{ρ}).

Together with (4.27) this leads to

Re ∑_{ρ} \frac{1}{s − ρ} = Re \frac{Λ′(s)}{Λ(s)} = \frac{a}{s} + \frac{a}{s − 1} + \frac{1}{2} log(D_{E}^{m_{1}m_{2}}N(π_{1} × π_{2}))

+ \frac{L′(s, π_{1,∞} × π_{2,∞})}{L(s, π_{1,∞} × π_{2,∞})} + \frac{L′(s, π_{1,f} × π_{2,f})}{L(s, π_{1,f} × π_{2,f})}.

Let ϵ > 0, and set σ = 2 + ϵ. By Lemma 4.3, Lemma 4.4 and (4.29) it follows that there exists C > 0 such that

∑_{ρ} \frac{σ − Re(ρ)}{(σ − Re(ρ))^{2} + (Im(ρ) − T)^{2}} ≤ \frac{1}{2} log(D_{E}^{m_{1}m_{2}}N(π_{1} × π_{2}))
+ C(1 + log(|T | + c(π_{1} × π_{2}))
≤ C_{1} log(|T | + ν(π_{1} × π_{2}))

(4.31)

for all T ∈ ℝ and π_{i} ∈ Π_{cus}(GL_{m_{i}}(𝔸_{E})), i = 1, 2. Let T > 0. Then it follows from (4.31) that

N(T + 1; π_{1}, π_{2})−N(T; π_{1}, π_{2})

≤ 2(3 + ϵ) ∑_{ρ} \frac{σ − Re(ρ)}{(σ − Re(ρ))^{2} + (Im(ρ) − T)^{2}}

≤ C log(T + ν(π_{1} × π_{2}))

for all π_{i} ∈ Π_{cus}(GL_{m_{i}}(𝔸_{E})), i = 1, 2. This implies the proposition. □

5. Normalizing factors

In this section we consider the global normalizing factors of intertwining operators. Our main purpose is to estimate certain integrals involving the logarithmic derivatives of the normalizing factors. The behaviour of these integrals is crucial for the estimation of the spectral side. From now on we assume that the ground field is ℚ. Denote by 𝔸 the ring of adèles of ℚ.

Let M ∈ L. Then there exists a partition (n_{1}, …, n_{r}) of n such that

M = GL_{n_{1}} × ⋯ × GL_{n_{r}} .

Let Q, P ∈ P(M). Let v be a place of ℚ and let π_{v} ∈ Π(M(ℚ_{v})). Associated to P, Q and π_{v} is the local intertwining operator

J_{Q|P} (π_{v}, λ), λ ∈ 𝔞_{M,ℂ}^{∗},

between the induced representations I_{P} (π_{v,λ}) and I_{Q}(π_{v,λ}), which is defined by an integral over N_{Q}(ℚ_{v}) ∩ N\overline{P} (ℚ_{v}), and hence depends upon a choice of Haar measure on this group. By [A7] there exist meromorphic functions r_{Q|P} (π_{v}, λ), λ ∈ 𝔞_{M,ℂ}^{∗}, such that the normalized local intertwining operators

R_{Q|P} (π_{v}, λ) = r_{Q|P} (π_{v}, λ)^{−1}J_{Q|P} (π_{v}, λ)

satisfy the conditions of Theorem 2.1 of [A7]. There is a general construction of normalizing factors which works for any group [A7], [CLL]. For GL_{n}, however, the intertwining operators can be normalized by L-functions [A7, §4], [AC, p.87]. The normalizing factors are given as

(5.1)  r_{Q|P} (π_{v}, λ) = ∏ ; α∈∑_{P} ∩ ∑\overline{Q} r_{α}(π_{v}, λ(α̌)),

where r_{α}(π_{v}, s) is a meromorphic function of one complex variable and Σ_{P} (resp. Σ\overline{Q}) denotes the roots of (P, A_{M}) (resp. (\overline{Q}, A_{M})). Thus to define the normalizing factors, it is enough to define the functions r_{α}(π_{v}, s) for any root α of (G, A_{M}) and any π_{v} ∈ Π(M(ℚ_{v})). To this end note that π_{v} is equivalent to a representation π_{1,v} ⊗ ⋯ ⊗ π_{r,v} with π_{i,v} ∈ Π(GL_{n_{i}}(ℚ_{v})) and the root α corresponds to an ordered pair (i, j) of distinct integers between 1 and r. Fix a nontrivial additive character ψ_{v} of ℚ_{v}. Let L(s, π_{i,v} × π̃_{j,v}) and ϵ(s, π_{i,v} × π̃_{j,v}, ψ_{v}) be the Rankin-Selberg L-function and the ϵ-factor attached to (π_{i,v}, π̃_{j,v}) and ψ_{v}. Set

(5.2)  r_{α}(π_{v}, s) = \frac{L(s, π_{i,v} × π̃_{j,v})}{L(1 + s, π_{i,v} × π̃_{j,v})ϵ(s, π_{i,v} × π̃_{j,v}, ψ_{v})}.

It follows from Theorem 6.1 of [Sh1] that there are Haar measures on the group N_{Q}(ℚ_{v}) ∩ N\overline{P} (ℚ_{v}), depending on ψ_{v}, such that the normalizing factors (5.1) have all the right properties (see [A7, §4], [AC, p.87]). Now suppose that π ∈ Π_{dis}(M(𝔸)). Then the global normalizing factor r_{Q|P} (π, λ) is defined by the infinite product

r_{Q|P} (π, λ) = Π_{v}r_{Q|P} (π_{v}, λ),

which converges in a certain chamber. By (5.1) it follows that there exist meromorphic functions r_{α}(π, s) of one complex variable such that

(5.3)  r_{Q|P} (π, λ) = ∏ ; α∈∑_{P} ∩ ∑\overline{Q} r_{α}(π, λ(α̌)).

Let π = π_{1} ⊗ ⋯ ⊗ π_{r}. If α corresponds to (i, j) then by (5.2) we have

(5.4)  r_{α}(π, s) = \frac{L(s, π_{i} × π̃_{j})}{L(1 + s, π_{i} × π̃_{j})ϵ(s, π_{i} × π̃_{j})},

where L(s, π_{i} ×π̃_{j}) and ϵ(s, π_{i} ×π̃_{j}) are the global L-function and the ϵ-factor, respectively, considered in the previous section.

The main goal of this section is to study the multidimensional logarithmic derivatives of the normalizing factors that occur on the spectral side of the trace formula [A4]. By (5.3) this problem is reduced to the investigation of the logarithmic derivatives of the analytic functions r_{α}(π, s). Furthermore, by (5.4) each r_{α}(π, s) may be regarded as the normalizing factor attached to a standard maximal parabolic subgroup in GL_{m} with m ≤ n. So let m_{1}, m_{2} ∈ ℕ with m_{1} + m_{2} ≤ n. Given π_{i} ∈ Π_{dis}(GL_{m_{i}}(𝔸)), i = 1, 2, set

(5.5)  r(π_{1} ⊗ π_{2}, s) = \frac{L(s, π_{1} × π̃_{2})}{L(1 + s, π_{1} × π̃_{2})ϵ(s, π_{1} × π̃_{2})}.

We shall now study the logarithmic derivatives of these functions. For this purpose we need some preparation. Suppose that π_{i}, i = 1, 2, is given in the form (4.20) and assume that k_{1} ≤ k_{2}. Set k = k_{1} + k_{2} − 2. For j = 0, …, k let the integers a_{j} be defined by

(5.6)  a_{i} = \{ \begin{matrix}i + 1 : & i ≤ k_{1} − 1; \\ k_{1} : & k_{1} − 1 ≤ i ≤ k_{2} − 1; \\ k − i + 1 : & i ≥ k_{2} − 1. \end{matrix}

Note that a_{i} = a_{k−i}, i = 0, ..., k. It follows from (4.20) that

(5.7)  L(s, π_{1} × π̃_{2}) = ∏_{i=0}^{k} L(s + k/2 − i, δ_{1} × δ̃_{2})^{a_{i}}.

Define a polynomial of one variable x by

p(x) = ∏_{i=0}^{k} ((x + k/2 − i)(1 − x − k/2 + i))^{a_{i}} .

Then p(x) has real coeefficients and satisfies p(x) = p(1 − x). Let a be the order of the pole of L(s, δ_{1} × δ̃_{2}) at s = 1. Note that a ≤ 1. Set

(5.8)  Λ(s) = p(s)^{a}N(π_{1} × π̃_{2})^{s/2}L(s, π_{1} × π̃_{2}).

Then Λ(s) satisfies the functional equation

(5.9)  Λ(s) = W (π_{1} × π̃_{2})N(π_{1} × π̃_{2})^{1/2}\overline{Λ(1 − \overline{s})}.

Furthermore Λ(s) is an entire function of order 1. Therefore it can be written as Weierstrass product of the form

Λ(s) = e^{A+Bs} ∏_{ρ}(1 − s/ρ)e^{s/ρ}

with A, B ∈ ℂ and ρ runs over the zeros of Λ(s). Taking the logarithmic derivative and applying the functional equation (5.9) to the right hand side, we get

(\frac{Λ(s)}{Λ(s + 1)})′ · \frac{Λ(s + 1)}{Λ(s)} = \frac{Λ′(s)}{Λ(s)} + \frac{\overline{Λ′(−\overline{s})}}{\overline{Λ(−\overline{s})}}

= 2 Re(B) + 2 Re ∑_{ρ} \frac{1}{ρ} + ∑_{ρ} \{\frac{1}{s − ρ} − \frac{1}{s + \overline{ρ}}\} .

By (4.30) it follows that the first two terms on the right hand side cancel and hence we get

(5.10)  (\frac{Λ(s)}{Λ(s + 1)})′ · \frac{Λ(s + 1)}{Λ(s)} = 2 ∑_{ρ} \frac{Re(ρ)}{(s − ρ)(s + \overline{ρ})}.

Therefore, combining (4.18), (5.5) and (5.8), we obtain

\begin{aligned}\frac{r′(π_{1} ⊗ π_{2}, s)}{r(π_{1} ⊗ π_{2}, s)} = & log N(π_{1} × π̃_{2}) \\ & + a \{\frac{p′(s + 1)}{p(s + 1)} − \frac{p′(s)}{p(s)}\} + 2 ∑_{ρ} \frac{Re(ρ)}{(s − ρ)(s + \overline{ρ})}. \end{aligned}

In particular, if s = iλ, λ ∈ ℝ, then it follows from the definition of p(s) that

\frac{r′(π_{1} ⊗ π_{2}, iλ)}{r(π_{1} ⊗ π_{2}, iλ)} = log N(π_{1} × π̃_{2})

+ 2a ∑_{i=0}^{k} \{\frac{a_{i}(k/2 − i + 1)}{λ^{2} + (k/2 − i + 1)^{2}} − \frac{a_{i}(k/2 − i − 1)}{λ^{2} + (k/2 − i − 1)^{2}} \}

+ 2 ∑_{ρ} \frac{Re(ρ)}{Re(ρ)^{2} + (Im(ρ) − λ)^{2}} .

Proposition 5.1. There exists C > 0 such that

∫_{−T}^{T} |\frac{r′(π_{1} ⊗ π_{2}, iλ)}{r(π_{1} ⊗ π_{2}, iλ)}| dλ ≤ CT log(T + ν(π_{1} × π̃_{2}))

for all T > 0 and π_{i} ∈ Π_{dis}(GL_{m_{i}}(𝔸)), i = 1, 2.

Proof. By the above formula it suffices to estimate the integral

∫_{−T}^{T} ∑_{ρ} \frac{| Re(ρ)|}{Re(ρ)^{2} + (Im(ρ) − λ)^{2}} dλ.

We split the series as follows

∑_{ρ} = ∑_{| Im(ρ)|≤T +1} + ∑_{| Im(ρ)|>T +1} .

To estimate the integral of the first sum, observe that for all β ∈ ℝ^{+} and γ ∈ ℝ we have

∫_{−T}^{T} \frac{β}{β^{2} + (γ − λ)^{2}} dλ ≤ ∫_{−∞}^{∞} \frac{dλ}{1 + λ^{2}} = π.

Hence by Proposition 4.5 we get

∫_{−T}^{T} ∑_{| Im(ρ)|≤T +1} \frac{| Re(ρ)|}{Re(ρ)^{2} + (Im(ρ) − λ)^{2}} dλ ≤ πN(T + 1, π_{1}, π̃_{2})
≤ CT log(T + ν(π_{1} × π̃_{2})).

It remains to consider the integral of the second sum. Observe that by (5.7) the zeros ρ of Λ(s) satisfiy | Re(ρ)| ≤ k/2 + 1. Set

σ = k + 3, C = 2(k + 2)^{2}.

Then the following inequality holds for all λ ∈ ℝ with |λ| ≤ T, all β ∈ ℝ^{×} with |β| ≤ k/2+1 and all γ ∈ ℝ with |γ| > T + 1:

\frac{|β|}{β^{2} + (γ − λ)^{2}} ≤ C \{\frac{σ − β}{(σ − β)^{2} + (γ − T)^{2}} + \frac{σ − β}{(σ − β)^{2} + (γ + T)^{2}} \} .

Thus we get

∑_{| Im(ρ)|>T +1} \frac{| Re(ρ)|}{Re(ρ)^{2} + (Im(ρ) − λ)^{2}}

≤ C\{∑_{ρ} \frac{σ − Re(ρ)}{(σ − Re(ρ))^{2} + (Im(ρ) − T)^{2}}

+ ∑_{ρ} \frac{σ − Re(ρ)}{(σ − Re(ρ))^{2} + (Im(ρ) + T)^{2}} \}.

Combining (5.7) and (4.31), it follows that for σ = k + 3 there exists C_{1} > 0 such that

∑_{ρ} \frac{σ − Re(ρ)}{(σ − Re(ρ))^{2} + (Im(ρ) − T)^{2}} ≤ C_{1} log(|T | + ν(π_{1} × π̃_{2}))

for all T ∈ ℝ and π_{i} ∈ Π_{dis}(GL_{m_{i}}(𝔸)), i = 1, 2. Combining these observation we get

∫_{−T}^{T} ∑_{| Im(ρ)|>T +1} \frac{| Re(ρ)|}{Re(ρ)^{2} + (Im(ρ) − λ)^{2}} ≤ CT log(T + ν(π_{1} × π̃_{2})).

This completes the proof of the proposition. □

The next proposition will be important for the determination of the asymptotic behaviour of the spectral side.

Proposition 5.2. There exists C > 0 such that

\begin{matrix}∫_{−∞}^{∞} |r′(π_{1} ⊗ π_{2}, iλ)r(π_{1} ⊗ π_{2}, & iλ)^{−1}|e^{−tλ^{2}} dλ \\ ≤ C log(1 + ν(π_{1} × π̃_{2}))\frac{1 + | log t|}{\sqrt{t}} \end{matrix}

for all 0 < t ≤ 1 and π_{i} ∈ Π_{dis}(GL_{m_{i}}(𝔸)), i = 1, 2.

Proof. By Proposition 5.1 it follows that we have

∫_{0}^{λ} |r′(π_{1} ⊗ π_{2}, iu)r(π_{1} ⊗ π_{2}, iu)^{−1}|du ≤ Cλ^{2}

as |λ| → ∞. Hence, using integration by parts, it follows that the integral on the left hand side of the claimed inequality equals

2t ∫_{−∞}^{∞} ∫_{0}^{λ} |r′(π_{1} ⊗ π_{2}, iu)r(π_{1} ⊗ π_{2}, iu)^{−1}|du λe^{−tλ^{2}} dλ.

Applying Proposition 5.1 we get

∫_{−∞}^{∞} |r′(π_{1} ⊗ π_{2}, iλ)r(π_{1} ⊗ π_{2}, iλ)^{−1}|e^{−tλ^{2}} dλ

≤ Ct ∫_{−∞}^{∞} log (|λ| + ν(π_{1} × π̃_{2})) λ^{2}e^{−tλ^{2}}dλ

≤ C_{1} log(1 + ν(π_{1} × π̃_{2}))\frac{1 + | log t|}{\sqrt{t}}

for all 0 < t ≤ 1 and π_{i} ∈ Π_{dis}(GL_{m_{i}}(𝔸)), i = 1, 2. □

Let M ∈ L and let Q, P ∈ P(M). Our next goal is to estimate the corresponding integrals involving the generalized logarithmic derivatives of the global normalizing factors r_{Q|P} (π, λ). For this purpose we will use the notion of a (G, M) family introduced by Arthur in Section 6 of [A5]. For the convenience of the reader we recall the definition of a (G, M) family and explain some of its properties.

For each P ∈ P(M), let c_{P} (λ) be a smooth function on i𝔞_{M}^{∗}. Then the set

{c_{P} (λ) | P ∈ P(M)}

is called a (G, M) family if the following holds: Let P, P′ ∈ P(M) be adjacent parabolic groups and suppose that λ belongs to the hyperplane spanned by the common wall of the chambers of P and P′. Then

c_{P} (λ) = c_{P′}(λ).

Let

(5.11)  θ_{P} (λ) = vol (𝔞_{P}^{G}/ℤ(Δ_{P}^{∨}))^{−1} ∏_{α∈Δ_{P}} λ(α^{∨}), λ ∈ i𝔞_{P}^{∗} ,

where ℤ(Δ_{P}^{∨}) is the lattice in 𝔞_{P}^{G} generated by the co-roots

{α^{∨} | α ∈ Δ_{P} }.

Let {c_{P} (λ)} be a (G, M) family. Then by Lemma 6.2 of [A5], the function

(5.12)  c_{M}(λ) = ∑_{P ∈P(M)} c_{P} (λ)θ_{P} (λ)^{−1}

extends to a smooth function on i𝔞_{M}^{∗}. The value of c_{M}(λ) at λ = 0 is of particular importance in connection with the spectral side of the trace formula. It can be computed as follows. Let p = dim(A_{M}/A_{G}). Set λ = tΛ, t ∈ ℝ, Λ ∈ 𝔞_{M}^{∗}, and let t tend to 0. Then

(5.13)  c_{M}(0) = \frac{1}{p!} ∑_{P ∈P(M)} (lim_{t→0} (\frac{d}{dt})^{p} c_{P} (tΛ)) θ_{P} (Λ)^{−1}

[A5, (6.5)]. This expression is of course independent of Λ.

For any (G, M) family {c_{P} (λ) | P ∈ P(M)} and any L ∈ L(M) there is associated a natural (G, L) family which is defined as follows. Let Q ∈ P(L) and suppose that P ⊂ Q. The function

λ ∈ i𝔞_{L}^{∗} ↦ c_{P} (λ)

depends only on Q. We will denote it by c_{Q}(λ). Then

{c_{Q}(λ) | Q ∈ P(L)}

is a (G, L) family. We write

c_{L}(λ) = ∑_{Q∈P(L)} c_{Q}(λ)θ_{Q}(λ)^{−1}

for the corresponding function (5.12).

Let Q ∈ P(L) be fixed. If R ∈ P^{L}(M), then Q(R) is the unique group in P(M) such that Q(R) ⊂ Q and Q(R) ∩ L = R. Let c_{R}^{Q} be the function on i𝔞_{M}^{∗} which is defined by

c_{R}^{Q}(λ) = c_{Q(R)}(λ).

Then {c_{R}^{Q}(λ) | R ∈ P^{L}(M)} is an (L, M) family. Let c_{M}^{Q}(λ) be the function (5.12) associated to this (L, M) family.

We consider now special (G, M) families defined by the global normalizing factors. Fix P ∈ P(M), π ∈ Π_{dis}(M(𝔸)) and λ ∈ i𝔞_{M}^{∗}. Define

(5.14)  ν_{Q}(P, π, λ, Λ) := r_{Q|P} (π, λ)^{−1}r_{Q|P} (π, λ + Λ), Q ∈ P(M).

This set of functions is a (G, M) family [A4, p.1317]. It is of a special form. By (5.3) we have

ν_{Q}(P, π, λ, Λ) = ∏_{α∈Σ_{Q}∩Σ\overline{P}} r_{α}(π, λ(α^{∨}))^{−1}r_{α}(π, λ(α^{∨}) + Λ(α^{∨})).

Suppose that L ∈ L(M), L_{1} ∈ L(L) and S ∈ P(L_{1}). Let

{ν_{Q_{1}}^{S}(P, π, λ, Λ) | Q_{1} ∈ P^{L_{1}}(L)}

be the associated (L_{1}, L) family and let ν_{L}^{S}(P, π, λ, Λ) be the function (5.12) defined by this family. Set

ν_{L}^{S}(P, π, λ) := ν_{L}^{S}(P, π, λ, 0).

If α is any root in Σ(G, A_{M}), let α_{L}^{∨} denote the projection of α^{∨} onto 𝔞_{L}. If F is a subset of Σ(G, A_{M}), let F_{L}^{∨} be the disjoint union of all the vectors α_{L}^{∨}, α ∈ F. Then by Proposition 7.5 of [A4] we have

\begin{aligned}ν_{L}^{S}(P, π, λ) = & ∑_{F} vol(𝔞_{L}^{L_{1}}/ℤ(F_{L}^{∨})) \\ & · (∏_{α∈F} r_{α}(π, λ(α^{∨}))^{−1}r′_{α}(π, λ(α^{∨}))) , \end{aligned} ; (5.15)

where F runs over all subsets of Σ(L_{1}, A_{M}) such that F_{L}^{∨} is a basis of 𝔞_{L}^{L_{1}}. Let t > 0. Then by (5.15) we get

\begin{matrix}∫_{i𝔞_{L}^{∗}/𝔞_{G}^{∗}} | & ν_{L}^{S}(P, π, λ)|e^{−t∥λ∥^{2}} dλ ≤ ∑_{F} vol(𝔞_{L}^{L_{1}}/ℤ(F_{L}^{∨})) \\ · ∫_{i𝔞_{L}^{∗}/𝔞_{G}^{∗}} ∏_{α∈F} |r_{α}π, λ(α∨))^{−1}r_{α}′(π, λ(α∨))|e^{−t∥λ∥^{2}} dλ. \end{matrix}

Fix any subset F of Σ(L_{1}, A_{M}) such that F_{L}^{∨} is a basis of 𝔞_{L}^{L_{1}}. Let

{ω̃_{α} | α ∈ F }

be the basis of (𝔞_{L}^{L_{1}})^{∗} which is dual to F_{L}^{∨}. We can write λ ∈ i𝔞_{L}^{∗}/i𝔞_{G}^{∗} as

λ = ∑_{α∈F} z_{α}ω̃_{α} + λ_{1}, z_{α} ∈ iℝ, λ_{1} ∈ i𝔞_{L_{1}}^{∗}/i𝔞_{G}^{∗}.

Observe that λ(α^{∨}) = z_{α}. Let l_{1} = dim(A_{L_{1}}/A_{G}). Then there exists C > 0, independent of π, such that for all t > 0 we have

\begin{aligned}∫i𝔞_{L}^{∗}/𝔞_{G}^{∗} ∏_{α∈F} & |r_{α}(π, λ(α^{∨}))^{−1}r′_{α}(π, λ(α^{∨}))|e^{−t∥λ∥^{2}} dλ \\ & ≤ Ct^{−l_{1}/2} ∏_{α∈F} ∫_{iℝ} |r_{α}(π, z_{α})^{−1}r′_{α}(π, z_{α})|e^{−tz_{α}^{2}} dz_{α}. \end{aligned} ; (5.16)

Suppose that M = GL_{n_{1}} × ⋯ × GL_{n_{r}}. Then π = π_{1} ⊗ ⋯ ⊗ π_{r} with π_{i} ∈ Π_{dis}(GL_{n_{i}}(𝔸)). Now recall that a given root α ∈ Σ(G, A_{M}) corresponds to an ordered pair (i, j) of distinct integers i and j between 1 and r. Then it folows from (5.4) and (5.5) that r_{α}(π, s) = r(π_{i} ⊗ π_{j}, s). Let l = dim(A_{L}/A_{G}) and k = dim(A_{L}/A_{L_{1}}). Then by Proposition 5.2 and (5.16) it follows that there exists C > 0

\begin{matrix}∫_{i𝔞_{L}^{∗}/𝔞_{G}^{∗}} |ν_{L}^{S}(P, π, & λ)|e^{−t∥λ∥^{2}} dλ \\ ≤ C ∏_{i,j} log(1 + ν(π_{i} × π̃_{j}))\frac{(1 + | log t|)^{k}}{t^{l/2}} \end{matrix} ; (5.17)

for all 0 < t ≤ 1 and all π ∈ Π_{dis}(M(𝔸)).

Next we shall estimate the numbers ν(π_{i} × π̃_{j}). For π_{∞} ∈ Π(GL_{m}(ℝ)), let the complex numbers μ_{j}(π_{∞}), j = 1, ..., m, be defined by (4.14) and set

c(π_{∞}) = (∑_{j=1}^{m} |μ_{j}(π_{∞})|^{2})^{1/2} .

Given an open compact subgroup K_{f} of GL_{m}(𝔸_{f}), set

Π(GL_{m}(𝔸))_{K_{f}} := {π ∈ Π(GL_{m}(𝔸)) | H_{π_{f}}^{K_{f}} ≠ 0},

where π = π_{∞} ⊗ π_{f} and H_{π_{f}} denotes the Hilbert space of the representation π_{f}.

Lemma 5.3. Let K_{f,i} ⊂ GL_{m_{i}}(𝔸_{f}), i = 1, 2, be two open compact subgroups. There exists C > 0 such that

ν(π_{1} × π_{2}) ≤ C(1 + c(π_{1,∞}) + c(π_{2,∞}))

for all π_{i} ∈ Π(GL_{m_{i}}(𝔸))_{K_{f,i}}, i = 1, 2.

Proof. First consider c(π_{1} × π_{2}) which is defined by (4.21). It follows from Lemma 4.2 that there exists C > 0 such that

c(π_{1} × π_{2}) ≤ C(c(π_{1,∞}) + c(π_{2,∞}))

for all π_{i} ∈ Π(GL_{m_{i}}(𝔸)), i = 1, 2. It remains to estimate N(π_{1} × π_{2}). For this we first observe that, as the epsilon factor is a product of local epsilon factors, we can factor N(π_{1} × π_{2}) as

N(π_{1} × π_{2}) = ∏_{p} N(π_{1,p} × π_{2,p}),

where p runs over the finite places of ℚ. This is a finite product. In fact, if p is unramified for both π_{1} and π_{2}, then N(π_{1,p} × π_{2,p}) = 1. Moreover there is an integer f(π_{1,p} × π_{2,p}) such that

N(π_{1,p} × π_{2,p}) = p^{f(π_{1,p}×π_{2,p})}

(see e.g. [MS]). Since we fix the ramification, there is a finite set S of finite places of ℚ, such that

N(π_{1} × π_{2}) = ∏_{p∈S} p^{f(π_{1,p}×π_{2,p})}

for all π_{i} ∈ Π(GL_{m_{i}}(𝔸))_{K_{f,i}}, i = 1, 2. This reduces our problem to the estimation of f(π_{1,p} × π_{2,p}). Let f(π_{i,p}) be the conductor of π_{i,p}, i = 1, 2. Then f(π_{i,p}) ≥ 0 and by Theorem 1 of [BH] and Corollary (6.5) of [BHK] we have

(5.18)  0 ≤ f(π_{1,p} × π_{2,p}) ≤ m_{1}f(π_{1,p}) + m_{2}f(π_{2,p}).

Let m ∈ ℕ and let K_{p} be an open compact subgroup of GL_{m}(ℚ_{p}). By Lemma 2.2 of [MS] there exists C_{p} > 0 such that f(π_{p}) ≤ C_{p} for all π_{p} ∈ Π(GL_{m}(ℚ_{p})) with π_{p}^{K_{p}} ≠ 0. Together with (5.18) this implies that there exists C > 0 such that

N(π_{1} × π_{2}) ≤ C

for all π_{i} ∈ Π(GL_{m_{i}}(𝔸))_{K_{f,i}}, i = 1, 2. This completes the proof of the lemma. □

We continue with the estimation of c(π_{∞}). Given π_{∞} ∈ Π(GL_{m}(ℝ), ξ_{0}), let λ_{π_{∞}} be the Casimir eigenvalue of the restriciton of π_{∞} to GL_{m}(ℝ)^{1}. Furthermore for σ ∈ Π(O(m)) let λ_{σ} denote the Casimir eigenvalue of σ. We note that if [π_{∞}|_{O(m)} : σ] > 0, then −λ_{π_{∞}} + λ_{σ} ≥ 0 [DH, Lemma 2.6] .

Lemma 5.4. There exists C > 0 such that

c(π_{∞}) ≤ C(1 − λ_{π_{∞}} + λ_{σ})^{1/2}

for all π_{∞} ∈ Π(GL_{m}(ℝ), ξ_{0}) and σ ∈ Π(O(m)) with [π_{∞}|_{O(m)} : σ] > 0.

Proof. Write π_{∞} as Langlands quotient π_{∞} = J_{R}^{GL_{m}}(τ, s), where τ is a discrete series representation of M_{R}(ℝ) and the parameters s_{1}, ..., s_{r} ∈ ℂ satisfy Re(s_{1}) ≥ Re(s_{2}) ≥ ⋯ ≥ Re(s_{r}). We may assume that the central character of τ is trivial on A_{R}(ℝ)^{0} and hance, we can regard τ as a discrete series representation of M_{R}(ℝ)^{1}. Let 𝔪_{R}^{1} denote the Lie algebra of M_{R}(ℝ)^{1}. Note that 𝔪_{R}^{1} is the direct sum of a finite number of copies of 𝔰l(2, ℝ). Let 𝔱 ⊂ 𝔪_{R}^{1} be the standard compact Cartan subalgebra equipped with the canonical norm. Then 𝔥 = 𝔱 ⊕ 𝔞_{R} is a Cartan subalgebra of 𝔤l_{m}(ℝ). Let Λ_{τ} ∈ i𝔱^{∗} be the Harish-Cahndra parameter of τ. It follows from the definition of the parameters μ_{j}(π_{∞}) in terms of the Langlands parameters that there exists C > 0 such that

c(π_{∞})^{2} ≤ C(∥ Λ_{τ} ∥^{2} + ∥ s ∥^{2})

for all π_{∞} ∈ Π(GL_{m}(ℝ), ξ_{0}). Let γ : Z(𝔤l_{m}(ℂ)) → I(𝔥_{ℂ}) be the Harish-Chandra homomorphism. By Proposition 8.22 of [Kn] the infinitesimal character χ of the induced representation I_{R}^{GL_{m}}(τ, s) with respect to 𝔥 is given by χ(Z) = (Λ_{τ} + s)(γ(Z)), Z ∈ Z(𝔤l_{m}(ℂ)). Since π_{∞} is an irreducible quotient of I_{R}^{GL_{m}}(τ, s), it has the same infinitesimal character. Let H_{1}, ..., H_{r} be an orthonormal basis of 𝔞_{R} and H_{r+1}, ..., H_{m} an orthonormal basis of 𝔱. Then

γ(Ω) = ∑_{i=1}^{r} H_{i}^{2} − ∑_{j=r+1}^{m} H_{j}^{2}− ∥ ρ ∥^{2}

[Wa1, p.168]. Hence, the Casimir eigenvalue λ_{π} of π_{∞} is given by

λ_{π_{∞}} = (Λ_{τ} + s)(γ(Ω)) = ∑_{i=1}^{r} s_{i}^{2}+ ∥ Λ_{τ} ∥^{2} − ∥ ρ ∥^{2} .

Since π_{∞} is unitary, it follows from Theorem 3.3 of Chapter XI of [BW] that there exists C > 0, independent of π_{∞}, such that ∥ Re(s) ∥≤ C. Hence there exists C_{1} > 0 such that

∥ Λ_{τ} ∥^{2} + ∥ s ∥^{2}≤ C_{1} − λ_{π_{∞}}+ ∥ Λ_{τ} ∥^{2}

for all π_{∞} ∈ Π(GL_{m}(ℝ), ξ_{0}). Now let σ ∈ Π(O(m)) and suppose that π_{∞} ∈ Π(GL_{m}(ℝ), ξ_{0}) is such that [π_{∞}|_{O(m)} : σ] > 0. Since σ occurs in π_{∞}, it also occurs in I_{R}^{GL_{m}}(τ, s). Using Frobenius reciprocity as in [Kn, p.208], it follows that there exists ω ∈ Π(O(m) ∩ M_{R}(ℝ)) such that

[τ|_{O(m)∩M_{R}(ℝ)} : ω] > 0 and [σ|_{O(m)∩M_{R}(ℝ)} : ω] > 0.

Let λ_{σ} and λ_{ω} denote the Casimir eigenvalues of σ and ω, respectively. By [Mu2, (5.15)], the second inequality implies λ_{ω} ≤ λ_{σ}. On the other hand, by [Wa2, p.398], the first inequality implies

∥ Λ_{τ} ∥^{2}≤ λ_{ω}+ ∥ ρ_{R} ∥^{2} .

Combining our estimations the lemma follows. □

Now let K_{f} be an open compact subgroup of G(𝔸_{f}). Set

K_{M,f} = K_{f} ∩ M(𝔸_{f}).

Then K_{M,f} is an open compact subgroup of M(𝔸_{f}). There exist open compact subgroups K_{f,i} of GL_{n_{i}}(𝔸_{f}), i = 1, ..., r, such that K_{f,1} × ⋯ × K_{f,r} is a subgroup of finite index of K_{f}. Set

Π(M(𝔸), ξ_{0})_{K_{f}} = {π ∈ Π(M(𝔸), ξ_{0}) | H_{π_{f}}^{K_{M,f}} ≠ {0}},

where π = π_{∞} ⊗ π_{f}. Let π ∈ Π(M(𝔸), ξ_{0})_{K_{f}}. Then π = π_{1} ⊗ ⋯ ⊗ π_{r} and π_{i} belongs to Π(GL_{n_{i}}(𝔸), ξ_{0})_{K_{f,i}} and by Lemma 5.3 it follows that there exists C > 0 such that

(5.19)  ∏_{i,j} log(1 + ν(π_{i} × π̃_{j})) ≤ C ∏_{i,j} log(2 + c(π_{i,∞}) + c(π_{j,∞}))

for all π = π_{1} ⊗ ⋯ ⊗ π_{r} ∈ Π(M(𝔸), ξ_{0})_{K_{f}}.

Let K_{M,∞} = O(n_{1}) × ⋯ × O(n_{r}) be the standard maximal compact subgroup of M(ℝ). Let σ ∈ Π(O(n)). For π ∈ Π(M(𝔸), ξ_{0}) set

[π_{∞} : σ] = ∑_{τ∈Π(K_{M,∞})}[π_{∞}|_{K_{M,∞}} : τ][σ|_{K_{M,∞}} : τ].

Put

Π(M(𝔸), ξ_{0})_{K_{f} ,σ} = {π ∈ Π(M(𝔸), ξ_{0})_{K_{f}} | [π_{∞} : σ] > 0}

and

Π_{dis}(M(𝔸), ξ_{0})_{K_{f} ,σ} = Π_{dis}(M(𝔸), ξ_{0}) ∩ Π(M(𝔸), ξ_{0})_{K_{f} ,σ}.

Suppose that π ∈ Π(M(𝔸), ξ_{0})_{K_{f} ,σ}. Let τ ∈ Π(K_{M,∞}) be such that [σ|_{K_{M,∞}} : τ] > 0 and [π_{∞}|_{K_{M,∞}} : τ] > 0.

Let λ_{π_{∞}} and λ_{τ} denote the Casimir eigenvalues of the restriction of π_{∞} to M(ℝ)^{1} and of τ, respectively. Note that λ_{π_{∞}} = ∑_{i} λ_{π_{i,∞}} and λ_{τ} = ∑_{i} λ_{τ_{i}}, where τ = ⊗_{i}τ_{i}. Then it follows from (5.19) and Lemma 5.4 that there exists C > 0 such that

∏_{i,j} log(1 + ν(π_{i} × π̃_{j})) ≤ C ∏_{i,j} log(2 − λ_{π_{i,∞}} + λ_{τ_{i}} − λ_{π_{j,∞}} + λ_{τ_{j}})
≤ C(log(2 − λ_{π_{∞}} + λ_{τ}))^{r^{2}}

for all π ∈ Π(M(𝔸), ξ_{0})_{K_{f} ,σ}. Since there are only finitely many τ that occur in σ|_{K_{M,∞}}, we get

(5.20)  ∏_{i,j} log(1 + ν(π_{i} × π̃_{j})) ≤ C_{1}(log(2 + |λ_{π_{∞}}|))^{r^{2}}

for all π ∈ Π(M(𝔸), ξ_{0})_{K_{f} ,σ}. Combining (5.17)–(5.20) we obtain

Proposition 5.5. Let M ∈ L, L ∈ L(M) and P ∈ P(M). Let l = dim(A_{L}/A_{G}). Let K_{f} be an open compact subgroup of GL_{n}(𝔸_{f}) and let σ ∈ Π(O(n)). There exists C > 0 such that

∫_{i𝔞_{L}^{∗}/𝔞_{G}^{∗}} |ν_{L}^{S}(P, π, λ)|e^{−t∥λ∥^{2}} dλ ≤ C (log(2 + |λ_{π_{∞}}|))^{n^{2}} \frac{(1 + | log t|)^{l}}{t^{l/2}}

for all 0 < t ≤ 1 and π ∈ Π_{dis}(M(𝔸), ξ_{0})_{K_{f} ,σ}.

6. The spectral side

We shall use the noninvariant trace formula of Arthur [A1], [A2], applied to the heat kernel, to determine the growth of the discrete spectrum. To begin with, we explain the general structure of the spectral side of the Arthur trace formula. The spectral side is a sum of distributions

∑_{χ∈𝔛}J_{χ}(f), χ ∈ C_{0}^{∞}(G(𝔸)^{1}).

Here 𝔛 is the set of cuspidal datas which consists of Weyl group orbits of pairs (M_{B}, ρ_{B}), where M_{B} is the Levi component of a parabolic subgroup and ρ_{B} is a cuspidal automorphic representation of M_{B}(𝔸). The distributions J_{χ} are described by Theorem 8.2 of [A4]. Let C^{1}(G(𝔸)^{1}) be the space of integrable rapidly decreasing functions on G(𝔸)^{1} [MS, §1.3]. In [MS, Theorem 0.1] it has been proved that the spectral side of the trace formula for GL_{n} is absolutely convergent for all f ∈ C^{1}(G(𝔸)^{1}). In this case the expression of the spectral side simplifies.

To describe this in more detail, we need to introduce some notation. Let M ∈ L and P, Q ∈ P(M). Let A^{2}(P) and A^{2}(Q) be the corresponding spaces of automorphic functions (see §1.5). Let W (𝔞_{P} , 𝔞_{Q}) be the set of all linear isomorphisms from 𝔞_{P} to 𝔞_{Q} which are restrictions of elements of the Weyl group W (A_{0}). The theory of Eisenstein series associates to each s ∈ W (𝔞_{P} , 𝔞_{Q}) an intertwining operator

M_{Q|P} (s, λ) : A^{2}(P) → A^{2}(Q), λ ∈ 𝔞_{P,ℂ}^{∗},

which, for Re(λ) in a certain chamber, can be defined by an absolutely convergent integral and admits an analytic continuation to a meromorphic function of λ ∈ 𝔞_{P,ℂ}^{∗} [La]. Set

M_{Q|P} (λ) := M_{Q|P} (1, λ).

Fix P ∈ P(M) and λ ∈ i𝔞_{M}^{∗}. For Q ∈ P(M) and Λ ∈ i𝔞_{M}^{∗} define

𝔐_{Q}(P, λ, Λ) = M_{Q|P} (λ)^{−1}M_{Q|P} (λ + Λ).

Then

(6.1)  {𝔐_{Q}(P, λ, Λ) | Λ ∈ i𝔞_{M}^{∗}, Q ∈ P(M)}

is a (G, M) family with values in the space of operators on A^{2}(P) [A4, p.1310]. Let L ∈ L(M). Then, as explained in the previous section, the (G, M) family (6.1) has an associated (G, L) family

{𝔐_{Q_{1}}(P, λ, Λ) | Λ ∈ i𝔞_{L}^{∗}, Q_{1} ∈ P(L)}

and

𝔐_{L}(P, λ, Λ) = ∑_{Q_{1}∈P(L)} 𝔐_{Q_{1}}(P, λ, Λ)θ_{Q_{1}}(Λ)^{−1}

extends to a smooth function on i𝔞_{L}^{∗}. Put

𝔐_{L}(P, λ) = 𝔐_{L}(P, λ, 0).

This operator depends only on the intertwining operators. It equals

𝔐_{L}(P, λ) =

lim_{Λ→0} ( ∑_{Q_{1}∈P(L)} vol(𝔞_{Q_{1}}^{G}/ℤ(Δ_{Q_{1}}^{∨}))M_{Q|P} (λ)^{−1} \frac{M_{Q|P} (λ + Λ)}{∏_{α∈Δ_{Q_{1}}} Λ(α^{∨})}) ,

where λ and Λ are constrained to lie in i𝔞_{L}^{∗}, and for each Q_{1} ∈ P(L), Q is a group in P(M_{P}) which is contained in Q_{1}. Then 𝔐_{L}(P, λ) is an unbounded operator which acts on the Hilbert space \overline{A}^{2}(P). For π ∈ Π(M(𝔸)^{1}) let A_{π}^{2}(P) be the subspace of A^{2}(P) determined by π (see §1.5). Let ρ_{π}(P, λ) be the induced representation of G(𝔸) in \overline{A}_{π}^{2}(P).

Let W^{L}(𝔞_{M})_{reg} be the set of elements s ∈ W (𝔞_{M}) such that {H ∈ 𝔞_{M} | sH = H} = 𝔞_{L}. For any function f ∈ C^{1}(G(𝔸)^{1}) and s ∈ W^{L}(𝔞_{M})_{reg} set

\begin{aligned}J_{M,P}^{L} & (f, s) \\ & = ∑_{π∈Π_{dis}(M(𝔸)^{1})}∫_{i𝔞_{L}^{∗}/i𝔞_{G}^{∗}} tr(𝔐_{L}(P, λ)M_{P |P} (s, 0)ρ_{π}(P, λ, f)) dλ. \end{aligned} ; (6.2)

By Theorem 0.1 of [MS] this integral-series is absolutely convergent with respect to the trace norm. Furthermore for M ∈ L and s ∈ W^{L}(𝔞_{M})_{reg} set

a_{M,s} = |P(M)|^{−1}|W_{0}^{M}||W_{0}|^{−1}| det(s − 1)_{𝔞_{M}^{L}}|^{−1}.

Then for any f in C^{1}(G(𝔸)^{1}), the spectral side J_{spec}(f) of the Arthur trace formula is given by

(6.3)  J_{spec}(f) = ∑_{M∈L} ∑_{L∈L(M)} ∑_{P ∈P(M)} ∑_{s∈W^{L}(𝔞_{M})_{reg}} a_{M,s}J_{M,P}^{L} (f, s).

Note that all sums in this expression are finite.

We shall now evaluate the spectral side at a function ϕ_{t}, t > 0, which is given in terms of the heat kernel of a Bochner-Laplace operator. Then our main purpose is to determine the behaviour of J_{spec}(ϕ_{t}) as t → 0.

Let G(ℝ)^{1} = G(𝔸)^{1}∩G(ℝ). By definition G(ℝ)^{1} consisits of all g ∈ G(ℝ) with | det(g)| = 1. Hence G(ℝ)^{1} is semisimple and

G(ℝ) = G(ℝ)^{1} · A_{G}(ℝ)^{0}.

Let

X = G(ℝ)^{1}/K_{∞}

be the associated Riemannian symmetric space. Given σ ∈ Π(K_{∞}), let Ẽ_{σ} → X be the associated homogeneous vector bundle. Let Ω_{G(ℝ)^{1}} be the Casimir element of G(ℝ)^{1} and let Δ̃_{σ} be the operator in L^{2}(Ẽ_{σ}) which is induced by −R(Ω_{G(ℝ)^{1}}) ⊗ Id. Let

(6.4)  H_{t}^{σ} ∈ (C^{1}(G(ℝ)^{1}) ⊗ End(V_{σ}))^{K_{∞}×K_{∞}}

be the kernel of the heat operator e^{−tΔ̃_{σ}} where C^{1}(G(ℝ)) is Harish-Chandra’s space of integrable rapidly decreasing functions. Set

h_{t}^{σ} = tr H_{t}^{σ}.

We extend h_{t}^{σ} to a function on G(ℝ) by

h_{t}^{σ}(g · z) = h_{t}^{σ}(g), g ∈ G(ℝ)^{1}, z ∈ A_{G}(ℝ)^{0}.

Then h_{t}^{σ} satisfies

h_{t}^{σ}(gz) = h_{t}^{σ}(g), g ∈ G(ℝ), z ∈ A_{G}(ℝ)^{0}.

Let χ_{σ} be the character of σ. Then h_{t}^{σ} also satisfies

h_{t}^{σ} = χ_{σ} ∗ h_{t}^{σ} ∗ \overline{χ_{σ}}.

Let K_{f} be an open compact subgroup of G(𝔸_{f}) and let 1_{K_{f}} be the charcteristic function of K_{f} in G(𝔸_{f}). Set

χ_{K_{f}} = vol(K_{f})^{−1}1_{K_{f}} .

Define the function ϕ_{t} on G(𝔸) by

(6.5)  ϕ_{t}(g) = h_{t}^{σ}(g_{∞})χ_{K_{f}} (g_{f})

for any point

g = g_{∞}g_{f}, g_{∞} ∈ G(ℝ), g_{f} ∈ G(𝔸_{f}),

in G(𝔸). Then ϕ_{t} satisfies ϕ_{t}(gz) = ϕ_{t}(g) for z ∈ A_{G}(ℝ)^{0}, g ∈ G(𝔸). It follows from (6.4) and the definition of C^{1}(G(𝔸)^{1}) that the restriction ϕ_{t}^{1} of ϕ_{t} to G(𝔸)^{1} belongs to C^{1}(G(𝔸)^{1}).

Let π be any unitary representation of G(𝔸) which is trivial on A_{G}(ℝ)^{0}. Then we can define

π(ϕ_{t}) = ∫_{G(𝔸)/A_{G}(ℝ)^{0}} ϕ_{t}(g)π(g) dg.

Suppose that π = π_{∞} ⊗ π_{f}, where π_{∞} and π_{f} are unitary representations of G(ℝ) and G(𝔸_{f}), respectively. Then π_{∞} is trivial on A_{G}(ℝ)^{0}. So we can set

π_{∞}(ϕ_{t}) = ∫_{G(ℝ)/A_{G}(ℝ)^{0}} π_{∞}(g_{∞})h_{t}^{σ}(g_{∞}) dg_{∞}.

Let Π_{K_{f}} denote the orthogonal projection of the Hilbert space H_{π_{f}} of π_{f} onto the subspace H_{π_{f}}^{K_{f}} of K_{f}-invariant vectors. Then

π(ϕ_{t}) = π_{∞}(h_{t}^{σ}) ⊗ Π_{K_{f}} .

Now let π ∈ Π(M(𝔸)^{1}). We identify π with a representation of M(𝔸) which is trivial on A_{M}(ℝ)^{0}. Let I_{P}^{G}(π_{λ}), λ ∈ 𝔞_{M,ℂ}^{∗}, be the induced representation of G(𝔸). Let π = π_{∞} ⊗ π_{f}. Then

I_{P}^{G}(π_{λ}) = I_{P}^{G}(π_{∞,λ}) ⊗ I_{P}^{G}(π_{f,λ}).

Let H_{P} (π_{∞})_{σ} denote the σ-isotypical subspace of the Hilbert space H_{P} (π_{∞}) of the induced representation. Then H_{P} (π_{∞})_{σ} is an invariant subspace of I_{P}^{G}(π_{∞,λ}, h_{t}^{σ}). Let λ_{π} be the Casimir eigenvalue of the restriction of π_{∞} to M(ℝ)^{1}. By Proposition 8.22 of [Kn] it follows that

I_{P}^{G}(π_{∞,λ}, h_{t}) ↾ H_{P} (π_{∞})_{σ} = e^{−t(−λ_{π}+∥λ∥^{2})} Id .

Now observe that there is a canonical isomorphism

j_{P} : H_{P} (π) ⊗ Hom_{M(𝔸)} (π, I_{M(ℚ)A_{M} (ℝ)^{0}}^{M(𝔸)}(ξ_{0})) → \overline{A}_{π}^{2}(P),

which intertwines the induced representations. Let Π_{K_{f} ,σ} denote the orthogonal projection of \overline{A}_{π}^{2}(P) onto A_{π}^{2}(P)_{K_{f} ,σ}. Then it follows that

(6.6)  ρ_{π}(P, λ, ϕ_{t}) = e^{−(−λ_{π}+∥λ∥^{2})}Π_{K_{f} ,σ}.

Suppose that λ ∈ (𝔞_{P}^{G})_{ℂ}^{∗}. Then ρ_{π}(P, λ, g) is trivial on A_{G}(ℝ)^{0}. This implies ρ_{π}(P, λ, ϕ_{t}) = ρ_{π}(P, λ, ϕ_{t}^{1}), where ϕ_{t}^{1} is the restriciton of ϕ_{t} to G(𝔸)^{1}. Together with (6.6) we get

\begin{aligned}J_{M,P}^{L} (ϕ_{t}^{1}, s) = & ∑_{π∈Π_{dis}(M(𝔸)^{1})}e^{tλ_{π}} \\ & · ∫_{i𝔞_{L}^{∗}/𝔞_{G}^{∗}} e^{−t∥λ∥^{2}} tr(𝔐_{L}(P, λ)M_{P |P} (s, 0)Π_{K_{f} ,σ}) dλ. \end{aligned} ; (6.7)

To study this integral-series, we introduce the normalized intertwining operators

(6.8)  N_{Q|P} (π, λ) := r_{Q|P} (π, λ)^{−1}M_{Q|P} (π, λ), λ ∈ 𝔞_{M,ℂ}^{∗},

where r_{Q|P} (π, λ) are the global normalizing factors considered in the previous section. Let P ∈ P(M) and λ ∈ i𝔞_{M}^{∗} be fixed. For Q ∈ P(M) and Λ ∈ i𝔞_{M}^{∗} define

(6.9)  𝔑_{Q}(P, π, λ, Λ) = N_{Q|P} (π, λ)^{−1}N_{Q|P} (π, λ + Λ),

Then as functions of Λ ∈ i𝔞_{M}^{∗},

{𝔑_{Q}(P, π, λ, Λ) | Q ∈ P(M)}

is a (G, M) family. The verification is the same as in the case of the unnormalized intertwining operators [A4, p.1310]. For L ∈ L(M), let

{𝔑_{Q_{1}}(P, π, λ, Λ) | Λ ∈ i𝔞_{L}^{∗}, Q_{1} ∈ P(L)}

be the associated (G, L) family.

Let 𝔐_{Q_{1}}(P, π, λ, Λ) be the restriction of 𝔐_{Q_{1}}(P, λ, Λ) to \overline{A}_{π}^{2}(P). Then by (6.8) and (5.14) it follows that

(6.10)  𝔐_{Q_{1}}(P, π, λ, Λ) = 𝔑_{Q_{1}}(P, π, λ, Λ)ν_{Q_{1}}(P, π, λ, Λ)

for all Λ ∈ i𝔞_{L}^{∗} and all Q_{1} ∈ P(L).

For Q ⊃ P let L̂_{P}^{Q} ⊂ 𝔞_{P}^{Q} be the lattice generated by {ω̃^{∨} | ω̃ ∈ Δ̂_{P}^{Q}}. Define

θ̂_{P}^{Q}(λ) = vol(𝔞_{P}^{Q}/L̂_{P}^{Q})^{−1} ∏_{ω̃∈Δ̂_{P}^{Q}} λ(ω̃^{∨}).

For S ∈ F(L) put

𝔑′_{S}(P, π, λ)

= lim_{Λ→0} ∑_{{R|R⊃S}}(−1)^{dim(A_{S}/A_{R})}θ̂_{S}^{R}(Λ)^{−1}𝔑_{R}(P, π, λ, Λ)θ_{R}(Λ)^{−1}. (6.11)

Let 𝔐_{L}(P, π, λ) be the restriction of 𝔐_{L}(P, λ) to \overline{A}_{π}^{2}(P). Then by (6.10) and Lemma 6.3 of [A5] we get

(6.12)  𝔐_{L}(P, π, λ) = ∑_{S∈F(L)} 𝔑′_{S}(P, π, λ)ν_{L}^{S}(P, π, λ).

Let 𝔑′_{S}(P, π, λ)_{K_{f} ,σ} denote the restriction of 𝔑′_{S}(P, π, λ) to A_{π}^{2}(P)_{K_{f} ,σ}. Then by (6.7) we get

\begin{matrix}J_{M,P}^{L} & (ϕ_{t}^{1}, s) = ∑_{π∈Π_{dis}(M(𝔸)^{1})}e^{tλ_{π}} \\ · ∑_{S∈F(L)} ∫_{i𝔞_{L}^{∗}/𝔞_{G}^{∗}} e^{−t∥λ∥^{2}}ν_{L}^{S}(P, π, λ) tr(M_{P |P} (s, 0)𝔑′_{S}(P, π, λ)_{K_{f} ,σ}) dλ. \end{matrix} ; (6.13)

Next we shall estimate the norm of 𝔑′_{S}(P, π, λ)_{K_{f} ,σ}. For a given place v of ℚ let J_{Q|P} (π_{v}, λ) be the intertwining operator between the induced representations I_{P}^{G}(π_{v,λ}) and I_{Q}^{G}(π_{v,λ}). Let

R_{Q|P} (π_{v}, λ) = r_{Q|P} (π_{v}, λ)^{−1}J_{Q|P} (π_{v}, λ), λ ∈ 𝔞_{M,ℂ}^{∗},

be the normalized local intertwining operator. These operators satisfy the conditions (R_{1}) − (R_{8}) of Theorem 2.1 of [A7]. Assume that K_{f} = ∏_{p<∞} K_{p}. For any place v denote by H_{P} (π_{v}) the Hilbert space of the induced representation I_{P}^{G}(π_{v}). If p < ∞ let R_{Q|P} (π_{p}, λ)_{K_{p}} be the restriction of R_{Q|P} (π_{p}, λ) to the subspace of K_{p}-invariant vectors H_{P} (π_{p})^{K_{p}} in H_{P} (π_{p}). Let R_{Q|P} (π_{∞}, λ)_{σ} denote the restriction of R_{Q|P} (π_{∞}, λ) to the σisotypical subspace of I_{P}^{G}(π_{∞}) in H_{P} (π_{∞}). It was proved in [Mu2, (6.24)] that there exist a finite set of places S_{0}, including the Archimedean one, and constants C > 0 and q ∈ ℕ, such that

\begin{aligned}∥ 𝔑′_{S}(P, π, λ)_{K_{f} ,σ} ∥≤ C( ∑_{p∈S_{0}\\{∞\}} ∑_{k=1}^{q} & ∥ D_{λ}^{k}R_{Q|P} (π_{p}, λ)_{K_{p}} ∥ \\ & ∑_{k=1}^{q} ∥ D_{λ}^{k}R_{Q|P} (π_{∞}, λ)_{σ} ∥ ) \end{aligned}

for all λ ∈ i𝔞_{M}^{∗}, σ ∈ Π(K_{∞}) and π ∈ Π(M(𝔸)). By Proposition 0.2 of [MS] it follows that there exists C > 0 such that

(6.14)  ∥ 𝔑′_{S}(P, π, λ)_{K_{f} ,σ} ∥≤ C

for all λ ∈ i𝔞_{M}^{∗} and π ∈ Π_{dis}(M(𝔸)^{1}). Observe that M_{P |P} (s, 0) is unitary. Let l = dim(A_{L}/A_{G}). Using (6.13), (6.14) and Proposition 5.5 it follows that there exists C > 0 such that

|J_{M,P}^{L} (ϕ_{t}^{1}, s)| ≤ C \frac{(2 + | log t|)^{l}}{t^{l/2}}

· ∑_{π∈Π_{dis}(M(𝔸)^{1})}dim A_{π}^{2}(P)_{K_{f} ,σ}(log(1 + |λ_{π}|))^{n^{2}}e^{tλ_{π}}

(6.15)

for all 0 < t ≤ 1. The series can be estimated using Proposition 3.5. Let X_{M} = M(ℝ)/K′_{M,∞} and let m = dim X_{M}. It follows from Proposition 3.5 that for every ϵ > 0 there exists C > 0 such that the series is bounded by Ct^{−m/2−ϵ} for 0 < t ≤ 1. Together with (6.15) we obtain the following proposition.

Proposition 6.1. Let m = dim X_{M} and l = dim A_{L}/A_{G}. For every ϵ > 0 there exists C > 0 such that

|J_{M,P}^{L} (ϕ_{t}^{1}, s)| ≤ Ct^{−(m+l)/2−ϵ}

for all 0 < t ≤ 1.

Now we distinguish two cases. First assume that M = G. Then L = P = G and s = 1. Let R1_{dis} be the restriciton of the regular representation R^{1} of G(𝔸)^{1} in L^{2}(G(ℚ)\G(𝔸)^{1}) to the discrete subspace. Then J_{G,G}^{G}(ϕ_{t}^{1}, 1) = Tr R1_{dis}(ϕ_{t}^{1}). Let R_{dis} be the regular representation of G(𝔸) in L2_{dis}(A_{G}(ℝ)^{0}G(ℚ)\G(𝔸)). Then the operator R_{dis}(ϕ_{t}) is isomorphic to R1_{dis}(ϕ_{t}^{1}). Thus

J_{G,G}^{G}(ϕ_{t}^{1}, 1) = Tr R_{dis}(ϕ_{t}).

Given π ∈ Π_{dis}(G(𝔸), ξ_{0}), let m(π) denote the multiplicity with which π occurs in the regular representation of G(𝔸) in L^{2}(𝔸_{G}(ℝ)^{0}G(ℚ)\G(𝔸)). Then using Corollary 2.2 in [BM] we get

J_{G,G}^{G}(ϕ_{t}^{1}, 1)

= ∑_{π∈Π_{dis}(G(𝔸),ξ_{0})}m(π) dim(H_{π_{f}}^{K_{f}} ) dim(H_{π_{∞}} ⊗ V_{σ})^{O(n)}e^{tλ_{π}}. (6.16)

Now assume that M ≠ G is a proper Levi subgroup. Let P = MN. Let X = G(ℝ)^{1}/K_{∞}. Then

X ≅ X_{M} × A_{M}(ℝ)^{0}/A_{G}(ℝ)^{0} × N(ℝ).

Since l = dim A_{L}/A_{G} ≤ dim A_{M}/A_{G}, it follows that m + l ≤ dim X − 1. Thus together with Proposition 6.1 we get

Theorem 6.2. Let d = dim X. For every open compact subgroup K_{f} of G(𝔸_{f}) and every σ ∈ Π(O(n)) the spectral side of the trace formula, evaluated at ϕ_{t}^{1}, satisfies

J_{spec}(ϕ_{t}^{1})

= ∑_{π∈Π_{dis}(G(𝔸),ξ_{0})}m(π) dim(H_{π_{f}}^{K_{f}} ) dim(H_{π_{∞}} ⊗ V_{σ})^{O(n)}e^{tλ_{π}}

+ O(t^{−(d−1)/2})

(6.17)

as t → 0^{+}.

This theorem can be restated in a slightly different way as follows. There exist arithmetic subgroups Γ_{i} ⊂ G(ℝ), i = 1, ..., m, such that

A_{G}(ℝ)^{0}G(ℚ)\G(𝔸)/K_{f} ≅ ⨆_{i=1}^{m}(Γ_{i}\G(ℝ)^{1})

(cf. [Mu1, section 9]). Let Δ_{σ,i} be the operator induced by the negative of the Casimir operator in C^{∞}(Γ_{i}\G(ℝ)^{1}, σ), i = 1, ..., m. Let

λ_{0} ≤ λ_{1} ≤ λ_{2} ≤ ⋯

be the L^{2}-eigenvalues of Δ_{σ} = ⊕_{i=1}^{m}Δ_{σ,i}, where each eigenvalue is counted with its multiplicity. Let d = dim X. If we proceed in the same way as in the proof of Lemma 3.2, then it follows that (6.17) is equivalent to

(6.18)  J_{spec}(ϕ_{t}^{1}) = ∑_{i} e^{−tλ_{i}} + O(t^{−(d−1)/2})

as t → 0^{+}.

Let Γ(N) ⊂ SL_{n}(ℤ) be the principal congruence subgroup of level N. Let μ_{0} ≤ μ_{1} ≤ ⋯ be the eigenvalues, counted with multiplicity, of Δ_{σ} acting in L^{2}(Γ(N)\ SL_{n}(ℝ), σ). Then it follows from (6.18) and (3.10) that

(6.19)  J_{spec}(ϕ_{t}^{1}) = φ(N) ∑_{i} e^{−tμ_{i}} + O(t^{−(d−1)/2})

as t → 0^{+}.

Our next purpose is to study J_{spec} as a functional on the Schwartz space. Let K_{f} be an open compact subgroup of G(𝔸_{f}) and let σ ∈ Π(K_{∞}). Denote by C^{1}(G(𝔸)^{1}; K_{f}, σ) the set of all h ∈ C^{1}(G(𝔸)^{1}) which are bi-invariant under K_{f} and transform under K_{∞} according to σ. Let Δ_{G} be the Laplace operator of G(ℝ)^{1}. Then we have

Proposition 6.3. For every open compact subgroup K_{f} of G(𝔸_{f}) and every σ ∈ Π(K_{∞}) there exist C > 0 and k ∈ ℕ such that

|J_{spec}(f)| ≤ C ∥ (Id +Δ_{G})^{k}f ∥_{L^{1}(G(𝔸)^{1})}

for all f ∈ C^{1}(G(𝔸)^{1}; K_{f}, σ).

Proof. This follows essentially from the proof of Theorem 0.2 in [Mu2] combined with Proposition 0.2 of [MS]. We include some details. Let M ∈ L, L ∈ L(M) and P ∈ P(M). By (6.3) it suffices to estimate J_{M,P}^{L} (f, s). Since M_{P |P} (s, 0) is unitary, it follows from (6.2) that

|J_{M,P}^{L} (f, s)| ≤ ∑_{π∈Π_{dis}(M(𝔸)^{1})}∫_{i𝔞_{L}^{∗}/𝔞_{G}^{∗}} ∥ 𝔐_{L}(P, λ)ρ_{π}(P, λ, f) ∥_{1} dλ,

where ∥ · ∥_{1} denotes the trace norm for operators in the Hilbert space \overline{A}_{π}^{2}(P). Using (6.12) it follows that the right hand side is bounded by

∑_{π∈Π_{dis}(M(𝔸)^{1})}∫_{i𝔞_{L}^{∗}/i𝔞_{G}^{∗}} ∥ 𝔑′_{S}(P, π, λ)ρ_{π}(P, λ, f) ∥_{1} |ν_{L}^{S}(P, π, λ)| dλ

The function ν_{L}^{S}(P, π, λ) can be estimated by Theorem 5.4 of [Mu2]. This reduces our problem to the estimation of the trace norm of the operator 𝔑′_{S}(P, π, λ)ρ_{π}(P, λ, f). Let K_{f} be an open compact subgroup of G(𝔸_{f}) and let σ ∈ Π(K_{∞}). Denote by Π_{K_{f} ,σ} the orthogonal projection of the Hilbert space \overline{A}_{π}^{2}(P) onto the finite-dimensional subspace A_{π}^{2}(P)_{K_{f} ,σ}. Let f ∈ C^{1}(G(𝔸)^{1}; K_{f}, σ). Then

ρ_{π}(P, λ, f) = Π_{K_{f} ,σ} ∘ ρ_{π}(P, λ, f) ∘ Π_{K_{f} ,σ}

for all π ∈ Π(M(A)^{1}). Let

D = Id +Δ_{G}.

For any k ∈ ℕ let ρ_{π}(P, λ, D^{2k})_{K_{f} ,σ} denote the restriction of the operator ρ_{π}(P, λ, D^{2k}) to the subspace A_{π}^{2}(P)_{K_{f} ,σ}. Then we get

\begin{aligned}∥ 𝔑′_{S} & (P, π, λ)ρ_{π}(P, λ, f) ∥_{1} \\ & ≤∥ 𝔑′_{S}(P, π, λ)_{K_{f} ,σ} ∥ · ∥ ρ_{π}(P, λ, D^{2k})_{K_{f} ,σ}^{−1} ∥_{1} \\ & · ∥ ρ_{π}(P, λ, D^{2k}f) ∥, \end{aligned} ; (6.20)

By (6.9) of [Mu2] we get

(6.21)  ∥ ρ_{π}(P, λ, D^{2k})_{K_{f} ,σ}^{−1} ∥≤ C \frac{dim A_{π}^{2}(P)_{K_{f} ,σ}}{(1+ ∥ λ ∥^{2} +λ_{π}^{2})^{k}} ,

and since ρ_{π}(P, λ) is unitary, we have

(6.22)  ∥ ρ_{π}(P, λ, D^{2k}f) ∥≤∥ D^{2k}f ∥_{L^{1}(G(𝔸)^{1})} .

Together with (6.14) it follows that there exists C > 0 such that

\begin{aligned}∥ 𝔑′_{S} & (P, π, λ)ρ_{π}(P, λ, f) ∥_{1} \\ & ≤ C ∥ D^{2k}f ∥_{L^{1}(G(𝔸)^{1})} (1+ ∥ λ ∥)^{−k/2} \frac{dim A_{π}^{2}(P)_{K_{f} ,σ}}{(1 + λ_{π}^{2})^{k/2}} \end{aligned} ; (6.23)

for all λ ∈ i𝔞_{M}^{∗} and π ∈ Π_{dis}(M(𝔸)^{1}). Let d = dim G(ℝ)^{1}/K_{∞}. By Theorem 5.4 of [Mu2] there exists k_{0} ∈ ℕ such that for k ≥ k_{0} we have

(6.24)  ∫_{i𝔞_{L}^{∗}/𝔞_{G}^{∗}} |ν_{L}^{S}(P, π, λ)|(1+ ∥ λ ∥^{2})^{−k/2} dλ ≤ C_{k}(1 + λ_{π}^{2})^{8d^{2}}

for all π ∈ Π_{dis}(M(𝔸)^{1}) with A_{π}^{2}(P)_{K_{f} ,σ} ≠ 0. Furthermore, by Proposition 3.4 we have

(6.25)  ∑_{π∈Π_{dis}(M(𝔸)^{1})} \frac{dim A_{π}^{2}(P)_{K_{f} ,σ}}{(1 + λ_{π}^{2})^{k/2}} < ∞

for k > m/2 + 1, where m = dim M(ℝ)^{1}/K_{M,∞}. Combining (6.23)- (6.25), it follows that for each k > m/2 + 16d^{2} + 1 there exists C_{k} > 0 such that

\begin{aligned}∑_{π∈Π_{dis}(M(𝔸)^{1})}∫_{i𝔞_{L}^{∗}/i𝔞_{G}^{∗}} ∥ 𝔑′_{S}(P, π, λ)ρ_{π}(P, λ, f) & ∥_{1} |ν_{L}^{S}(P, π, λ)| dλ \\ & ≤ C_{k} ∥ D^{2k}f ∥_{L^{1}(G(𝔸)^{1})} . \end{aligned}

This completes the proof. □

Now we return to the function ϕ_{t} defined by (6.5). It follows from the definition that the restriction ϕ_{t}^{1} of ϕ_{t} belongs to C^{1}(G(𝔸)^{1}, K_{f}, σ). We shall now modify ϕ_{t} in the following way. Let φ ∈ C_{0}^{∞}(ℝ) be such that φ(u) = 1, if |u| ≤ 1/2, and φ(u) = 0, if |u| ≥ 1. Let d(x, y) denote the geodesic distance of x, y ∈ X and set

r(g_{∞}) := d(g_{∞}K_{∞}, K_{∞}).

Given t > 0, let φ_{t} ∈ C_{0}^{∞}(G(ℝ)^{1}) be defined by

φ_{t}(g_{∞}) = φ(r^{2}(g_{∞})/t^{1/2}).

Then supp φ_{t} is contained in the set {g_{∞} ∈ G(ℝ)^{1} | r(g_{∞}) < t^{1/4}}. Extend φ_{t} to G(ℝ) by

φ_{t}(g_{∞}z) = φ_{t}(g_{∞}), g_{∞} ∈ G(ℝ)^{1}, z ∈ A_{G}(ℝ)^{0},

and then to a function on G(𝔸) by multiplying φ_{t} by the characteristic function of K_{f}. Put

(6.26)  ϕ̃_{t}(g) = φ_{t}(g)ϕ_{t}(g), g ∈ G(𝔸).

Then the restriction ϕ̃_{t}^{1} of ϕ̃_{t} to G(𝔸)^{1} belongs to C_{c}^{∞}(G(𝔸)^{1}).

Proposition 6.4. There exist C, c > 0 such that

|J_{spec}(ϕ_{t}^{1}) − J_{spec}(ϕ̃_{t}^{1})| ≤ Ce^{−c/\sqrt{t}}

for 0 < t ≤ 1.

Proof. Let ψ_{t} = ϕ_{t} − ϕ̃_{t} and f_{t} = 1 − φ_{t}. Let ψ_{t}^{1} denote the restriction of ψ_{t} to G(𝔸)^{1}. Then by Proposition 6.3 there exists k ∈ ℕ such that

|J_{spec}(ϕ_{t}^{1}) − J_{spec}(ϕ̃_{t}^{1})| = |J_{spec}(ψ_{t}^{1})| ≤ C_{k} ∥ (Id +Δ_{G})^{k}ψ_{t}^{1} ∥_{L^{1}(G(𝔸)^{1})} .

In order to estimate the L^{1}-norm of ψ_{t}^{1}, recall that by definition

ψ_{t}(g_{∞}g_{f}) = f_{t}(g_{∞})h_{t}^{σ}(g_{∞})χ_{K_{f}} (g_{f}).

Hence

∥ (Id +Δ_{G})^{k}ψ_{t}^{1} ∥_{L^{1}(G(𝔸)^{1})}=∥ (Id +Δ_{G})^{k}(f_{t}h_{t}^{σ}) ∥_{L^{1}(G(ℝ)^{1})} .

Let 𝔤(ℝ)^{1} be the Lie algebra of G(ℝ)^{1} and let X_{1}, ..., X_{a} be an orthonormal basis of 𝔤(ℝ)^{1}. Then Δ_{G} = − ∑_{i} X_{i}^{2}. Denote by ∇ the canonical connection on G(ℝ)^{1}. Then it follows that there exists C > 0 such that

|(Id +Δ_{G})^{k}f(g)| ≤ C ∑_{l=0}^{2k} ∥ ∇^{l}f(g) ∥, g ∈ G(ℝ)^{1},

for all f ∈ C^{∞}(G(ℝ)^{1}). By Proposition 2.1 there exist constants C, c > 0 such that

(6.27)  ∥ ∇^{j}h_{t}^{σ}(g) ∥≤ Ct^{−(a+j)/2}e^{−cr^{2}(g)/t}, g ∈ G(ℝ)^{1},

for j ≤ 2k and 0 < t ≤ 1. Let χ_{t} be the characteristic function of the set ℝ − (−t^{1/4}, t^{1/4}). Recall that f_{t}(g) = (1 − φ)(r^{2}(g)/t^{1/2}) and (1 − φ)(u) is constant for |u| ≥ 1. This implies that there exist constants C, c > 0 such that

(6.28)  ∥ ∇^{j}f_{t}(g) ∥≤ Ct^{−k}χ_{t}(r(g)), g ∈ G(ℝ)^{1},

for j ≤ 2k and 0 < t ≤ 1. Combining (6.27) and (6.28) we obtain

\begin{aligned}∑_{l=0}^{2k} ∥ ∇^{l}(f_{t}h_{t}^{σ})(g) ∥ & ≤ C_{1}t^{−a/2−2k}χ_{t}(r(g))e^{−cr^{2}(g)/t} \\ & ≤ C_{2}e^{−c_{1}/\sqrt{t}}e^{−c_{1}r^{2}(g)} \end{aligned}

for all g ∈ G(ℝ)^{1} and 0 < t ≤ 1. Finally note that for every c > 0, e^{−cr^{2}(g)} is an integrable function on G(ℝ)^{1}. This finishes the proof. □

7. Proof of the main theorem

In this section we evaluate the geometric side of the trace formula at the function ϕ̃_{t}^{1} and investigate its asymptotic behaviour as t → 0. Then we compare the geometric and the spectral side and prove our main theorem.

Let us briefly recall the structur of the geometric side J_{geo} of the trace formula [A1]. The coarse 𝔬-expansion of J_{geo}(f) is a sum of distributions

J_{geo}(f) = ∑_{𝔬∈O}J_{𝔬}(f), f ∈ C_{c}^{∞}(G(𝔸)^{1}),

which are parametrized by the set O of conjugacy classes of semisimple elements in G(ℚ). The distributions J_{𝔬}(f) are defined in [A1]. We shall use the fine 𝔬-expansion of the spectral side [A10] which expresses the distributions J_{𝔬}(f) in terms of weighted orbital integrals J_{M}(γ, f). To describe the fine 𝔬-expansion we have to introduce some notation. Suppose that S is a finite set of valuations of ℚ. Set

G(ℚ_{S})^{1} = G(ℚ_{S}) ∩ G(𝔸)^{1},

where

ℚ_{S} = ∏_{v∈S} ℚ_{v}.

Suppose that ω is a compact neighborhood of 1 in G(𝔸)^{1}. There is a finite set S of valuations of ℚ, which contains the Archimedean place, such that ω is the product of a compact neighborhood of 1 in G(ℚ_{S})^{1} with ∏_{v∉S} K_{v}. Let S_{ω}^{0} be the minimal such set. Let C_{ω}^{∞}(G(𝔸)^{1}) denote the space of functions in C_{c}^{∞}(G(𝔸)^{1}) which are supported on ω. For any finite set S ⊃ S_{ω}^{0} set

C_{ω}^{∞}(G(ℚ_{S})^{1}) = C_{ω}^{∞}(G(𝔸)^{1}) ∩ C_{c}^{∞}(G(ℚ_{S})^{1}).

Let us recall the notion of (M, S)-equivalence [A10, p.205]. For any γ ∈ M(ℚ) denote by γ_{s} (resp. γ_{u}) the semisimple (resp. unipotent) Jordan component of γ. Then two elements γ and γ′ in M(ℚ) are called (M, S)-equivalent if there exists δ ∈ M(ℚ) with the following two properties.

(i) γ_{s} is also the semisimple Jordan component of δ^{−1}γ′δ.

(ii) γ_{u} and (δ^{−1}γ′δ)_{u}, regarded as unipotent elements in M_{γ_{s}}(ℚ_{S}), are M_{γ_{s}}(ℚ_{S})-conju-gate.

Denote by (M(ℚ))_{M,S} the set of (M, S)-equivalence classes in M(ℚ). Note that (M, S)- equivalent elements γ and γ′ in M(ℚ) are, in particular, M(ℚ_{S})-conjugate. Given γ ∈ M(ℚ), let

J_{M}(γ, f), f ∈ C_{c}^{∞}(G(ℚ_{S})^{1}),

be the weighted orbital integral associated to M and γ [A11]. We observe that J_{M}(γ, f) depends only on the M(ℚ_{S})-orbit of γ. Then by Theorem 9.1 of [A10] there exists a finite set S_{ω} ⊃ S_{ω}^{0} of valuations of ℚ such that for all S ⊃ S_{ω} and any f ∈ C_{ω}^{∞}(G(ℚ_{S})^{1}), we have

(7.1)  J_{geo}(f) = ∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1} ∑_{γ∈(M(ℚ))_{M,S}} a^{M}(S, γ)J_{M}(γ, f).

This is the fine 𝔬-expansion of the geometric side of the trace formula. The interior sum is finite.

Recall that the restriction ϕ̃_{t}^{1} of ϕ̃_{t} to G(𝔸)^{1} belongs to C_{c}^{∞}(G(𝔸)^{1}) and hence, J_{geo} can be evaluated at ϕ̃_{t}^{1}. By construction of ϕ̃_{t}^{1} there exists a compact neighborhood ω of 1 in G(𝔸)^{1} and a finite set S ⊃ S_{ω} of valuations of ℚ such that

ϕ̃_{t}^{1} ∈ C_{ω}^{∞}(G(ℚ_{S})^{1}), 0 < t ≤ 1.

Hence we can apply (7.1) to evaluate J_{geo}(ϕ̃_{t}^{1}). In this way our problem is reduced to the investigation of the weighted orbital integrals J_{M}(γ, ϕ̃_{t}^{1}). Actually for γ ∈ M(ℚ) we may replace ϕ̃_{t}^{1} by ϕ̃_{t}.

To begin with we establish some auxiliary results. Given h ∈ G(ℝ), let

C_{h} = {g^{−1}hg | g ∈ G(ℝ)}

be the conjugacy class of h in G(ℝ).

Lemma 7.1. Let k ∈ K_{∞}. Then C_{k} ∩ K_{∞} is the K_{∞}-conjugacy class of k.

Proof. Let 𝔤 and 𝔨 denote the Lie algebras of G(ℝ) and K_{∞}, respectively. Let θ be a Cartan involution of 𝔤 with fixed point set 𝔨 and let 𝔭 be the (−1)-eigenspace of θ. Then the map

(k′, X) ∈ K_{∞} × 𝔭 ⟼ k′ exp(X) ∈ G(ℝ)

is an analytic isomorphism of analytic manifolds. If k_{1} ∈ K_{∞}, then k_{1} is a θ-invariant semisimple element. Therefore, its centralizer G_{k_{1}} is a reductive subgroup and the restriction of θ to G_{k_{1}} is a Cartan involution. Thus the restriction of the above Cartan decomposition to the centralizer of k_{1} yields a Cartan decomposition of G_{k_{1}}(ℝ). Let g ∈ G(ℝ) such that g^{−1}kg ∈ K_{∞}. Write g = k′ exp(X) with k′ ∈ K_{∞} and X ∈ 𝔭. Since g^{−1}kg is θ-invariant, we get

exp(−X)k′^{−1}kk′ exp(X) = exp(X)k′^{−1}kk′ exp(−X).

Hence exp(2X) ∈ G_{k′^{−1}kk′}(ℝ). From the Cartan decomposition of the latter group we conclude that exp(2X) = exp(Y) for some Y ∈ 𝔭_{k′^{−1}kk′}, and hence X ∈ 𝔭_{k′^{−1}kk′}. This implies that g^{−1}kg = k^{′−1}kk′. □

It follows from Lemma 7.1 that C_{k} ∩ K_{∞} is a submanifold of C_{k}.

Lemma 7.2. Let k ∈ K_{∞} − {±1}. Then C_{k} ∩ K_{∞} is a proper submanifold of C_{k}.

Proof. Let the notation be as in the previous lemma. First note that the tangent space of C_{k} at k is given by

T_{k}C_{k} ≅ (Ad(k) − Id)(𝔤).

Furthermore

Ad(k)(𝔨) ⊂ 𝔨, Ad(k)(𝔭) ⊂ 𝔭.

Hence we get

T_{k}(C_{k} ∩ K_{∞}) = T_{k}C_{k} ∩ 𝔨 = (Ad(k) − Id)(𝔨),

and so the normal space N_{k} to C_{k} ∩ K_{∞} in C_{k} at k is given by

N_{k} ≅ (Ad(k) − Id)(𝔭).

Suppose that Ad(k) = Id on 𝔭. Since 𝔨 = [𝔭, 𝔭], it follows that Ad(k) = Id on 𝔤. Hence k belongs to the center of G^{0}, which implies that k = ±1. Thus if k ≠ ±1, we have dim N_{k} > 0. □

Next we recall the notion of an induced space of orbits [A11, p.255]. Given an element γ ∈ M(ℚ_{S}), let γ^{G} be the union of those conjugacy classes in G(ℚ_{S}) which for any P ∈ P(M) intersect γN_{P} (ℚ_{S}) in an open set. There are only finitely many such conjugacy classes.

Proposition 7.3. Let d = dim G(ℝ)^{1}/K_{∞}. Let M ∈ L and γ ∈ M(ℚ). Then

lim_{t→0} t^{d/2}J_{M}(γ, ϕ̃_{t}) = 0

if either M ≠ G, or M = G and γ ≠ ±1.

Proof. By Corollary 6.2 of [A11] the distribution J_{M}(γ, ϕ̃_{t}) is given by the integral of ϕ̃_{t} over γ^{G} with respect to a measure dμ on γ^{G} which is absolutely continuous with respect to the invariant measure class. Thus J_{M}(γ, ϕ̃_{t}) is equal to a finite sum of integrals of the form

∫_{G_{γn}(ℚ_{S})\G(ℚ_{S})} ϕ̃_{t}(g^{−1}γng)dμ(g),

where n ∈ N_{P} (ℚ_{S}) for some P ∈ P(M). Now recall that by (6.5) and (6.26), ϕ̃_{t}(g) is the product of φ_{t}(g_{∞})h_{t}^{σ}(g_{∞}) with χ_{K_{f}} (g_{f}) for any g = g_{∞}g_{f}. Hence our problem is reduced to the investigation of the integral

∫_{G_{γn∞}(ℝ)\G(ℝ)}(φ_{t}h_{t}^{σ})(g_{∞}^{−1}γn_{∞}g_{∞}) dμ(g_{∞}).

Furthermore, by Proposition 2.1 there exists C > 0 such that

|h_{t}^{σ}(g_{∞})| ≤ Ct^{−d/2}, 0 < t ≤ 1.

Hence it suffices to show that

(7.2)  lim_{t→0} ∫_{G_{γn∞}(ℝ)\G(ℝ)} φ_{t}(g_{∞}^{−1}γn_{∞}g_{∞}) dμ(g_{∞}) = 0

if either M ≠ G, or M = G and γ ≠ ±1.

By definition of γ^{G}, the conjugacy class of γn in G(ℚ_{S}) has to intersect γN_{P} (ℚ_{S}) in an open subset. This implies that γn_{∞} ≠ ±1, if either M ≠ G, or M = G and γ ≠ ±1. Then it follows from Lemma 7.2 that C_{γn_{∞}} ∩ K_{∞} is a proper submanifold of C_{γn_{∞}}. The measure dμ(g_{∞}) is of the form f(g_{∞})dg_{∞} for some smooth function f on G(ℝ). Hence being a proper submanifold, C_{γn_{∞}} ∩ K_{∞} is a subset of C_{γn_{∞}} with measure zero with respect to dμ. Next observe that

∫_{G_{γn∞}(ℝ)\G(ℝ)} φ_{t}(g_{∞}^{−1}γn_{∞}g_{∞}) |f(g_{∞})| dg_{∞} < ∞.

Since supp φ_{t′} ⊂ supp φ_{t} for t′ < t, and 0 ≤ φ_{t} ≤ 1 for all t > 0, it follows that there exists C > 0 such that

| ∫_{G_{γn∞}(ℝ)\G(ℝ)} φ_{t}(g_{∞}^{−1}γn_{∞}g_{∞}) dμ(g_{∞})| ≤ C

for all 0 < t ≤ 1. Furthermore by definition of φ_{t} we have

lim_{t→0} φ_{t}(x) = 0

for all x ∈ C_{γn_{∞}} − (C_{γn_{∞}} ∩ K_{∞}). Since C_{γn_{∞}} ∩ K_{∞} has measure zero with respect to dμ, (7.2) follows by the dominated convergence theorem. □

We can now state the main result of this section.

Theorem 7.4. Let d = dim G(ℝ)^{1}/K_{∞}, let K_{f} be an open compact subgroup of G(𝔸_{f}) and let σ ∈ Π(O(n)) such that σ(−1) = Id if −1 ∈ K_{f}. Then

lim_{t→0} t^{d/2}J_{geo}(ϕ̃_{t}^{1}) = \frac{dim(σ)}{(4π)^{d/2}} vol(G(ℚ)\G(𝔸)^{1}/K_{f})(1 + 1_{K_{f}} (−1)).

Proof. By (7.1) and Proposition 7.3 if follows that

lim_{t→0} t^{d/2}J_{geo}(ϕ̃_{t}^{1}) = lim_{t→0} t^{d/2}(a^{G}(S, 1)ϕ̃_{t}^{1}(1) + a^{G}(S, −1)ϕ̃_{t}^{1}(−1)).

By Theorem 8.2 of [A10] we have

a^{G}(S, ±1) = vol(G(ℚ)\G(𝔸)^{1}).

Furthermore

ϕ̃_{t}^{1}(±1) = h_{t}^{σ}(±1)χ_{K_{f}} (±1).

Since σ satisfies σ(−1) = Id, if −1 ∈ K_{f}, it follows from (2.5) that h_{t}^{σ}(−1) = h_{t}^{σ}(1). Finally, by Lemma 2.3 we have

h_{t}^{σ}(±1) = \frac{dim(σ)}{(4π)^{d/2}}t^{−d/2} + O(t^{−(d−1)/2})

as t → 0. Combined with χ_{K_{f}} (±1) = 1_{K_{f}} (±1) vol(K_{f})^{−1}, the theorem follows. □

We shall now use the trace formula to prove the main results of this paper. Recall that the coarse trace formula is the identity

J_{spec}(f) = J_{geo}(f), f ∈ C_{c}^{∞}(G(𝔸)^{1}),

between distributions on G(𝔸)^{1} [A1]. Applied to ϕ̃_{t}^{1} we get the equality

J_{spec}(ϕ̃_{t}^{1}) = J_{geo}(ϕ̃_{t}^{1}), t > 0.

Put ε_{K_{f}} = 1, if −1 ∈ K_{f} and ε_{K_{f}} = 0 otherwise. Combining Theorem 6.2, Proposition 6.4 and Theorem 7.4, we obtain

∑_{π∈Π_{dis}(G(𝔸),ξ_{0})}m(π) dim(H_{π_{f}}^{K_{f}} ) dim(H_{π_{∞}} ⊗ V_{σ})^{O(n)}e^{tλ_{π}}

∼ \frac{dim(σ)}{(4π)^{d/2}} vol(G(ℚ)\G(𝔸)^{1}/K_{f})(1 + ε_{K_{f}})t^{−d/2}

(7.3)

as t → 0. Applying Karamat’s theorem [Fe, p.446], we obtain

∑_{π∈Π_{dis}(G(𝔸),ξ_{0})}m(π) dim(H_{π_{f}}^{K_{f}} ) dim(H_{π_{∞}} ⊗ V_{σ})^{O(n)}

∼ dim(σ)\frac{vol(G(ℚ)\G(𝔸)^{1}/K_{f})}{(4π)^{d/2}Γ(d/2 + 1)}(1 + ε_{K_{f}})λ^{d/2}

(7.4)

as λ → ∞. By Lemma 3.3 it follows that this asymptotic formula continues to hold if we replace the sum over Π_{dis}(G(𝔸), ξ_{0}) by the sum over Π_{cus}(G(𝔸), ξ_{0}). Finally note that by [Sk] we have m(π) = 1 for all π ∈ Π_{cus}(G(𝔸), ξ_{0}). This completes the proof of Theorem 0.2.

Now suppose that K_{f} is the congruence subgroup K(N) and Γ(N) ⊂ SL_{n}(ℤ) the principal congruence subgroup of level N. Then by (3.10) we have

vol(G(ℚ)\G(𝔸)^{1}/K(N)) = φ(N) vol(Γ(N)\ SL_{n}(ℝ)).

Furthermore, ε_{K(N)} = 1 if and only if −1 ∈ Γ(N). If −1 is contained in Γ(N), then the fibre of the canonical map

Γ(N)\ SL_{n}(ℝ) → Γ(N)\ SL_{n}(ℝ)/ SO(n)

is equal to SO(n)/{±1}. Otherwise the fibre is equal to SO(n). We normalize the Haar measure on SL_{n}(ℝ) so that vol(SO(n)) = 1. Then in either case we have

vol(Γ(N)\ SL_{n}(ℝ))(1 + ε_{K(N)}) = vol(Γ(N)\ SL_{n}(ℝ)/ SO(n)).

Let X = SL_{n}(ℝ)/ SO(n) and let λ_{0} ≤ λ_{1} ≤ ⋯ be the eigenvalues, counted with multiplicity, of the Bochner-Laplace operator Δ_{σ} acting in L^{2}(Γ(N)\ SL_{n}(ℝ), σ).

Combining (6.18), Proposition 6.4, Theorem 7.4 and the above observations, we get

∑_{i} e^{−tλ_{i}} = dim(σ)\frac{vol(Γ(N)\X)}{(4π)^{d/2}}t^{−d/2} + o(t^{−d/2})

as t → 0. Using again Karamata’s theorem [Fe, p.446], we get

N Γ(N)_{dis}(λ, σ) = dim(σ)\frac{vol(Γ(N)\X)}{(4π)^{d/2}Γ(d/2 + 1)}λ^{d/2} + o(λ^{d/2})

as λ → ∞. By Proposition 3.6 it follows that the same asymptotic formula holds if we replace N Γ(N)_{dis}(λ, σ) by N Γ(N)_{cus}(λ, σ). This is exactly the statement of Theorem 0.1.

References

[A1] J. Arthur, A trace formula for reductive groups I: terms associated to classes in G(ℚ), Duke. Math. J. 45 (1978), 911–952.

[A2] J. Arthur, A trace formula for reductive groups II: applications of a truncation operator, Comp. Math. 40 (1980), 87–121.

[A3] J. Arthur, On a family of distributions obtained from Eisenstein series I: Applications of the Paley-Wiener theorem, Amer. J. Math. 104 (1982), 1243–1288.

[A4] J. Arthur, On a family of distributions obtained from Eisenstein series II: Explicit formulas, Amer. J. Math. 104 (1982), 1289–1336.

[A5] J. Arthur, The trace formula in invariant form, Annals of Math. 114 (1981), 1–74.

[A6] J. Arthur, The invariant trace formula. II. Global theory, J. Amer. Math. Society 1 (1988), 501– 554.

[A7] J. Arthur, Intertwining operators and residues. I. Weighted characters, J. Funct. Analysis 84 (1989), 19–84.

[A8] J. Arthur, On the Fourier transforms of weighted orbital integrals, J. reine angew. Math. 452 (1994), 163–217.

[A9] J. Arthur, Automorphic Representations and Number Theory, In: 1980 Seminar on Harmonic Analysis, Canadian Math. Soc., Conference Proceedings, Volume 1, AMS, Providence, RI, 1981.

[A10] J. Arthur, On a family of distributions obtained from orbits, Canadian J. Math. 38 (1986), 179– 214.

[A11] J. Arthur, The local behaviour of weighted orbital integrals, Duke Math. J. 56 (1988), 223–293.

[AC] J. Arthur and L. Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Annals Math. Studies 120, Princeton Univ. Press, Princeton, NJ, 1989.

[BM] D. Barbasch and H. Moscovici, L^{2}-index and the Selberg trace formual, J. Funct. Analysis 53 (1983), 151–201.

[BGV] N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, Springer, Berlin, 1996.

[BW] A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Math. Stud. 94, Princeton Univ. Press, 1980.

[Bu] C.J. Bushnell, Induced representations of locally profinite groups, Journal of Algebra 134 (1990), 104–114.

[BH] C.J. Bushnell and G. Henniart, An upper bound on conductors for pairs, J. Number Theory 65 (1997), 183–196.

[BHK] C.J. Bushnell, G.M. Henniart, and P.C. Kutzko, Local Rankin-Selberg convolutions for GL_{n}: Explicit conductor formula, J. Amer. Math. Soc. 11 (1998), 703–730.

[Cha] I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, New York, 1984.

[CLL] L. Clozel, J.P. Labesse, and R. Langlands, Morning seminar on the trace formula, mimeoraphed notes, IAS, Princeton, 1984.

[CLY] S.Y. Cheng, P. Li, and S.-T. Yau, On the upper estimate of the heat kernel of a complete Riemannian manifold, Amer. J. Math. 103 (1981), 1021–1063.

[DH] A. Deitmar and W. Hoffmann, On limit multiplicities for spaces of automorphic forms, Canad. J. Math. 51 (1999), 952-976.

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

[Do2] H. Donnelly, Asymptotitc expansions for the compact quotients of properly discontinuous group actions, Illinois J. Math. 23 (1979), 485–496.

[Ef] I. Efrat, The Selberg trace formula for PSL_{2}(ℝ), Mem. Amer. Math. Soc. 65 (1987), no. 359.

[Fe] W. Feller, An introduction to probability theory and its applications, 2nd ed., Vol. II, Wiley, New York, 1971.

[He] D. Hejhal, The Selberg trace formula for PSL(2, ℝ), Vol. 2, Lecture Notes in Math. 1001, Springer-Verlag, Berlin.

[J] H. Jacquet, Principal L-functions of the linear group, Proc. Symp. Pure Math. 33 (1979), 63-86.

[JPS] H. Jacquet, I.I. Piatetski-Shapiro, and J.A. Shalika, Rankin-Selberg convolutions, Amer. J. Math. 105 (1983), 367-464.

[JS1] H. Jacquet and J.A. Shalika, Euler products and classification of automorphic representations I, Amer. J. Math. 103 (1981), 499-558.

[JS3] H. Jacquet, J.A. Shalika, The Whittaker models of induced representations, Pacific J. Math. 109 (1983), 107–120.

[Kn] A.W. Knapp, Representation theory of semisimple groups, Princeton University Press, Princeton, New Jersey 1986.

[La] R.P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Math. 544, Springer–Verlag, Berlin–Heidelberg–New York, 1976.

[La1] R.P. Langlands, On the classification of irreducible representations of real algebraic groups, Mathematical Surveys and Monographs 31, Amer. Math. Soc. 1989.

[LRS] W. Luo, Z. Rudnick, and P. Sarnak, On the generalized Ramanujan conjecture for GL(n). Proc. Symp. Pure Math. 66, Vol. II, pp. 301–310, Amer. Math. Soc., Providence, Rhode Island, 1999.

[MW] C. Moeglin et J.-L. Waldspurger, Le spectre résiduel de GL(n), Ann. scient. Éc. Norm. Sup., 4^{e} série, t. 22 (1989), 605–674.

[Mia] R. Miatello, The Minakshisundaram-Pleijel coefficients for the vector valued heat kernel on compact locally symmetric spaces of negative curvature, Trans. Amer. Math. Soc. 260 (1980), 1–33.

[Mil] St. Miller, On the existence and temperedness of cusp forms for SL_{3}(ℤ). J. Reine Angew. Math. 533 (2001), 127–169.

[Mu1] W. Müller, The trace class conjecture in the theory of automorphic forms. II, Geom. funct. anal. 8 (1998), 315–355.

[Mu2] W. Müller, On the spectral side of the Arthur trace formula, Geom. funct. anal. 12 (2002), 669– 722.

[Mu3] W. Müller, Eigenvalue estimates for locally symmetric spaces of finite volume. Symposium ”Partial Differential Equations” (Holzhau, 1988), 179–196, Teubner-Texte Math., 112, Teubner, Leipzig, 1989.

[MS] W. Müller and B. Speh, with appendix by E. Lapid, Absolute convergence of the spectral side of the Arthur trace formula for GL_{n}, Preprint, 2003.

[N] E. Nelson, Analytic vectors, Ann. of Math. 70 (1959), 572–615.

[PS] R. Phillips and P. Sarnak, On cusp forms for cofinite subgroups of PSL(2,ℝ), Invent. Math. 80 (1985), 339–364.

[Rez] A. Reznikov, Eisenstein matrix and existence of cusp forms in rank one symmetric spaces. Geom. Funct. Anal. 3 (1993), no. 1, 79–105.

[RS] Z. Rudnick and P. Sarnak, Zeros of principal L-functions and random matrix theory. Duke Math. J. 81 (1996), 269–322.

[Sa] P. Sarnak, On cusp forms, in: The Selberg trace formula and related topics, ed. by D. Hejhal et al., Contemporary Math. vol. 53, Amer. Math. Soc., 1984, pp. 393–407.

[Se] A. Selberg, Harmonic analysis, in ”Collected Papers”, Vol. I, Springer-Verlagf, Berlin-Heidelberg-New York (1989), 626–674.

[Sh1] F. Shahidi, Fourier transforms of intertwining operators and Plancherel measures for GL_{n}, Amer. J. Math 106 (1984), 67–111.

[Sh2] F. Shahidi, On certain L-functions, Amer. J. Math. 103 (1981), 297–356.

[Sh3] F. Shahidi, Local coefficients and normalization of intertwining operators for GL_{n}, Comp. Math. 48 (1983), 271–295.

[Sh4] F. Shahidi, Local coefficients as Artin factors for real groups, Duke Math. J. 52 (1985), 973–1007.

[Sh5] F. Shahidi, A proof of Langlands’ conjecture on Plancherel measures; Complementary series for p-adic groups, Annals Math. 132 (1990), 273–330.

[Sh6] F. Shahidi, On multiplicativity of local factors, Israel Math. conf. Proc. Vol. 3 (1990), 279–289.

[Sk] J.A. Shalika, The multiplicity one theorem for GL_{n}, Annals Math. 100 (1974), 171-193.

[Ve] A.B. Venkov, Spectral theory of automorphic forms and its applications, Kluwer Academic Pub., Soviet Series 51, 1990.

[Wa1] G. Warner, Harmonic analysis on semi-simple Lie groups I, Springer-Verlag, Berlin-Heidelberg-New York, 1972.

[Wa2] G. Warner, Harmonic analysis on semi-simple Lie groups II, Springer-Verlag, Berlin-Heidelberg-New York, 1972.

Universität Bonn, Mathematisches Institut, Beringstrasse 1, D – 53115 Bonn, Germany

E-mail address: mueller@math.uni-bonn.de