ON THE SPECTRAL SIDE OF THE ARTHUR TRACE FORMULA

WERNER MÜLLER

0. Introduction

Let G be a connected reductive algebraic group defined over ℚ and let G(𝔸) be the group of points of G with values in the ring of adèles of ℚ. Then G(ℚ) embeds diagonally as a discrete subgroup of G(𝔸). Let G(𝔸)^{1} be the intersection of the kernels of the maps x ↦ |χ(x)|, x ∈ G(𝔸), where χ ranges over the group X(G)_{ℚ} of characters of G defined over ℚ. Then the (noninvariant) trace formula of Arthur is an identity

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

between distributions on G(𝔸)^{1}. The left hand side is the geometric side and the right hand side the spectral side of the trace formula. The distributions J_{𝔬} are parametrized by semisimple conjugacy in G(ℚ) and are closely related to weighted orbital integrals on G(𝔸)^{1}.

In this paper we are concerned with the spectral side of the trace formula. The distribution J_{χ} are defined in terms of truncated Eisenstein series. They are parametrized by the set of cuspidal data 𝔛 which consists of the Weyl group orbits of pairs (M_{B}, r_{B}), where M_{B} is the Levi component of a standard parabolic subgroup and r_{B} is an irreducible cuspidal automorphic representation of M_{B}(𝔸)^{1}. In [A4], Arthur has derived an explicit formula for the distributions J_{χ} which expresses them in terms of generalized logarithmic derivatives of intertwining operators. So far, the resulting integral-series is only known to converge conditionally. This suffices, for example, for the comparison of trace formulas which, at present, is the main application of the trace formula. However, with regard to potential applications of the trace formula in spectral theory and geometry it would be highly desirable to know that the spectral side of the trace formula is absolutely convergent. This would also simplify the applications of the trace formula in the theory of automorphic forms [Lb].

Date: October 30, 2002.

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

Key words and phrases. trace formula, Automorphic forms.

The problem of the absolute convergence of the spectral side of the trace formula is the main issue of the present paper. We will not settle the problem, but we shall reduce it to a question about local components of automorphic representations.

To describe the results in more detail we have to introduce some notation. We fix a Levi component M_{0} of a minimal parabolic subgroup P_{0} of G. Let P be a parabolic subgroup of G, defined over ℚ, with unipotent radical N_{P}. Let M_{P} be the unique Levi component of P which contains M_{0}. We denote the split component of the center of M_{P} by A_{P} and its Lie algebra by 𝔞_{P}. For parabolic groups P ⊂ Q there is a natural surjective map 𝔞_{P} → 𝔞_{Q} whose kernel we will denote by 𝔞_{P}^{Q}. Let A^{2}(P) be the space of square integrable automorphic forms on N_{P} (𝔸)M_{P} (ℚ)\G(𝔸). Let Q be another parabolic subgroup of G, defined over ℚ, with Levi component M_{Q}, split component A_{Q} and corresponding Lie algebra 𝔞_{Q}. 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}^{∗}. Set

M_{Q|P} (λ) := M_{Q|P} (1, λ).

Let Π(M_{P} (𝔸)^{1}) be the set of equivalence classes of irreducible unitary representations of M_{P} (𝔸)^{1}. Let χ ∈ 𝔛 and π ∈ Π(M_{P} (𝔸)^{1}). Then (χ, π) singles out a certain subspace A_{χ,π}^{2}(P) of A^{2}(P) (see §1.6). Let \overline{A}_{χ,π}^{2}(P) be the Hilbert space completion of A_{χ,π}^{2}(P) with respect to the canonical inner product. For each λ ∈ 𝔞_{P,ℂ}^{∗} we have an induced representation ρ_{χ,π}(P, λ) of G(𝔸) in \overline{A}_{χ,π}^{2}(P).

For each Levi subgroup L let P(L) be the set of all parabolic subgroups with Levi component L. If P is a parabolic subgroup, let Δ_{P} denote the set of simple roots of (P, A_{P}). Let L be a Levi subgroup which contains M_{P}. Set

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}_{χ,π}^{2}(P). In the special case that L = M and dim 𝔞_{L}^{G} = 1, the operator 𝔐_{L}(P, λ) has a simple description. Let P be a parabolic subgroup with Levi component M. Let α be the unique simple root of (P, A_{P}) and let ω̃ be the element in (𝔞_{M}^{G})^{∗} such that ω̃(α^{∨}) = 1. Let \overline{P} be the opposite parabolic group of P. Then

𝔐_{L}(P, zω̃) = − vol(𝔞_{M}^{G}/ℤα^{∨})M_{\overline{P} |P} (zω̃)^{−1} · \frac{d}{dz} M_{\overline{P} |P} (zω̃).

Let f ∈ C_{c}^{∞}(G(𝔸)^{1}). Then Arthur [A4, Theorem 8.2] proved that J_{χ}(f) equals the sum over Levi subgroups M containing M_{0}, over L containing M, over π ∈ Π(M(𝔸)^{1}), and over s ∈ W^{L}(𝔞_{M})_{reg}, a certain subset of the Weyl group, of the product of

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

a factor to which we need not pay too much attention , and of

(0.1)  ∫_{i𝔞_{L}^{∗}/i𝔞_{G}^{∗}} ∑_{P ∈P(M)} tr(𝔐_{L}(P, λ)M_{P |P} (s, 0)ρ_{χ,π}(P, λ, f)) dλ.

So far, it is only known that ∑_{χ∈𝔛} |J_{χ}(f)| < ∞ and the goal is to show that the integral–sum obtained by summing (0.1) over χ ∈ 𝔛 and π ∈ Π(M(𝔸)^{1}) is absolutely convergent with respect to the trace norm.

Given π ∈ Π(M(𝔸)) with π = ⊗_{v}π_{v}, let J_{Q|P} (π_{v}, λ) be the local intertwining operator between the induced representations I_{P}^{G}(π_{v,λ}) and I_{Q}^{G}(π_{v,λ}). By [CLL, §15] and [A7] there exist normalizing factors r_{Q|P} (π_{v}, λ) such that the normalized 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].

If v < ∞, let K_{v} ⊂ G(ℚ_{v}) be an open compact subgroup. Denote by R_{Q|P} (π_{v}, λ)_{K_{v}} the restriction of R_{Q|P} (π_{v}, λ) to the subspace H_{P} (π_{v})^{K_{v}} of K_{v}-invariant vectors in the Hilbert space H_{P} (π_{v}) of the induced representation. If v = ∞, let K_{∞} ⊂ G(ℝ) be a maximal compact subgroup. Given π ∈ Π(M(ℝ)) and σ ∈ Π(K_{∞}), let R_{Q|P} (π, λ)_{σ} be the restriction of R_{Q|P} (π, λ) to the σ-isotypical subspace of H_{P} (π). Let λ_{π} and λ_{σ} denote the Casimir eigenvalues of π and σ, respectively.

For a given place v, let Π_{disc}(M(ℚ_{v})) be the subset consisting of all representations π_{v} ∈ Π(M(ℚ_{v})) such that there exists an automorphic representation π in the discrete spectrum of M(𝔸) whose local component at the place v is π_{v}. Finally, let C^{1}(G(𝔸)^{1}) be the space of integrable rapidly decreasing functions on G(𝔸)^{1} (see §1.3 for its definition). Then our main result is the following theorem, which reduces the problem of the absolute convergence of the spectral side of the Arthur trace formula to a problem about local components of automorphic representations. Theorem 0.1. Suppose that for every M ∈ L(M_{0}), Q, P ∈ P(M) and every place v the following holds.

1) If v < ∞, then for every open compact subgroup K_{v} ⊂ G(ℚ_{v}) and every invariant differential operator D_{λ} on i𝔞_{M}^{∗} there exists C > 0 such that

(0.2)  ∥ D_{λ}R_{Q|P} (π_{v}, λ)_{K_{v}} ∥≤ C

for all λ ∈ i𝔞_{M}^{∗} and all π_{v} ∈ Π_{disc}(M(ℚ_{v})).

2) If v = ∞, then for all invariant differential operators D_{λ} on i𝔞_{M}^{∗} there exist C > 0 and N ∈ ℕ such that

(0.3)  ∥ D_{λ}R_{Q|P} (π, λ)_{σ} ∥≤ C(1+ ∥ λ ∥ +|λ_{π}| + |λ_{σ}|)^{N}

for all λ ∈ i𝔞_{M}^{∗}, σ ∈ Π(K_{∞}) and π ∈ Π_{disc}(M(ℝ)).

Then for every f ∈ C^{1}(G(𝔸)^{1}), the spectral side of the trace formula is absolutely convergent.

We add some comments about the assumptions 1) and 2). It follows from results of Arthur [A5, p.51] and [A8, Lemma 2.1] that (0.2) and (0.3) hold uniformly for tempered representations π_{v}. On the other hand, to establish (0.2), (0.3) or (0.4) is not a problem of pure local harmonic analysis. One can not expect that these estimations will hold uniformly for all π_{v} ∈ Π(M(ℚ_{v})). Let, for example, dim 𝔞_{M}/𝔞_{G} = 1 and suppose that for each ϵ > 0 there exists π_{v} ∈ Π(M(ℚ_{v})) such that the normalized intertwining operator R_{\overline{P} |P} (π_{v}, λ) has a pole λ_{0} with | Re(λ_{0})| ≤ ϵ. Then it is certainly not possible to obtain uniform estimates for derivatives of R_{\overline{P} |P} (π_{v}, λ) along the imaginary axis. An example where this actually occurs is GL_{n}.

Especially the uniformity in σ in (0.3) seems to be difficult to achieve. Of course, this condition can be relaxed in various ways. If we relax (0.3) by not requesting uniformity in σ, we get the following weaker version of Theorem 0.1 which suffices for many purposes. Let K = ∏_{v} K_{v} be a maximal compact subgroup of G(𝔸) which is admissible relative to M_{0} (see §1.2). Theorem 0.2. Suppose that in Theorem 0.1 in place of condition 2) the following condition holds:

2’) If v = ∞, then for all invariant differential operators D_{λ} on i𝔞_{M}^{∗} and all σ ∈ Π(K_{∞}) there exist C > 0 and N ∈ ℕ such that

(0.4)  ∥ D_{λ}R_{Q|P} (π, λ)_{σ} ∥≤ C(1+ ∥ λ ∥ +|λ_{π}|)^{N}

for all λ ∈ i𝔞_{M}^{∗} and π ∈ Π_{disc}(M(ℝ)).

Then for every K-finite f ∈ C^{1}(G(𝔸)^{1}), the spectral side of the trace formula is absolutely convergent.

At the moment we don’t know how to prove any of the conditions (0.2), (0.3) and (0.4) in general. However, for G = GL_{n}, considered as an algebraic group over a number field, we are able to prove (0.2) and (0.4). The method relies on work of Luo, Rudnick and Sarnak [LRS] who established nontrivial bounds towards the generalized Ramanujan conjecture. For GL_{n} any local component of a cuspidal automorphic representation is equivalent to a full induced representation I_{P}^{G}(τ, s) where τ is tempered and the parameters s = (s_{1}, ..., s_{r}) satisfy s_{1} > s_{2} > ⋯ > s_{r} and |s_{i}| < 1/2. If π_{v} is unramified, it follows from Theorem 2 of [LRS] that the s_{i}’s satisfy the nontrivial bound

(0.5)  |s_{i}| < \frac{1}{2} − \frac{1}{n^{2} + 1}, i = 1, ..., r.

Using the method of [LRS], one can show that (0.5) holds also for the ramified components. Furthermore, using the work of Mœglin and Waldspurger [MW] on the residual spectrum, one can show that similar nontrivial bounds exist for the continuous parameters of any local component of an automorphic representation in the discrete spectrum of GL_{m}(𝔸). These bounds are the essential ingredients in the proof of (0.2) and (0.4) in the case of GL_{n}. Details will appear in a forthcoming paper with B. Speh [MS].

Now we shall explain the main steps of the proof of Theorem 0.1. First observe that M_{P |P} (s, 0) is unitary. Therefore, in order to estimate the trace norm of (0.1), it suffices to estimate the integral

(0.6)  ∫_{i𝔞_{L}^{∗}/i𝔞_{G}^{∗}} ∥ 𝔐_{L}(P, λ)ρ_{χ,π}(P, λ, f) ∥_{1} dλ.

To deal with this integral, we introduce a certain normalization of intertwining operators. For π ∈ Π(M(𝔸)) let A_{π}^{2}(P) be the space of square integrable automorphic forms of type π (see §1). Let M_{Q|P} (π, λ) denote the restriction of the intertwining operator M_{Q|P} (λ) to A_{π}^{2}(P). Let π = ⊗_{v}π_{v} and let r_{Q|P} (π_{v}, λ) be the normalizing factor for the local intertwining operator considered above. Suppose π = ⊗π_{v} occurs in the discrete spectrum of M(𝔸), which is equivalent to A_{π}^{2}(P) ≠ 0, then the Euler product

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

converges absolutely in some chamber and r_{Q|P} (π, λ) admits a meromorphic continuation to 𝔞_{M,ℂ}^{∗}. Using this meromorphic function, we introduce the normalized global intertwining operator by

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

By definition, the operator N_{Q|P} (π, λ) is equivalent to the direct sum of finitely many copies of ⊗_{v}R_{Q|P} (π_{v}, λ).

Let 𝔐_{L}(P, π, λ) be the restriction of 𝔐_{L}(P, λ) to the subspace A_{π}^{2}(P). It follows from Arthur’s theory of (G, M) families [A4, p.1329] that

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

where the sum runs over all parabolic subgroups S containing L, the operator 𝔑′_{S}(P, π, λ) is built out of normalized intertwining operators on the local groups G(ℚ_{v}) and ν_{L}^{S}(P, π, λ) is a scalar valued function which is defined in terms of normalizing factors. This reduces the estimation of the integral (0.6) to two separate problems, one involving 𝔑′_{S}(P, π, λ) and the other one ν_{L}^{S}(P, π, λ).

First we are dealing with ν_{L}^{S}(P, π, λ). By Proposition 7.5 of [A4], ν_{L}^{S}(P, π, λ) can be expressed in terms of logarithmic derivatives of normalizing factors associated with maximal parabolic subgroups in certain Levi subgroups. Therefore we may assume that dim(𝔞_{P} /𝔞_{G}) = 1. Let α be the unique simple root of (P, A). Then there exists a meromorphic function r̃_{\overline{P} |P} (π, z) of one variable such that r_{\overline{P} |P} (π, λ) = r̃_{\overline{P} |P} (π, λ(α^{∨})), and our problem is to derive estimates, which are uniform with respect to π, of integrals of the form

(0.8)  ∫_{ℝ} |r̃_{\overline{P} |P} (π, iu)^{−1} \frac{d}{du}r̃_{\overline{P} |P} (π, iu)| (1 + u^{2})^{−N}du.

To deal with this integral, we note that r̃_{\overline{P} |P} (π, z) is a meromorphic function of order n = 16 dim G + 2. This follows from (0.7), since by Theorem 0.1 of [Mu3], the matrix coefficients of M_{Q|P} (π, λ) are meromorphic functions of order ≤ n and by Theorem 2.1 of [A7], the normalized local intertwining operators R_{Q|P} (π_{v}, λ) are rational functions of q_{v}^{−λ(α^{∨})}, if v < ∞, and of λ(α^{∨}), if v = ∞. Thus there exist entire functions r_{i}(π, z), i = 1, 2, of order ≤ n such that r̃_{\overline{P} |P} (π, z) = r_{1}(π, z)/r_{2}(π, z). Using the representation of r_{i}(π, z) as a Weierstraß product, we reduce the estimation of the integral (0.8) to the the estimation of the number of poles, counted with their order, of r̃_{\overline{P} |P} (π, z) in a circle of radius R > 0. By (0.7), this problem is closely related to the estimation of the number of poles, counted with their order, of matrix coefficients of M_{Q|P} (π, λ) in a circle of radius R > 0. The latter problem has been settled in [Mu3, Proposition 6.6]. Together with Proposition 7.5 of [A4], these estimates imply estimates for the corresponding integrals involving ν_{L}^{S}(P, π, λ). In this way we get Theorem 5.4 which is our main technical result.

Next consider 𝔑′_{S}(P, π, λ). Given an open compact subgroup K_{f} of G(𝔸_{f}) and σ ∈ Π(K_{∞}), let A_{π}^{2}(P)_{K_{f} ,σ} be the subspace of A_{π}^{2}(P) consisting of all automorphic forms which are K_{f}-invariant and transform under K_{∞} according to σ. Let 𝔑′_{S}(P, π, λ)_{K_{f} ,σ} be the restriction of 𝔑′_{S}(P, π, λ) to the subspace A_{π}^{2}(P)_{K_{f} ,σ}. Now observe that for any h ∈ C^{1}(G(𝔸)^{1}) there exists an open compact subgroup K_{f} of G(𝔸_{f}) such that h is left and right invariant under K_{f}. Then the estimation of ∥ 𝔑′_{S}(P, π, λ)ρ_{χ,π}(P, λ, h) ∥_{1} can be reduced to the estimation of ∥ 𝔑′_{S}(P, π, λ)_{K_{f} ,σ} ∥ where σ runs over Π(K_{∞}). By Arthur’s theory of (G, M)-families, the estimation of the norm of the finite rank operators 𝔑′_{S}(P, π, λ)_{K_{f} ,σ} can be reduced to the estimation of derivatives of finitely many normalized local intertwining operators R_{Q|P} (π_{v}, λ)_{K_{v}}, v < ∞, and R_{Q|P} (π_{∞}, λ)_{σ}. Combined with Theorem 5.4 this implies Theorem 0.1. The proof of Theorem 0.2 is similar.

The paper is organized as follows. In §1 we collect some preliminary facts. In §2 we discuss briefly normalized local and global intertwining operators. The local normalizing factors are studied in some detail in §3. We recall the definition of the normalizing factors and we prove some results that we need in the next section. In §4 we investigate the poles of the global normalizing factors. This section is mainly based on results obtained in [Mu3]. In §5 we establish Theorem 5.4 which is the main result about generalized logarithmic derivatives of global normalizing factors. In §6 we study the absolute convergence of the spectral side of the trace formula and we prove our main results, Theorem 0.1 and Theorem 0.2. In §7 we discuss the example of GL_{n} and we sketch a method to prove (0.2) and (0.4).

Acknowledgments Most of this work was done during the author’s stay at the Institute for Advanced Study in Princeton. The author is very grateful to the IAS for hospitality and financial support by a grant from the Ellentuck Fund. He also would like to thank J. Cogdell, D. Ramakrishnan and F. Shahidi for some very useful discussions. Especially he is very grateful to W. Hoffmann and the referee for their comments, suggestions and corrections which helped to improve the paper considerably.

1. Preliminaries

We shall follow partially the notation introduced by Arthur [A1]-[A4].

1.1. Let G be a reductive algebraic group defined over ℚ. As in [A4], [A5], we shall fix a subgroup M_{0} of G, defined over ℚ, which is a Levi component of some minimal parabolic subgroup of G defined over ℚ. In this paper, a parabolic subgroup will mean a parabolic subgroup of G, defined over ℚ, and a Levi subgroup of G will mean a subgroup of G which contains M_{0} and is the Levi component of some parabolic subgroup of G. It is a reductive subgroup of G which is 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), respectively. 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.

Suppose that M ⊂ M_{1} ⊂ L are Levi subgroups of G. If Q ∈ P^{L}(M_{1}) and R ∈ P^{M_{1}}(M), there is a unique group Q(R) ∈ P^{L}(M) which is contained in Q and whose intersection with M_{1} is R.

Let A_{P} be the split component of the center of M_{P}. A_{P} is defined over ℚ. Let X(M_{P})_{ℚ} be the group of characters of M_{P} defined over ℚ. Then

𝔞_{P} = Hom(X(M_{P})_{ℚ}, ℝ)

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

𝔞_{P}^{∗} = X(M_{P})_{ℚ} ⊗ ℝ.

We shall often denote A_{P}, 𝔞_{P} and 𝔞_{P}^{∗} by A_{M}, 𝔞_{M} and 𝔞_{M}^{∗}, respectively, since they depend only on M. Also, we shall write A_{0} = A_{M_{0}}, 𝔞_{0} = 𝔞_{M_{0}} and 𝔞_{0}^{∗} = 𝔞_{M_{0}}^{∗}.

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

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 𝔞_{P}^{Q} is dual to 𝔞_{P}^{∗} /𝔞_{Q}^{∗}.

The group M_{Q}∩P is a parabolic subgroup of M_{Q} with unipotent radical

N_{P}^{Q} = N_{P} ∩ M_{Q}.

Let Δ_{P}^{Q} be the set of simple roots of (M_{Q} ∩P, A_{P}). Then Δ_{P}^{Q} is a subset of Δ_{P}. We may identify 𝔞_{Q} with the subspace

{H ∈ 𝔞_{P} | α(H) = 0, α ∈ Δ_{P}^{Q}}.

Furthermore, to Δ_{P}^{Q} one can associate a basis {α^{∨} | α ∈ Δ_{P}^{Q}} of 𝔞_{P}^{Q} and Δ̂_{P}^{Q} is defined to be the corresponding dual basis of (𝔞_{P}^{Q})^{∗} [A2]. Then Δ_{P}^{Q} and Δ̂_{P}^{Q} are naturally embedded subsets of 𝔞_{0}^{∗}. Let

𝔞_{P}^{+} = {H ∈ 𝔞_{P} | α(H) > 0 for all α ∈ Δ_{P} },

and

(𝔞_{P}^{∗})^{+} = {Λ ∈ 𝔞_{P}^{∗} | Λ(α^{∨}) > 0 for all α ∈ Δ_{P} }.

We shall denote the restricted Weyl group of (G, A_{0}) by W_{0}. It acts on 𝔞_{0} and 𝔞_{0}^{∗} in the usual way. For every s ∈ W_{0} we shall fix a representative w_{s} in the intersection of G(ℚ) with the normalizer of A_{0}. w_{s} is determined modulo M_{0}(ℚ). If P_{1} and P_{2} are parabolic subgroups, let W (𝔞_{P_{1}}, 𝔞_{P_{2}}) denote the set of distinct isomorphisms from 𝔞_{P_{1}} onto 𝔞_{P_{2}} obtained by restricting elements of W_{0} to 𝔞_{P_{1}}. P_{1} and P_{2} are said to be associated if W (𝔞_{P_{1}}, 𝔞_{P_{2}}) is not empty.

1.2. We fix an embedding of G into GL_{n}, defined over ℚ. For a given place v of ℚ, let G(ℚ_{v}) be the group of ℚ_{v}-rational points of G. Let 𝔸 be the ring of adèles of ℚ and let G(𝔸) be the corresponding adèlevalued group. If f stands for the set of finite places of ℚ and 𝔸_{f} is the corresponding ring of finite adèles, then

G(𝔸) = G(ℝ) × G(𝔸_{f}).

For any prime p, let

G(ℤ_{p}) = GL_{n}(ℤ_{p}) ∩ G(ℚ_{p}).

This is an open compact subgroup of G(ℚ_{p}). We shall fix a maximal compact subgroup

K = ∏_{v} K_{v}

of G(𝔸) which is admissible relative to M_{0} in the sense of [A5]. For any such K the following properties hold:

1) For almost all primes p, one has K_{p} = G(ℤ_{p}).

2) For every finite p, K_{p} is a special maximal compact subgroup. This implies that G(ℚ_{p}) = P_{0}(ℚ_{p}) · K_{p} for all P_{0} ∈ F(M_{0}).

3) The Lie algebras of K_{ℝ} and A_{0}(ℝ) are orthogonal with respect to the Killing form.

Given P ∈ F(M_{0}), let

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

This is a closed subgroup of M_{P} (𝔸), and M_{P} (𝔸) is the direct product of M_{P} (𝔸)^{1} and A_{P} (ℝ)^{0}. By the assumptions on K, G(𝔸) = P (𝔸)K. Therefore, any x ∈ G(𝔸) can be written as

(1.1) x = namk, n ∈ N_{P} (𝔸), m ∈ M(𝔸)^{1}, a ∈ A_{P} (ℝ)^{0}, k ∈ K.

Let

H_{P} : G(𝔸) → 𝔞_{P}

be the associated height function as defined in [A2]. Then H_{P} (x) is the image of a ∈ A_{P} (ℝ)^{0} in the decomposition (1.1) with respect to the isomorphism A_{P} (ℝ)^{0} ≅ 𝔞_{P}.

We shall fix a Euclidean norm ∥ · ∥ on 𝔞_{0} which is invariant under the action of the Weyl group of (G, A_{0}). On each space 𝔞_{P}^{Q} , P ⊂ Q, we take as Haar measure the Euclidean measure associated to the restriction of ∥ · ∥ to 𝔞_{P}^{Q}. We then normalize the Haar measures on K, G(𝔸), N_{P} (𝔸), M_{P} (𝔸), A_{P} (ℝ)^{0}, M_{P} (𝔸)^{1}, etc. as in [A2].

1.3. Let Ξ and σ be the functions that enter the definition of Harish-Chandra’s Schwartz space on G(ℝ) [Wa2, p.156] and extend them to functions on G(𝔸) in the obvious way. For any place v, let G(ℚ_{v})^{1} denote the intersection of G(ℚ_{v}) with G(𝔸)^{1}. Let U(𝔤(ℝ)^{1} ⊗ ℂ) be the universal enveloping algebra of the complexification of the Lie algebra of G(ℝ)^{1}. Let K_{f} be an open compact subgroup of G(𝔸_{f})^{1}. Then the double coset space K_{f}\G(𝔸)^{1}/K_{f} is a discrete union of countably many copies of G(ℝ)^{1}. In particular it is a differentiable manifold. Suppose that Ω is a subset of G(𝔸)^{1} such that K_{f} · Ω · K_{f} = Ω and K_{f}\Ω/K_{f} is the disjoint union of finitely many copies of G(ℝ)^{1}. Let C^{1}(G(𝔸)^{1}; Ω, K_{f}) be the space of all functions h: G(𝔸)^{1} → ℂ satisfying the following conditions:

(1) h is bi-invariant under K_{f}, supp h ⊂ Ω, and h : K_{f}\Ω/K_{f} → ℂ is a smooth function.

(2) For all D_{1}, D_{2} ∈ U(𝔤(ℝ)^{1} ⊗ ℂ) and all r ∈ ℕ, we have

\begin{aligned}∥ h & ∥_{D_{1},D_{2},r} \\ & := sup_{x∈G(𝔸)^{1}} ((1 + σ(x))^{r}Ξ^{−2}(x)|D_{1} ∗ h ∗ D_{2}(x)|) < ∞. \end{aligned}

Let C^{1}(G(𝔸)^{1}; Ω, K_{f}) be equipped with the topology defined by the semi-norms ∥ · ∥_{D_{1},D_{2},r}. Let C^{1}(G(𝔸)^{1}) be the topological direct limit over all pairs (Ω, K_{f}) of the spaces C^{1}(G(𝔸)^{1}; Ω, K_{f}).

1.4. Let H be any algebraic group over ℚ and let F be a local field. We shall denote by Π(H(𝔸)) (resp. Π(H(F)), Π(K), etc.) the set of equivalence classes of irreducible unitary representations of H(𝔸) (resp. H(F), K, etc.).

1.5. Given a unitary character ξ of A_{P} (ℝ)^{0} , let L^{2}(M_{P} (ℚ)\M_{P} (𝔸))_{ξ} be the space of all measurable functions ϕ on M_{P} (ℚ)\M_{P} (𝔸) such that ϕ(xm) = ξ(x)ϕ(m) for all x ∈ A_{P} (ℝ)^{0}, m ∈ M_{P} (𝔸), and ϕ is square integrable on M_{P} (ℚ)\M_{P} (𝔸)^{1}. Let Π_{disc}(M_{P} (𝔸))_{ξ} denote the subspace of all π ∈ Π(M_{P} (𝔸)) which are equivalent to a subrepresentation of the regular representation of M_{P} (𝔸) on L^{2}(M_{P} (ℚ)\M_{P} (𝔸))_{ξ}. Set

Π_{disc}(M_{P} (𝔸)) = ⋃_{ξ∈Π(A_{P} (ℝ)^{0})}Π_{disc}(M_{P} (𝔸))_{ξ}.

Recall that Π(M_{P} (𝔸)^{1}) can be canonically identified with the set of orbits under the action of i𝔞_{P}^{∗} defined by

π → π_{λ} = e^{λ(H_{P} (·))}π, π ∈ Π(M_{P} (𝔸)), λ ∈ i𝔞_{P}^{∗} .

Since M_{P} (𝔸) is the direct product of M_{P} (𝔸)^{1} and A_{P} (ℝ)^{0}, any representation of M_{P} (𝔸)^{1} corresponds to a representation of M_{P} (𝔸) which is trivial on A_{P} (ℝ)^{0}. We identify these two representations and in this way we obtain an embedding of Π(M_{P} (𝔸)^{1}) in Π(M_{P} (𝔸)).

Given π ∈ Π(M_{P} (𝔸)) with π = ⊗_{v}π_{v}, set π_{f} = ⊗_{v<∞}π_{v}. For an open compact subgroup K_{f} ⊂ G(𝔸_{f}), let

K_{M,f} = M_{P} (𝔸_{f}) ∩ K_{f}.

Set

(1.2) Π_{disc}(M_{P} (𝔸); K_{f}) = {π ∈ Π_{disc}(M_{P} (𝔸)) | π_{f}^{K_{M,f}} ≠ {0}} .

Let Π_{disc}(M_{P} (𝔸)^{1}; K_{f}) be the intersection of Π_{disc}(M_{P} (𝔸); K_{f}) with the subspace Π_{disc}(M_{P} (𝔸)^{1}) of Π_{disc}(M_{P} (𝔸)).

1.6. Let P = NM be a parabolic subgroup and let ϕ be a measurable, locally integrable function on N(ℚ)\G(𝔸). Then the constant term ϕ_{P} of ϕ along P is defined for almost every g by

(1.3)  ϕ_{P} (g) = ∫_{N(ℚ)\N(𝔸)} ϕ(ng) dn.

This is a measurable, locally integrable function on N(𝔸)\G(𝔸).

1.7. Let P be a parabolic subgroup. Then we denote by A^{2}(P) the space of automorphic forms on N_{P} (𝔸)M_{P} (ℚ)\G(𝔸) which are square integrable on M_{P} (ℚ)\M_{P} (𝔸)^{1} × K. This is the space of smooth functions

ϕ: N_{P} (𝔸)M_{P} (ℚ)\G(𝔸) → ℂ

which satisfy the following conditions:

i) The span of the set of functions

x ↦ (zϕ)(xk), x ∈ G(𝔸),

indexed by k ∈ K and z ∈ Z(𝔤_{ℂ}), is finite dimensional. ii)

∥ ϕ ∥^{2}= ∫_{K} ∫_{M_{P} (ℚ)\M_{P} (𝔸)^{1}} |ϕ(mk)|^{2} dm dk < ∞.

Furthermore, an automorphic form ϕ ∈ A^{2}(P) is called cuspidal, if the following additional condition holds:

iii) For all standard parabolic subgroups Q ⫋ P, ϕ_{Q} = 0.

The subspace of all cuspidal automorphic forms in A^{2}(P) will be denoted by A_{0}^{2}(P).

1.8. Given π ∈ Π_{disc}(M_{P} (𝔸))_{ξ}, let A_{π}^{2}(P) be the subspace of A^{2}(P) consisting of all functions ϕ 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 L^{2}(M_{P} (ℚ)\M_{P} (𝔸))_{ξ}. If π ∈ Π(M_{P} (𝔸)) is not contained in Π_{disc}(M_{P} (𝔸)), we put A_{π}^{2}(P) = 0. Let K_{f} be an open compact subgroup of G(𝔸_{f}). Then we denote by A_{π}^{2}(P)_{K_{f}} the subspace of all K_{f}-invariant functions in A_{π}^{2}(P). Furthermore, if σ ∈ Π(K_{∞}), then we denote by A_{π}^{2}(P)_{K_{f} ,σ} the σ-isotypical subspace of A_{π}^{2}(P)_{K_{f}}.

1.9. Let 𝔛 be the set of W_{0} conjugacy classes of pairs (M_{B}, r_{B}), where B is a parabolic subgroup and r_{B} is an irreducible cuspidal automorphic representation of M_{B}(𝔸)^{1}. Let

L^{2}(M_{P} (ℚ)\M_{P} (𝔸)^{1}) = ⨁_{χ∈𝔛}L^{2}(M_{P} (ℚ)\M_{P} (𝔸)^{1})_{χ}

be the decomposition of L^{2}(M_{P} (ℚ)\M_{P} (𝔸)^{1}) introduced by Arthur in [A2, Section 3]. Given χ ∈ 𝔛, let A_{χ,π}^{2}(P) be the subspace of A_{π}^{2}(P) consisting of all function ϕ such that for each x ∈ G(𝔸), the restriction of ϕ_{x} to M_{P} (𝔸)^{1} belongs to L^{2}(M_{P} (ℚ)\M_{P} (𝔸)^{1})_{χ}.

If we identify Π(M_{P} (𝔸)^{1}) with a subset of Π(M_{P} (𝔸)), then A_{χ,π}^{2}(P) is well defined for any π ∈ Π(M_{P} (𝔸)^{1}). This is a space of functions on N_{P} (𝔸)M_{P} (𝔸)A_{P} (ℝ)^{0}\G(𝔸). The direct sum

⨁_{π∈Π(M_{P} (𝔸)^{1})}A_{χ}^{2}(P, π)

is the space that was denoted by A^{2}(P, χ) in [Mu3].

1.10. Let \overline{A}^{2}(P) be the Hilbert space completion of A^{2}(P). For any λ ∈ 𝔞_{P,ℂ}^{∗} we have an induced representation ρ(P, λ) of G(𝔸) on \overline{A}^{2}(P) which is defined by

(1.4)  (ρ(P, λ, y)ϕ)(x) = ϕ(xy)e^{(λ+ρ_{P})(H_{P} (xy))}e^{−(λ+ρ_{P})(H_{P} (x))},

for elements x, y ∈ G(𝔸) and ϕ ∈ \overline{A}^{2}(P). The Hilbert space completions \overline{A}_{π}^{2}(P) and \overline{A}_{χ,π}^{2}(P) of the subspaces A_{π}^{2}(P) and A_{χ,π}^{2}(P), respectively, are invariant under ρ(P, λ) and we shall denote the restriction of ρ(P, λ) to A_{π}^{2}(P) (resp. A_{χ,π}^{2}(P)) by ρ_{π}(P, λ) (resp. ρ_{χ,π}(P, λ)).

1.11. Given any irreducible unitary representation π of M_{B}(𝔸)^{1}, let λ_{π} be the eigenvalue of the Casimir operator of M_{B}(ℝ), acting in the Gårding space H_{π_{∞}}^{∞} of the Archimedean constituent π_{∞} of π. For χ ∈ 𝔛 and (M_{B}, r_{B}) ∈ χ, the Casimir eigenvalue λ_{r_{B}} depends only on the class χ and we denote it by λ_{χ}.

2. Normalized intertwining operators

Let M, M_{1} ∈ L(M_{0}), P ∈ P(M) and P_{1} ∈ P(M_{1}). For each s ∈ W (𝔞_{M}, 𝔞_{M_{1}}), ϕ ∈ A^{2}(P), and λ ∈ 𝔞_{P,ℂ}^{∗} such that Re(λ) ∈ (𝔞_{P}^{∗})^{+} + ρ_{P}, let M_{P_{1}|P} (s, λ)ϕ be defined by

\begin{aligned}M & _{P_{1}|P} (s, λ)ϕ(x) = e^{−(sλ+ρ_{P_{1}})(H_{P_{1}}(x))} \\ & ∫_{N_{1}(𝔸)∩w_{s}N(𝔸)w_{s}^{−1}\N_{1}(𝔸)} ϕ(w_{s}^{−1}n_{1}x)e^{(λ+ρ_{P})(H_{P} (w_{s}^{−1}n_{1}x))} dn_{1} \end{aligned} ; (2.1)

for x ∈ G(𝔸). The integral is absolutely convergent for λ as above and admits an analytic continuation to a meromorphic function of λ ∈ 𝔞_{P,ℂ}^{∗} with values in the space of linear operators from A^{2}(P) to A^{2}(P_{1}). This operator is the global intertwining operator

M_{P_{1}|P} (s, λ): A^{2}(P) → A^{2}(P_{1}).

Let π ∈ Π_{disc}(M(𝔸)) and χ ∈ 𝔛. Then M_{P_{1}|P} (s, λ) maps the subspace A_{π}^{2}(P) (resp. A_{χ,π}^{2}(P)) to A_{sπ}^{2}(P_{1}) (resp. A_{χ,sπ}^{2}(P_{1})). The main functional equations are

(2.2)  M_{P_{2}|P} (ts, λ) = M_{P_{2}|P_{1}}(t, sλ)M_{P_{1}|P} (s, λ)

for t ∈ W (𝔞_{1}, 𝔞_{2}) and s ∈ W (𝔞, 𝔞_{1}).

By (1.4) and (1.5) of [A4], most of the considerations concerning intertwining operators can be reduced to the case where P_{1} and P have the same Levi component M, and s = 1.

Thus, from now on we shall assume that P, Q ∈ P(M) and we put

M_{Q|P} (λ) := M_{Q|P} (1, λ), λ ∈ 𝔞_{M,ℂ}^{∗}.

Given π ∈ Π(M(𝔸)), let

M_{Q|P} (π, λ): A_{π}^{2}(P) → A_{π}^{2}(Q)

be the restriction of M_{Q|P} (λ) to A_{π}^{2}(P). We shall now express this operator in terms of local intertwining operators. Let π_{λ} be the representation of P (𝔸) which is defined by

π_{λ}(nm) = e^{λ(H_{M} (m))}π(m), n ∈ N_{P} (𝔸), m ∈ M_{P} (𝔸).

Let (I_{P}^{G}(π_{λ}), H_{P} (π)) be the induced representation of G(𝔸). Similarly let (I_{Q}^{G}(π_{λ}), H_{Q}(π)) be the representation of G(𝔸) induced from Q(𝔸). Let ξ be a unitary character of A_{M}(ℝ)^{0} and suppose that π ∈ Π_{disc}(M(𝔸))_{ξ}. We extend ξ by 1 to a character of M(ℚ)A_{M}(ℝ)^{0}. Then there is a canonical isomorphism

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

of G(𝔸)-modules where G(𝔸) acts on the left by I_{P}^{G}(π_{λ})⊗Id. A similar isomorphism j_{Q} exists with respect to Q. Let H_{P}^{0} (π) (resp. H_{Q}^{0}(π)) be the subspace of elements which are right K-finite and left Z(𝔤_{ℂ})-finite. Using (2.3), it follows that M_{Q|P} (π, λ) induces an intertwining operator

J_{Q|P} (π, λ): H_{P}^{0} (π) → H_{Q}^{0}(π)

such that

j_{Q} ∘ (J_{Q|P} (π, λ) ⊗ Id) = M_{Q|P} (π, λ) ∘ j_{P} .

It follows from (2.1) that for Re(λ) ∈ (𝔞_{P}^{∗})^{+} + ρ_{P}, this operator is defined by the following convergent integral

\begin{aligned}J_{Q|P} (π, λ)ϕ(x) & = e^{−(λ+ρ_{Q})(H_{Q}(x))} \\ & · ∫_{N_{Q}(𝔸)∩N_{P} (𝔸)\N_{Q}(𝔸)} ϕ(nx)e^{(λ+ρ_{P})(H_{P} (nx))} dn. \end{aligned} ; (2.4)

where x ∈ G(𝔸) and ϕ ∈ H_{P}^{0} (π).

Let v be any place of ℚ and let (π_{v}, V_{v}) ∈ Π(M(ℚ_{v})). Given λ ∈ 𝔞_{M,ℂ}^{∗}, let π_{v,λ} be the representation of P (ℚ_{v}) on V_{v} defined by

π_{v,λ}(n_{v}m_{v}) = π_{v}(m_{v})e^{λ(H_{M} (m_{v}))}, n_{v} ∈ N(ℚ_{v}), m_{v} ∈ M(ℚ_{v}).

Let (I_{P}^{G}(π_{v,λ}), H_{P} (π_{v})) denote induced representation. The Hilbert space is the space of measurable functions

ϕ_{v} : N(ℚ_{v}) \ G(ℚ_{v}) → V_{v}

such that

1. ϕ_{v}(m_{v}x_{v}) = π(m_{v})ϕ_{v}(x_{v}), m_{v} ∈ M(ℚ_{v}), x_{v} ∈ G(ℚ_{v});

2. ∥ ϕ_{v} ∥^{2}= ∫_{K_{v}} ∥ ϕ_{v}(k) ∥_{V_{v}}^{2} dk < ∞.

Let H_{P}^{0} (π_{v}) ⊂ H_{P} (π_{v}) be the subspace of K_{v}-finite functions. Then the local intertwining operator

J_{Q|P} (π_{v}, λ) : H_{P}^{0} (π_{v}) → H_{Q}^{0}(π_{v})

is defined by

\begin{aligned}J_{Q|P} & (π_{v}, λ)ϕ_{v}(x_{v}) = e^{−(λ+ρ_{Q})(H_{Q}(x_{v}))} \\ & · ∫_{N_{Q}(ℚ_{v})∩N_{P} (ℚ_{v})\N_{Q}(ℚ_{v})} ϕ_{v}(n_{v}x_{v})e^{(λ+ρ_{P})(H_{P} (n_{v}x_{v}))}dn_{v}. \end{aligned} ; (2.5)

The integral converges absolutely for Re(λ) ∈ (𝔞_{P}^{∗})^{+} + ρ_{P} and can be continued to a meromorphic function of λ ∈ 𝔞_{M,ℂ}^{∗} with values in the space of linear operators from H_{P}^{0} (π_{v}) to H_{Q}^{0}(π_{v}) [Sh1].

Now let π ∈ Π(M(𝔸)). Then π is a restricted tensor product

π = ⊗_{v}π_{v}

where almost all (π_{v}, V_{v}) are unramified, i.e., dim V_{v}^{K_{v}∩M(ℚ_{v})} = 1 for almost all v. Moreover, we have

(I_{P}^{G}(π_{λ}), H_{P} (π)) ≅ (⊗_{v}I_{P}^{G}(π_{v,λ}), ⊗_{v}H_{P} (π_{v})) .

Let ϕ ∈ H_{P}^{0} (π) and suppose that ϕ = ⊗_{v}ϕ_{v}. Observe that each ϕ_{v} belongs to H_{P}^{0} (π_{v}) and for almost all v, ϕ_{v} is K_{v}–invariant. Comparing (2.4) and (2.5), it follows that

(2.6)  J_{Q|P} (π, λ)ϕ = ⊗_{v}(J_{Q|P} (π_{v}, λ)ϕ_{v})

whenever the product on the right converges.

It is possible to normalize local intertwining operators. Let v be any valuation of ℚ and let π_{v} ∈ Π(M(ℚ_{v})). It is proved in [A7], [CLL] that there exist scalar valued meromorphic functions r_{Q|P} (π_{v}, λ) of λ ∈ 𝔞_{P,ℂ}^{∗} such that the normalized intertwining operators

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

satisfy the conditions (R_{1}) − (R_{8}) of Theorem 2.1 of [A7]. We recall some of the properties satisfied by the normalized intertwining operators.

(R.1) If S ∈ P(M), then

(2.8)  R_{S|P} (π_{v}, λ) = R_{S|Q}(π_{v}, λ)R_{Q|P} (π_{v}, λ).

(R.2)

(2.9)  R_{Q|P} (π_{v}, λ)^{∗} = R_{P |Q}(π_{v}, −\overline{λ}).

(R.3) Let L ∈ L(M), S ∈ P(L), and Q, Q′ ∈ P^{L}(M). Then

(2.10)  (R_{S(Q′)|S(Q)}(π_{v}, λ)ϕ)_{k} = R_{Q′|Q}(π_{v}, λ)ϕ_{k}

for any ϕ ∈ H_{P}^{0} (S(R)(ℚ_{v})) and k ∈ K_{v}. (R.4) Let v be a finite place. Suppose that π_{v} is unramified, and that

K_{v} is very special. Let ϕ_{v} ∈ H_{P} (π_{v}) be a function such that ϕ_{v}(k) = ϕ_{v}(1) for all k ∈ K_{v}. Then in the compact picture of the induced representation, one has

(2.11)  R_{Q|P} (π_{v}, λ)ϕ_{v} = ϕ_{v}.

The functions r_{Q|P} (π_{v}, λ) are called normalizing factors. They satisfy similar properties. We recall some of them. Given P ∈ P(M), let Σ_{P}^{r} be the set of reduced roots of (P, A_{M}).

(r.1) For β ∈ Σ_{P}^{r}, let M_{β} ∈ L(M) be such that

𝔞_{M_{β}} = {H ∈ 𝔞_{M} | β(H) = 0}

and let P_{β} be the unique group in P^{M_{β}}(M) whose simple root is β. Then

(2.12)  r_{Q|P} (π_{v}, λ) = ∏_{β∈Σ_{Q}^{r}∩Σr ; \overline{P}} r_{\overline{P}_{β}|P_{β}}(π_{v}, λ),

Note that each r_{\overline{P}_{β}|P_{β}}(π_{v}, λ) depends only on the projection λ(β^{∨}).

(r.2) If π_{v} is an irreducible constituent of an induced representation I_{R}^{M}(σ_{v}), where σ_{v} ∈ Π_{2}(M_{1}(ℚ_{v})), R ∈ P^{M}(M_{1}), and M_{1} ⊂ M, then

(2.13)  r_{Q|P} (π_{v}, λ) = r_{Q(R)|P (R)}(σ_{v}, λ).

(r.3)

(2.14)  r_{Q|P} (π_{v}, λ)r_{P |Q}(π_{v}, λ) = J_{Q|P} (π_{v}, λ)J_{P |Q}(π_{v}, λ).

(r.4)

(2.15)  \overline{r_{Q|P} (π_{v}, λ)} = r_{P |Q}(π_{v}, −\overline{λ}).

(r.5) If v is a finite place of ℚ, then r_{Q|P} (π_{v}, λ) is a rational function

in the variables \{q_{v}^{−λ(β̃)} | β ∈ Σ_{Q}^{r} ∩ Σr ; \overline{P} \}, where the β̃’s are suitably normalized ”coroots”. If v = ∞, then r_{Q|P} (π_{v}, λ) is a rational function in the variables \{λ(β^{∨}) | β ∈ Σ_{Q}^{r} ∩ Σr ; \overline{P} \}.

Now we return to global intertwining operators. Let π ∈ Π_{disc}(M(𝔸)). For ϕ ∈ H_{P}^{0} (π) with ϕ = ⊗_{v}ϕ_{v} set

(2.16)  R_{Q|P} (π, λ)ϕ = ⊗_{v}(R_{Q|P} (π_{v}, λ)ϕ_{v}).

Since ϕ_{v} is K_{v}- invariant for almost all v, it follows from (2.11) that the right hand side is actually a finite product and therefore, it converges for all λ ∈ 𝔞_{M,ℂ}^{∗} which are not poles of the local intertwining operators. In this way we get a a meromorphic operator valued function

R_{Q|P} (π, λ) : H_{P}^{0} (π) → H_{Q}^{0}(π)

of λ ∈ 𝔞_{M,ℂ}^{∗}. Using the isomorphism (2.3) and the corresponding one for Q, we obtain a meromorphic operator valued function

N_{Q|P} (π, λ): A_{π}^{2}(P) → A_{π}^{2}(Q)

of λ ∈ 𝔞_{M,ℂ}^{∗} such that

(2.17)  j_{Q} ∘ N_{Q|P} (π, λ) = (R_{Q|P} (π, λ) ⊗ Id) ∘ j_{P} .

Furthermore, put

(2.18)  r_{Q|P} (π, λ) = ∏_{v} r_{Q|P} (π_{v}, λ).

By (R.4) it follows that for ϕ as above, we have

J_{Q|P} (π_{v}, λ)ϕ_{v} = r_{Q|P} (π_{v}, λ)ϕ_{v}

for almost all v. Therefore, the infinite product (2.18) converges in the domain of absolute convergence of the infinite product (2.6) and for λ in this domain we have

(2.19)  M_{Q|P} (π, λ) = r_{Q|P} (π, λ)N_{Q|P} (π, λ).

Since both M_{Q|P} (π, λ) and N_{Q|P} (π, λ) are meromorphic functions of λ ∈ 𝔞_{M,ℂ}^{∗}, it follows that r_{Q|P} (π, λ) admits a meromorphic continuation to 𝔞_{M,ℂ}^{∗}. The meromorphic function r_{Q|P} (π, λ) is the global normalizing factor and N_{Q|P} (π, λ) is the normalized global intertwining operator.

Using (2.12), (2.14), (2.15) and the functional equations (2.2), it follows that r_{Q|P} (π, λ) has the following properties

(1)

(2.20)  r_{Q|P} (π, λ)r_{P |Q}(π, λ) = 1.

(2)

(2.21)  \overline{r_{Q|P} (π, λ)} = r_{P |Q}(π, −\overline{λ}).

(3) For each β ∈ Σ_{Q}^{r} ∩ Σr ; \overline{P} let P_{β} be as in (2.12). Then

(2.22)  r_{Q|P} (π, λ) = ∏_{β∈Σ_{Q}^{r}∩Σr ; \overline{P}} r_{\overline{P}_{β}|P_{β}}(π, λ).

Note that r_{\overline{P}_{β}|P_{β}}(π, λ) depends only on the projection λ(β^{∨}).

3. Local normalizing factors

In this section we shall investigate the local normalizing factors in more detail. In particular, we shall study their logarithmic derivatives. To begin with, we recall the construction of the normalizing factors.

First assume that v is a finite valuation. Then the existence of normalizing factors such that Theorem 2.1 of [A7] holds has been verified by Langlands in [CLL], Lecture 15. Let π_{v} ∈ Π(M(ℚ_{v})). The local normalizing factors r_{Q|P} (π_{v}, λ) have to satisfy (2.1)-(2.3) in [A7]. Therefore, it suffices to define them when dim(𝔞_{M}/𝔞_{G}) = 1 and π_{v} is square integrable modulo A_{G}. Assume for the moment that these conditions are satisfied. Let P ∈ P(M) and let α be the unique simple root of (P, A_{M}). Then Langlands has shown that there exists a rational function V_{P} (π_{v}, z) of one variable such that

(3.1)  r_{\overline{P} |P} (π_{v}, λ) = V_{P} (π_{v}, q_{v}^{−λ(α̃)}),

where α̃ ∈ 𝔞_{M} is independent of π_{v}. We recall the definition of V_{P} (π_{v}, z). Suppose that P_{v} is a parabolic subgroup of G defined over ℚ_{v} and let M_{v} be a Levi component of P_{v} over ℚ_{v}. Denote by A_{M_{v}} the split component of the center of M_{v}. Set

𝔞_{M_{v}} = Hom(X(M_{v})_{ℚ_{v}}, ℝ)

and

𝔞_{M_{v}}^{∗} = X(M_{v})_{ℚ_{v}} ⊗ ℝ.

Let

H_{M_{v}} : M_{v}(ℚ_{v}) → 𝔞_{M_{v}}

be defined by

q_{v}^{⟨H_{Mv} (m_{v}),χ⟩} = |χ(m_{v})|_{v}, χ ∈ X(M_{v})_{ℚ_{v}}, m_{v} ∈ M_{v}(ℚ_{v}).

Given π ∈ Π(M_{v}(ℚ_{v})) and λ ∈ i𝔞_{M_{v}}^{∗}, let π_{λ} denote the representation defined by

π_{λ}(m_{v}) = π(m_{v})e^{λ(H_{Mv} (m_{v}))}, m_{v} ∈ M_{v}(ℚ_{v}).

Let

𝔞_{M_{v},π}^{∨} = {λ ∈ i𝔞_{M_{v}}^{∗} | π_{λ} ≅ π}

denote the stabilizer of π with respect to this action of i𝔞_{M_{v}}^{∗}. Then 𝔞_{M_{v},π}^{∨} is a lattice in i𝔞_{M_{v}}^{∗} and the orbit 𝔬_{π} of π is equal to i𝔞_{M_{v}}^{∗}/𝔞_{M_{v},π}^{∨}. Let

𝔞_{M_{v},ℚ_{v}} = H_{M_{v}}(M_{v}(ℚ_{v})), 𝔞̃_{M_{v},ℚ_{v}} = H_{M_{v}}(A_{M_{v}}(ℚ_{v})).

Then 𝔞_{M_{v},ℚ_{v}} and 𝔞̃_{M_{v},ℚ_{v}} are lattices in 𝔞_{M_{v}}. Given a real vector space V and a closed subgroup V_{1} of V,

V_{1}^{∨} = Hom(V_{1}, 2πiℤ) ⊂ iV^{∗}.

let us agree to set Let 𝔞_{M_{v},π} ⊂ 𝔞_{M_{v}} be the dual lattice to 𝔞_{M_{v},π}^{∨}. Then

𝔞̃_{M_{v},ℚ_{v}} ⊂ 𝔞_{M_{v},π} ⊂ 𝔞_{M_{v},ℚ_{v}}.

Set

L_{M_{v}} = (𝔞_{M_{v},ℚ_{v}} + 𝔞_{G_{v}}) /𝔞_{G_{v}}, L̃_{M_{v}} = (𝔞̃_{M_{v},ℚ_{v}} + 𝔞_{G_{v}}) /𝔞_{G_{v}},

and

L(π) = (𝔞_{M_{v},π} + 𝔞_{G_{v}}) /𝔞_{G_{v}}.

Then L_{M_{v}}, L̃_{M_{v}}, and L(π) are lattices in 𝔞_{M_{v}}^{G_{v}} = 𝔞_{M_{v}}/𝔞_{G_{v}}.

Suppose that P_{v} is a maximal parabolic subgroup, that is dim 𝔞_{M_{v}}^{G_{v}} = 1. Then there exists α(π) ∈ 𝔞_{M_{v}} such that

L(π) = \frac{log q}{2π} ℤ(α(π)).

In [Si1] Silberger has shown that for a supercuspidal representation π there exists a rational function Ũ_{P_{v}}(π, z) such that the Plancherel measure μ(π, λ) satisfies

(3.2)  μ(π, λ) = Ũ_{P_{v}} (π, q^{−λ(α(π))}) .

Let α̃ ∈ 𝔞_{M_{v}} be such that

L_{M_{v}} = \frac{log q}{2π} ℤ(α̃).

Since L(π) ⊂ L_{M_{v}}, there exists k(π) ∈ ℤ such that α(π) = k(π)α̃. Let

(3.3)  U_{P_{v}}(π, z) = Ũ_{P_{v}} (π, z^{k(π)}) .

Then

μ(π, λ) = U_{P_{v}} (π, q^{−λ(α̃)}) .

Now suppose that P_{v} is arbitrary, but π is still supercuspidal. For each reduced root α ∈ Σ_{r}(P_{v}, A_{v}) let A_{α} denote the largest subtorus of A_{v} which lies in the kernel of the root character of α. Let M_{α} denote the centralizer of A_{α}. Let ^{∗}P_{α} = P_{v} ∩ M_{α}. Then ^{∗}P_{α} = M_{v}N_{α}.

Let μ_{α}(π, λ) be the Plancherel measure with respect to (M_{α},^{∗}P_{α}). According to [HC3], Theorem 24, there exist constants γ = γ(G/M) and γ_{α} = γ(M_{α}/M), α ∈ Σ_{r}(P_{v}, A_{v}), such that

(3.4)  γ^{−2}μ(π, λ) = ∏_{α∈Σ_{r}(P_{v},A_{v})} γ_{α}^{−2}μ_{α}(π, λ),

Hence if {α̃ | α ∈ Δ_{P_{v}}} is a set of generators of the lattice L_{M_{v}}, then μ(π, λ) is a rational function in the variables {q^{−λ(α̃)} | α ∈ Δ_{P_{v}}}. Finally, by Theorem 1 of [Si2], this can be extended to all discrete series representations of M_{v}(ℚ_{v}).

Now let P = MN be a maximal parabolic subgroup of G defined over ℚ. Then X(M)_{ℚ} ⊂ X(M)_{ℚ_{v}} induces an embedding 𝔞_{M}^{∗} ⊂ 𝔞_{M_{v}}^{∗} and by the above, it follows that there exists α̃ ∈ 𝔞_{M} and a rational function U_{P} (π, z) such that

(3.5)  μ(π, λ) = U_{P} (π, q^{−λ(α̃)}) ,

for all π ∈ Π_{2}(M(ℚ_{v})), λ ∈ 𝔞_{M,ℂ}^{∗}. As shown by Langlands [CLL], the rational function U_{P} (π, z) has the form

U_{P} (π, z) = a\frac{∏_{i=1}^{r}(1 − α_{i}z)(1 − \overline{α}_{i}^{−1}z)}{∏_{i=1}^{r}(1 − β_{i}z)(1 − \overline{β}_{i}^{−1}z)},

where the α_{i}’s and β_{i}’s satisfy |α_{i}| ≤ 1, |β_{i}| ≤ 1, i = 1, ..., r, and a is a certain constant. Then the rational function V_{P} (π, z) in (3.1) is defined by

(3.6)  V_{P} (π, z) = b\frac{∏_{i=1}^{r}(1 − β_{i}z)}{∏_{i=1}^{r}(1 − α_{i}z)}

for a suitable constant b. In particular, it follows that 2r is the number of poles of U_{P} (π, z). For our applications we need a bound for r. This is done in the following lemma. Lemma 3.1. Let M ∈ L(M_{0}) be such that dim(𝔞_{M}/𝔞_{G}) = 1. There exists C > 0 such that for all P ∈ P(M) and all π ∈ Π_{2}(M(ℚ_{v})) the number of poles of the rational function V_{P} (π, z) is less than or equal to C.

Proof. Let P_{v} be a maximal parabolic subgroup of G defined over ℚ_{v} and let π ∈ Π(M_{v}(ℚ_{v})) be supercuspidal. By Theorem 1.6 of [Si1], the rational function Ũ_{P_{v}}(π, z) in (3.2) has at most 4 poles. Now observe that L̃_{M_{v}} ⊂ L(π) ⊂ L_{M_{v}} and L_{M_{v}}/L̃_{M_{v}} is finite. This implies that the number of poles of the rational function U_{P_{v}}(π, z) defined by (3.3) is bounded by a constant which is independent of π. The general case is reduced to this one using the product formula (3.4) and Theorem 1 of [Si2]. □

Using (2.1)-(2.3) of [A7], the local normalizing factors can be defined for all M ∈ L(M_{0}), P, Q ∈ P(M) and π_{v} ∈ Π(M(ℚ_{v})).

Next suppose that v = ∞. In this case the existence of normalizing factors such that Theorem 2.1 in [A7] holds has been established by Arthur [A7]. The definition is as follows. Let ^{L}M be the L-group of M and let ρ = ρ̃_{Q|P} be the contragredient representation of the adjoint representation ρ_{Q|P} of ^{L}M on the complex vector space ^{L}𝔫_{Q} ∩^{L}𝔫_{P} \^{L}𝔫_{Q}. Let L(s, π, ρ) be the L-factor attached to π and ρ = ρ̃_{Q|P}. Then Arthur has shown in [A7] that the functions

(3.7)  r_{Q|P} (π, λ) := \frac{L(0, π_{λ}, ρ)}{L(1, π_{λ}, ρ)}

satisfy all properties required by normalizing factors. We briefly recall the definition of the L-function and refer to [A7, pp.33-35] for more details.

To any π ∈ Π(M(ℝ)) and λ ∈ 𝔞_{P,ℂ}^{∗}, there corresponds a map

ϕ_{λ} : W_{ℝ} → ^{L}M

from the Weil group of ℝ to the L-group of M, which is uniquely determined by π_{λ} up to conjugation by ^{L}M^{0} [L3]. Let

(3.8)  ρ · ϕ_{λ} = ⨁_{τ} τ_{λ}

be the decomposition of ρ · ϕ_{λ} into irreducible representations of W_{ℝ}. Then by definition

L(s, π_{λ}, ρ) = L(s, ρ · ϕ_{λ}) = ∏_{τ} L(s, τ_{λ}).

So it remains to describe the L-factors L(0, τ_{λ})L(1, τ_{λ})^{−1}. To this end let T ⊂ M be a maximal torus over ℝ whose real split component is A_{M}. Let ⟨λ, λ^{∨}⟩ denote the canonical pairing X^{∗}(T) × X_{∗}(T) → ℤ between the space X^{∗}(T) of characters and the space X_{∗}(T) of oneparameter subgroups of T. Let Σ_{P} (G, T) be the set of roots of (G, T) which restrict to roots of (P, A_{M}). The Galois group Gal(ℂ/ℝ) acts on Σ_{P} (G, T). Let \overline{σ} be the action of the nontrivial element of Gal(ℂ/ℝ). The eigenspaces of ρ̃_{Q|P} (ϕ_{λ}(ℂ^{∗})) are the root spaces of {−α^{∨}|α ∈ Σ_{P} (G, T)} and the irreducible constituents τ_{λ} of ρ̃_{Q|P} · ϕ_{λ} correspond to orbits of \overline{σ} in Σ_{P} (G, T). Furthermore, the map ϕ: W_{ℝ} → ^{L}M determines elements μ, ν ∈ X^{∗}(T) ⊗ ℂ with μ − ν ∈ X^{∗}(T). Let

Γ_{ℂ}(z) := 2(2π)^{−z}Γ(z) and Γ_{ℝ}(z) = π^{−z/2}Γ(z/2).

If a two-dimensional constituent τ_{λ} corresponds to a pair \{α, \overline{σ}α\} of complex roots, then τ_{λ} is induced from the quasi-character

z ↦ z^{⟨μ+λ,α^{∨}⟩}\overline{z}^{⟨ν+λ,α^{∨}⟩}

of ℂ^{∗}. Replacing α^{∨} by \overline{σ}α^{∨} if necessary, we can assume that ⟨\overline{σ}μ − μ, α^{∨}⟩ is a nonpositive integer. Then

(3.9)  \frac{L(0, τ_{λ})}{L(1, τ_{λ})} = \frac{Γ_{ℂ}(⟨μ + λ, α^{∨}⟩)}{Γ_{ℂ}(⟨μ + λ, α^{∨}⟩ + 1)}.

The one-dimensional constituents τ_{λ} correspond to real roots α_{0} in Σ_{P} (G, T). There is at most one of these. If α_{0} exists, then τ_{λ} is induced from the quasi-character of ℝ^{∗}

x ↦ (\frac{x}{|x|})^{−N_{0}} |x|^{⟨μ+λ,α_{0}^{∨}⟩},

where N_{0} ∈ {0, 1}. In this case

(3.10)  \frac{L(0, τ_{λ})}{L(1, τ_{λ})} = \frac{Γ_{ℝ}(⟨μ + λ, α_{0}^{∨}⟩ + N_{0})}{Γ_{ℝ}(⟨μ + λ, α_{0}^{∨}⟩ + N_{0} + 1)}.

Remark: It has been conjectured by Langlands [L1, p.282] that for any local field, intertwining operators can be normalized by L-functions. For GL(n) this was proved by Shahidi [Sh2]. Namely, let P be a standard maximal parabolic subgroup of GL(n). Then 𝔞_{P,ℂ}^{∗} ≅ ℂ^{2} and M_{P} = GL(n_{1}) × GL(n_{2}) for some decomposition n = n_{1} + n_{2}. Let F be a local non-Archimedean field and let ψ be a non-trivial additive character of F. Let π_{1} ⊗ π_{2} be an irreducible unitary representation of M_{P} (F) = GL(n_{1}, F) × GL(n_{2}, F). Let L(z, π_{1} × π̌_{2}) and ε(z, π_{1} × π̌_{2}, ψ) be the Rankin-Selberg L-function and the ε-factor defined by Jacquet, Piatetski-Shapiro and Shalika [JPS]. Then the normalizing factor r_{\overline{P} |P} (π, s), s = (s_{1}, s_{2}), can be chosen to be

r_{\overline{P} |P} (π, s) = \frac{L(s_{1} − s_{2}, π_{1} × π̌_{2})}{ε(s_{1} − s_{2}, π_{1} × π̌_{2}, ψ)L(1 + s_{1} − s_{2}, π_{1} × π̌_{2})}.

In [Sh4], this has been generalized to quasi-split groups and generic representations π.

We can now estimate the logarithmic derivatives of the normalizing factors. First we consider the case of a finite place v. Let q_{v} be the number of elements of the residue field of ℚ_{v} (which in our case is a prime number).

Lemma 3.2. Let α ∈ \overline{(𝔞_{M}^{∗})^{+}}. There exist C, c > 0 such that for every finite valuation v and every π_{v} ∈ Π(M(ℚ_{v})) we have

(3.11)  |r_{Q|P} (π_{v}, zα)^{−1} · \frac{d}{dz} r_{Q|P} (π_{v}, zα)| ≤ Cq_{v}^{−2}

for Re(z) ≥ c.

Proof. First we assume that dim 𝔞_{M}/𝔞_{G} = 1 and π_{v} ∈ Π(M(ℚ_{v})) is square integrable modulo A_{M}(ℚ_{v}). Let α be the unique simple root of (P, A). Then r_{\overline{P} |P} (π_{v}, λ) is given by (3.1). Let λ = zα, z ∈ ℂ. Then λ(α^{∨}) = z and by (3.6), it follows that

\begin{aligned}r_{\overline{P} |P} (π_{v}, zα)^{−1} \frac{d}{dz} r_{\overline{P} |P} (π_{v}, zα) = & log(q_{v}) q_{v}^{−z} \\ & · ∑_{i=1}^{r} \{\frac{β_{i}}{1 − β_{i}q_{v}^{−z}} − \frac{α_{i}}{1 − α_{i}q_{v}^{−z}} \} . \end{aligned}

Recall that the α_{i}’s and β_{i}’s satisfy |α_{i}| ≤ 1, |β_{i}| ≤ 1, i = 1, ..., r. Moreover, by Lemma 3.1 there exists C_{1} ≥ 0, which is independent of π_{v} ∈ Π(M(ℚ_{v})), such that r ≤ C_{1}. Therefore, for Re(z) > 3 we obtain

(3.12) |r_{\overline{P} |P} (π_{v}, zα)^{−1} \frac{d}{dz} r_{\overline{P} |P} (π_{v}, zα)| ≤ C_{1} \frac{log(q_{v})q_{v}^{− Re(z)}}{1 − q_{v}^{− Re(z)}} < C_{2}q_{v}^{−2}.

Now let M ∈ L(M_{0}) be arbitrary, but still assume that π_{v} is square integrable modulo A_{M}(ℚ_{v}). Let P, Q ∈ P(M). For each β ∈ Σ_{P}^{r} let M_{β} ∈ L(M) be such that

𝔞_{M_{β}} = {H ∈ 𝔞_{M} | β(H) = 0}.

Then dim 𝔞_{M}/𝔞_{M_{β}} = 1. Let P_{β} be the unique group in P^{M_{β}}(M) whose simple root is β. Furthermore, let α ∈ (𝔞_{M}^{∗})^{+}, ν ∈ 𝔞_{M,ℂ}^{∗} and z ∈ ℂ. Then by (2.12) we get

\begin{matrix}r_{Q|P} (π_{v}, zα + ν)^{−1} · \frac{d}{dz} r_{Q|P} (π_{v}, zα + ν) \\ = ∑_{β∈Σ_{P}^{r} ∩Σr ; \overline{Q}} r_{\overline{P}_{β}|P_{β}}(π_{v}, ⟨zα + ν, β^{∨}⟩β)^{−1} · \frac{d}{dz} r_{\overline{P}_{β}|P_{β}}(π_{v}, ⟨zα + ν, β^{∨}⟩β). \end{matrix}

By assumption we have ⟨α, β^{∨}⟩ ≥ 0 for every β ∈ Σ_{P}^{r}. If ⟨α, β^{∨}⟩ = 0, then the corresponding logarithmic derivative vanishes. Suppose that a := ⟨α, β^{∨}⟩ > 0. Let c_{0} > 0 be such that ∥ β ∥≤ c_{0} for all β ∈ Σ_{P}^{r}. Then

Re (z⟨α, β^{∨}⟩ + ⟨ν, β^{∨}⟩) ≥ a Re(z) − c_{0} ∥ ν ∥

and it follows from (3.7) that there exist C, c > 0, depending on α, c_{0} and ∥ ν ∥, such that

(3.13)  |r_{Q|P} (π_{v}, zα + ν)^{−1} · \frac{d}{dz} r_{Q|P} (π_{v}, zα + ν)| ≤ Cq_{v}^{−2}

for all π_{v} ∈ Π_{2}(M(ℚ_{v})) and Re(z) ≥ c.

Next assume that π_{v} is tempered. Then π_{v} is an irreducible constituent of an induced representation I_{R}^{M}(τ_{v}), where R ∈ P^{M}(M_{1}), M_{1} ⊂ M and τ_{v} ∈ Π_{2}(M_{R}(ℚ_{v})). Then I_{P}^{G}(π_{v,λ}) is canonically isomorphic to a subrepresentation of I_{P (R)}^{G}(τ_{v,λ}) and by (2.13) we have

r_{Q|P} (π_{v}, λ) = r_{Q(R)|P (R)}(τ_{v}, λ),

where P (R) ⊂ P, Q(R) ⊂ Q. Now recall that there is a canonical inclusion 𝔞_{P}^{∗} ⊂ 𝔞_{P (R)}^{∗} and with respect to this inclusion, we have (𝔞_{P}^{∗})^{+} ⊂ \overline{(𝔞_{P (R)}^{∗})^{+}}. Thus α can be identified with an element of \overline{(𝔞_{P (R)}^{∗})^{+}}. Hence (3.11) holds for all tempered π_{v} ∈ Π(M(ℚ_{v})).

Now let π_{v} be an arbitrary representation in Π(M(ℚ_{v})). Then π_{v} is the Langlands quotient J_{R}^{M}(τ_{v}, μ) of a representation I_{R}^{M}(τ_{v}, μ), where M_{R} is an admissible Levi subgroup of M, τ_{v} ∈ Π(M_{R}(ℚ_{v})) is a tempered representation, and μ is a point in the chamber of (𝔞_{R}^{M})^{∗} = 𝔞_{R}^{∗}/𝔞_{M}^{∗} attached to R [Si3]. Set Λ = μ + λ. Then, as explained in [A7, p.30], we have

(3.14)  r_{Q|P} (π_{v}, λ) = r_{Q(R)|P (R)}(τ_{v}, Λ).

Let ρ_{v} ∈ 𝔞_{M}^{∗} be defined by

δ_{P} (a)^{1/2} = q_{v}^{ρ_{v}(H(a))}, a ∈ A_{M}.

Then it follows from Theorem 3.3 of Chapter XI of [BW] that

⟨μ, β^{∨}⟩ ≤ ⟨ρ_{v}, β^{∨}⟩, β ∈ Φ(R, A_{R}).

Since μ belongs to 𝔞_{R}^{∗}/𝔞_{M}^{∗}, it follows that ∥ μ ∥≤∥ ρ_{v} ∥. Let α ∈ (𝔞_{P}^{∗})^{+}. As observed above, α can be identified with an element of \overline{(𝔞_{P (R)}^{∗})^{+}}. Hence, combining (3.13) and (3.14) the desired estimation (3.11) follows. □

Next we consider the infinite place. Let π ∈ Π(M(ℝ)) and let ϕ: W_{ℝ} → LM be the map associated to π. Let μ, ν ∈ X^{∗}(T) ⊗ ℂ be the elements determined by the map ϕ (see [L3], [A7, p.34]). To indicate the dependence on π, we shall write μ_{π} and ν_{π}. We note that there is a canonical injection of the space

𝔞_{M,ℂ}^{∗} = X^{∗}(M)_{ℚ} ⊗ ℂ

into X^{∗}(T) ⊗ ℂ. Lemma 3.3. Let β ∈ (𝔞_{M}^{∗})^{+}. There exist C, c > 0 such that

(3.15)  |r_{Q|P} (π, zβ)^{−1} · \frac{d}{dz} r_{Q|P} (π, zβ)| ≤ C

for for all π ∈ Π(M(ℝ)) and all z ∈ ℂ with Re(z) ≥ c.

Proof. First assume that π ∈ Π(M(ℝ)) is tempered. As explained above, the normalizing factor r_{Q|P} (π, λ) is a product of finitely many meromorphic functions each of which is either of the form (3.9) or (3.10). So it suffices to consider the logarithmic derivative of the Gamma factors. Recall that for Re(z) > 0 the following formula holds

\frac{Γ′(z + 1)}{Γ(z + 1)} = \frac{1}{2z} + log z − ∫_{0}^{∞} \{\frac{1}{2} − \frac{1}{u} + \frac{1}{e^{u} − 1}\} e^{−uz} du

[Wh, p.248]. Let 0 ≤ a ≤ 1 and Re(z) > 2. Then we get

|\frac{Γ′(z)}{Γ(z)} − \frac{Γ′(z + a)}{Γ(z + a)}| ≤ \frac{2}{Re(z)} + log |1 + \frac{a}{z − 1}| + \frac{π}{2}

+ 2 ∫_{0}^{∞} |\frac{1}{2} − \frac{1}{u} + \frac{1}{e^{u} − 1}|e^{−u Re(z)/2} du.

Hence there exists C > 0 such that

(3.16)  |\frac{Γ′(z)}{Γ(z)} − \frac{Γ′(z + a)}{Γ(z + a)}| ≤ C for Re(z) > 2.

Let β ∈ (𝔞_{M}^{∗})^{+}, ν ∈ 𝔞_{M,ℂ}^{∗} and α ∈ Σ_{P} (G, T). Since α ↾ 𝔞_{M} is a root of (P, A), it follows that ⟨β, α^{∨}⟩ > 0. Let α ∈ Σ_{P} (G, T) be such that ⟨\overline{σ}μ_{π} − μ_{π}, α^{∨}⟩ ≤ 0. Then we have Re⟨μ_{π}, α^{∨}⟩ ≥ 0. Hence

Re⟨μ_{π} + zβ + ν, α^{∨}⟩ ≥ ⟨β, α^{∨}⟩ Re(z)− ∥ α ∥ · ∥ ν ∥ .

Using (3.16) together with (3.9) and (3.10), it follows that there exist constants C, c > 0 such that

(3.17)  |\frac{L(1, τ_{zβ+ν})}{L(0, τ_{zβ+ν})} · \frac{d}{dz} (\frac{L(0, τ_{zβ+ν})}{L(1, τ_{zβ+ν})}) | ≤ C

for Re(z) ≥ c(1+ ∥ ν ∥) and all π ∈ Π(M(ℝ)), where τ_{λ} and π_{λ} are related by (3.8).

Now let π be an arbitrary representation in Π(M(ℝ)). Then there exist a parabolic subgroup R of M, a tempered representation τ of M_{R}(ℝ) and a point ξ in the positive chamber of (𝔞_{R}^{∗}/𝔞_{M}^{∗}) attached to R such that π is equivalent to the Langlands quotient J_{R}(τ, ξ) [L3]. Set Λ = ξ + λ. Then, as explained in [A7, p.30], we have

r_{Q|P} (π, λ) = r_{Q(R)|P (R)}(τ, Λ).

Moreover, by Theorem 5.2 of Chapter IV of [BW] it follows that

| Re⟨ξ, α^{∨}⟩| ≤ 4 ∥ ρ_{P} ∥ for all α ∈ Δ_{P} .

Together with (3.17) this implies the claimed result. □

4. Poles of global normalizing factors

Let M ∈ L(M_{0}) and P, Q ∈ P(M). Let π ∈ Π_{disc}(M(𝔸)) with π = ⊗_{v}π_{v}. Then by §2 the infinite product

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

is absolutely convergent in some chamber and admits an analytic extension to a meromorphic function of λ ∈ 𝔞_{M,ℂ}^{∗}. In this section we shall study the poles of r_{Q|P} (π, λ).

Recall that a function f: ℂ^{N} → ℂ is called a meromorphic function of order p ≥ 0, if f can be written as a quotient f = g_{1}/g_{2} of two entire functions g_{i} : ℂ^{N} → ℂ, i = 1, 2, satisfying

|g_{i}(z)| ≤ Ce^{c∥z∥^{p}}, z ∈ ℂ^{N}, i = 1, 2,

for certain constants C, c > 0. With this definition we have the following proposition. Proposition 4.1. Let n = dim G(ℝ)/K_{∞}. For all π ∈ Π_{disc}(M(𝔸)), the normalizing factor r_{Q|P} (π, λ) is a meromorphic function of λ ∈ 𝔞_{M,ℂ}^{∗} of order ≤ n + 2.

Proof. By (2.22) we may assume that dim(𝔞_{M}/𝔞_{G}) = 1. Let P ∈ P(M) and let α be the unique simple root of (P, A_{M}). Let π ∈ Π_{disc}(M(𝔸)). Then A_{π}^{2}(P) ≠ {0} and we have to consider the intertwining operator M_{\overline{P} |P} (π, λ). Recall that M_{\overline{P} |P} (π, λ) is unitary for λ ∈ i𝔞_{M}^{∗}. In particular, M_{\overline{P} |P} (π, λ) is regular at λ = 0. Put

M(π, λ) = M_{P |\overline{P}} (π, 0)M_{\overline{P} |P} (π, λ), λ ∈ 𝔞_{M,ℂ}^{∗}.

Next consider the normalized intertwining operator N_{\overline{P} |P} (π, λ) which is defined by (2.19). It follows from (2.8), (2.9), (2.16) and (2.17) that

N_{P |\overline{P}} (π, 0)^{∗}N_{P |\overline{P}} (π, 0) = Id .

Hence N_{P |\overline{P}} (π, λ) is regular at λ = 0 and N_{P |\overline{P}} (π, 0) is invertible. Put

N(π, λ) = N_{P |\overline{P}} (π, 0)N_{\overline{P} |P} (π, λ), λ ∈ 𝔞_{M,ℂ}^{∗}.

Furthermore by (2.20) and (2.21) we get

(4.1)  |r_{\overline{P} |P} (π, λ)| = 1, λ ∈ i𝔞_{M}^{∗}.

Thus r_{\overline{P} |P} (π, λ) is also regular at λ = 0 and r_{\overline{P} |P} (π, 0) ≠ 0. By (2.19) we get

(4.2)  M(π, λ) = r_{P |\overline{P}} (π, 0)r_{\overline{P} |P} (π, λ)N(π, λ).

Now observe that there exists an open compact subgroup K_{f} ⊂ G(𝔸_{f}) such that A_{π}^{2}(P)_{K_{f}} ≠ {0}. Hence there exists σ ∈ Π(K_{∞}) such that A_{π}^{2}(P)_{K_{f} ,σ} ≠ {0} (cf. section 1.8 for the definition). Put

d = dim A_{π}^{2}(P)_{K_{f} ,σ}

and

c(π, σ) = r_{P |\overline{P}} (π, 0)^{−d}.

Then |c(π, σ)| = 1. Let M(π, λ)_{K_{f} ,σ} (resp. N(π, λ)_{K_{f} ,σ}) denote the restriction of M(π, λ) (resp. N(π, λ)) to the subspace A_{π}^{2}(P)_{K_{f} ,σ}. Then we have det N(π, λ)_{K_{f} ,σ} ≢ 0 and by (4.2) we get

(4.3)  r_{\overline{P} |P} (π, λ)^{d} = c(π, σ) \frac{det M(π, λ)_{K_{f} ,σ}}{det N(π, λ)_{K_{f} ,σ}} .

Thus it suffices to prove that both the numerator and the denominator on the right hand side are meromorphic functions of order ≤ n + 2. As for the numerator, it follows from Theorem 0.1 of [Mu3] that det M(π, λ)_{K_{f} ,σ} is a meromorphic function of λ ∈ 𝔞_{M,ℂ}^{∗} of order ≤ n+2. In fact, in [Mu3] we only dealt with the case of the trivial character ξ. However, all the results of [Mu3] can be extend without any difficulty to the case of a nontrivial character ξ. It remains to consider the denominator. By (2.11), (2.16) and (2.17) there exists a finite set S_{π} of finite places of ℚ such that

det N(π, λ)_{K_{f} ,σ} = det (R_{P |\overline{P}} (π_{∞}, 0)_{σ}R_{\overline{P} |P} (π_{∞}, λ)_{σ})

· ∏_{v∈S_{π}} det (R_{P |\overline{P}} (π_{v}, 0)_{K_{v}}R_{\overline{P} |P} (π_{v}, λ)_{K_{v}}) , (4.4)

where R_{\overline{P} |P} (π_{∞}, λ)_{σ} denotes the restriction of R_{\overline{P} |P} (π_{∞}, λ) to the σisotypical subspace H_{P} (π_{∞})_{σ} of H_{P} (π_{∞}) and R_{\overline{P} |P} (π_{v}, λ)_{K_{v}} denotes the restriction of R_{\overline{P} |P} (π_{v}, λ) to the subspace H_{P} (π_{v})^{K_{v}} of K_{v}-invariant functions. By Theorem 2.1 of [A7], R_{\overline{P} |P} (π_{∞}, λ) is a rational function of λ(α^{∨}) and for each finite place v, R_{\overline{P} |P} (π_{v}, λ) is a rational function of q_{v}^{−λ(α^{∨})}. Therefore det(R_{P |\overline{P}} (π_{∞}, 0)_{σ}R_{\overline{P} |P} (π_{∞}, λ)_{σ}) is a rational functions of λ(α^{∨}) and for each v < ∞, det(R_{P |\overline{P}} (π_{v}, 0)_{K_{v}}R_{\overline{P} |P} (π_{v}, λ)_{K_{v}}) is a rational function of q_{v}^{−λ(α^{∨})}. Since the function z ∈ ℂ ↦ q^{−z} is entire and of order 1, it follows that det N(π, λ)_{σ,K_{f}} is a meromorphic function of λ ∈ 𝔞_{M,ℂ}^{∗} of order ≤ 1. By (4.3) it follows that r_{\overline{P} |P} (π, λ)^{d} and hence r_{\overline{P} |P} (π, λ) is a meromorphic function of λ ∈ 𝔞_{M,ℂ}^{∗} of order ≤ n + 2. □

Remark. Assume that G is a quasi-split connected reductive group over a number field F with ring of adèles 𝔸_{F}. Let P = MN be a maximal parabolic subgroup of G. Let π be a globally generic cuspidal representation of M(𝔸_{F}). Then it follows from [Sh4] that the intertwining operator M_{\overline{P} |P} (π, λ) can be normalized by automorphic L-functions. Furthermore in [GS], Gelbart and Shahidi proved that the L-functions occurring in the normalizing factor are meromorphic functions of order 1. Therefore, one should expect that the normalizing factor r_{Q|P} (π, λ) is of order 1 in general.

Now assume that dim 𝔞_{M}/𝔞_{G} = 1. Our next goal is to estimate the number of poles of r_{\overline{P} |P} (π, λ) in a circle of radius R > 0. For this purpose we have to introduce some notation.

Let Π_{disc}(M(𝔸); K_{f}) be the space of representations defined by (1.2). For every π ∈ Π_{disc}(M(𝔸); K_{f}) we have A_{π}^{2}(P)^{K_{f}} ≠ {0}.

Let π be an irreducible unitary representation of M(ℝ) and let I_{P}^{G}(π) be the induced representation of G(ℝ). Recall that among all K_{∞}- types τ_{Λ′} occurring in I_{P}^{G}(π), the minimal K_{∞}-types of I_{P}^{G}(π) are those τ_{Λ} for which |Λ′ + 2ρ_{K}|^{2} is minimizing at Λ′ = Λ. Let W_{P} (π) be the set of minimal K_{∞}-types of I_{P}^{G}(π). Then W_{P} (π) is a non empty finite subset of Π(K_{∞}). Let λ_{π} be the Casimir eigenvalues of π and for any τ ∈ Π(K_{∞}), let λ_{τ} be the Casimir eigenvalue of τ. Put

(4.5)  Λ_{π} := min_{τ∈W_{P} (π)} \sqrt{λ_{π}^{2} + λ_{τ}^{2}}.

If π ∈ Π(M(𝔸)), put

Λ_{π} := Λ_{π_{∞}}.

For a given pole η of r_{\overline{P} |P} (π, λ), let m(η) denote its order. Set

n_{P} (π, R) = ∑_{|η|≤R} m(η),

where the sum runs over all poles of r_{\overline{P} |P} (π, λ). Proposition 4.2. Let m = dim G and let K_{f} be an open compact subgroup of G(𝔸_{f}). There exists C > 0 such that for all R > 0 and all π ∈ Π_{disc}(M(𝔸); K_{f}) we have

n_{P} (π, R) ≤ C(1 + R^{2} + Λ_{π}^{2})^{8m}.

Proof. Let π ∈ Π_{disc}(M(𝔸); K_{f}). Then there exists σ ∈ Π(K_{∞}) such that A_{π}^{2}(P)_{K_{f} ,σ} ≠ {0}. Put

\overline{N}(π, λ) := N_{P |\overline{P}} (π, λ)N_{\overline{P} |P} (π, 0), λ ∈ 𝔞_{M,ℂ}^{∗}.

Then

\overline{N}(π, λ)N(π, λ) = Id

and by (4.3) we get

(4.6) r_{\overline{P} |P} (π, λ)^{d} = c(π, σ) det(M(π, λ)_{K_{f} ,σ}) · det(\overline{N}(π, λ)_{K_{f} ,σ}).

Thus it suffices to estimate the number of poles of the functions occurring on the right. It follows from Proposition 6.6 and Lemma 6.1 of [Mu3], that the number of poles, counted with their order, of det M(π, λ)_{K_{f} ,σ} in the disc |λ| ≤ R is bounded by

C(1 + R^{2} + λ_{π}^{2} + λ_{σ}^{2})^{8m},

where C > 0 is independent of π and σ. As noted above, in [Mu3] we only dealt with the case of the trivial character ξ. However, everything can be extended to a nontrivial character ξ without any difficulty.

It remains to consider det \overline{N}(π, λ)_{K_{f} ,σ}. For any place v let

\overline{R}(π_{v}, λ) = R_{P |\overline{P}} (π_{v}, λ)R_{\overline{P} |P} (π_{v}, 0).

By (2.16) and (2.17) we have

\overline{N}(π, λ) ∘ j_{P} = j_{P} ∘ ((⊗_{v}\overline{R}(π_{v}, λ)) ⊗ Id) ,

and there exists a finite set S_{π} of finite places, which depends only on π and K_{f}, such that

\overline{R}(π_{v}, λ)_{K_{v}} = Id

for all v ∉ S_{π} ∪ {∞}. Thus

(4.7) det (\overline{N}(π, λ)_{K_{f} ,σ}) = det (\overline{R}(π_{∞}, λ)_{σ}) · ∏_{v∈S_{π}} det (\overline{R}(π_{v}, λ)_{K_{v}}) .

Let n_{P} (π_{v}, R), v < ∞, (resp. n_{P} (π_{∞}, R)) denote the number of poles, counted with the order, of det (\overline{R}(π_{v}, λ)_{K_{v}}) (resp. det (\overline{R}(π_{∞}, λ)_{σ})) in the disc |λ| < R. Then we have to estimate n_{P} (π_{v}, R) for any v ≤ ∞.

Let v < ∞ and let π_{v} be any irreducible unitary representation of M(ℚ_{v}). Let α̃ ∈ 𝔞_{M} be as (3.5). By Theorem 2.2.2 of [Sh1] there exists a polynomial Q_{v}(z), Q_{v}(0) = 1, such that

Q_{v} (q_{v}^{−λ(α̃)}) J_{P |\overline{P}} (π_{v}, λ)

is a holomorphic and non-zero operator. Moreover, the degree of the polynomial Q_{v} is independent of π_{v}. Let

d_{v} = dim H_{P} (π_{v})^{K_{v}}.

Then it follows from (2.7) and the definition of \overline{R}(π_{v}, λ) that

r_{P |\overline{P}} (π_{v}, λ)^{d_{v}}Q_{v} (q_{v}^{−λ(α̃)})^{d_{v}} det (\overline{R}(π_{v}, λ)_{K_{v}})

is a holomorphic function on 𝔞_{M,ℂ}^{∗}. By (r.5) there exist polynomials P_{1}(z) and P_{2}(z) such that

r_{P |\overline{P}} (π_{v}, λ) = \frac{P_{1}(q_{v}^{−λ(α̃)})}{P_{2}(q_{v}^{−λ(α̃)})}.

Thus it suffices to estimate the number of zeros of P_{1}(q_{v}^{−λ(α̃)}) and Q_{v}(q_{v}^{−λ(α̃)}), respectively, in a circle of radius R > 0. First observe that for every z ∈ ℂ the number of solutions of q_{v}^{−s} = z in the disc |s| ≤ R is bounded by 1 + (2π)^{−1} log(q_{v})R. Furthermore, the degree of the polynomial Q_{v} is bounded by some constant c_{v} > 0 which is independent of π_{v}. Using Lemma 3.1 and (2.1)–(2.3) of [A7], it follows that the degree of the polynomial P_{1}(z) is also bounded by a constant which is independent of π_{v}. This implies that there exists C_{v} > 0 such that

n_{P} (π_{v}, R) ≤ C_{v} dim (H_{P} (π_{v})^{K_{v}}) (1 + R)

for all π_{v} ∈ Π(M(ℚ_{v})) and R > 0. It remains to estimate the dimension of H_{P} (π_{v})^{K_{v}}. Suppose that π_{v} is the component at v of a representation π ∈ Π_{disc}(M(𝔸)); K_{f}). Then there exists ξ ∈ Π(A_{M}(ℝ)^{0}) such that π ∈ Π_{disc}(M(𝔸))_{ξ}. Let H_{P} (π)_{σ}^{K_{f}} be the σ-isotypical subspace of H_{P} (π)^{K_{f}}. By (2.3) it follows that

H_{P} (π)_{σ}^{K_{f}} ⊗ Hom_{M(𝔸)}(π, I_{M(ℚ)AM(ℝ)^{0}}^{M(𝔸)}(ξ)) ≅ A_{π}^{2}(P)_{K_{f} ,σ}.

Moreover we have

H_{P} (π)_{σ}^{K_{f}} ≅ H_{P} (π_{∞})_{σ} ⊗ ⨂_{v<∞} H_{P} (π_{v})^{K_{v}}

and dim H_{P} (π_{v})^{K_{v}} = 1 for v ∉ S_{π}. Thus it follows that

dim H_{P} (π_{v})^{K_{v}} ≤ dim A_{π}^{2}(P)_{K_{f} ,σ}.

The right hand side can be estimated by Lemma 6.1 of [Mu3]. It follows that

(4.8)  n_{P} (π_{v}, R) ≤ C_{v}(1 + λ_{π}^{2} + λ_{σ}^{2})^{3m}(1 + R).

Now let v = ∞ and let π_{∞} ∈ Π(M(ℝ)). Set

\overline{J}(π_{∞}, λ) = J_{P |\overline{P}} (π_{∞}, λ)J_{\overline{P} |P} (π_{∞}, 0).

Then

(4.9)  \overline{R}(π_{∞}, λ) = (r_{P |\overline{P}} (π_{∞}, λ)r_{\overline{P} |P} (π_{∞}, 0))^{−1} \overline{J}(π_{∞}, λ).

Let K_{M,∞} = K_{∞} ∩ M(ℝ) and let

σ|_{K_{M,∞}} = ⨁_{τ∈Π(K_{M,∞})} n_{τ}τ.

Set

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

By Corollary 4.7 of [VW], there exist complex numbers a_{i}(π_{∞}), i = 1, ..., r, and b_{i}(π_{∞}, σ), i = 1, ..., r[σ : π_{∞}], with r = r(π_{∞}) depending only on π_{∞}, and a constant C ∈ ℂ, such that

\begin{matrix}det & \overline{J}(π_{∞}, λ) \\ = C \frac{∏_{i=1}^{r(π_{∞})} Γ (⟨λ, α^{∨}⟩/(4⟨ρ_{P} , α^{∨}⟩) − a_{i}(π_{∞}))^{[σ:π_{∞}]}}{∏_{i=1}^{r(π_{∞})[σ:π_{∞}]} Γ (⟨λ, α^{∨}⟩/(4⟨ρ_{P} , α^{∨}⟩) − b_{i}(π_{∞}))} . \end{matrix} ; (4.10)

Lemma 4.3. There exists c > 0 such that r(τ) ≤ c for all τ ∈ Π(M(ℝ)).

Proof. Let b_{τ} be the polynomial which is associated to τ by Theorem 1.5 of [VW]. Then r(τ) is the degree of b_{τ} [VW, p.228]. So we have to estimate the degree of b_{τ}. The polynomial b_{τ} is obtained from a more general polynomial b_{τ,λ} occurring in Theorem 2.2 of [VW] by choosing λ = 4ρ_{P}. The polynomial b_{τ,λ} is associated to τ and a finite dimensional representation (η, F) of G satisfying the conditions (1)–(3) in [VW, p.210]. Then λ is the action of 𝔞_{M} on F^{𝔫}. It follows from the constructions on pp. 217-219 in [VW], that the polynomial b_{τ,λ} is the product of the denominator of β and the denominator of the element ž_{ν}, defined on p.219. Let Ω ∈ Z(𝔤_{ℂ}) be the Casimir element, and let χ_{Λ} be the infinitesimal character of τ. Then it follows that b_{τ,λ} equals

b_{τ,λ}(ν) = ∏_{μ∈Π(F)−{λ}} (χ_{Λ+ν}(Ω) − χ_{Λ+ν+λ−μ}(Ω))^{r(μ)}

· ∏_{μ∈Π(F)−{λ}} (χ_{Λ+ν+λ}(Ω) − χ_{Λ+ν+μ}(Ω))^{r(μ)}

Here Π(F) denotes the set of weights of F with respect to a fixed Cartan subalgebra 𝔥 of 𝔤 and r(μ) is the multiplicity of a given weight μ. From this description of b_{τ,λ} it follows that

r(τ) ≤ 2(dim F − 1).

Finally, b_{τ} is obtained by choosing F to be the representation described in example 2.1 of [VW]. □

Now recall that the poles of the Gamma function Γ(z) are simple poles at z = 0, −1, −2, ... and 1/Γ(z) is entire. Then it follows from (4.10) together with Lemma 4.3 that there exists a constant C_{1} > 0, independent of π_{∞}, such that the number of zeros, counted with their order, of det \overline{J}(π_{∞}, λ) in the disc |λ| ≤ R is bounded by

(4.11)  C_{1}[σ : π_{∞}](1 + R).

By Theorem 8.1 of [Kn] and remark 1 following Theorem 8.4 in [Kn], we have

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

Furthermore, by Weyl’s dimension formula, there exists C_{2} > 0 such that dim σ ≤ C_{2}(1+λ_{σ}^{2})^{p}, where p = 1/2 dim K. Thus (4.11) is bounded by

C_{2}(1 + λ_{σ}^{2})^{p}(1 + R).

It remains to consider the normalizing factor r_{P |\overline{P}} (π_{∞}, λ). It is given by formula (3.7). Let ϕ_{λ} : W_{ℝ} → ^{L}M be the map associated to (π_{∞})_{λ}. Let q be the number of irreducible constituents occurring in the decomposition (3.8) of ρ · ϕ_{λ}. Then q is bounded independently of π and it follows from the description of the L-factors in §3 that r_{P |\overline{P}} (π_{∞}, λ) is a product of q meromorphic functions of the form (3.9) or (3.10). From the form of these functions it follows immediately that the number of poles, counted with their order, of r_{P |\overline{P}} (π_{∞}, λ) in the disc |λ| ≤ R is bounded by C(1 + R). Putting our estimated together, we have proved that there exists C > 0, depending on K_{f}, such that

(4.12)  n_{P} (π, R) ≤ C(1 + R^{2} + λ_{π}^{2} + λ_{σ}^{2})^{8m}

for all R ≥ 0, and all π ∈ Π_{disc}(M(𝔸)) and σ ∈ Π(K_{∞}) such that A_{π}^{2}(P)_{K_{f} ,σ} ≠ {0}.

Let π ∈ Π_{disc}(M(𝔸); K_{f}). Let τ be a minimal K_{∞}-type of I_{P}^{G}(π_{∞}). Choose σ ∈ Π(K_{∞}) such that σ = τ. Then (4.12) applied to σ together with the definition of Λ_{π} implies the proposition. □

Corollary 4.4. Let m = dim G and n = 16m + 2. There exists C > 0, depending on K_{f}, such that for each π ∈ Π_{disc}(M(𝔸); K_{f}) we have

∑_{ρ≠0} \frac{m(ρ)}{|ρ|^{n}} ≤ C(1 + Λ_{π}^{2})^{8m},

where ρ runs over the poles of r_{\overline{P} |P} (π, λ).

5. Logarithmic derivatives of global normalizing factors

In this section we shall study generalized logarithmic derivatives of the global normalizing factors. First we assume that M ∈ L(M_{0}) is such that dim 𝔞_{M}/ dim 𝔞_{G} = 1. Let P ∈ P(M) and let α be the unique simple root of (P, A). Let π ∈ Π_{disc}(M(𝔸)) with π = ⊗_{v}π_{v}. By property (r.5) satisfied by the local normalizing factors, it follows that for each place v, there exists a meromorphic function r̃_{\overline{P} |P} (π_{v}, z) of one complex variable z such that the local normalizing factor r_{\overline{P} |P} (π_{v}, λ) is given by

r_{\overline{P} |P} (π_{v}, λ) = r̃_{\overline{P} |P} (π_{v}, λ(α^{∨})).

Let

r̃_{\overline{P} |P} (π, z) := ∏_{v} r̃_{\overline{P} |P} (π_{v}, z).

The infinite product is absolutely convergent in the half-plane Re(z) > ρ_{P} (α^{∨}), admits a meromorphic continuation to ℂ and the global normalizing factor is given by

r_{\overline{P} |P} (π, λ) = r̃_{\overline{P} |P} (π, λ(α^{∨})), λ ∈ 𝔞_{M,ℂ}^{∗}.

Our present goal is to estimate the logarithmic derivative of r̃_{\overline{P} |P} (π, z) along the imaginary axis.

To begin with, observe that by Lemma 3.2 and Lemma 3.3 there exist C, c > 0 such that

\begin{aligned}|r̃_{\overline{P} |P} & (π, z)^{−1} \frac{d}{dz} r̃_{\overline{P} |P} (π, z)| \\ & ≤ ∑_{v≤∞} |r̃_{\overline{P} |P} (π_{v}, z)^{−1} \frac{d}{dz} r̃_{\overline{P} |P} (π_{v}, z)| ≤ C \end{aligned} ; (5.1)

for all π ∈ Π_{disc}(M(𝔸)) and Re(z) ≥ c. Using (2.20) and (2.21), it follows that the function r̃_{\overline{P} |P} (π, z) satisfies

(5.2)  r̃_{\overline{P} |P} (π, z)\overline{r̃_{\overline{P} |P} (π, −\overline{z})} = 1.

Hence we get

|r̃_{\overline{P} |P} (π, z)^{−1} \frac{d}{dz} r̃_{\overline{P} |P} (π, z)| = |r̃_{\overline{P} |P} (π, −\overline{z})^{−1} (\frac{dr̃_{\overline{P} |P}}{dz} ) (π, −\overline{z})|,

and together with (5.1) we obtain the following proposition. Proposition 5.1. There exist C, c > 0 such that

|r̃_{\overline{P} |P} (π, z)^{−1} \frac{d}{dz} r̃_{\overline{P} |P} (π, z)| ≤ C

for all π ∈ Π_{disc}(M(𝔸)) and all z ∈ ℂ with | Re(z)| ≥ c.

In order to get estimates for the logarithmic derivative on the imaginary axis, we shall use the partial fraction decomposition of the meromorphic function r̃_{\overline{P} |P} (π, z)^{−1}(d/dz)(r̃_{\overline{P} |P} (π, z)), which allows us to treat the sum of the principal parts separately. Let n = 16 dim G + 2. Then it follows from Corollary 4.4 that r̃_{\overline{P} |P} (π, z) is a meromorphic function of order ≤ n. Thus there exist entire functions r_{1}(π, z) and r_{2}(π, z) of order ≤ n such that

r̃_{\overline{P} |P} (π, z) = \frac{r_{1}(π, z)}{r_{2}(π, z)}.

Furthermore, observe that by (5.2) a complex number η is a zero of r̃_{\overline{P} |P} (π, z) if and only if −\overline{η} is a pole of r̃_{\overline{P} |P} (π, z). Thus by Hadamard’s factorization theorem there exists a polynomial Q(z) of degree ≤ n such that

(5.3) r̃_{\overline{P} |P} (π, z) = e^{Q(z)} \begin{matrix}∏_{η} & [(1 − \frac{z}{η}) exp (∑_{k=1}^{n} \frac{1}{k} (\frac{z}{η} )^{k})] & a(η) \\ ∏_{η} & [(1 + \frac{z}{\overline{η}}) exp (∑_{k=1}^{n} \frac{1}{k} (−\frac{z}{\overline{η}} )^{k})] & a(η) \end{matrix} ,

where η runs over all zeros of r̃_{\overline{P} |P} (π, z) and a(η) denotes the order of the zero η.

Let D(π) denote the set of all poles and zeros of r̃_{\overline{P} |P} (π, z). Given η ∈ D(π), we denote by m(η) the order of η, i.e., m(η) is the integer such that (z − η)^{−m(η)} r̃_{\overline{P} |P} (π, z) is holomorphic in a neighborhood of η and does not vanish at z = η. For η ∈ ℂ^{×} we define the function h_{η}(z) by

h_{η}(z) = −\frac{1}{η} ∑_{k=0}^{n−1} (\frac{z}{η} )^{k} , z ∈ ℂ.

Then it follows from Corollary 4.4 that the series

(5.4)  f(π, z) = ∑_{{η∈D(π) \atop |η|≤1}} \frac{m(η)}{z − η} + ∑_{{η∈D(π) \atop |η|>1}} m(η) \{\frac{1}{z − η} − h_{η}(z)\}

is absolutely convergent on compact subsets of ℂ \ D(π) and the resulting function f(π, z) is a meromorphic function on ℂ whose set of poles equals D(π). Differentiating (5.3), we get

r̃_{\overline{P} |P} (π, z)^{−1} \frac{d}{dz} r̃_{\overline{P} |P} (π, z) = f(π, z) − ∑_{{η∈D(π) \atop |η|≤1}} m(η)h_{η}(z) + Q′(z).

Thus there is a polynomial g(π, z) of degree ≤ n − 1 such that

(5.5)  r̃_{\overline{P} |P} (π, z)^{−1} \frac{d}{dz} r̃_{\overline{P} |P} (π, z) = f(π, z) + g(π, z).

We begin with the investigation of g(π, z). Proposition 5.2. Let m = dim G. There exist C, c > 0 such that

|g(π, z)| ≤ C(1 + |z|^{2} + Λ_{π}^{2})^{18m}

for all π ∈ Π_{disc}(M(𝔸), K_{f}) and all z ∈ ℂ with | Re(z)| ≤ c.

Proof. Let c > 0 be the constant occurring in Proposition 5.1. First assume that | Re(z)| = c. By Proposition 5.1 it suffices to estimate f(π, z). Referring again to Proposition 5.1, it follows that D(π) is contained in the strip | Re(z)| ≤ c. Hence we may assume that c > 0 has been chosen so that for all π ∈ Π_{disc}(M(𝔸), K_{f}), the zeros and poles of r̃_{\overline{P} |P} (π, z) are contained in the strip | Re(z)| < c − δ, where δ > 0 is independent of π. Hence the poles of f(π, z) are contained in | Re(z)| < c − δ. Let η ∈ ℂ be a pole of f(π, z). Then for | Re(z)| ≥ c we get

(5.6)  |z − η| ≥ | Re(z − η)| ≥ c − | Re(η)| ≥ δ.

Furthermore, from the definition of f(π, z) it follows that

\begin{aligned}|f(π, z)| ≤ & ∑_{|η|≤2|z|} \frac{|m(η)|}{|z − η|} + ∑_{1≤|η|≤2|z|} |m(η)||h_{η}(z)| \\ & + ∑_{|η|>2|z|} |m(η)||\frac{1}{z − η} − h_{η}(z)|. \end{aligned}

Using (5.6) and Proposition 4.2, we can estimate the first sum as follows

\begin{aligned}∑_{|η|≤2|z|} \frac{|m(η)|}{|z − η|} ≤ \frac{1}{δ} ∑_{|η|≤2|z|} |m(η)| & ≤ \frac{2}{δ}n_{P} (π, 2|z|) \\ & ≤ C(1 + |z|^{2} + Λ_{π}^{2})^{8m}. \end{aligned}

Again by Proposition 4.2, we obtain for the second sum

\begin{aligned}∑_{1≤|η|≤2|z|} |m(η)||h_{η}(z)| & ≤ ∑_{1≤|η|≤2|z|} |m(η)| ∑_{k=0}^{n−1} \frac{|z|^{k}}{|η|^{k+1}} \\ & ≤ C|z|^{n−1}n_{P} (π, 2|z|) \\ & ≤ C_{1}|z|^{n−1}(1 + |z|^{2} + Λ_{π}^{2})^{8m}. \end{aligned} ; (5.7)

Finally, by Corollary 4.4 we get

\begin{aligned}∑_{|η|>2|z|} |m(η)||\frac{1}{z − η} − h_{η}(z)| & ≤ 2|z|^{n} ∑_{|η|>2|z|} \frac{|m(η)|}{|η|^{n+1}} \\ & ≤ C(1 + Λ_{π}^{2})^{8m}|z|^{n} \\ & ≤ C(1 + |z|^{2} + Λ_{π}^{2})^{8m}. \end{aligned} ; (5.8)

Putting our estimates together, it follows that there exists C > 0 such that

|f(π, z)| ≤ C(1 + |z|^{2} + Λ_{π}^{2})^{18m}

for | Re(z)| ≥ c. Hence by Proposition 5.1, there exists C > 0 such that

(5.9)  |g(π, z)| ≤ C(1 + |z|^{2} + Λ_{π}^{2})^{18m}

for all π ∈ Π_{disc}(M(𝔸), K_{f}) and all z ∈ ℂ with | Re(z)| = c. Now the proposition follows from the Phragmen-Lindelöf theorem. □

Note that Proposition 5.2 gives an upper bound for g(π, z) on the imaginary axis.

We shall now investigate f(π, z). From the definition of f(π, z) by (5.3) it is clear that the growth of f(π, z) along the imaginary axis depends on the distance of the poles and zeros of r̃_{\overline{P} |P} (π, z) from the imaginary axis. Therefore, without any further information about the distribution of the poles and zeros we cannot expect to get any estimates for f(π, iλ) as |λ| → ∞. However, what we can hope for is to obtain estimates for integrals involving f(π, iλ).

To this end, we decompose f(π, z) as follows

\begin{aligned}f(π, z) = & ∑_{{η∈D(π) \atop |η|≤2|z|}} m(η)\frac{1}{z − η} − ∑_{{η∈D(π) \atop 1≤|η|≤2|z|}} m(η)h_{η}(z) \\ & + ∑_{{η∈D(π) \atop |η|>2|z|}} m(η) \{\frac{1}{z − η} − h_{η}(z)\} . \end{aligned}

As for the second and the third sum, we observe that the estimations (5.7) and (5.8) are uniform in z ∈ ℂ. It remains to consider the first sum which we denote by f_{1}(π, z). Let

D_{±}(π) = {η ∈ D(π) | ±m(η) > 0}.

Then the map η → −\overline{η} is a bijection of D_{+}(π) onto D_{−}(π), and therefore, f_{1}(π, z) can be written as

f_{1}(π, z) = ∑_{{η∈D_{+}(π) \atop |η|≤2|z|}} m(η) \{\frac{1}{z − η} − \frac{1}{z + \overline{η}} \} .

In particular, for λ ∈ ℝ \ {0} we get

f_{1}(π, iλ) = − ∑_{{η∈D_{+}(π) \atop |η|≤2|λ|}} m(η)\frac{2 Re(η)}{Re(η)^{2} + (λ − Im(η))^{2}} .

Let ζ ∈ C^{∞}(ℝ) be such that 0 ≤ ζ ≤ 1, ζ(u) = 0 for |u| ≥ 3 and ζ(u) = 1 for |u| ≤ 2. Then it follows that

|f_{1}(π, iλ)| ≤ ∑_{{η∈D_{+}(π) \atop |η|≤2|λ|}} m(η)\frac{2| Re(η)|}{Re(η)^{2} + (λ − Im(η))^{2}}

≤ ∑_{η∈D_{+}(π)} ζ (\frac{|η|}{|λ|}) m(η)\frac{2| Re(η)|}{Re(η)^{2} + (λ − Im(η))^{2}} .

Thus we have proved that for λ ∈ ℝ\{0} the following inequality holds

\begin{aligned}|f(π, iλ)| ≤ & ∑_{η∈D_{+}(π)} ζ (\frac{|η|}{|λ|}) m(η)\frac{2| Re(η)|}{Re(η)^{2} + (λ − Im(η))^{2}} \\ & + C(1 + λ^{2} + Λ_{π}^{2})^{8m}. \end{aligned} ; (5.10)

Put

F (λ) := \begin{cases}∑_{η∈D_{+}(π)} ζ (\frac{|η|}{|λ|}) m(η)\frac{2| Re(η)|}{Re(η)^{2} + (λ − Im(η))^{2}} , & λ ≠ 0; \\ 0, & λ = 0. \end{cases}

Note that 0 ∉ D(π). Therefore, on any finite interval [−a, a], F (λ) is the sum of finitely many smooth and nonnegative functions. Hence F (λ) is a smooth and nonnegative function. We shall now estimate the integral of F (u) over a finite interval. Using Proposition 4.2 and the properties of ζ, we obtain

\begin{aligned}∫_{0}^{λ} F (u) du ≤ ∑_{{η∈D_{+}(π) \atop |η|≤3λ}} & m(η) ∫_{0}^{λ} \frac{2| Re(η)|}{Re(η)^{2} + (u − Im(η))^{2}} du \\ & ≤ 2πn_{P} (π, 3λ) \\ & ≤ C(1 + λ^{2} + Λ_{π}^{2})^{8m}. \end{aligned} ; (5.11)

Let N ≥ 8m + 2 and R > 0. Using integration by parts and (5.11), we obtain

| ∫_{−R}^{R} F (u)(1+u^{2})^{−N} du|

= | ∫_{−R}^{R} (∫_{0}^{u} F (t) dt) \frac{d}{du}(1 + u^{2})^{−N} du|

+ | ∫_{−R}^{R} F (t) dt|(1 + R^{2})^{−N}

≤ C(1 + Λ_{π}^{2})^{8m} ∫_{ℝ}(1 + u^{2})^{8m}|\frac{d}{du}(1 + u^{2})^{−N} | du

≤ C_{N}(1 + Λ_{π}^{2})^{8m}.

Here we have used that by (5.11) the boundary term is bounded by a constant independent of R. Since F ≥ 0, this inequality implies that F (u) is integrable with respect to the measure (1 + u^{2})^{−N}du. Putting our estimates together, we obtain the following theorem. Theorem 5.3. Let M ∈ L(M_{0}) and assume that dim 𝔞_{M}/𝔞_{G} = 1. Let P ∈ P(M) and let m = dim G(ℝ). For every N ≥ 8m + 2 there exists C_{N} > 0 such that for all π ∈ Π_{disc}(M(𝔸); K_{f}) the following inequality holds

∫_{ℝ} |r̃_{\overline{P} |P} (π, iu)^{−1} \frac{d}{du}r̃_{\overline{P} |P} (π, iu)|(1 + u^{2})^{−N} du ≤ C_{N}(1 + Λ_{π}^{2})^{18m}.

Now suppose that M ∈ L(M_{0}) is arbitrary. Then we have to consider the multidimensional logarithmic derivatives of the normalizing factors defined by Arthur in [A4]. 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

(5.12)  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

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

which is defined on the complement of a finite set of hyperplanes, 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

(5.14)  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 (5.13).

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 (5.13) 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_{disc}(M(𝔸)) and λ ∈ i𝔞_{M}^{∗}. Define

(5.15) 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. Given β ∈ Σ_{P}^{r} ∩ Σr ; \overline{Q}, set

r_{β}(π, z) = r̃_{\overline{P}_{β}|P_{β}}(π, z), z ∈ ℂ.

Then by (2.22) we have

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

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 (5.13) defined by this family. Set

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

If β is any root in Σ^{r}(G, A_{M}), let β_{L}^{∨} denote the projection of β^{∨} onto 𝔞_{L}. If F is a subset of Σ^{r}(G, A_{M}), let F_{L}^{∨} be the disjoint union of all the vectors β_{L}^{∨}, β ∈ 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}^{∨})) \\ & · (∏_{β∈F} r_{β}(π, λ(β^{∨}))^{−1}r′_{β}(π, λ(β^{∨}))) , \end{aligned} ; (5.16)

where F runs over all subsets of Σ^{r}(L_{1}, A_{M}) such that F_{L}^{∨} is a basis of 𝔞_{L}^{L_{1}}. Let N ∈ ℕ. Then by (5.16) we get

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

Here F runs over all subsets of Σ^{r}(L_{1}, A_{M}) such that F_{L}^{∨} is a basis of 𝔞_{L}^{L_{1}}. Fix such a subset F. Let

{ω̃_{β} | β ∈ F }

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

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

Observe that λ(β^{∨}) = z_{β}. Suppose that N > 2 dim(A_{L_{1}}/A_{G})+2. Then there exists C_{N} > 0, independent of π, such that

\begin{aligned}∫i𝔞_{L}^{∗}/𝔞_{G}^{∗} ∏_{β∈F} & |r_{β}(π, λ(β^{∨}))^{−1}r′_{β}(π, λ(β^{∨}))|(1+ ∥ λ ∥^{2})^{−N} dλ \\ & ≤ C_{N} ∏_{β∈F} ∫_{iℝ} |r_{β}(π, z_{β})^{−1}r′_{β}(π, z_{β})|(1 + |z_{β}|^{2})^{−N/2} dz_{β}. \end{aligned}

Combined with Theorem 5.3 we obtain Theorem 5.4. Let M ∈ L(M_{0}), L ∈ L(M), L_{1} ∈ L(L) and S ∈ P(L_{1}). Let m = dim G(ℝ)/K_{∞}. For every N ≥ 8m + 2 there exists C_{N} > 0 such that

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

for all π ∈ Π_{disc}(M(𝔸), K_{f}) and any minimal K_{∞}-type σ of I_{P}^{G}(π_{∞}).

6. Absolute convergence of the spectral side

In this section we prove Theorem 0.1 and Theorem 0.2. For this purpose we have to study the multidimensional logarithmic derivatives of the global intertwining operators that are the main ingredients of the spectral side. First we explain the structure of the spectral side in more detail. Let M ∈ L(M_{0}). 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 above, 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).

For s ∈ W (𝔞_{M}^{∗}) let

M(P, s) = M_{P |P} (s, 0).

The spectral side is a sum of distributions

∑_{χ∈𝔛}J_{χ}

on G(𝔸)^{1}. By Theorem 8.2 of [A4], the distribution J_{χ} can be described as follows. Let χ ∈ 𝔛, π ∈ Π(M(𝔸)^{1}) and h ∈ C_{c}^{∞}(G(𝔸)^{1}). Note that 𝔐_{L}(P, λ) and ρ_{χ,π}(P, λ, h) both act in the Hilbert space H_{P} (π)_{χ}. Let W^{L}(𝔞_{M})_{reg} be the set of elements s ∈ W (𝔞_{M}) such that {H ∈ 𝔞_{M} | sH = H} = 𝔞_{L}. Then J_{χ}(f) equals the sum over M ∈ L(M_{0}), L ∈ L(M), π ∈ Π(M(𝔸)^{1}) and s ∈ W^{L}(𝔞_{M})_{reg} of the product of

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

with

∫_{i𝔞_{L}^{∗}/i𝔞_{G}^{∗}} |P(M)|^{−1} ∑_{P ∈P(M)} tr (𝔐_{L}(P, λ)M(P, s)ρ_{χ,π}(P, λ, h)) dλ.

Our goal is to determine the conditions under which the integral-series obtained by summing this expression over χ ∈ 𝔛, is absolutely convergent. Since M(P, s) is unitary, we have to estimate the integral

(6.2)  ∫_{i𝔞_{L}^{∗}/i𝔞_{G}^{∗}} ∥ 𝔐_{L}(P, λ)ρ_{χ,π}(P, λ, h) ∥_{1} dλ,

where ∥ · ∥_{1} denotes the trace norm.

We shall now assume that h ∈ C^{1}(G(𝔸)^{1}). Let N_{Q|P} (π, λ), P, Q ∈ P(M), be the normalized intertwining operator which by (2.19) is defined as

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

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

(6.3)  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 operator [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 H_{P} (π)_{χ}. Then by (2.19) and (5.15) it follows that

(6.4)  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}. (6.5)

Let 𝔐_{L}(P, π, λ) be the restriction of 𝔐_{L}(P, λ) to H_{P} (π)_{χ}. Then by Lemma 6.3 of [A5] we have

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

Hence the integral (6.2) can be estimated by

∑_{S∈F(L)} ∫_{i𝔞_{L}^{∗}/i𝔞_{G}^{∗}} ∥ 𝔑′_{S}(P, π, λ)ρ_{χ,π}(P, λ, h) ∥_{1} |ν_{L}^{S}(P, π, λ)| dλ.

We shall now study the integral in more detail. Let Ω and Ω_{K} be the Casimir operators of G(ℝ) and K_{∞} respectively. Set

Δ = Id −Ω + 2Ω_{K}.

Then Δ acts on A_{χ,π}^{2}(P) through each of the representations ρ_{χ,π}(P, λ). Let K_{f} be an open compact subgroup of G(𝔸_{f}) and let σ ∈ Π(K_{∞}). Then the operators

ρ_{χ,π}(P, λ, Δ), λ ∈ i𝔞_{P}^{∗} ,

have A_{χ,π}^{2}(P)_{K_{f}} and A_{χ,π}^{2}(P)_{K_{f} ,σ} as invariant subspaces. We shall denote the restriction of ρ_{χ,π}(P, λ, Δ) to A_{χ,π}^{2}(P)_{K_{f}} and A_{χ,π}^{2}(P)_{K_{f} ,σ}, respectively, by ρ_{χ,π}(P, λ, Δ)_{K_{f}} and ρ_{χ,π}(P, λ, Δ)_{K_{f} ,σ}, respectively. Recall that by (2.3), ρ_{χ,π}(P, λ) is equivalent to I_{P}^{G}(π_{λ})⊗Id. Let λ_{π} and λ_{σ} denote the Casimir eigenvalues of π_{∞} and σ, respectively. Then it follows from Proposition 8.22 of [Kn] that

(6.7)  ρ_{χ,π}(P, λ, Δ)_{K_{f} ,σ} = (1+ ∥ λ ∥^{2} −λ_{π} + 2λ_{σ}) Id .

To estimate the right hand side we use the following lemma.

Lemma 6.1. For all π ∈ Π_{disc}(M(𝔸)^{1}; K_{f}) and σ ∈ Π(K_{∞}), one has

(6.8)  −λ_{π} + λ_{σ} ≥ 0, if dim A_{π}^{2}(P)_{K_{f} ,σ} ≠ {0}.

Proof. The lemma is a consequence of a more general result. Let π_{∞} ∈ Π(M(ℝ)) and suppose that σ ∈ Π(K_{∞}) occurs in I_{P (ℝ)}^{G(ℝ)}(π_{∞})|_{K_{∞}}. Let

π_{∞}|_{K_{∞}∩M(ℝ)} = ∑_{ω∈Π(K_{∞}∩M(ℝ))}n_{ω}ω.

Then

[I_{P (ℝ)}^{G(ℝ)}(π_{∞})|_{K_{∞}} : σ] = ∑_{ω∈Π(K_{∞}∩M(ℝ))}n_{ω}[σ|_{K_{∞}∩M(ℝ)} : ω]

[Kn, p.208]. Hence there exists ω ∈ Π(K_{∞} ∩ M(ℝ)) such that

[σ|_{K_{∞}∩M(ℝ)} : ω] > 0 and [π_{∞}|_{K_{∞}∩M(ℝ)} : ω] > 0.

By [Mu2, (5.15)], the first condition implies that the Casimir eigenvalues λ_{ω} of ω and λ_{σ} of σ satisfy λ_{ω} ≤ λ_{σ_{∞}}. On the other hand, since ω occurs in π_{∞}|_{K_{∞}∩M(ℝ)} it follows that −λ_{π_{∞}} + λ_{ω} ≥ 0 [DH, Lemma 2.6]. This completes the proof. □

Using (6.7) and (6.8), it follows that

\begin{aligned}∥ ρ_{χ,π}(P, λ, Δ)_{K_{f} ,σ} ∥^{2} & ≥ (1+ ∥ λ ∥)^{2} + (−λ_{π} + 2λ_{σ})^{2} \\ & ≥ \frac{1}{4}(1+ ∥ λ ∥^{2} +λ_{π}^{2} + λ_{σ}^{2}). \end{aligned} ; (6.9)

Let S ∈ F(L) be fixed. Given an open compact subgroup K_{f} of G(𝔸_{f}) and σ ∈ Π(K_{∞}), 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}. Lemma 6.2. Let K_{f} be an open compact subgroup of G(𝔸_{f}) and let h ∈ C^{1}(G(𝔸)^{1}) be bi-invariant under K_{f}. Suppose that there exist N ∈ ℕ and C > 0 such that

(6.10)  ∥ 𝔑′_{S}(P, π, λ)_{K_{f} ,σ} ∥≤ C(1+ ∥ λ ∥^{2} +λ_{π}^{2} + λ_{σ}^{2})^{N}

for all π ∈ Π_{disc}(M(𝔸), K_{f}), σ ∈ Π(K_{∞}) and λ ∈ i𝔞_{L}^{∗}. Then for every k ∈ ℕ there exists C_{k} > 0 such that

∫_{i𝔞_{L}^{∗}/i𝔞_{G}^{∗}} ∥ 𝔑′_{S}(P, π, λ)ρ_{χ,π}(P, λ, h) ∥_{1} |ν_{L}^{S}(P, π, λ)| dλ ≤ C_{k}(1 + Λ_{π})^{−k}

for all χ ∈ 𝔛 and π ∈ Π(M(𝔸)^{1}).

Proof. Since h is bi-invariant under K_{f}, ρ_{χ,π}(P, λ, h) maps the Hilbert space \overline{A}_{χ,π}^{2}(P) into the subspace \overline{A}_{χ,π}^{2}(P)_{K_{f}}. Moreover \overline{A}_{χ,π}^{2}(P)_{K_{f}} is an invariant subspace for ρ_{χ,π}(P, λ, h). Hence ρ_{χ,π}(P, λ, h) = 0, unless π belongs to Π_{disc}(M(𝔸), K_{f}). So we may assume that π belongs to Π_{disc}(M(𝔸), K_{f}). Then for each k ∈ ℕ we get

\begin{matrix}∥ 𝔑′_{S}(P, & π, λ)ρ_{χ,π}(P, λ, h) ∥_{1} \\ =∥ 𝔑′_{S}(P, π, λ)_{K_{f}} ρ_{χ,π}(P, λ, h)_{K_{f}} ∥_{1} \\ ≤∥ 𝔑′_{S}(P, π, λ)_{K_{f}} ρ_{χ,π}(P, λ, Δ^{2k})_{K_{f}}^{−1} ∥_{1} \\ · ∥ ρ_{χ,π}(P, λ, Δ^{2k}h) ∥ . \end{matrix} ; (6.11)

Furthermore, using (6.9) and (6.10) we get

\begin{aligned}∥ 𝔑′_{S} & (P, π, λ)_{K_{f}} ρ_{χ,π}(P, λ, Δ^{2k})_{K_{f}}^{−1} ∥_{1} \\ & ≤ ∑_{σ∈Π(K_{∞})} ∥ 𝔑′_{S}(P, π, λ)_{K_{f} ,σ} ∥ · ∥ ρ_{χ,π}(P, λ, Δ^{2k})_{K_{f} ,σ}^{−1} ∥_{1} \\ & ≤ 2C ∑_{σ∈Π(K_{∞})} \frac{dim A_{χ,π}^{2}(P)_{K_{f} ,σ}}{(1+ ∥ λ ∥^{2} +λ_{π}^{2} + λ_{σ}^{2})^{k−N}} . \end{aligned} ; (6.12)

By Lemma 6.1 of [Mu3] there exist C_{1} > 0 and N_{1} ∈ ℕ such that

dim A_{χ}^{2}(P)_{K_{f} ,σ} ≤ C_{1}(1 + λ_{χ}^{2} + λ_{σ}^{2})^{N_{1}}

for all χ ∈ 𝔛 and σ ∈ Π(K_{∞}). Actually in [Mu3] we considered the space A^{2}(P, χ, σ), where σ is an irreducible representation of K. The two spaces are not equal, but they are closely related. Moreover λ_{χ} was denoted by μ_{χ} in [Mu3]. If A_{χ,π}^{2}(P) ≠ 0, it follows from Langlands’ construction of A_{χ}^{2}(P) in terms of iterated residues of cuspidal Eisenstein series that

(6.13)  |λ_{χ} − λ_{π}| ≤ c

with c > 0 independent of χ and π (see (4.21) of [Mu3]). Hence there exist C_{2} > 0 and N_{1} ∈ ℕ such that

(6.14)  dim A_{χ,π}^{2}(P)_{K_{f} ,σ} ≤ C_{2}(1 + λ_{π}^{2} + λ_{σ}^{2})^{N_{1}}

for all χ ∈ 𝔛, π ∈ Π_{disc}(M(𝔸), K_{f}) and σ ∈ Π(K_{∞}). Set

N_{2} = \frac{1}{2}(N + N_{1}).

Now observe that there exists n_{0} ∈ ℕ such that

∑_{σ_{∞}∈Π(K_{∞})}(1 + λ_{σ})^{−n} < ∞.

the right hand side is finite for n ≥ n_{0}. Let Λ_{π} be the number defined by (4.5). Then by (6.12) and (6.14) it follows that for every k > 2(n_{0}+N_{2}) there exists C_{k} > 0 such that

\begin{aligned}∥ 𝔑′_{S}(P, π, λ)_{K_{f}} & ρ_{χ,π}(P, λ, Δ^{4k})_{K_{f}}^{−1} ∥_{1} \\ & ≤ C_{k}(1+ ∥ λ ∥^{2} +Λ_{π}^{2})^{N_{2}−k} \\ & ≤ C_{k}(1+ ∥ λ ∥^{2})^{(N_{2}−k)/2}(1 + Λ_{π}^{2})^{(N_{2}−k)/2} \end{aligned} ; (6.15)

for all π ∈ Π_{disc}(M(𝔸), K_{f}) and χ ∈ 𝔛. Next observe that for λ ∈ i𝔞_{L}^{∗} the operator ρ_{χ,π}(P, λ, g) is unitary. Hence it follows that

(6.16)  ∥ ρ_{χ,π}(P, λ, Δ^{4k}h) ∥≤∥ Δ^{4k}h ∥_{L^{1}(G(𝔸)^{1})}

for all π ∈ Π_{disc}(M(𝔸), K_{f}) and χ ∈ 𝔛. Combing (6.11), (6.15) and (6.16), it follows that for every n ∈ ℕ there exists C_{n} > 0 such that

∥ 𝔑′_{S}(P, π, λ)ρ_{χ,π}(P, λ, h) ∥_{1}≤ C_{n}(1+ ∥ λ ∥^{2})^{−n}(1 + Λ_{π}^{2})^{−n}

for all χ ∈ 𝔛 and π ∈ Π_{disc}(M(𝔸), K_{f}). Combined with Theorem 5.4 the claimed estimation of the integral follows. □

Proof of Theorem 0.1: Let h ∈ C^{1}(G(𝔸)^{1}) be bi-invariant under K_{f}. As observed in the proof of Lemma 6.2, it follows that ρ_{χ,π}(P, λ, h) = 0, unless π ∈ Π_{disc}(M(𝔸), K_{f}). Let L2_{disc}(M(ℚ)A_{P} (ℝ)^{0}\M(𝔸)) be the largest closed subspace of the Hilbert space L^{2}(M(ℚ)A_{P} (ℝ)^{0}\M(𝔸)) which decomposes discretely under the regular representation of M(𝔸). Then

L2_{disc}(M(ℚ)A_{P} (ℝ)^{0}\M(𝔸)) = ⨁_{π∈Π(M(𝔸))}m(π)H_{π},

and each multiplicity m(π) is finite. Thus, if the assumption (6.10) of Lemma 6.2 is satisfied, it follows from Lemma 6.2 that for every n ∈ ℕ there exists C_{n} > 0 such that

\begin{matrix}∑_{χ∈𝔛} ∑_{π∈Π(M(𝔸)^{1})}∫_{i𝔞_{L}^{∗}/i𝔞_{G}^{∗}} ∥ 𝔑′_{S}( & P, π, λ)ρ_{χ,π}(P, λ, h) ∥_{1} |ν_{L}^{S}(P, π, λ)| dλ \\ ≤ C_{n} ∑_{π∈Π_{disc}(M(𝔸)^{1},K_{f})}m(π)(1 + Λ_{π})^{−n}. \end{matrix} (6.17)

It remains to investigate the sum on the right hand side. Let

K_{M,f} = K_{f} ∩ M(𝔸_{f}).

Then 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. Section 9 of [Mu1]). Therefore we get

\begin{matrix}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)) \end{matrix} ; (6.18)

as M(ℝ)-modules. For each i, i = 1, ..., l, let L2_{disc}(A_{P} (ℝ)^{0}Γ_{M,i}\M(ℝ)) be the discrete subspace of the regular representation of M(ℝ) in L^{2}(A_{P} (ℝ)^{0}Γ_{M,i}\M(ℝ)). Then it follows from (6.18) that

L2_{disc}(A_{P} (ℝ)^{0}M(ℚ)\M(𝔸))^{K_{M,f}}

≅ ⨁_{i=1}^{l} L2_{disc}(A_{P} (ℝ)^{0}Γ_{M,i}\M(ℝ)) (6.19)

as M(ℝ) modules. For i, 1 ≤ i ≤ l, and π_{∞} ∈ Π(M(ℝ)) denote by m_{Γ_{M,i}}(π_{∞}) the multiplicity of π_{∞} in the regular representation of M(ℝ) in L2_{disc}(A_{P} (ℝ)^{0}Γ_{M,i}\M(ℝ)). Then by (6.19) we get

∑_{π∈Π_{disc}(M(𝔸)^{1},K_{f})}m(π)(1 + Λ_{π})^{−n}

≤ ∑_{i=1}^{l} ∑_{π_{∞}∈Π(M(ℝ))}m_{Γ_{M,i}}(π_{∞})(1 + Λ_{π_{∞}})^{−n}

(6.20)

Let σ ∈ Π(K_{∞}) be a minimal K_{∞}-type occurring in I_{P}^{G}(π_{∞}) with Casimir eigenvalue λ_{σ}. Let K_{M,∞} = M(ℝ) ∩ K_{∞}. By (5.15) of [Mu2] we have that λ_{σ} ≥ λ_{τ} for any irreducible constituent τ ∈ Π(K_{M,∞}) of σ_{∞}|K_{M,∞}. Thus the right hand side of (6.20) is bounded by

∑_{i=1}^{l} ∑_{τ∈Π(K_{M,∞})} ∑_{π_{∞}∈Π(M(ℝ))}m_{Γ_{M,i}}(π_{∞})\frac{dim(H(π_{∞}) ⊗ V_{τ})^{K_{M,∞}}}{(1 + λ_{π_{∞}}^{2} + λ_{τ}^{2})^{n/2}} .

By Corollary 0.3 of [Mu2] this sum is finite for n sufficiently large. Thus we proved Proposition 6.3. Let K_{f} be an open compact subgroup of G(𝔸_{f}) and let h ∈ C^{1}(G(𝔸)^{1}) be bi-invariant under K_{f}. Suppose that there exist N ∈ ℕ and C > 0 such that

(6.21)  ∥ 𝔑′_{S}(P, π, λ)_{K_{f} ,σ} ∥≤ C(1+ ∥ λ ∥^{2} +λ_{π}^{2} + λ_{σ}^{2})^{N}

for all π ∈ Π_{disc}(M(𝔸), K_{f}), σ ∈ Π(K_{∞}) and λ ∈ i𝔞_{L}^{∗}. Then

(6.22) ∑_{χ∈𝔛} ∑_{π∈Π(M(𝔸)^{1})}∫_{i𝔞_{L}^{∗}/i𝔞_{G}^{∗}} ∥ 𝔐_{L}(P, λ)ρ_{χ,π}(P, λ, h) ∥_{1} dλ < ∞

Let h ∈ C^{1}(G(𝔸)^{1}). Then there exists an open compact subgroup K_{f} of G(𝔸_{f}) such that h is bi-invariant under K_{f}. Using the observations made at the beginning of this section, it follows that (6.22) implies that the spectral side of the trace formula is absolutely convergent.

We shall now continue by investigating condition (6.21) in detail. To calculate 𝔑′_{S}(P, π, λ), let Λ ∈ i𝔞_{M}^{∗}. By [A5, p.37] 𝔑′_{S}(P, π, λ) equals

\frac{1}{q!} ∑_{\{R|R⊃S\}}(−1)^{q}θ̂_{S}^{R}(Λ)^{−1} (lim_{t→0} (\frac{d}{dt})^{q} 𝔑_{R}(P, π, λ, tΛ)) θ_{R}(Λ)^{−1},

where q = dim(A_{S}/A_{R}). Since N_{Q|P} (π, λ) is unitary for λ ∈ i𝔞_{M}^{∗}, it follows from (6.3) that we have to estimate the norm of

(6.23)  lim_{t→0} (\frac{d}{dt})^{q} N_{Q|P} (π, λ + tΛ)_{σ}, λ ∈ i𝔞_{M}^{∗}.

To this end, we may use (2.16) and (2.17) to replace N_{Q|P} (π, λ) by

R_{Q|P} (π, λ) = ⊗_{v}R_{Q|P} (π_{v}, λ).

Next note that any compact open subgroup K_{f} = ∏_{v<∞} K_{v} of G(𝔸_{f}) is such that K_{v} is a hyperspecial compact subgroup for almost all v. Hence, by (2.11) there exists a finite set of places S_{0}, including the Archimedean one, such that we have

R_{Q|P} (π_{v}, λ)_{K_{v}} = Id, v ∉ S_{0}, π ∈ Π_{disc}(M(𝔸), K_{f}).

Let D_{Λ} denote the directional derivative on i𝔞_{M}^{∗} in the direction of Λ. Then it follows that there exists C > 0 such that

\begin{matrix}∥ 𝔑′_{S}(P, π, λ)_{K_{f} ,σ} ∥≤ & C( ∑_{v∈S_{0}\\{∞\}} ∑_{k=1}^{q} ∥ D_{Λ}^{k}R_{Q|P} (π_{v}, λ)_{K_{v}} ∥ \\ + ∑_{k=1}^{q} ∥ D_{Λ}^{k}R_{Q|P} (π_{∞}, λ)_{σ} ∥) \end{matrix} ; (6.24)

for all λ ∈ i𝔞_{M}^{∗}, σ ∈ Π(K_{∞}) and π ∈ Π(M(𝔸)). Together with Proposition 6.3 this implies Theorem 0.1.

□

Proof of Theorem 0.2: The proof of Theorem 0.2 is similar to the proof of Theorem 0.1. We only have to modify some of the arguments.

Given an open compact subgroup K_{f} of G(𝔸_{f}) and σ ∈ Π(K_{∞}), let Π_{K_{f} ,σ} denote the orthogonal projection of the Hilbert space \overline{A}_{χ,π}^{2}(P) onto the finite dimensional subspace A_{χ,π}^{2}(P)_{K_{f} ,σ}. Let h ∈ C^{1}(G(𝔸)^{1}) be K-finite. Then there exists an open compact subgroup K_{f} of G(𝔸_{f}) such that h is left and right invariant under K_{f}. Furthermore, there exist σ_{1}, ..., σ_{m} ∈ Π(K_{∞}) such that

(6.25)  ρ_{χ,π}(P, λ, h) = ∑_{i,j=1}^{m} Π_{K_{f} ,σ_{i}} ∘ ρ_{χ,π}(P, λ, h) ∘ Π_{K_{f} ,σ_{j}}

for all π ∈ Π(M(𝔸)^{1}) and χ ∈ 𝔛. Let k ∈ ℕ. Then by (6.25) we get

\begin{aligned}∥ 𝔑′_{S} & (P, π, λ)ρ_{χ,π}(P, λ, h) ∥_{1} \\ & ≤ ∑_{i=1}^{m} ∥ 𝔑′_{S}(P, π, λ)_{K_{f} ,σ_{i}} ∥ · ∥ ρ_{χ,π}(P, λ, Δ^{2k})_{K_{f} ,σ_{i}}^{−1} ∥_{1} \\ & · ∥ ρ_{χ,π}(P, λ, Δ^{2k}h) ∥ . \end{aligned} ; (6.26)

Here we assume, of course, that A_{χ,π}^{2}(P)_{K_{f} ,σ_{i}} ≠ 0, i = 1, ..., m. Then it follows from (6.9) that

(6.27)  ∥ ρ_{χ,π}(P, λ, Δ^{2k})_{K_{f} ,σ_{i}}^{−1} ∥_{1}≤ C \frac{dim A_{χ,π}^{2}(P)_{K_{f} ,σ_{i}}}{(1+ ∥ λ ∥^{2} +λ_{π}^{2})^{k}}

for i = 1, ..., m. Given σ ∈ Π(K_{∞}), let

Π_{disc}(M(𝔸)^{1})_{K_{f} ,σ}
= {π ∈ Π_{disc}(M(𝔸)^{1}; K_{f}) | [I_{P}^{G}(π_{∞})|_{K_{∞}} : σ] > 0}.

Then we proceed as above to show that for every n ∈ ℕ there exists C_{n} > 0 such that

\begin{aligned}∑_{χ∈𝔛} ∑_{π∈Π(M(𝔸)^{1})} & ∫_{i𝔞_{L}^{∗}/i𝔞_{G}^{∗}} ∥ 𝔑′_{S}(P, π, λ)ρ_{χ,π}(P, λ, h) ∥_{1} |ν_{L}^{S}(P, π, λ)| dλ \\ & ≤ C_{n} ∑_{i=1}^{m} ∑_{π∈Π_{disc}(M(𝔸)^{1})_{K_{f} ,σ_{i}}} m(π)(1 + λ_{π}^{2})^{−n}. \end{aligned} (6.28)

To estimate the right hand side, we fix σ ∈ Π(K_{∞}). Then as in (6.20) we get

∑_{π∈Π_{disc}(M(𝔸)^{1})_{K_{f} ,σ}} \frac{m(π)}{(1 + λ_{π}^{2})^{n}}

≤ ∑_{i=1}^{l} ∑_{π_{∞}∈Π(M(ℝ))}m_{Γ_{M,i}}(π_{∞})\frac{dim(H(π_{∞}) ⊗ V_{σ})^{K_{M,∞}}}{(1 + λ_{π_{∞}}^{2})^{n}} .

It follows from Theorem 0.1 of [Mu1] that for sufficiently large n, this series is convergent. This completes the proof of Theorem 0.2.

□

We observe that for tempered representations, the existence of estimates like (0.2), (0.3) and (0.4) follows from results of Arthur [A5, p.51] and [A8, Lemma 2.1]. Let Π_{temp}(M(𝔸)^{1}) be the subspace of all π in Π(M(𝔸)^{1}) such that the local constituents π_{v} of π are tempered for all v. Then we obtain Proposition 6.4. For every M ∈ L(M_{0}), L ∈ L(M) and P ∈ P(M) we have

∑_{χ∈𝔛} ∑_{π∈Π_{temp}(M(𝔸)^{1})}∫_{i𝔞_{L}^{∗}/i𝔞_{G}^{∗}} ∥ 𝔐_{L}(P, π, λ)ρ_{χ,π}(P, λ, h) ∥_{1} dλ < ∞.

7. The example of GL_{n}

In this section we shall briefly discuss the case where G = GL_{n}. Let P_{0} be the subgroup of upper triangular matrices of G. This is the minimal standard parabolic subgroup of G. Its Levi subgroup M_{0} is the group of diagonal matrices. Let P be a parabolic subgroup of G defined over ℚ, and let M be the unique Levi component of P which contains M_{0}. Then

M ≅ GL_{n_{1}} × ⋯ × GL_{n_{r}} .

We shall identify 𝔞_{M} with ℝ^{r}. Let e_{1}, ..., e_{r} denote the standard basis of (ℝ^{r})^{∗}. Then the roots Σ_{P} are given by

Σ_{P} = {e_{i} − e_{j} | 1 ≤ i < j ≤ r}.

Let v be a place of ℚ. Fix a nontrivial continuous character ψ_{v} of the additive group ℚ_{v}^{+} of ℚ_{v} and equip ℚ_{v} with the Haar measure which is selfdual with respect to ψ_{v}. Given irreducible unitary representations π_{1,v} and π_{2,v} of GL_{n_{1}}(ℚ_{v}) and GL_{n_{2}}(ℚ_{v}), respectively, let L(s, π_{1,v}×π_{2,v}) and ϵ(s, π_{1,v} × π_{2,v}, ψ_{v}) denote the Rankin-Selberg L-factor and the ϵfactor defined by Jacquet, Piateski-Shapiro, and Shalika [JPS], [JS1].

Let P_{1}, P_{2} ∈ P(M). Then there exist permutations σ_{1}, σ_{2} ∈ S_{r} such that the set of roots of (P_{i}, A_{i}) is given by

Σ_{P_{k}} = {e_{i} − e_{j} | σ_{k}(i) < σ_{k}(j)}.

Put

I(σ_{1}, σ_{2}) = {(i, j) | 1 ≤ i, j ≤ r, σ_{1}(i) < σ_{1}(j), σ_{2}(i) > σ_{2}(j)}.

Then

Σ_{P_{1}} ∩ Σ_{\overline{P}_{2}} = \{e_{i} − e_{j} | (i, j) ∈ I(σ_{1}, σ_{2})\}.

Let π_{v} = π_{1,v} ⊗ ⋯ ⊗ π_{r,v}, where π_{i,v} ∈ Π(GL_{n_{i}}(ℚ_{v})), i = 1, ..., r. Given s = (s_{1}, ..., s_{r}) ∈ ℂ^{r}, set

\begin{aligned}r_{P_{2}|P_{1}} & (π_{v}, s) := \\ & ∏_{(i,j)∈I(σ_{1},σ_{2})} \frac{L(s_{i} − s_{j}, π_{i,v} × π̃_{j,v})}{L(1 + s_{i} − s_{j}, π_{i,v} × π̃_{j,v})ϵ(s_{i} − s_{j}, π_{i,v} × π̃_{j,v}, ψ_{v})}. \end{aligned}

As explained in [AC, p.87], the meromorphic functions r_{P_{2}|P_{1}}(π, s) satisfy all the properties of Theorem 2.1 of [A7] and they are the natural choice of normalizing factors in the case of GL_{n}. We note that they do not coincide with the normalizing factors used in the previous sections. They differ, however, only by a factor which can be expressed in terms of the ϵ-factors.

Now let π_{1} and π_{2} be automorphic representations of GL_{n_{1}}(𝔸) and GL_{n_{2}}(𝔸), respectively. Then the global Rankin-Selberg L-function L(s, π_{1} × π_{2}) is defined by

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

where the product is over all places v of ℚ and π_{i} = ⊗_{i}π_{i,v}. The product converges absolutely in a half-plane Re(s) ≫ 0. If π_{1} and π_{2} belong to the discrete spectrum of GL_{n_{1}}(𝔸) and GL_{n_{2}}(𝔸), respectively, then L(s, π_{1} × π_{2}) admits a meromorphic extension to the whole complex plane.

To define the global ϵ-factor ϵ(s, π_{1} × π_{2}) one has to pick a non-trivial continuous character ψ : 𝔸^{+} → ℂ^{×} of the additive group 𝔸^{+} of 𝔸. Then ψ = ⊗_{v}ψ_{v} and ϵ(s, π_{1,v} × π_{2,v}, ψ_{v}) = 1 for almost all places v. Hence the product

ϵ(s, π_{1} × π_{2}, ψ) = ∏_{v} ϵ(s, π_{1,v} × π_{2,v}, ψ_{v})

exists for all s ∈ ℂ and defines an entire function. The global ϵ-factor is independent of ψ and therefore, will be denoted by ϵ(s, π_{1} × π_{2}).

Let π ∈ Π_{disc}(M(𝔸)). Then π = π_{1} ⊗ ⋯ ⊗ π_{r} with π_{i} ∈ Π_{disc}(GL_{n_{i}}(𝔸)) and for s ∈ ℂ^{r}, the global normalizing factor is defined by

r_{P_{2}|P_{1}}(π, s) := ∏_{(i,j)∈I(σ_{1},σ_{2})} \frac{L(s_{i} − s_{j}, π_{i} × π̃_{j})}{L(1 + s_{i} − s_{j}, π_{i} × π̃_{j})ϵ(s_{i} − s_{j}, π_{i} × π̃_{j})}.

Theorem 5.3 is closely related to the estimation of the winding numbers

∫_{1}^{λ} \frac{L′(1 + it, π_{1} × π̃_{2})}{L(1 + it, π_{1} × π̃_{2})} dt

with upper bounds depending on the Casimir eigenvalues of π_{1,∞} and π_{2,∞} in the same way as in Theorem 5.3. In the present case, such estimates can be obtained using standard methods of analytic number theory. In fact, the bounds can be improved considerably.

Next we discuss the conditions (0.2) and (0.4). As mentioned in the introduction, for GL_{n} it is possible to prove that (0.2) and (0.4) hold. We shall briefly indicate the main steps of the proof. Let ρ be a representation of GL_{m}(ℚ_{v}) and s ∈ ℂ. Then we denote by ρ[s] the representation of GL_{m}(ℚ_{v}) defined by

ρ[s](g) = | det g|^{s}ρ(g), g ∈ GL_{m}(ℚ_{v}).

Let π be a cuspidal automorphic representation of GL_{m}(𝔸). Then it is known that each local component π_{v} of π is generic [Sk] and therefore, by [JS2] it follows that π_{v} is equivalent to a full induced representation, i.e.,

(7.1)  π_{v} ≅ I_{P}^{GL_{m}}(τ_{1}[t_{1}] ⊗ ⋯ ⊗ τ_{r}[t_{r}]),

where P is a standard parabolic subgroup of GL_{m} with Levi component GL_{m_{1}} × ⋯ × GL_{m_{r}}, τ_{i} is a tempered representation of GL_{m_{i}}(ℚ_{v}) and the t_{i}’s are real numbers satisfying

t_{1} > t_{2} > ⋯ > t_{r}, |t_{i}| < 1/2, i = 1, ..., r.

For π_{v} unramified, Luo, Rudnick and Sarnak [LRS] proved that the parameters t_{i} satisfy the following nontrivial bound:

(7.2)  max_{i} |t_{i}| < \frac{1}{2} − \frac{1}{m^{2} + 1}.

Using the same method, one can show that (7.2) holds at all places. Now let P be a standard parabolic subgroup of GL_{n} with Levi component GL_{n_{1}} × ⋯×GL_{n_{r}} and let π_{v} be the local v-component of a cuspidal automorphic representation of M(𝔸). Then π_{v} = ⊗_{i}π_{i,v} and each π_{i,v} is a full induced representation of the form (7.1) with parameters t_{ij} satisfying (7.2). Using induction in stages, it follows that for each i there exist a parabolic subgroup R_{i} of GL_{n_{i}}(ℚ_{v}) of type (n_{i1}, ..., n_{il_{i}}), a discrete series representation δ_{i,v} of M_{R_{i}}(ℚ_{v}) and t_{i} = (t_{i1}, ..., t_{il_{i}}) ∈ ℝ^{l_{i}} satisfying

(7.3)  max_{i} |t_{ij}| < \frac{1}{2} − \frac{1}{n_{i}^{2} + 1},

such that π_{i,v} ≅ I_{R_{i}}^{M}(δ_{i,v}, t_{i}). Put l = l_{1} + ⋯ + l_{r},

δ_{v} = ⊗_{i}δ_{i,v}, t = (t_{11}, ..., t_{1l_{1}}, ..., t_{r1}, ..., t_{rl_{r}}).

Generalizing property (R.2) of [A8, p.172], we get

(7.4)  R_{Q|P} (π_{v}, s) = R_{Q(R)|P (R)}(δ_{v}, s + t), s ∈ ℂ^{l},

where s is identified with an element in ℂ^{l} with respect to the embedding which corresponds to the canonical embedding 𝔞_{M}^{∗} ⊂ 𝔞_{P (R)}^{∗}. This leads to an immediate reduction of the problem. We can assume that π_{v} is square integrable. However, now we have to estimate the norm of the derivatives of R_{Q|P} (π_{v}, s)_{K_{v}} (resp. R_{Q|P} (π_{v}, s)_{σ_{v}}) in the domain

{s ∈ ℂ^{r} | | Re(s_{i})| < 1/2 − 1/(n^{2} + 1), i = 1, ..., r}.

The important point is that for π_{v} square integrable, R_{Q|P} (π_{v}, s) is holomorphic in the domain

{s ∈ ℂ^{r} | Re(s_{i} − s_{j}) > −1, 1 ≤ i < j ≤ r}

[MW]. Using the product formula for normalized intertwining operators, the above problem can be further reduced to the case where P is maximal parabolic and Q = \overline{P}. Then M = GL_{n_{1}} × GL_{n_{2}}, π_{v} = π_{1,v} ⊗ π_{2,v}, and we may regard the intertwining operator as a function R_{\overline{P} |P} (π_{v}, s) of one complex variable. Now we distinguish two cases.

1. v < ∞.

Let K_{v} ⊂ GL_{n}(ℚ_{v}) be an open compact subgroup. We may assume that K_{v} is a congruence subgroup. Then we have to estimate the norm of derivatives of R_{\overline{P} |P} (π_{v}, s)_{K_{v}} in the strip | Re(s)| < 1 − 2/(n^{2} + 1). Let

K_{M,v} = K_{v} ∩ M(ℚ_{v}).

Then K_{M,v} is an open compact subgroup of M(ℚ_{v}). Let 1 denote the trivial representation of K_{M,v} and let Π_{2}(M(ℚ_{v}); K_{M,v}) be the set of all π_{v} ∈ Π_{2}(M(ℚ_{v})) such that [π_{v}|_{K_{M,v}} : 1] > 0. By Theorem 10 of [HC2], Π_{2}(M(ℚ_{v}); K_{M,v}) is a compact subset of Π_{2}(M(ℚ_{v})). Furthermore, 𝔞_{M}^{∗} ≅ ℝ^{2} acts on Π_{2}(M(ℚ_{v})) by

π_{1,v} ⊗ π_{2,v} ↦ π_{1,v}[iu_{1}] ⊗ π_{2,v}[iu_{2}], (u_{1}, u_{2}) ∈ ℝ^{2}.

The stabilizer of a given representation π_{v} is a lattice L ⊂ ℝ^{2} so that the orbit 𝔬_{π_{v}} of π_{v} is a compact torus ℝ^{2}/L. Thus there exist δ_{1}, ..., δ_{l} ∈ Π_{2}(M(ℚ_{v}); K_{M,v}) such that

Π_{2}(M(ℚ_{v}); K_{M,v}) = 𝔬_{δ_{1}} ⊔ ⋯ ⊔ 𝔬_{δ_{l}}.

Since

R_{\overline{P} |P} (π_{1,v}[iu_{1}] ⊗ π_{2,v}[iu_{2}], s) = R_{\overline{P} |P} (π_{1,v} ⊗ π_{2,v}, s + i(u_{1} + u_{2})),

it suffices to consider a fixed discrete series representation π_{v}. Now recall that R_{\overline{P} |P} (π_{v}, s) is holomorphic in the strip | Re(s)| < 1. Furthermore by Theorem 2.1 of [A7], R_{\overline{P} |P} (π_{v}, s)_{K_{v}} is a finite rank matrix whose entries are rational functions of p_{v}^{−s}. Hence for every u ∈ ℝ, R_{\overline{P} |P} (π_{v}, u + iw)_{K_{v}} is a periodic function of w ∈ ℝ. From these observations it follows immediately that for every k ∈ ℕ_{0} there exists C > 0 such that

(7.5)  ∥ D_{s}^{k}R_{\overline{P} |P} (π_{v}, s)_{K_{v}} ∥≤ C

for all s ∈ ℂ in the strip | Re(s)| ≤ 1 − 2/(n^{2} + 1).

2. v = ∞.

Let σ_{v} ∈ Π(O(n)). Then we have to estimate the norm of derivatives of R_{\overline{P} |P} (π_{v}, s)_{σ_{v}} in the strip | Re(s)| < 1 − 2/(n^{2} + 1). First note that

M(ℝ) ≅ (ℝ^{∗})^{2} × (SL_{n_{1}}(ℝ) × SL_{n_{2}}(ℝ)).

Furthermore the set of discrete series representations of SL_{n_{i}}(ℝ) containing a fixed SO(n_{i})-type is finite [Wa2, p.398]. Hence in the same way as above, it follows that we can fix the discrete series representation π_{v}. Again R_{\overline{P} |P} (π_{v}, s) is holomorphic in the strip | Re(s)| < 1 and by Theorem 2.1 of [A7], R_{\overline{P} |P} (π_{v}, s)_{σ_{v}} is a rational function of s ∈ ℂ. This implies that for every k ∈ ℕ_{0} there exist C > 0 and N ∈ ℕ such that

(7.6)  ∥ D_{s}^{k}R_{\overline{P} |P} (π_{v}, s)_{σ_{v}} ∥≤ C(1 + |s|)^{N}

for all s ∈ ℂ with | Re(s)| < 1 − 2/(n^{2} + 1).

Combining (7.5) and (7.6) with the various steps of the reduction it follows that (0.2) and (0.4) hold for all local components π_{v} of cuspidal automorphic representations.

It remains to deal with local components of automorphic forms in the residual spectrum. For this purpose we use the description of the residual spectrum given by Mœglin and Waldspurger [MW]. First we recall the notion of a Speh representation [MW, I.5]. Let k|m, d = m/k and R a standard parabolic subgroup of GL_{m} of type (d, ..., d). Let δ be a discrete series representation of GL_{d}(ℚ_{v}). Then the induced representation

I_{R}^{GL_{m}}(δ[(k − 1)/2] ⊗ δ[(k − 3)/2 ⊗ ⋯ ⊗ δ[(1 − k)/2])

has a unique irreducible quotient which we denote by J(δ, k). It follows from Theorem D of [Ta] and [Vo] that for every π_{v} ∈ Π(GL_{m}(ℚ_{v})) there exist a standard parabolic subgroup P of type (m_{1}, ..., m_{r}), k_{i}|m_{i}, discrete series representations δ_{i} of GL_{d_{i}}(ℚ_{v}), d_{i} = m_{i}/k_{i}, and real numbers t_{1}, ..., t_{r} satisfying |t_{i}| < 1/2 such that

π_{v} ≅ I_{P}^{GL_{m}}(J(δ_{1}, k_{1})[t_{1}] ⊗ ⋯ ⊗ J(δ_{r}, k_{r})[t_{r}]).

Now suppose that π_{v} is a local component of an automorphic representation π in the residual spectrum of GL_{m}(𝔸). By [MW] there exist a standard parabolic subgroup Q of GL_{m} of type (d, ..., d) and a cuspidal automorphic representation μ of GL_{d}(𝔸) such that π_{v} is the unique irreducible quotient of the induced representation

I_{Q}^{GL_{m}}(μ_{v}[(k − 1)/2] ⊗ μ_{v}[(k − 3)/2] ⊗ ⋯ ⊗ μ_{v}[(1 − k)/2]),

where μ_{v} is the v-component of μ. As explained above, μ_{v} is equivalent to an induced representation of the form (7.1) with parameters t_{i} satisfying (7.2). Using induction in stages, it follows that

μ_{v} ≅ I_{R}^{GL_{d}}(δ_{1}[t_{1}] ⊗ ⋯ ⊗ δ_{r}[t_{r}]),

where R is a standard parabolic subgroup of GL_{d} of type (d_{1}, ..., d_{r}), δ_{i} is a discrete series representations of GL_{d_{i}}(ℚ_{v}), i = 1, ..., r, and the parameters t_{i} satisfy t_{1} ≥ t_{2} ≥ ⋯ ≥ t_{r} and (7.2). Then it follows from Proposition I.9 and Lemma I.8 of [MW] that there is a standard parabolic subgroup P of GL_{m} of type (kd_{1}, ..., kd_{r}) such that

(7.7)  π_{v} ≅ I_{P}^{GL_{m}}(J(δ_{1}, k)[t_{1}] ⊗ ⋯ ⊗ J(δ_{r}, k)[t_{r}])

and

(7.8)  max_{i} |t_{i}| < \frac{1}{2} − \frac{1}{m^{2} + 1}.

This is the extension of the results of [LRS] to local components of automorphic representations in the discrete spectrum.

Now we can proceed in the same way as in the cuspidal case. The only difference is that we have to deal with the slightly more general Speh representations in place of the discrete series representations. In this way one can establish (0.2) and (0.4). This implies that for GL_{n} the spectral side of the Arthur trace formula is absolutely convergent. Details will appear in [MS].

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

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