ON THE SINGULARITIES OF RESIDUAL INTERTWINING OPERATORS

WERNER MÜLLER

Abstract. Let G be a reductive algebraic group defined over ℚ. Let P, P′ be parabolic subgroups of G, defined over ℚ, and let t ∈ W (𝔞_{P} , 𝔞_{P′}). In this paper we study the intertwining operator M_{P′|P} (t, λ), λ ∈ 𝔞_{P,ℂ}^{∗}, acting in corresponding spaces of automorphic forms. One of the main results states that each matrix coefficient of M_{P′|P} (t, λ) is a meromorphic function of order ≤ n + 1, where n = dim G. Using this result, we further investigate the rank one intertwining operators, in particular, we study the distribution of their poles.

0. Introduction

The main purpose of this paper is to study the analytic properties of the intertwining operators arising in the theory of Eisenstein series [L1]. Our primary interest in this problem comes from the Arthur trace formula [A8]. The spectral side of the Arthur trace formula consists of a sum of distributions J_{χ}, which are indexed by a certain set of cuspidal data 𝔛, and by the main result of [A4], each distribution J_{χ} can be expressed in terms of generalized logarithmic derivatives of intertwining operators. This is the motivation for our investigation of the intertwining operators. Using the results of the present paper, we will try to tackle the problem of absolute convergence of the spectral side of the Arthur trace formula in a forthcoming paper. To state our results, we have to introduce some notation. Let G be a reductive algebraic group defined over ℚ, which for the introduction we take to be a connected semi-simple algebraic group. Let 𝔸 be the ring of adèles of ℚ. We denote by G(𝔸) the adèlized group, i.e., G(𝔸) is the restricted direct product of the groups G(ℚ_{v}) where v runs over all valuations of ℚ. We fix a Levi component M_{0} of a minimal parabolic subgroup P_{0} of G, defined over ℚ, and a maximal compact subgroup K of G(𝔸) which is admissible with respect to M_{0} [A5]. 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}. Let A^{2}(P) be the space of square integrable automorphic forms on

Date: June, 1998.

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

Key words and phrases. Automorphic forms, spectral decomposition.

N_{P} (𝔸)M_{P} (ℚ)A_{P} (ℝ)^{0}\G(𝔸).

This is the space of square integrable functions on this quotient which are K-finite and Z(𝔤)-finite, Z(𝔤) being the center of the universal enveloping algebra of 𝔤_{ℂ}. For every ϕ ∈ A^{2}(P) and every λ ∈ 𝔞_{P,ℂ}^{∗} with Re(λ) in a certain chamber, the Eisenstein series E(P, ϕ, λ, x) is defined by the absolutely convergent series (1.5), and one of the main results of Langlands’ theory is that E(P, ϕ, λ, x) can be continued analytically to a meromorphic function of λ ∈ 𝔞_{P,ℂ}^{∗}. 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 of the form Ad(g) for some g ∈ G(ℚ). P and Q are said to be associated if W (𝔞_{P} , 𝔞_{Q}) is not empty. Suppose that this is the case. Then 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 (see section 1). It is also part of Langlands’ theory that each intertwining operator M_{Q|P} (s, λ) admits an analytic continuation to a meromorphic function of λ ∈ 𝔞_{P,ℂ}^{∗}. The intertwining operators play an important role in the harmonic analysis of the spaces G(ℚ)\G(𝔸). In particular, generalized logarithmic derivatives of intertwining operators occur on the spectral side of the Arthur trace formula [A8]. Let Π(K) be the set of equivalence classes of irreducible unitary representations of K. We fix σ ∈ Π(K) and cuspidal data χ ∈ 𝔛 (see section 1). This singles out a finite dimensional subspace A^{2}(P, χ, σ) of A^{2}(P). The intertwining operator M_{Q|P} (s, λ) maps A^{2}(P, χ, σ) to A^{2}(Q, χ, σ). We shall call a holomorphic function f : ℂ^{n} → ℂ^{m} to be an entire function of order r ≥ 0, if there exist C, c > 0 such that

∥ f(z) ∥≤ Ce^{c∥z∥^{r}}, z ∈ ℂ^{n}.

Then our first main results are the following two theorems.

Theorem 0.1. Let n = dim G(ℝ)/K_{∞}. Let P, Q be associated parabolic subgroups and let t ∈ W (𝔞_{P} , 𝔞_{Q}). Let χ ∈ 𝔛 and σ ∈ Π(K). Let M_{Q|P} (t, λ) be the intertwining operator mapping the space A^{2}(P, χ, σ) to A^{2}(Q, χ, σ), λ ∈ 𝔞_{P,ℂ}^{∗}. There exist an entire function F (λ) of order ≤ n + 2 and an entire function M̂_{Q|P} (t, λ) of order ≤ n + 2 with values in the linear operators from A^{2}(P, χ, σ) to A^{2}(Q, χ, σ) such that

M_{Q|P} (t, λ) = \frac{M̂_{Q|P} (t, λ)}{F (λ)}, λ ∈ 𝔞_{P,ℂ}^{∗}.

Theorem 0.2. Let Q ⊃ P be parabolic subgroups. Let χ ∈ 𝔛 and σ ∈ Π(K). There exists an entire function q(λ) ≠ 0 of λ ∈ 𝔞_{P,ℂ}^{∗} of order ≤ n + 2 such that the following holds:

1) For each ϕ ∈ A^{2}(P, χ, σ) and each x ∈ G(𝔸), the function defined by

Ê^{Q}(P, ϕ, λ, x) = q(λ)E^{Q}(P, ϕ, λ, x)

is an entire function of λ ∈ 𝔞_{P,ℂ}^{∗} of order ≤ n + 2.

2) For every compact subset W of

N_{Q}(𝔸)M_{Q}(ℚ)A_{Q}(ℝ)^{0}\G(𝔸),

there exist C, c > 0 such that

sup_{x∈W} |Ê^{Q}(P, ϕ, λ, x)| ≤ C ∥ ϕ ∥ e^{c∥λ∥^{n+2}}

for all λ ∈ 𝔞_{P,ℂ}^{∗} and all ϕ ∈ A^{2}(P, χ, σ).

For SL(2), the statements of Theorem 0.1 and Theorem 0.2 were proved by Selberg [Se]. In [Mu1], we treated the case of the rank one cuspidal intertwining operators and rank one cuspidal Eisenstein series for general G. The proof given in [Mu1] is based on an extension of Colin de Verdiere’s method of the analytic continuation of Eisenstein series for SL(2). For this method to work, the assumption that we are dealing with cuspidal automorphic forms, is essential and the proof cannot be extended to the noncuspidal case. To prove Theorems 0.1 and 0.2 in general, we use Langlands’ description of the residual spectrum in terms of iterated residues of cuspidal Eisenstein series. We procced in several steps. In section 2, we use results of [Mu1] and the factorization of the intertwining operator into a product of intertwining operators induced from rank one intertwining operators to show that each cuspidal intertwining operator is a meromorphic operator valued function of order ≤ n + 2. Then, in section 3, we use this result together with the inner product formula for truncated Eisenstein series [L2] to establish Theorem 0.2 for all cuspidal Eisenstein series. Finally, in section 4, we use Langlands’ description of the space A^{2}(P, χ, σ) in terms of iterared residues of cuspidal Eisenstein series to express the general intertwining operator and noncuspidal Eisenstein series by iterated residues of cuspidal Eisenstein series and intertwining operators. Combining everything, this leads to the proof of Theorems 0.1 and 0.2. In section 5 we study the rank one intertwining operators more closely. Let P = NM be a parabolic subgroup and suppose that the split component A_{P} has dimension one. This is a maximal parabolic subgroup. Then any parabolic subgroup Q which is associated to P is either conjugate to P or its opposite group P^{−} = N^{−}M. If P and P^{−} are conjugate, then we put

M(z) = M_{P |P} (−1, zλ_{0}), z ∈ ℂ,

where λ_{0} is the normalized simple root of (P, A_{P}). Then M(z) is an operator in A^{2}(P, χ, σ). In the opposite case, if P and P^{−} are not conjugate, we form a block matrix M(z) from the intertwining operators M_{P |P^{−}}(1, −zλ_{0}) and M_{P^{−}|P} (1, zλ_{0}). The resulting operator acts then in A^{2}(P, χ, σ) ⊕ A^{2}(P^{−}, χ, σ). It follows from Theorem 0.1 that det M(z) is a meromorphic operator valued function of z ∈ ℂ of order ≤ n + 2. By recent results of Franke [Fr], it follows that det M(z) has only finitely many poles in the half-plane Re(z) ≥ 0 and all these poles are real. Denote them by u_{1}, ..., u_{l}. Let m_{j} be the order of the pole u_{j}. Using these facts together with Hadamard’s factorization theorem, we obtain the following product formula

det M(z) = det M(0)q^{z} ∏_{j=1}^{l}(\frac{z + u_{j}}{z − u_{j}} )^{m_{j}} ∏_{η} \frac{z + \overline{η}}{z − η} , z ∈ ℂ, (0.1)

where η runs over all poles, counted with their orders, of det M(z) in the half-plane Re(z) < 0. For the cuspidal rank one intertwining operators, this formula was proved in [Mu1, Section 6]. As in [Mu1], formula (0.1) can be used to compute the logarithmic derivative of det M(z). The formula that we obtain in this way shows that (d/dz) log det M(iλ), λ ∈ ℝ, is essentially of the form Σ_{−} − Σ_{+} where Σ_{∓} is a sum over all poles of det M(z) in the half-plane ∓ Re(z) > 0. The point is that all terms in Σ_{∓} are positive and Σ_{+} is a finite sum. In section 6 we estimate the number of poles of det M(z) in a circle of radius R > 0. The estimates obtained in this way are uniform in the Casimir eigenvalues of the Archimedean constituent σ_{∞} of σ, and χ.

These facts will be important for the estimation of the generalized logarithmic derivatives of the intertwining operators occurring in the Arthur trace formula which we will consider in a forthcoming paper. Theorems 0.1 and 0.2 may also have applications to the study of automorphic L-functions. In [GS] and [Ra], for example, Theorems 0.1 and 0.2 have been established in special cases and were used to prove that certain automorphic L-functions are bounded in vertical strips.

Acknowledgement I would like to thank J. Franke, G. Harder and W. Hoffmann for very helpful discussions on various parts of this paper. I also would like to thank the referee for his comments and remarks on an earlier version of this paper.

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, which contains M_{0}. 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). 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. Notice that M_{Q(R)} = M_{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}, 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 adèles -valued 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 ∈ 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

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

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. By Π(G(𝔸)) (resp. Π(M_{P} (𝔸)), Π(M_{P} (𝔸)^{1}), Π(K), etc.) we shall denote the set of equivalence classes of irreducible unitary representations of G(𝔸) (resp. M_{P} (𝔸), M_{P} (𝔸)^{1}, K, etc.). We note that the representations Π(M_{P} (𝔸)^{1}) can be naturally identified with the orbits of i𝔞_{P}^{∗} under the action of

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

1.4. 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 by

ϕ_{P} (g) = ∫_{ℕ(ℚ)\N(𝔸)} ϕ(ng) dn.  (1.2)

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

1.5. 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)A_{P} (ℝ)^{0}\G(𝔸) which are square integrable on M_{P} (ℚ)\M_{P} (𝔸)^{1}×K. This is the space of functions

ϕ: N_{P} (𝔸)M_{P} (ℚ)A_{P} (ℝ)^{0}\G(𝔸) → ℂ

which satisfy the following conditions:

i) For any x ∈ G(𝔸), the function m ↦ ϕ(mx), m ∈ M(𝔸), is Z(𝔪_{ℂ})-finite.

ii) The span of the set of functions ϕ_{k}(x) = ϕ(xk), x ∈ G(𝔸),

indexed by k ∈ K, is a finite dimensional space. iii)

∥ 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:

iv) 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 A2_{cusp}(P).

1.6. The space A^{2}(P) has several natural decompositions into invariant subspaces. Let L2_{disc}(M_{P} (ℚ)A_{P} (ℝ)^{0}\M_{P} (𝔸)) be the largest closed subspace of L^{2}(M_{P} (ℚ)A_{P} (ℝ)^{0}\M_{P} (𝔸)) on which the regular representation of M_{P} (𝔸) decomposes discretely. Then each ϕ ∈ A^{2}(P) has the property that for any x ∈ G(𝔸), the function

ϕ_{x}(m) = ϕ(mx), m ∈ M_{P} (𝔸),  (1.3)

belongs to L2_{disc}(M_{P} (ℚ)A_{P} (ℝ)^{0}\M_{P} (𝔸)). Let Π^{0}(M_{P} (𝔸)) be the set of all π ∈ Π(M_{P} (𝔸)) which are trivial on A_{P} (ℝ)^{0}. For any π ∈ Π^{0}(M_{P} (𝔸)), let L^{2}(M_{P} (ℚ)A_{P} (ℝ)^{0}\M_{P} (𝔸))(π) be the isotypical component of π, and let A^{2}(P)(π) be the subspace of all ϕ ∈ A^{2}(P) such that for each x ∈ G(𝔸), the function ϕ_{x}, defined by (1.3), belongs to L^{2}(M_{P} (ℚ)A_{P} (ℝ)^{0}\M_{P} (𝔸))(π). Then we get the following direct sum decomposition

A^{2}(P) = ⨁_{π∈Π^{0}(M_{P} (𝔸))}A^{2}(P)(π).

Another decomposition can be described in terms of the cuspidal data which means that one keeps track of the source of an iterated residue of a cuspidal Eisenstein series. In [A2], Arthur introduced a decomposition

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

indexed by a certain set 𝔛. We recall that 𝔛 is 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}. Given χ ∈ 𝔛, let A^{2}(P, χ) be the closed subspace of A^{2}(P) consisting of those ϕ such that for all x ∈ G(𝔸), the function ϕ_{x}, defined by (1.3), belongs to L^{2}(M_{P} (ℚ)\M_{P} (𝔸)^{1})_{χ}. Then

A^{2}(P) = ⨁_{χ∈𝔛}A^{2}(P, χ).

For our purpose it is important to single out a finite dimensional subspace of automorphic forms. To do so we fix a finite dimensional unitary representation σ of K on a complex vector space V. Then we denote by A^{2}(P, χ, σ) the subspace of all ϕ ∈ A^{2}(P, χ) such that for all x ∈ G(𝔸), the function

ϕ_{x}(k) = ϕ(xk), k ∈ K,

belongs to the isotypical subspace L^{2}(K)(σ) of σ in L^{2}(K). Then A^{2}(P, χ, σ) is a finite dimensional subspace of A^{2}(P, χ). To any χ ∈ 𝔛, there corresponds a class P_{χ} of associated parabolic subgroups. Suppose that P ∈ P_{χ}. Then L^{2}(M_{P} (ℚ)\M_{P} (𝔸)^{1})_{χ} is a subspace of the space of cusp forms L2_{cusp}(M_{P} (ℚ)\M_{P} (𝔸)^{1}). It is known that

L2_{cusp}(M_{P} (ℚ)\M_{P} (𝔸)^{1}) = ⨁_{ρ} V_{ρ},

where ρ ranges over all irreducible unitary representations of M_{P} (𝔸)^{1} and each V_{ρ} is an M_{P} (𝔸)^{1}- invariant subspace of L2_{cusp}(M_{P} (ℚ)\M_{P} (𝔸)^{1}), isomorphic under the action of M_{P} (𝔸)^{1} to a finite number of copies of ρ. Then

L^{2}(M_{P} (ℚ)\M_{P} (𝔸)^{1})_{χ} = L2_{cusp}(M_{P} (ℚ)\M_{P} (𝔸)^{1})_{χ} = ⨁_{{ρ|(M_{P} ,ρ)∈χ}} V_{ρ}.

Therefore, if P ∈ P_{χ} then A^{2}(P, χ) consists only of cusp functions. So, let A2_{cusp}(P, χ) denote the subspace of cusp functions in A^{2}(P, χ), and similarly by A2_{cusp}(P, χ, σ) we denote the subspace of cusp functions in A^{2}(P, χ, σ). Then it follows from the above remark that

A2_{cusp}(P, χ, σ) = \{ \begin{matrix}A^{2}(P, χ, σ) ; & if P ∈ P_{χ}, \\ \{0\} ; & if P ∉ P_{χ}. \end{matrix}  (1.4)

1.7. Next we recall some facts about Eisenstein series [L1]. Let ϕ ∈ A^{2}(P), x ∈ G(𝔸) and λ ∈ 𝔞_{P,ℂ}^{∗} with Re(λ) ∈ (𝔞_{P}^{∗})^{+} + ρ_{P} . Then the series

E(ϕ, λ, x) = ∑_{γ∈P (ℚ)\G(ℚ)}ϕ(γx)e^{(λ+ρ_{P})(H_{P} (γx))}  (1.5)

converges absolutely. This is the Eisenstein series attached to ϕ. As a function of λ, E(ϕ, λ, x) has an analytic continuation to a meromorphic function of λ ∈ 𝔞_{P,ℂ}^{∗} [L1]. If it is necessary to indicate that the Eisenstein series is associated to P, we will denote it by E(P, ϕ, λ, x).

Let P′ be any other parabolic subgroup which is associated to P. Each s ∈ W (𝔞_{P} , 𝔞_{P′}) has a representative w_{s} in G(ℚ). Given t ∈ W (𝔞_{P} , 𝔞_{P′}), ϕ ∈ A^{2}(P) and λ ∈ 𝔞_{P,ℂ}^{∗} with Re(λ) ∈ (𝔞_{P}^{∗})^{+} + ρ_{P}, define M_{P′|P} (t, λ)ϕ to be

\begin{aligned}(M & _{P′|P} (t, λ)ϕ)(x) = e^{−(tλ+ρ_{P′})(H_{P′}(x))} \\ & ∫_{N′(𝔸)∩w_{t}N(𝔸)w_{t}^{−1}\N′(𝔸)} ϕ(w_{t}^{−1}n′x)e^{(λ+ρ_{P})(H_{P} (w_{t}^{−1}n′x))} dn′. \end{aligned} (1.6)

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′). This is the global intertwining operator

M_{P′|P} (t, λ): A^{2}(P) → A^{2}(P′).

If χ ∈ 𝔛 and σ ∈ Π(K), then M_{P′|P} (t, λ) maps A^{2}(P, χ, σ) to A^{2}(P′, χ, σ). The main functional equations are

\begin{aligned}E(ϕ, λ, x) & = E(M_{P′|P} (t, λ)ϕ, tλ, x) \\ M_{P_{2}|P} (ts, λ) & = M_{P_{2}|P_{1}}(t, sλ)M_{P_{1}|P} (s, λ). \end{aligned}  (1.7)

for t ∈ W (𝔞_{1}, 𝔞_{2}) and s ∈ W (𝔞, 𝔞_{1}). Furthermore, suppose that ϕ belongs to A2_{cusp}(P). Then the constant term E_{P′}(P, ϕ, λ, x) of E(P, ϕ, λ, x) along P′ is given by

E_{P′}(P, ϕ, λ, x) = ∑_{t∈W (𝔞_{P} ,𝔞_{P′})}e^{(tλ+ρ_{P′})(H_{P′}(x))}(M_{P′|P} (t, λ)ϕ)(x) (1.8)

[MoWa].

2. Cuspidal intertwining operators

Let P ∈ P(M), P′ ∈ P(M′) and s ∈ W (𝔞_{P} , 𝔞_{P′}) and we assume that P ∈ P_{χ}. Then

L2_{cusp}(M_{P} (ℚ)\M_{P} (𝔸)^{1})_{χ} ≠ {0}.

Our main purpose in this section is to study the intertwining operator

M_{P′|P} (s, λ): A2_{cusp}(P, χ, σ) → A2_{cusp}(P′, χ, σ),

for λ ∈ 𝔞_{P,ℂ}^{∗}. First assume that the relative rank of P is equal to one, that is dim 𝔞_{P}^{G} = 1. Then any element of P(M) is conjugate either to P or to the opposite group P^{−} = N^{−}M. Moreover, P and P^{−} are conjugate if and only if −1 ∈ W (𝔞_{P}). Let P_{1}, ..., P_{r}, P_{1} = P, be a set of representatives for P(M)/W (𝔞_{P}). Then r = 1 or r = 2. Set P = P(M) and put

A2_{cusp}(P, χ, σ) = ⨁_{j=1}^{r} A2_{cusp}(P_{j}, χ, σ).  (2.1)

Let α_{i} be the unique simple root of (P_{i}, A_{i}), i = 1, ..., r, and put λ_{i} = α_{i}/|α_{i}|. If r = 2, then λ_{1} = −λ_{2} and we put λ = λ_{1}.

If r = 1, then W (𝔞_{P}) = {±1}. In this case, we set

M(z) = M_{P |P} (−1, zλ), z ∈ ℂ.  (2.2)

If r = 2, then W (𝔞_{P_{i}}) = {1}, i = 1, 2. Then we put

M(z) = (\begin{matrix}0 & M_{P_{1}|P_{2}}(1, −zλ) \\ M_{P_{2}|P_{1}}(1, zλ) & 0 \end{matrix}) , z ∈ ℂ. (2.3)

In either case, M(z) defines a linear operator

M(z): A2_{cusp}(P, χ, σ) → A2_{cusp}(P, χ, σ)

which is a meromorphic function of z ∈ ℂ. Moreover, by (1.7) it satisfies the functional equation

M(z)M(−z) = Id, z ∈ ℂ.  (2.4)

We also have

M(z)^{∗} = M(\overline{z}), z ∈ ℂ.  (2.5)

The poles of M(z) in the half-plane 0 ≤ Re(z) are all simple and contained in the interval (0, ||ρ_{P} ||]. Given ϕ ∈ A^{2}(P, χ, σ), ϕ = {ϕ_{i} | i = 1, ⋯, r}, put

E(ϕ, z) = ∑_{i=1}^{r} E(P_{i}, ϕ_{i}, zλ_{i}), z ∈ ℂ.

Then the functional equation (1.7) for the Eisenstein series takes the following form

E(ϕ, z) = E(M(z)ϕ, −z), z ∈ ℂ.  (2.6)

The poles of E(ϕ, z) coincide with the poles of M(z) (cf. [HC, Chapter IV, Theorem 7], for the analogous statement in the real case). Furthermore, it follows from (1.8) that the constant term of E(ϕ, z) along P_{i}, i = 1, ⋯, r, is given by

\begin{aligned}E_{P_{i}}(ϕ, z, x) & = e^{(zλ_{i}+ρ_{i})(H_{i}(x))}ϕ_{i}(x) \\ & + e^{(−zλ_{i}+ρ_{i})(H_{i}(x))}(M(z)ϕ)_{i}(x), x ∈ G(𝔸), \end{aligned}

and if P′ ∉ P, then E_{P′}(ϕ, z) = 0.

In [Mu1] we investigated the rank one cuspidal intertwining operators in the context of real Lie groups. In particular, it follows from the results of [Mu1, Section 5] that all matrix coefficients of any cuspidal rank one intertwining operator are meromorphic functions of finite order. We need to extend this result to the adèlic setting. For this purpose we use the well known correspondence between automorphic forms on the adèlic group G(𝔸) and on the real reductive group G(ℝ) [BJ]. Let K_{f} be an open compact subgroup of G(𝔸_{f}). Then the double coset space

G(ℚ)\G(𝔸)/G(ℝ)K_{f}

is known to be finite [Bo1, Section 5]. Let C ⊂ G(A_{f}) be a set of representatives of the double cosets. Then the groups

Γ_{c} = (G(ℝ) × cK_{f}c^{−1}) ∩ G(ℚ), c ∈ C,  (2.7)

are arithmetic subgroups of G(ℝ), and the action of G(ℝ) on the space of double cosets G(ℚ)\G(𝔸)/K_{f} induces the following decomposition into G(ℝ)-orbits

G(ℚ)\G(𝔸)/K_{f} ≅ ⨆_{c∈C}(Γ_{c}\G(ℝ)).  (2.8)

Now let

φ: G(ℚ)Z_{G}(ℝ)^{0}\G(𝔸) → ℂ

be an automorphic form [MoWa] which is K_{f}-invariant. Given c ∈ C, let φ_{c} be the function on G(ℝ) which is defined by

φ_{c}(x) = φ(cx), x ∈ G(ℝ).

Then φ_{c} is Γ_{c}-invariant and

φ_{c}: Γ_{c}Z_{G}(ℝ)^{0}\G(ℝ) → ℂ

is an automorphic form on G(ℝ). As proved by Borel and Jacquet [BJ], the map ϕ ⟼ (ϕ_{c})_{c∈C} induces an isomorphism

A(G(ℚ)Z_{G}(ℝ)^{0}\G(𝔸))^{K_{f}} ≅ ⨁_{c∈C} A(Γ_{c}Z_{G}(ℝ)^{0}\G(ℝ)) (2.9)

of the space of K_{f}-invariant automorphic forms on G(𝔸) and the direct sum of the spaces of Γ_{c}-automorphic forms on G(ℝ). In particular, this correspondence may be applied to Eisenstein series. The following computations were kindly communicated to us by W. Hoffmann. Let P be a standard parabolic subgroup and suppose that ϕ ∈ A^{2}(P) is right K_{f}-invariant. Then we have to compute E(P, ϕ, λ, cx) as a function of x ∈ G(ℝ). Since N is unipotent, it satisfies strong approximation [Bo1], i.e., we have that for any compact open subgroup U of N(𝔸_{f}),

N(𝔸) = N(ℚ)N(ℝ)U  (2.10)

and therefore,

N(ℚ)\N(𝔸) ≅ (N(ℚ) ∩ (N(ℝ) × U))\(N(ℝ) × U).

Given δ ∈ G(ℚ) and c ∈ C, let ϕ_{δc} be the function on G(ℝ), defined by ϕ_{δc}(x) = ϕ(δcx), x ∈ G(ℝ). Let ^{δ}P = δP δ^{−1}. Then an easy calculation using (2.10) shows that ϕ_{δc} is a square integrable automorphic form

ϕ_{δc}: δ^{−1}N(ℝ)A(ℝ)^{0}δ(Γ_{c} ∩^{δ}P (ℝ))\G(ℝ) → ℂ.

Let E(Γ_{c},^{δ}P (ℝ), ϕ_{δc}, λ, x) be the Eisenstein series on Γ_{c}\G(ℝ) associated to these data. Using that

H_{P} (δx) = Ad(δ)H_{^{δ}P} (x) + H_{P} (δ),

we get

\begin{matrix}E( & P, ϕ, λ, cx) \\ = ∑_{δ∈P (ℚ)\G(ℚ)/Γ_{c}} e^{(λ+ρ_{P})(H_{P} (δc))}E(Γ_{c},^{δ}P (ℝ), ϕ_{δc},^{δ}λ, x), \end{matrix} (2.11)

where ^{δ}λ = λ ∘ Ad(δ). Next we compare the constant terms. Let P′ be any other parabolic group and let φ be an automorphic form on G(ℚ)\G(𝔸)/K_{f}. Then the constant term φ_{P′} is defined by (1.2). Given c ∈ C, φ_{c} is an automorphic form on Γ_{c}\G(ℝ) and the constant term (φ_{c})_{P′} of φ_{c} along P′ is defined as

(φ_{c})_{P′}(x) = ∫_{Γ_{c}∩N′(ℝ)\N′(ℝ)} φ_{c}(n_{∞}x)dn_{∞}.

To compare the constant terms we use (2.10) where we choose U to be cK_{f}c^{−1} ∩ N(𝔸_{f}). Then we obtain

φ_{P′}(cx) = (φ_{c})_{P′}(x), x ∈ G(ℝ).  (2.12)

If we apply this equality to (2.11), we get

\begin{aligned}E & _{P′}(P, ϕ, λ, cx) \\ & = ∑_{δ∈P (ℚ)\G(ℚ)/Γ_{c}} e^{(λ+ρ_{P})(H_{P} (δc))}E_{P′}(Γ_{c},^{δ}P (ℝ), ϕ_{δc},^{δ}λ, x) \end{aligned} (2.13)

for all x ∈ G(ℝ). Now assume that the automorphic form ϕ belongs to A2_{cusp}(P, χ, σ). Then there exists a compact open subgroup K_{f} ⊂ G(𝔸_{f}) such that each ϕ ∈ A2_{cusp}(P, χ, σ) is right K_{f}-invariant. Being a cuspidal automorphic function, the constant term E_{P′}(P, ϕ, λ, x) is given by (1.8). Furthermore, by (2.12) each ϕ_{δc} is a cuspidal automorphic function on Γ_{c}\G(ℝ). To describe the constant terms on the right hand side of (2.13), we consider an arithmetic subgroup Γ of G(ℝ) and a parabolic subgroup P. Let A2_{cusp}(Γ, P) be the space of cuspidal automorphic forms on (Γ ∩ P (ℝ))N(ℝ)A(ℝ)^{0}\G(ℝ) [L1]. Recall that this is the subspace of L2_{cusp}((Γ ∩ P (ℝ))N(ℝ)A(ℝ)^{0}\G(ℝ)) consisting of all ϕ such that

1) ϕ is right K_{∞}-finite.

2) For each x ∈ G(ℝ), the function m → ϕ(mx) of m ∈ M(ℝ) is Z(𝔪_{ℂ})-finite.

Let P_{1} and P_{2} be two associated parabolic subgroups. For each t ∈ W (𝔞_{P_{1}}, 𝔞_{P_{2}}), there is an associated intertwining operator

C_{P_{2}|P_{1}}^{Γ}(t, λ): A2_{cusp}(Γ, P_{1}) → A2_{cusp}(Γ, P_{2})  (2.14)

which is a meromorphic function of λ ∈ 𝔞_{P_{1},ℂ}^{∗} [L1]. Then the constant terms of the Eisenstein series on the right hand side of (2.13) are given by

\begin{aligned}E & _{P′}(Γ_{c},^{δ}P (ℝ), ϕ_{δc},^{δ}λ, x) \\ & = ∑_{w∈W (𝔞_{^{δ}P} ,𝔞_{P′})}e^{(w^{δ}λ+ρ_{P′})(H_{P′}(x))} (C_{P′|^{δ}P}^{Γ_{c}} (w,^{δ}λ)ϕ_{δc}) (x). \end{aligned} (2.15)

Next observe that for each w ∈ W (𝔞_{^{δ}P} , 𝔞_{P′}) there exists t ∈ W (𝔞_{P} , 𝔞_{P′}) such that w = t ∘ Ad(δ^{−1}). Hence, the right hand side of (2.15) can be written as a sum over W (𝔞_{P} , 𝔞_{P′}). Collecting all terms on the right hand side of (2.13) that correspond to a given t ∈ W (𝔞_{P} , 𝔞_{P′}), gives

\begin{aligned}(M & _{P′|P} (t, λ)ϕ)(cx) \\ & = ∑_{δ∈P (ℚ)\G(ℚ)/Γ_{c}} e^{(λ+ρ_{P})(H_{P} (δc))} (C_{P′|^{δ}P}^{Γ_{c}} (^{δ}t,^{δ}λ)ϕ_{δc}) (x). \end{aligned} (2.16)

The intertwining operators (2.14) are not exactly those that we considered in [Mu1], but they are closely related to these operators. To describe the relation, let Γ and P be as above. Let σ_{∞}: K_{∞} → GL(V_{σ}) be an irreducible unitary representation of K_{∞} and let ψ: Z(𝔪_{ℂ}) → ℂ be a character. Then we denote by A2_{cusp}(Γ, P, ψ, σ_{∞}) the subspace of all ϕ ∈ A2_{cusp}(Γ, P) such that

1) For each x ∈ G(ℝ), the function k → ϕ(xk) of k ∈ K_{∞} is contained in the σ_{∞}-isotypical subspace of L^{2}(K_{∞}).

2) For each x ∈ G(ℝ), the function m → ϕ(mx) of m ∈ M(ℝ) is an eigenfunction of Z(𝔪_{ℂ}) with character ψ.

This is a finite-dimensional space of automorphic forms on Γ\G(ℝ) and A2_{cusp}(Γ, P) is the direct sum of all such subspaces. The intertwining operator (2.14) preserves this decomposition, i.e., for all σ_{∞} and ψ, (2.14) induces an operator

C_{P_{2}|P_{1}}^{Γ}(t, λ): A2_{cusp}(Γ, P_{1}, ψ, σ_{∞}) → A2_{cusp}(Γ, P_{2},^{t}ψ, σ_{∞}).

Now let σ_{∞} be the Archimedean component of σ. Since ϕ belongs to A2_{cusp}(P, χ, σ), there exists a character ψ : Z(𝔪_{ℂ}) → ℂ such that ϕ_{δc} ∈ A2_{cusp}(Γ_{c}, δ_{P} , ψ, σ_{∞}). Next we introduce the corresponding space of vector valued automorphic forms. For P = NM, let K_{∞}^{M} = M(ℝ) ∩ K. Set

X_{M} = A_{P} (ℝ)^{0}\M(ℝ)/K_{∞}^{M}.

Let Γ_{M} ⊂ M(ℝ) be the image of Γ ∩ P (ℝ) under the canonical projection P (ℝ) → P (ℝ)/N(ℝ) ≅ M(ℝ). Then Γ_{M} is an arithmetic subgroup of M(ℝ). Let L2_{cusp}(Γ_{M}A_{P} (ℝ)^{0}\M(ℝ), ψ, σ_{∞}) be the space of all cuspidal functions

φ: Γ_{M}A_{P} (ℝ)^{0}\M(ℝ) → V_{σ}

which satisfy:

1) φ(mk) = σ_{∞}(k)^{−1}φ(m) for all m ∈ M(ℝ) and k ∈ K_{∞}^{M}.

2) Dφ = ψ(D)φ for all D ∈ Z(𝔪_{ℂ}).

Then there is a canonical isomorphism

A2_{cusp}(Γ, P, ψ, σ_{∞}) ≅ L2_{cusp}(Γ_{M}A_{P} (ℝ)^{0}\M(ℝ), ψ, σ_{∞}) ⊗ V_{σ}^{∗},

and with respect to this isomorphism, the intertwining operator (2.14) takes the form

C_{P_{2}|P_{1}}^{Γ}(t, λ) = c_{P_{2}|P_{1}}^{Γ}(t, λ) ⊗ Id_{V_{σ}}

where c_{P_{2}|P_{1}}^{Γ}(t, λ) is the corresponding intertwining operator in the space of vector valued automorphic forms. The c_{P_{2}|P_{1}}^{Γ}(t, λ) are the intertwining operators which have been studied in [Mu1]. Now we return to formula (2.16). Suppose that P is a parabolic subgroup of relative rank one and that P′ is associated to P. Observe that the set P (ℚ)\G(ℚ)/Γ_{c} parametrizes conjugacy classes of parabolic subgroups mod Γ_{c}. In [Mu1, p.484] we have put together the intertwining operators, associated to representatives of Γ_{c}-conjugacy classes of parabolic subgroups, into a block matrix called C_{c}(s), s ∈ ℂ. The resulting operator acts on a space of cuspidal automorphic forms which was denoted in [Mu1] by E_{cus}(σ_{∞}, O). Here O stands for an orbit type of associated characters ψ. From the definition of C_{c}(s) follows that each of the operators c_{P′|^{δ}P}^{Γ_{c}} (^{δ}t,^{δ}λ) occurs as an entry of the block matrix C_{c}(λ), λ ∈ 𝔞_{P,ℂ}^{∗}. Therefore, we can apply the results of [Mu1]. It follows from [Mu1, Theorem 4.30] and [Mu1, Section 5] that for all ϕ, ψ ∈ E_{cus}(σ, O), ⟨C_{c}(λ)ϕ, ψ⟩ is a meromorphic function of λ ∈ 𝔞_{P,ℂ}^{∗} of order ≤ n + 2, where n = dim G(ℝ)/K_{∞}. Combined with (2.16) we obtain the following theorem. Theorem 2.1. Let n = dim G(ℝ)/K_{∞}. Then for all elements ϕ and ψ of A2_{cusp}(P, χ, σ), the function ⟨M(z)ϕ, ψ⟩ is a meromorphic function of z ∈ ℂ of order ≤ n + 2. Remark. Theorem 2.1 can also be proved directly without using the expression of the adèlic intertwining operators in terms of the intertwining operators for the real group. Since G(ℚ)\G(𝔸)/K_{∞}K_{f} is the disjoint union of locally symmetric spaces Γ_{c}\X, the cut-off Laplacian Δ_{T}, introduced in [Mu1], can also be defined in the adèlic setting (see [DH] for details). Once Δ_{T} exists, one can proceed as in [Mu1]. Now we shall extend Theorem 2.1 to cuspidal intertwining operators of any rank. Let P ∈ P(M), P_{1} ∈ P(M_{1}). Suppose that P and P_{1} are associated and let t ∈ W (𝔞_{P} , 𝔞_{P_{1}}). Let w_{t} ∈ G(ℚ) be a representative of t. Then M_{1} = w_{t}Mw_{t}^{−1} and tP = w_{t}P w_{t}^{−1} is a parabolic subgroup which belongs to P(M_{1}). The restriction of t to 𝔞_{P} ⊂ 𝔞_{0} defines an element in W (𝔞_{P} , 𝔞_{tP}). Let

t: A^{2}(P) → A^{2}(tP)

be the linear operator defined by

(tϕ)(x) = ϕ(w_{t}^{−1}x), x ∈ G(𝔸).  (2.17)

By Lemma 1.1 of [A5] there exists T_{0} ∈ 𝔞_{0} such that

H_{P_{0}}(w_{t}^{−1}) = T_{0} − t^{−1}T_{0}.

Then by (1.4) and (1.5) in [A4], one has

tM_{P_{1}|P} (s, λ) = M_{tP_{1}|P} (ts, λ)e^{−(sλ+ρ_{P_{1}})(T_{0}−t^{−1}T_{0})},  (2.18)

and

M_{P_{1}|P} (s, λ)t^{−1} = M_{P_{1}|P} (st^{−1}, tλ)e^{(λ+ρ_{P})(T_{0}−t^{−1}T_{0})}  (2.19)

for s ∈ W (𝔞_{P} , 𝔞_{P_{1}}). Thus as far as the order of the matrix coefficients of M_{P_{1}|P} (s, λ) are concerned, we can always assume that P_{1} and P have the same Levi component M, that is P, P_{1} ∈ P(M). Let C be the set of Weyl chambers in 𝔞_{P}. Then the Weyl group W (𝔞_{P}) acts both on P(M) and C, and

P(M)/W (𝔞_{P}) ≅ C/W (𝔞_{P}).

In particular, for each chamber C ∈ C, we get a parabolic group P_{C} = N_{C}M. Set Φ(C) := Φ_{P_{C}} and Δ(C) := Δ_{P_{C}}. Let C = {C_{1}, … C_{r}} and set P_{i} = P_{C_{i}}, i = 1, …, r. This set contains a set of representatives for P(M)/W (𝔞_{P}). Let 1 ≤ i, j ≤ r and t ∈ W (𝔞_{P}) be given. Then there exists k, 1 ≤ k ≤ r, such that tC_{i} = C_{k}. By the functional equation (1.7), we have

M_{P_{j}|P_{i}}(t, λ) = M_{P_{j}|P_{k}}(1, tλ)M_{P_{k}|P_{i}}(t, λ).  (2.20)

First consider M_{P_{k}|P_{i}}(t, λ). Since tC_{i} = C_{k} we have P_{k} = tP_{i}. Hence, by (2.19) it follows that

M_{P_{k}|P_{i}}(t, λ) = M_{P_{k}|P_{k}}(1, tλ)te^{(λ+ρ_{P})(T_{0}−t^{−1}T_{0})}.

But M_{P_{k}|P_{k}}(1, λ) = Id. Thus

M_{P_{k}|P_{i}}(t, λ) = e^{(λ+ρ_{P})(T_{0}−t^{−1}T_{0})}t.

Therefore, there exist C, c > 0 such that for all ϕ ∈ A2_{cusp}(P_{i}, χ, σ), ψ ∈ A2_{cusp}(P_{k}, χ, σ) and t ∈ W (𝔞_{P}), one has

|⟨M_{P_{k}|P_{i}}(t, λ)ϕ, ψ⟩| ≤ C ∥ ϕ ∥ ∥ ψ ∥ e^{c∥λ∥}, λ ∈ 𝔞_{P,ℂ}^{∗}. (2.21)

Definition 2.2. Let V be a finite dimensional complex vector space with norm ∥ · ∥ and let α ∈ ℝ_{+}. An entire function

f: ℂ^{N} → V

is said to be of order α, if there exist C, c > 0 such that

∥ f(Λ) ∥≤ C exp(c ∥ Λ ∥^{α}), Λ ∈ ℂ^{N}.

Then (2.21) can be reformulated as follows: Proposition 2.3. Let 1 ≤ i, j ≤ r and t ∈ W (𝔞_{P}) be such that t(C_{i}) = C_{j}. Then M_{P_{j}|P_{i}}(t, λ) is an entire operator valued function of order 1. If remains to consider M_{P_{j}|P_{i}}(1, λ). To this end we proceed as in [HC, Chapter V, 8]. First we recall some notation. Two chambers C_{1} and C_{2} are called adjacent if C_{1} ≠ C_{2} and Cl(C_{1}) ∩ Cl(C_{2}) contains a semiregular element. Let C ∈ C be given and let α ∈ Δ(C). Then there exists a unique chamber C^{α} which is adjacent to C and −α ∈ Δ(C^{α}). Given C_{i}, C_{j} ∈ C, there exists a sequence of chambers C_{i_{1}}, …, C_{i_{N}} such that

1) C_{i_{1}} = C_{i}, C_{i_{N}} = C_{j}.
2) C_{i_{k}} and C_{i_{k+1}} are adjacent for all k, 1 ≤ k ≤ N.

3) If α_{k} ∈ Δ(C_{i_{k}}) is such that C_{i_{k+1}} = C_{i_{k}}^{α_{k}}, 1 ≤ k ≤ N, then

α_{k} ∈ Φ(C_{l}) for l ≤ k and −α_{k} ∈ Φ(C_{l}) for l > k.

Using the functional equation (1.7), it follows by induction that

\begin{aligned}M_{P_{j}|P_{i}} & (1, λ) = M_{P_{i_{N}} |P_{i_{N−1}}}(1, λ) ⋯ \\ & ⋯ M_{P_{i_{3}}|P_{i_{2}}}(1, λ)M_{P_{i_{2}}|P_{i_{1}}}(1, λ), λ ∈ 𝔞_{P,ℂ}^{∗}. \end{aligned}  (2.22)

Hence we need only to consider M_{P_{j}|P_{i}}(1, λ) for adjacent chambers C_{j} and C_{i}. Assume that i = 1 and j = 2, so that C_{2} = C_{1}^{α}, where α ∈ Δ(C_{1}). Then P_{1} = N_{1} · M and P_{2} = N_{2} · M. Define (P′, A′) = (P_{1}, A_{P})_{{α}}. Then also (P′, A′) = (P_{2}, A_{P})_{{α}}. Thus P_{1}, P_{2} ⊂ P′ and M ⊂ M′. Let P_{i}^{∗} = M′∩P_{i}, i = 1, 2. Then P_{i}^{∗} are parabolic subgroups of M′ and M_{P_{i}^{∗}} = M. Moreover, we have

N_{i} = N_{i}^{∗}N′ and A = A^{∗}.  (2.23)

Let σ_{M′} be the restriction of σ to K^{M′} = M′(𝔸) ∩ K. Given χ ∈ 𝔛, let A2_{cusp}(P_{i}^{∗}, χ, σ_{M′}) be the corresponding space of automorphic forms on N_{i}^{∗}(𝔸)M^{∗}(ℚ)A^{∗}(ℝ)^{0}\M′(𝔸), and for λ^{∗} ∈ 𝔞_{P_{1}^{∗},ℂ}^{∗}, let

M_{P_{2}^{∗}|P_{1}^{∗}}(1, λ^{∗}): A2_{cusp}(P_{1}^{∗}, χ, σ_{M′}) → A2_{cusp}(P_{2}^{∗}, χ, σ_{M′}), (2.24)

be the associated intertwining operator. Let ϕ ∈ A2_{cusp}(P_{i}, χ, σ). Then for any k ∈ K, the function

ϕ_{k}(m′) = ϕ(m′k), m′ ∈ M′(𝔸),

belongs to A2_{cusp}(P_{i}^{∗}, χ, σ_{M′}), and using the definition (1.6) of the intertwining operator combined with (2.23), it follows that

(M_{P_{2}|P_{1}}(1, λ)ϕ)_{k} = M_{P_{2}^{∗}|P_{1}^{∗}}(1, λ^{∗})(ϕ_{k}),  (2.25)

where λ^{∗} is the projection of λ onto (𝔞_{M_{1}^{∗}}^{M′})_{ℂ}^{∗} = (𝔞_{M}^{M′})_{ℂ}^{∗}. Since P_{1}^{∗} and P_{2}^{∗} are parabolic subgroups of M′ of relative rank one, it follows from Theorem 2.1 that there exist an entire function F_{21}(z) of order ≤ n_{1} + 2, n_{1} = dim M′(ℝ)/K^{M′}, and an entire function M̂_{P_{2}^{∗}|P_{1}^{∗}}(z), of order ≤ n_{1} + 2, with values in the linear operators from

A2_{cusp}(P_{1}^{∗}, χ, σ_{M′}) to A2_{cusp}(P_{2}^{∗}, χ, σ_{M′}) such that

M_{P_{2}^{∗}|P_{1}^{∗}}(1, λ^{∗}) = \frac{M̂_{P_{2}^{∗}|P_{1}^{∗}}(λ^{∗})}{F_{21}(λ^{∗})}, λ^{∗} ∈ 𝔞_{P_{1}^{∗},ℂ}^{∗}.

Given ϕ ∈ A2_{cusp}(P_{1}, χ, σ), k ∈ K and λ ∈ (𝔞_{M}^{M′})_{ℂ}^{∗}, put

(M̂_{P_{2}|P_{1}}(λ)ϕ)(mk) = (M̂_{P_{2}^{∗}|P_{1}^{∗}}(λ^{∗})ϕ_{k}) (m) m ∈ M′(𝔸), (2.26)

where again λ^{∗} is the projection of λ onto (𝔞_{M}^{M′})_{ℂ}^{∗}. Then M̂_{P_{2}|P_{1}}(λ)ϕ can be identified with an element of A2_{cusp}(P_{2}, χ, σ). In this way we obtain a holomorphic function of linear operators

M̂_{P_{2}|P_{1}}(λ): A2_{cusp}(P_{1}, χ, σ) → A2_{cusp}(P_{2}, χ, σ), λ ∈ 𝔞_{P,ℂ}^{∗},

such that

M_{P_{2}|P_{1}}(1, λ) = \frac{M̂_{P_{2}|P_{1}}(λ)}{F_{21}(λ)}, λ ∈ 𝔞_{P,ℂ}^{∗}.

Let ϕ ∈ A2_{cusp}(P_{1}, χ, σ) and ψ ∈ A2_{cusp}(P_{2}, χ, σ). Then by (2.26), we have

⟨M̂_{P_{2}|P_{1}}(λ)ϕ, ψ⟩ = ∫_{K}⟨M̂_{P_{2}^{∗}|P_{1}^{∗}}(λ^{∗})ϕ_{k}, ψ_{k}⟩ dk.

Hence, we get

|⟨M̂_{P_{2}|P_{1}}(λ)ϕ, ψ⟩|
≤∥ M̂_{P_{2}^{∗}|P_{1}^{∗}}(λ^{∗}) ∥ \sqrt{∫_{K} ∥ ϕ_{k} ∥^{2} dk} \sqrt{∫_{K} ∥ ψ_{k} ∥^{2} dk}
≤∥ M̂_{P_{2}^{∗}|P_{1}^{∗}}(λ^{∗}) ∥ ∥ ϕ ∥ ∥ ψ ∥ .

Since M̂_{P_{2}^{∗}|P_{1}^{∗}}(λ^{∗}) is of order ≤ n_{1} + 2, where n_{1} ≤ n, it follows that M̂_{P_{2}|P_{1}}(λ) is an entire operator valued function of order ≤ n_{1} + 2. Now observe that for any i, j, the length N of the chain of adjacent chambers connecting C_{i} and C_{j} is bounded by the number of chambers. Therefore the above estimates combined with (2.22) and Proposition 2.3 imply the following theorem Theorem 2.4. Let n = dim G(ℝ)/K_{∞}. Let M be a Levi subgroup. For all P_{1}, P_{2} ∈ P(M) and t ∈ W (𝔞_{M}), there exist an entire function F_{P_{1},P_{2}}(t, λ) of λ ∈ 𝔞_{M,ℂ}^{∗} of order ≤ n + 2 and an entire operator valued function.

M̂_{P_{2}|P_{1}}(t, λ): A2_{cusp}(P_{1}, χ, σ) → A2_{cusp}(P_{2}, χ, σ), λ ∈ 𝔞_{M,ℂ}^{∗},

of order ≤ n + 2 such that

M_{P_{2}|P_{1}}(t, λ) = \frac{M̂_{P_{2}|P_{1}}(t, λ)}{F_{P_{1},P_{2}}(t, λ)}, λ ∈ 𝔞_{M,ℂ}^{∗}.

3. Cuspidal Eisenstein series

In this section we study Eisenstein series E(P, ϕ, λ, x) attached to cuspidal automorphic forms ϕ and our main purpose is to show that λ ↦ E(P, ϕ, λ, x) is a meromorphic function of finite order. First we recall the definition of the truncation operator Λ^{T} introduced by Arthur [A7]. Fix P_{0} ∈ P(M_{0}). Let T ∈ 𝔞_{0}^{+} be sufficiently regular which means that α(T) ≫ 0 for all α ∈ Δ_{P_{0}}. For any such T one has the truncation operator Λ^{T} which acts on continuous functions h on G(ℚ)\G(𝔸). Let τ̂_{P} be the characteristic function of

{H ∈ 𝔞_{0} | ω(H) > 0, ω ∈ Δ̂_{P} }.

Then Λ^{T} h is defined as

\begin{aligned}Λ^{T} h(x) = & ∑_{\{P |P ⊃P_{0}\}}(−1)^{dim(A_{P} /A_{G})} \\ & ∑_{γ∈P (ℚ)\G(ℚ)}∫_{N_{P} (ℚ)\N_{P} (𝔸)} h(nγx)τ̂_{P} (H_{P} (γx) − T) dn. \end{aligned}

Let P be a standard parabolic subgroup, i.e., P_{0} ⊂ P. Fix χ ∈ 𝔛 and σ ∈ Π(K) as above. Let ϕ ∈ A2_{cusp}(P, χ, σ). Then the truncated Eisenstein series Λ^{T} E(ϕ, λ, x) is square integrable and Langlands [L2] has established the following inner product formula for truncated Eisenstein series.

\begin{aligned}⟨Λ^{T} E(ϕ, λ), Λ^{T} & E(ϕ, λ′)⟩ = ∑_{P_{1}⊃P_{0}} ∑_{t,t′∈W (𝔞_{P} ,𝔞_{P_{1}})}Vol(𝔞_{P_{1}/L_{P_{1}}}) \\ & × \frac{⟨M_{P_{1}|P} (t, λ)ϕ, M_{P_{1}|P} (t′, λ′)ϕ⟩}{∏_{α∈Δ_{P_{1}}}(tλ + t′\overline{λ′})(α^{∨})}e^{(tλ+t′\overline{λ′})(T)}. \end{aligned} (3.1)

where L_{P_{1}} ⊂ 𝔞_{P_{1}} is the lattice generated by the co-roots {α^{∨} | α ∈ Δ_{P_{1}}}. By Theorem 2.4 there exists an entire function p(λ) ≠ 0 on 𝔞_{P,ℂ}^{∗} of order ≤ n + 2 such that for all P_{1} ⊃ P_{0} and t ∈ W (𝔞_{P} , 𝔞_{P_{1}}), the operator valued function

M̃_{P_{1}|P} (t, λ) = p(λ)M_{P_{1}|P} (t, λ)

is an entire function of order ≤ n + 2. Given P_{1} ⊃ P_{0} and t, t′ ∈ W (𝔞_{P} , 𝔞_{P′)}, let

D_{P_{1},t,t′} = \{λ ∈ 𝔞_{P,ℂ}^{∗} | _{∏α∈Δ_{P_{1}}}⟨t′\overline{λ} + tλ, α⟩ = 0\}.  (3.2)

Put

D = ⋃_{P_{1}⊃P_{0}} ⋃_{t,t′∈W (𝔞_{P} ,𝔞_{P_{1}})}D_{P_{1},t,t′}.  (3.3)

This is the union of finitely many real subspaces V_{1}, …, V_{m} of 𝔞_{P,ℂ}^{∗} of codimension 1 or 2. Let L_{1}, …, L_{r} be those subspaces V_{i} which are complex subspaces of 𝔞_{P,ℂ}^{∗}. In particular we have codim_{ℝ}L_{i} = 2, 1 ≤ i ≤ r. Moreover, let L_{r+1}, …, L_{k} be the intersections of all pairs (V_{i}, V_{j}), 1 ≤ i, j ≤ m, i ≠ j, such that V_{i} ∩ V_{j} is a complex subspaces of 𝔞_{P,ℂ}^{∗} of codimension 1. For each i, 1 ≤ i ≤ k, let α_{i} ∈ 𝔞_{P}^{∗} be an defining equation for L_{i}:

L_{i} = {λ ∈ 𝔞_{P,ℂ}^{∗} | α_{i}(λ) = 0},

and let m_{i} be the minimum of all m ∈ ℕ such that α_{i}(λ)^{m}E(ϕ, λ) is holomorphic near L_{i} for all ϕ ∈ A2_{cusp}(P, χ, σ). Put

q(λ) = p(λ) ∏_{j=1}^{k} α_{j}(λ)^{m_{j}}.  (3.4)

Then q(λ) is entire of order ≤ n + 2. Put

Ê(ϕ, λ, x) = q(λ)E(ϕ, λ, x).  (3.5)

Let U ⊂ ℂ be an open subset and let F be a Fréchet space. We recall that a function f : U → F is called holomorphic, if for each z ∈ U, the function z′ ↦ (z′ − z)^{−1}(f(z′) − f(z)) has a limit in F if z′ tends to z. An equivalent condition is that for each continuous linear functional ℓ : F → ℂ, the function ℓ ∘ f is holomorphic. If U ⊂ ℂ^{N} is open, then f : U → F is called holomorphic, if f is holomorphic in each variable z_{1}, ..., z_{N} of ℂ^{N}.

Lemma 3.1. For every ϕ ∈ A2_{cusp}(P, χ, σ), λ ↦ Λ^{T} Ê(ϕ, λ) ∈ L^{2} is a holomorphic function of λ ∈ 𝔞_{P,ℂ}^{∗}.

Proof. For all P_{1} ⊃ P_{0} and t ∈ W (𝔞_{P} , 𝔞_{P_{1}}) put

M̂_{P_{1}|P} (t, λ) = q(λ)M_{P_{1}|P} (t, λ).  (3.6)

Then M̂_{P_{1}|P} (t, λ) is an entire function of λ ∈ 𝔞_{P,ℂ}^{∗} and it follows from (3.1) that

\begin{aligned}∥ Λ^{T} Ê(ϕ, λ) ∥^{2} & = ∑_{P_{1}⊃P_{0}} ∑_{t,t′∈W (𝔞_{P} ,𝔞_{P_{1}})}Vol(𝔞_{P_{1}}/L_{P_{1}}) \\ & × \frac{⟨M̂_{P_{1}|P} (t, λ)ϕ, M̂_{P_{1}|P} (t′, λ)ϕ⟩}{∏_{α∈Δ_{P_{1}}}(tλ + t′\overline{λ})(α^{∨})} e^{(tλ+t′\overline{λ})(T)}. \end{aligned} (3.7)

By (3.7), λ ↦ Λ^{T} Ê(ϕ, λ) is holomorphic on the open dense subset 𝔞_{P,ℂ}^{∗} \ D. Now let λ_{0} ∈ D and suppose that Λ^{T} Ê(ϕ, λ) is singular at λ_{0}. Fix P_{1} and t, t′ ∈ W (𝔞_{P} , 𝔞_{P_{1}}) such that λ_{0} ∈ D_{P_{1},t,t′}. As proved by Langlands [L1], the singularities of E(ϕ, λ) lie along hyperplanes defined by root equations. By (3.5) the same holds for Ê(ϕ, λ) and also for Λ^{T} Ê(ϕ, λ). Therefore there exist β ∈ Δ_{P} and c ∈ ℂ, such that

H = {λ ∈ 𝔞_{P,ℂ}^{∗} | λ(β^{∨}) = c}

is a singular hyperplane of Λ^{T} Ê(ϕ, λ) and λ_{0} ∈ H. Since Λ^{T} Ê(ϕ, λ) is holomorphic in 𝔞_{P,ℂ}^{∗} \ D, it follows that H ⊂ D. We may assume that H ⊂ D_{P_{1},t,t′}. Moreover, by the definition (3.4) of q(λ), there exists a unique root α_{0} ∈ Δ_{P_{1}} such that

H ⊂ \{λ ∈ 𝔞_{P,ℂ}^{∗} | ⟨tλ + t′\overline{λ}, α_{0}⟩ = 0\}

and this subspace has real codimension 1. This implies that we may choose the singular point λ_{0} ∈ H of Λ^{T} Ê(ϕ, λ) such that

\begin{matrix}⟨tλ_{0} + t′\overline{λ}_{0}, α_{0}⟩ & = 0 \\ ⟨tλ_{0} + t′\overline{λ}_{0}, α⟩ & ≠ 0 for all α ≠ α_{0}. \end{matrix}  (3.8)

Let ν ∈ 𝔞_{P,ℂ}^{∗} be a unit vector which is orthogonal to the subspace

\{λ ∈ 𝔞_{P,ℂ}^{∗} | ⟨tλ + t′\overline{λ}, α_{0}⟩ = 0\}

and consider the line

λ(u) = uν + λ_{0}, u ∈ ℝ.

For ϵ ≪ 1, we have

D ∩ {λ(u) | |u| < ϵ} = {λ_{0}}.

Since E(ϕ, λ) is a meromorphic function whose singularities lie along hyperplanes, there exists m ≥ 1 such that the function

F (ϕ, λ) = (⟨λ, β⟩ − c)^{m}Λ^{T} Ê(ϕ, λ)  (3.9)

is holomorphic in a neighborhood of λ_{0} and F (ϕ, λ_{0}) ≢ 0. We simply have to choose λ_{0} such that it is not contained in the intersection of any two singular hyperplanes. By assumption we have ⟨λ_{0}, β⟩ = c. Hence

⟨λ(u), β⟩ − c = u⟨ν, β⟩ + ⟨λ_{0}, β⟩ − c = u⟨ν, β⟩,

and it follows from (3.9) that there exists a ∈ L^{2}(G(ℚ)Z_{G}(ℝ)^{0}\G(𝔸)) such that

Λ^{T} Ê(ϕ, λ(u)) = \frac{a}{u^{m}} + O(u^{−m+1})

as u → 0. Therefore there exists c > 0 such that

∥ Λ^{T} Ê(ϕ, λ(u)) ∥^{2}= \frac{c}{u^{2m}} + O(u^{−2m+1})

as u → 0. On the other hand, since ⟨tλ_{0} + t′\overline{λ}_{0}, α_{0}⟩ = 0, we have

⟨tλ(u) + t′\overline{λ(u)}, α_{0}⟩ = u⟨tν + t′\overline{ν}, α_{0}⟩,

and by our choice of ν, ⟨tν + t′\overline{ν}, α_{0}⟩ ≠ 0. By (3.8), it follows that the right hand side of (3.7) is O(u^{−1}) as u → 0. But the left hand side is of order O(u^{−2m}), m ≥ 1, as u → 0. Therefore, we get a contradiction which proves that Λ^{T} Ê(ϕ, λ) is entire. □

Proposition 3.2. There exist C, c > 0 such that for all λ ∈ 𝔞_{P,ℂ}^{∗} and all ϕ ∈ A2_{cusp}(P, χ, σ), we have

∥ Λ^{T} Ê(ϕ, λ) ∥≤ C ∥ ϕ ∥ e^{c∥λ∥^{n+2}}.

Proof. Put

h(λ) = ∏_{P_{1}⊃P_{0}} ∏_{t,t′∈W (𝔞_{P} ,𝔞_{P_{1}})} ∏_{α∈Δ_{P_{1}}} ⟨tλ + t′\overline{λ}, α⟩.

Then it follows from (3.6), (3.7) and Theorem 2.4 that there exist C, c > 0 such that

|h(λ)|^{2} ∥ Λ^{T} Ê(ϕ, λ) ∥^{2}≤ C ∥ ϕ ∥ exp(c ∥ λ ∥^{n+2}), λ ∈ 𝔞_{P,ℂ}^{∗},(3.10)

for all ϕ ∈ A2_{cusp}(P, χ, σ). We have to show that (3.10) implies a similar estimate for ∥ Λ^{T} Ê(ϕ, λ) ∥. This is a consequence of the repeated application of the following lemma. □

Lemma 3.3. Let g(z) be an analytic function of z ∈ ℂ^{n}. Let t, t′ ∈ GL_{n}(ℝ) and v ∈ ℝ^{n}. Suppose that there exist constants C, c > 0 and m ∈ ℕ such that

|⟨tz + t′\overline{z}, v⟩| |g(z)|^{2} ≤ Ce^{c∥z∥^{m}}

for all z ∈ ℂ^{n}. Then there exist C_{1}, c_{1} > 0 such that

|g(z)| ≤ C_{1}e^{c_{1}∥z∥^{m}}, z ∈ ℂ^{n}.

Proof. To begin with, we reduce the proof to a similar estimate for a function of one complex variable. Since t, t′ and v are real, there exist f_{1}, f_{2} ∈ Hom(ℝ^{n}, ℝ) such that

⟨tz + t′\overline{z}, v⟩ = f_{1}(Re(z)) + if_{2}(Im(z)).  (3.11)

Suppose that f_{1} ≠ 0. The case f_{2} ≠ 0 can be treated in the same way. Let

f_{1}(z) = ∑_{i=1}^{n} a_{i}z_{i}.

We may assume that a_{n} ≠ 0. Let w ∈ ℂ^{n−1} be fixed and put

a = − ∑_{i=1}^{n−1} \frac{a_{i}}{a_{n}} Re(w_{i}).

Then for z ∈ ℂ, we have

\begin{matrix}| Re & (z) − a| |g(w, z)|^{2} ≤ \\ ≤ |a_{n}|^{−1}|⟨t(w, z) + t′\overline{(w, z)}, v⟩| |g(w, z)|^{2} \\ ≤ \frac{C}{|a_{n}|}e^{2^{m}c∥w∥^{m}}e^{2^{m}c|z|^{m}}, \end{matrix}  (3.12)

and it is sufficient to prove that a similar estimate holds for g(w, z). Thus we may consider an analytic function f(z) of z ∈ ℂ which satisfies

| Re(z) − a| |f(z)|^{2} ≤ Ce^{c|z|^{m}}, z ∈ ℂ,  (3.13)

for certain constants C, c > 0 and m ∈ ℕ. Then our goal is to estimate |f(z)|. By (3.13) it follows that

|f(z)| ≤ Ce^{c|z|^{m}} for | Re(z) − a| ≥ 1.

Therefore, it suffices to estimate |f(z)| for | Re(z)−a| ≤ 1. For z = x+ iy and |x−a| < 1, let Γ_{z} ⊂ ℂ be the rectangle with edges a−2+i(y−1), a + 2 + i(y − 1), a + 2 + i(y + 1) and a − 2 + i(y + 1). Then we have

f(z) = (2πi)^{−1} ∫_{Γ_{z}} f(ξ)(ξ − z)^{−1} dξ.

Furthermore, for ξ ∈ Γ_{z} we have

|f(ξ)(ξ − z)^{−1}| ≤ Ce^{c|ξ|^{m}/2}| Re(ξ) − a|^{−1/2}|ξ − z|^{−1}.

Since 1 ≤ |ξ − z| ≤ 4, there exist C_{1}, c′ > 0, which are independent of z and a, such that

Ce^{c|ξ|^{m}/2}|ξ − z|^{−1} ≤ C_{1}e^{c′|z|^{m}}.

Hence

|f(z)| ≤ C_{1}(2π)^{−1}e^{c′|z|^{m}} ∫_{Γ_{z}} | Re(ξ) − a|^{−1/2}|dξ|.

The last integral is bounded by a constant which is independent of z and a. Therefore, there exists C_{2} > 0, independent of z and a, such that

|f(z)| ≤ C_{2}e^{c′|z|^{m}}.  (3.14)

Now we apply (3.14) to the function f(z) = g(w, z) with the initial estimate (3.12). This leads to

\begin{aligned}|g(w, z)| & ≤ C_{1}e^{c_{1}(|a|^{m}+∥w∥^{m}+|z|^{m})} \\ & ≤ C_{2}e^{c_{2}∥(w,z)∥^{m}} \end{aligned}

with constants C_{2}, c_{2} > 0 which depend only on C, c, t, t′ and v.

□

Corollary 3.4. For every compact subset W ⊂ G(ℚ)Z_{G}(ℝ)^{0}\G(𝔸) there exist C, c > 0 such that

sup_{x∈W} |Ê(ϕ, λ, x)| ≤ C ∥ ϕ ∥ e^{c∥λ∥^{n+2}}

for all λ ∈ 𝔞_{P,ℂ}^{∗} and ϕ ∈ A2_{cusp}(P, χ, σ).

Proof. Let W ⊂ G(ℚ)Z_{G}(ℝ)^{0}\G(𝔸) be any compact subset. We may assume that W is K-invariant. There exists a truncation parameter T ∈ 𝔞_{0}^{∗} such that

Λ^{T} Ê(ϕ, λ, x) = Ê(ϕ, λ, x)  (3.15)

for all x ∈ W and λ ∈ 𝔞_{P,ℂ}^{∗}. Furthermore, there exists an open compact subgroup K_{f} of G(𝔸_{f}) such that each ϕ ∈ A2_{cusp}(P, χ, σ) is K_{f}-invariant. Then E(ϕ, λ, x) is also K_{f}-invariant for any ϕ ∈ A2_{cusp}(P, χ, σ). As in (2.8), let C ⊂ G(𝔸_{f}) be a set of representatives for the double cosets G(ℚ)\G(𝔸)/G(ℝ)K_{f}. For c ∈ C let Γ_{c} ⊂ G(ℝ) be the arithmetic subgroup (2.7). Then

W/K_{f} ≅ ⨆_{c∈C} W_{c}

where W_{c} is a compact subset of Γ_{c}Z_{G}(ℝ)^{0}\G(ℝ). We may assume that ϕ is an eigenfunction of the Casimir element Ω_{M} ∈ Z(𝔪_{ℂ}) with eigenvalue μ which means that for any x ∈ G(𝔸), the function ϕ_{x}(m) = ϕ(mx), m ∈ M(ℝ), satisfies

Ω_{M}ϕ_{x} = μϕ_{x}.

Let c ∈ C. Then as a function of x ∈ G(ℝ), we have

ΩE(ϕ, λ, cx) = r(λ)E(ϕ, λ, cx)  (3.16)

where Ω ∈ Z(𝔤_{ℂ}) is the Casimir element and r(λ) is a quadratic polynomial in λ which satisfies

|r(λ)| ≤ C(μ^{2}+ ∥ λ ∥^{2}), λ ∈ 𝔞_{P,ℂ}^{∗},  (3.17)

with C > 0 independent of μ and λ. Let Δ be the Laplacian of G(ℝ) with respect to the left invariant metric. Then

Δ = −Ω_{G} + 2Ω_{K_{∞}},

where Ω_{K_{∞}} ∈ Z((𝔨_{∞})_{ℂ}) is the Casimir element of K_{∞}. Let κ = σ_{∞}(Ω_{∞}). By (3.16) we get

ΔE(ϕ, λ, cx) = (−r(λ) + 2κ)E(ϕ, λ, cx).  (3.18)

Let φ ∈ C_{0}^{∞}(Γ_{c}Z_{G}(ℝ)^{0}\G(ℝ)) be such that φ ≡ 1 on W_{c}. We pick T ∈ 𝔞_{0} so that (3.15) holds on supp φ · K_{f}. We note that

Δ(φÊ) = φ(ΔÊ) + (Δφ)Ê − ⟨∇φ, ∇Ê⟩  (3.19)

For k ∈ ℕ let ∥ · ∥_{k} be the k-Sobolev norm on Γ_{c}Z_{G}(ℝ)^{0}\G(ℝ) and ∥ · ∥_{W_{c},k} the Sobolev norm on W_{c}. By the fundamental elliptic estimate and (3.19) we get

∥ Ê ∥_{W_{c},k} <∥ φÊ ∥_{k}≤
≤ C(∥ φÊ ∥_{k−2} + ∥ Δ(φÊ) ∥_{k−2})
≤ C(∥ φÊ ∥_{k−2} + ∥ φΔÊ ∥_{k−2})
+ C_{1} ∥ Ê ∥_{W_{1},k−1},

where W_{1} = supp φ. Using (3.17) and (3.18), we obtain

∥ Ê(ϕ, λ) ∥_{W_{c},k}≤ C_{2}(1+ ∥ λ ∥^{2}) ∥ Ê(ϕ, λ) ∥_{W_{1},k−1} .

By iteration, we get

∥ Ê(ϕ, λ) ∥_{W_{c},n}≤ C_{n}(1+ ∥ λ ∥^{2})^{n} ∥ Ê(ϕ, λ) ∥_{W_{n},0},

where W_{n} ⊃ W_{c} is some open relatively compact subset of the space Γ_{c}Z_{G}(ℝ)^{0}\G(ℝ). Suppose that our truncation parameter has been chosen so that (3.15) holds on W_{n} · K_{f}. Then by the Sobolev embedding theorem we obtain

\begin{aligned}sup_{x∈W_{c}} |Ê(ϕ, λ, cx)| & ≤ C ∥ Ê(ϕ, λ) ∥_{W_{c},n} \\ & ≤ C_{n+1}(1+ ∥ λ ∥^{2}) ∥ Λ^{T} Ê(ϕ, λ) ∥ . \end{aligned}

Applying Proposition 3.2, the claimed estimate follows.

□

4. Residual Eisenstein series and intertwining operators

The main purpose of this section is to extend Theorem 2.4 to all intertwining operators acting on A^{2}(P, χ, σ). Our approach to this problem is based on Langland’s description of the space A^{2}(P, χ, σ) in terms of iterated residues of cuspidal Eisenstein series [L1]. For the convenience of the reader we will summarize some of the basic features of this theory. For all details we refer the reader to [L1], [MoWa]. Let B be a parabolic subgroup. Given α ∈ Φ_{B} and c ∈ ℝ, we set

H(α, c) = {Λ ∈ 𝔞_{B}^{∗} | Λ(α^{∨}) = c}.  (4.1)

An affine subspace 𝔱 ⊂ 𝔞_{B}^{∗} is called admissible if 𝔱 is the intersection of hyperplanes of the form (4.1). Any such subspace 𝔱 can be written as

𝔱 = Λ + 𝔱̃,

where 𝔱̃ ⊂ 𝔞_{B}^{∗} is a linear subspace defined by root equations and Λ is vector in 𝔞_{B}^{∗} which is orthogonal to 𝔱̃. Let

𝔱_{ℂ} = Λ + 𝔱̃ ⊗ ℂ

be the complexified affine subspace of 𝔞_{B,ℂ}^{∗}. Then 𝔱_{ℂ} is called an admissible subspace of 𝔞_{B,ℂ}^{∗}. Now suppose that 𝔱_{1} ⊃ 𝔱_{2} are two admissible subspaces of 𝔞_{B}^{∗} and that 𝔱_{2} is of codimension one in 𝔱_{1}. Let F (Λ) be a meromorphic function on 𝔱_{1,ℂ} whose singularities lie along hyperplanes which are admissible as subspaces of 𝔞_{B,ℂ}^{∗}. Choose a unit vector ν in 𝔱̃_{1} which is normal to 𝔱̃_{2}. It is uniquely determined up to sign. Then we can define a meromorphic function Res_{(ν,𝔱_{2})}F (Λ) on 𝔱_{2,ℂ} by taking the residue with respect to 𝔱_{2,ℂ}. More precisely, let Λ_{0} ∈ 𝔱_{2,ℂ} be a point with the property that any singular hyperplane of F which contains Λ_{0} also contains 𝔱_{2,ℂ}. Let Γ be a sufficiently small positively oriented circle about the origin in the complex plane. Then we put

Res_{(ν,𝔱_{2})}F (Λ_{0}) = \frac{1}{2πi} ∫_{Γ} F (Λ_{0} + zν) dz.

The singularities of Res_{(ν,𝔱_{2})}F lie along the intersections of the singular hyperplanes of F ,which are different from 𝔱_{2,ℂ}, with 𝔱_{2,ℂ}. Let B ⊂ P be parabolic subgroups. Suppose that

𝔱_{0} ⊂ 𝔱_{1} ⊂ ⋯ ⊂ 𝔱_{r} = 𝔞_{B}^{∗}

is a sequence of admissible affine subspaces such that for each i, 1 ≤ i ≤ r, 𝔱_{i−1} is of codimension 1 in 𝔱_{i} and 𝔱̃_{0} = 𝔞_{P}^{∗}. Thus for each i there exists β ∈ Φ_{B} such that β^{∨} does not vanish on 𝔱̃_{i} and

𝔱̃_{i−1} = {Λ ∈ 𝔱̃_{i} | Λ(β^{∨}) = 0}.

For each i we choose a unit vector ν_{i} ∈ 𝔱̃_{i} which is orthogonal to 𝔱̃_{i−1}. Let S denote the sequence of affine subspaces together with the choices of unit normal vectors. Then we denote 𝔱_{0} by

𝔱_{S} = Λ_{S} + 𝔞_{P}^{∗} .

Let F be a meromorphic function on 𝔞_{B,ℂ}^{∗} whose singularities lie along admissible hyperplanes. Then we can define inductively meromorphic functions F_{i} on 𝔱_{i,ℂ} by

F_{r} = F and F_{i} = Res_{(ν_{i},𝔱_{𝔦})}F_{i+1}, i = 0, …, r − 1.

Set

F_{0} = Res_{S}F.

This is the iterated residue of F with respect to S. It is a meromorphic function on

𝔱_{S,ℂ} = Λ_{S} + 𝔞_{P,ℂ}^{∗}.

Sometimes we will denote the iterated residue Res_{S}F (Λ_{0}) also by

Res_{S,Λ→Λ_{0}}F (Λ).

Recall that by Langlands’ theory of Eisenstein series, A^{2}(P, χ, σ) is spanned by iterated residues of cuspidal Eisenstein series which are associated to M_{P} (𝔸). Furthermore A^{2}(P, χ, σ) can be identified with a space of automorphic forms on M_{P} (𝔸) [MoWa, p.37]. The Eisenstein series are attached to parabolic subgroups B^{∗} of M_{P}. For any such B^{∗}, there is a unique parabolic subgroup B ⊂ P such that B^{∗} = B∩M_{P} and M_{B} = M_{B^{∗}}. Considered as automorphic forms on G(𝔸), the Eisenstein series are associated to ϕ ∈ A2_{cusp}(B, χ, σ) and are defined by

E^{P} (B, ϕ, λ, x) = ∑_{γ∈B(ℚ)\P (ℚ)}e^{(λ+ρ_{B}^{P})(H_{B}(γx))}ϕ(γx),  (4.2)

where λ ∈ 𝔞_{B,ℂ}^{∗} and Re(λ) ∈ (𝔞_{B}^{∗})^{+} + ρ_{B}. Sometimes we will denote E^{P} (B, ϕ, λ, x) simply by E^{P} (ϕ, λ, x). For P = G, we obtain the usual Eisenstein series (1.5). Because of the relation with the standard Eisenstein series on M_{P} (𝔸), it is clear that each E^{P} (ϕ, λ, x) has an analytic continuation to a meromorphic function of λ ∈ 𝔞_{B,ℂ}^{∗} satisfying the same properties as the standard Eisenstein series do. Let B be any group in P_{χ} which is contained in P. Let

F_{B}: 𝔞_{B,ℂ}^{∗} → A^{2}(B, χ, σ)

be a meromorphic function whose singularities lie along hyperplanes. Then for fixed x, E^{P} (F_{B}(Λ), Λ, x) is a meromorphic function on 𝔞_{B,ℂ}^{∗} whose singularities lie along hyperplanes. Let S be as above. Then

Res_{S}E^{P} (F_{B}(Λ), Λ, x), Λ ∈ 𝔱_{S,ℂ},

is a meromorphic function on 𝔱_{S,ℂ}. Let (𝔞_{B}^{P})_{ℂ}^{∗} be the orthogonal complement of 𝔞_{P,ℂ}^{∗} in 𝔞_{B,ℂ}^{∗}, and let S((𝔞_{B}^{P})_{ℂ}^{∗}) be the symmetric algebra on (𝔞_{B}^{P})_{ℂ}^{∗}. If F_{B} is regular at a point Λ in 𝔱_{S,ℂ}, let

dF_{B}(Λ) = d_{S}F_{B}(Λ)

be the linear map from S((𝔞_{B}^{P})_{ℂ}^{∗}) to A^{2}(B, χ, σ) which is obtained by expanding the analytic function

λ → F_{B}(λ + Λ), λ ∈ (𝔞_{B}^{P})_{ℂ}^{∗},

as a Taylor series about λ = 0. Then Res_{S}E^{P} (F_{B}(Λ), Λ, x) depends only on dF_{B}(Λ). For each B ∈ P_{χ} with B ⊂ P, Langlands constructs a finite set of admissible affine subspaces

𝔱 = Λ_{𝔱} + 𝔞_{P}^{∗}

of 𝔞_{B}^{∗}, and for each 𝔱, a finite set S_{B}(P, 𝔱) of sequences of admissible affine subspaces with distinguished unit normals as above, satisfying 𝔱_{S} = 𝔱 for each S ∈ S_{B}(P, 𝔱) with the following properties: For any group B ∈ P_{χ}, B ⊂ P, let S_{B}(P) be the disjoint union over all 𝔱 of the sets S_{B}(P, 𝔱). Suppose that for each such B, F_{B} is a meromorphic function from 𝔞_{B,ℂ}^{∗} to A^{2}(B, χ, σ) whose singularities lie along hyperplanes, and which is regular at each point Λ_{S}, S ∈ S_{B}(P). Then the function

ϕ(x) = ∑_{{B∈P_{χ}|B⊂P }} ∑_{S∈S_{B}(P)} Res_{S,Λ→Λ_{S}}E^{P} (F_{B}(Λ), Λ, x) (4.3)

belongs to A^{2}(P, χ, σ) and A^{2}(P, χ, σ) is spanned by functions of this form. Recall that the condition B ∈ P_{χ} implies that A^{2}(B, χ, σ) coincides with A2_{cusp}(B, χ, σ) so that F_{B} takes values in A2_{cusp}(B, χ, σ). Thus for regular values Λ, E^{P} (F_{B}(Λ), Λ, x) can be identified with a cuspidal Eisenstein series for M_{P} (𝔸). Let

Φ = ⨁_{B} ⨁_{S∈S_{B}(P)}(d_{S}F_{B})(Λ_{S}).  (4.4)

Then Φ belongs to the vector space

⨁_{B} ⨁_{S∈S_{B}(P)} Hom(S((𝔞_{B}^{P})_{ℂ}^{∗}), A^{2}(B, χ, σ))  (4.5)

and ϕ depends only on the vector Φ. Let L(P, χ, σ) be the subspace of the vector space (4.5) consisting of all vectors of the form (4.4). Then Φ → ϕ defines a surjective map of L(P, χ, σ) onto A^{2}(P, χ, σ). Suppose that P_{1} is a parabolic subgroup which is associated to P and let t ∈ W (𝔞_{P} , 𝔞_{P_{1}}). There exists a parabolic subgroup B_{1} ∈ P_{χ} with B_{1} ⊂ P_{1} and a unique element t_{B} ∈ W (𝔞_{B}, 𝔞_{B_{1}}) such that the restriction of t_{B} to 𝔞_{P} equals t and such that t_{B}(α) ∈ Δ_{B_{1}} for every α ∈ Δ_{B} which vanishes on 𝔞_{P}. Suppose that S ∈ S_{B}(P). Then we can transform S by t_{B} into a sequence tS of admissible affine subspaces of 𝔞_{B_{1}}^{∗} with distinguished unit normals associated to B_{1} and P_{1}. It follows by symmetry that we may take for S_{B_{1}}(P_{1}) the collection

tS_{B}(P) = {tS | S ∈ S_{B}(P)}.

Now consider the intertwining operator

M_{P_{1}|P} (t, λ): A^{2}(P, χ, σ) → A^{2}(P_{1}, χ, σ), λ ∈ 𝔞_{P,ℂ}^{∗}.

Let ϕ ∈ A^{2}(P, χ, σ). Then ϕ is the image under the map (4.3) of a vector Φ in L(P, χ, σ) of the form (4.4) for a collection

{F_{B}(Λ) | B ∈ P_{χ}, B ⊂ P }

of functions as above. Let λ be a point in general position in 𝔞_{P,ℂ}^{∗}. Then

F̃_{B_{1}}(Λ_{1}) = M_{B_{1}|B}(t_{B}, t_{B}^{−1}Λ_{1} + λ)F_{B}(t_{B}^{−1}Λ_{1}), Λ_{1} ∈ 𝔞_{B_{1},ℂ}^{∗}, (4.6)

is a meromorphic function on 𝔞_{B_{1},ℂ}^{∗} with values in A^{2}(B_{1}, χ, σ). By choosing λ properly, we can assume that F̃_{B_{1}} is regular at each point Λ_{S_{1}}, S_{1} ∈ tS_{B}(P). Then by [A6, Lemma 2.1], one has

\begin{aligned}M & _{P_{1}|P} (t, λ)ϕ(x) = \\ & ∑_{\{B_{1}∈P_{χ}|B_{1}⊂P_{1}\}} ∑_{S_{1}∈S_{B_{1}}(P_{1})} Res_{S_{1},Λ_{1}→Λ_{S_{1}}}E^{P_{1}}(F̃_{B_{1}}(Λ_{1}), Λ_{1}, x). \end{aligned} (4.7)

Thus in order to investigate M_{P_{1}|P} (t, λ) as a meromorphic function of λ, we have to study the iterated residues on the right hand side of (4.7). We first establish the following lemma.

Lemma 4.1. Let f_{1} and f_{2} be two entire functions on ℂ^{n} of order m ∈ ℕ whose zero divisors are hyperplanes. Let f = f_{1}/f_{2} and let H be a singular hyperplane of f. Let F = Res_{H}f be the residue of f with respect to H. Then there exist entire functions F_{1} and F_{2} on H ≅ ℂ^{n−1} of order ≤ m such that F = F_{1}/F_{2}. The zero divisors of F_{i}, i = 1, 2, are hyperplanes in ℂ^{n−1}.

Proof. Suppose that H is given by the equation α(z) = c where α ∈ Hom(ℂ^{n}, ℂ) and c ∈ ℂ. Changing coordinates by z ⟼ Az + b, A ∈ U(n, ℂ), b ∈ ℂ^{n}, we may assume that H is defined by z_{1} = 0. Then there exist k_{i} ∈ ℕ and holomorphic functions g_{i} on ℂ^{n}, i = 1, 2, such that

f_{i}(z) = z_{1}^{k_{i}}g_{i}(z), z ∈ ℂ^{n}, and g_{i}(0, z_{2}, …, z_{n}) ≢ 0. (4.8)

Then

f(z) = z_{1}^{−l} \frac{g_{1}(z)}{g_{2}(z)},

where l = k_{2} − k_{1}. Since g_{2} ↾ H ≢ 0 and H is a singular hyperplane of f, we have l ≥ 1. Let Γ be a circle in the z_{1}-plane of sufficiently small radius ε with center at the origin. Let z′ = (z_{2}, …, z_{n}) ∈ H ≅ ℂ^{n−1} be such that g_{2}(0, z′) ≠ 0. Then by Cauchy′s formula we get

(Res_{H}f)(z′) = \frac{1}{2πi} ∫_{Γ} f(z_{1}, z′) dz_{1}

= \frac{1}{2πi} ∫_{Γ} z_{1}^{−l} \frac{g_{1}(z_{1}, z′)}{g_{2}(z_{1}, z′)} dz_{1}

= \frac{(−1)^{l}}{(l − 1)!} \frac{∂^{l−1}}{∂z_{1}^{l−1}} (\frac{g_{1}(z_{1}, z′)}{g_{2}(z_{1}, z′)}) |_{z_{1}=0}.

The right hand side can be written as F_{1}(z′)/F_{2}(z′) where F_{2}(z′) = g_{2}(0, z′)^{l} and F_{1}(z′) is a polynomial in (∂^{p}/∂z_{1}^{p}) (g_{j})(0, z′), j = 1, 2, 0 ≤ p ≤ l − 1. In particular, F_{1} and F_{2} are entire functions on ℂ^{n−1} and

Res_{H}f(z′) = \frac{F_{1}(z′)}{F_{2}(z′)}.

It remains to estimate F_{i}. Let p ∈ ℕ_{0}. Then it follows from (4.8) that

\frac{∂^{p}g_{j}}{∂z_{1}^{p}} (0, z′) = \frac{p!}{(k_{j} + p)!} \frac{∂^{k_{j}+p}f_{j}}{∂z_{1}^{k_{j}+p}} (0, z′).  (4.9)

By assumption, f_{j} is of order m. Hence, if we apply Cauchy′s formula to the right hand side of (4.9), it follows that there exist C_{p}, c_{p} > 0 such that

|\frac{∂^{p}g_{j}}{∂z_{1}^{p}} (0, z′)| ≤ C_{p}e^{c_{p}∥z′∥^{m}}, z′ ∈ ℂ^{n−1}.

This inequality implies that F_{1} and F_{2} are entire functions of order ≤ m.

□

Now we are ready to investigate the iterated residues in (4.7). Fix B ∈ P_{χ} with B ⊂ P. Let ϕ_{1}, …, ϕ_{k} be an orthonormal basis of A^{2}(B, χ, σ). Then

F_{B}(Λ) = ∑_{j=1}^{k} ψ_{j}(Λ)ϕ_{j},

where ψ_{j}, 1 ≤ j ≤ k, are meromorphic functions on 𝔞_{B,ℂ}^{∗} which are holomorphic at Λ_{S}. Thus

E^{P} (F_{B}(Λ), Λ, x) = ∑_{i=1}^{k} ψ_{j}(Λ)E^{P} (ϕ_{j}, Λ, x).

Let ϕ ∈ A^{2}(B, χ, σ) and let ψ be a meromorphic function on 𝔞_{B,ℂ}^{∗} whose singularities lie along hyperplanes and which is holomorphic at Λ_{S}. Then

Res_{S,Λ→Λ_{S}}[ψ(Λ)E^{P} (ϕ, Λ, x)]

depends only on the element d_{S}ψ ∈ S((𝔞_{B}^{P})_{ℂ}^{∗}) which is obtained from the Taylor expansion of the function λ → ψ(λ + Λ_{S}) of λ ∈ (𝔞_{B}^{P})_{ℂ}^{∗} at λ = 0. In fact, finitely many components of d_{S}ψ are sufficient. Hence, we can replace ψ by its Taylor polynomial of sufficiently high order. In particular, we can assume that F_{B}(Λ) is a holomorphic function and that there exist C, c > 0 such that

∥ F_{B}(Λ) ∥≤ Ce^{c∥Λ∥}, Λ ∈ 𝔞_{B,ℂ}^{∗},  (4.10)

for every B ∈ P_{χ} with B ⊂ P. Using the definition (4.6) of F̃_{B_{1}} and Theorem 2.4, it follows that there exist entire functions h(Λ) and M̂_{B_{1}|B}(t_{B}, Λ) of Λ ∈ 𝔞_{B,ℂ}^{∗} of order ≤ n+2 such that

F̃_{B_{1}}(Λ_{1}) = \frac{M̂_{B_{1}|B}(t_{B}, t_{B}^{−1}Λ_{1} + λ)F_{B}(t_{B}^{−1}Λ_{1})}{h(t_{B}^{−1}Λ_{1} + λ)}  (4.11)

for Λ_{1} ∈ 𝔞_{B_{1},ℂ}^{∗}. Since h(Λ) and M̂_{B_{1}|B}(t_{B}, Λ) are of order ≤ n + 2, we have

|h(t_{B}^{−1}Λ_{1} + λ)| ≤ Ce^{c(∥Λ_{1}∥^{n+2}+∥λ∥^{n+2})}  (4.12)

and

∥ M̂_{B_{1}|B}(t_{B}, t_{B}^{−1}Λ_{1} + λ) ∥≤ Ce^{c(∥Λ_{1}∥^{n+2}+∥λ∥^{n+2})}.  (4.13)

As explained above, the Eisenstein series E^{P_{1}}(ϕ_{1}, Λ_{1}, x) may be regarded as standard Eisenstein series on M_{P} (𝔸). Therefore, the results of section 3 can be also applied to E^{P_{1}}(ϕ_{1}, Λ_{1}, x). Thus there exists an entire function q(Λ_{1}) of Λ_{1} ∈ 𝔞_{B_{1},ℂ}^{∗} of order ≤ n + 2 such that for all ϕ_{1} ∈ A^{2}(B_{1}, χ, σ), the function of Λ_{1} ∈ 𝔞_{B_{1},ℂ}^{∗} defined by

Ê^{P_{1}}(ϕ_{1}, Λ_{1}, x) = q(Λ_{1})E^{P_{1}}(ϕ_{1}, Λ_{1}, x)

is entire and Corollary 3.4 holds for Ê^{P_{1}}(ϕ_{1}, Λ_{1}, x). Hence, by 4.11 we get

\begin{matrix}E^{P_{1}}( & F̃_{B_{1}}(Λ_{1}), Λ_{1}, x) \\ = \frac{Ê^{P_{1}}(M̂_{B_{1}|B}(t_{B}, t_{B}^{−1}Λ_{1} + λ)F_{B}(t_{B}^{−1}Λ_{1}), Λ_{1}, x)}{h(t_{B}^{−1}Λ_{1} + λ)q(Λ_{1})}. \end{matrix} (4.14)

By (4.12), the denominator is an entire function of (Λ_{1}, λ) ∈ 𝔞_{B_{1},ℂ}^{∗}×𝔞_{P,ℂ}^{∗} of order ≤ n+2. To estimate the numerator, we pick a compact subset W ⊂ G(ℚ)Z_{G}(ℝ)^{0}\G(𝔸). Then by Corollary 3.4 there exist C, c > 0 such that for all x ∈ W, we have

\begin{matrix}|Ê^{P_{1}}( & M̂_{B_{1}|B}(t_{B}, t_{B}^{−1}Λ_{1} + λ)F_{B}(t_{B}^{−1}Λ_{1}), Λ_{1}, x)| \\ ≤ C ∥ M̂_{B_{1}|B}(t_{B}, t_{B}^{−1}Λ_{1} + λ) ∥ · ∥ F_{B}(t_{B}^{−1}Λ_{1}) ∥ e^{c∥Λ_{1}∥^{n+2}}. \end{matrix}

By (4.10) and (4.13), the right hand side is bounded from above by

C_{1}e^{c_{1}(∥Λ_{1}∥^{n+2}+∥λ∥^{n+2})}

for certain constants C_{1}, c_{1} > 0. Hence for each x ∈ G(𝔸), the numerator on the right hand side of (4.14) is also an entire function of (Λ_{1}, λ) ∈ 𝔞_{B_{1},ℂ}^{∗}×𝔞_{P,ℂ}^{∗} of order ≤ n+2 and the estimates are uniform on compact subsets of G(ℚ)Z_{G}(ℝ)^{0}\G(𝔸). Now repeated application of Lemma 4.1 together with (4.7) implies that there exist an entire function F (λ) of order ≤ n + 2 on 𝔞_{P,ℂ}^{∗} and an entire function M̂_{P_{1}|P} (t, λ), λ ∈ 𝔞_{P,ℂ}^{∗}, with values in the bounded operators from A^{2}(P, χ, σ) to A^{2}(P_{1}, χ, σ) such that

M_{P_{1}|P} (t, λ) = \frac{M̂_{P_{1}|P} (t, λ)}{F (λ)}, λ ∈ 𝔞_{P,ℂ}^{∗}.  (4.15)

Moreover, the above estimates imply that for every compact subset W ⊂ G(ℚ)Z_{G}(ℝ)^{0}\G(𝔸) and every ϕ ∈ A^{2}(P, χ, σ), there exist C, c > 0 such that

sup_{x∈W} |M̂_{P_{1}|P} (t, λ)ϕ(x)| ≤ Ce^{c∥λ∥^{n+2}}, λ ∈ 𝔞_{P,ℂ}^{∗}.

Since A^{2}(P, χ, σ) is finite dimensional, it follows that for each W as above there exist C_{1}, c_{1} > 0 such that

sup_{x∈W} |M̂_{P_{1}|P} (t, λ)ϕ(x)| ≤ C_{1} ∥ ϕ ∥ e^{c_{1}∥λ∥^{n+2}}  (4.16)

for all λ ∈ 𝔞_{P,ℂ}^{∗} and ϕ ∈ A^{2}(P, χ, σ). In order to proceed we need the following auxiliary result. Lemma 4.2. Let ϕ_{1}, …, ϕ_{m} be linearly independent continuous functions on N_{P} (𝔸)M_{P} (ℚ)A_{P} (ℝ)^{0}\G(𝔸). Then there exist continuous functions ψ_{1}, …, ψ_{m} ∈ C_{c}^{0}(N_{P} (𝔸)M_{P} (ℚ)A_{P} (ℝ)^{0}\G(𝔸)) such that

det (⟨ϕ_{i}, ψ_{j}⟩_{i,j=1}^{m}) ≠ 0.

Proof. First we claim that there exist points x_{1}, …, x_{m} in the space N_{P} (𝔸)M_{P} (ℚ)A_{P} (ℝ)^{0}\G(𝔸) such that

det ((ϕ_{i}(x_{j}))_{i,j=1}^{m}) ≠ 0.  (4.17)

It is easy to construct x_{1}, …, x_{m} recursively. Since ϕ_{1} ≠ 0, there exists x_{1} such that ϕ_{1}(x_{1}) ≠ 0. Suppose that x_{1}, …, x_{k−1}, 2 ≤ k ≤ m, have been constructed such that

det ((ϕ_{i}(x_{j}))_{i,j=1}^{k−1}) ≠ 0.

Let x_{k} be any point in N_{P} (𝔸)M_{P} (ℚ)A_{P} (ℝ)^{0}\G(𝔸). Then

det ((ϕ_{i}(x_{j}))_{i,j=1}^{k}) = ∑_{j=1}^{k}(−1)^{j+k}A_{j}ϕ_{j}(x_{k})  (4.18)

where A_{j} is the determinant of the minor obtained by deleting the j-th line and the k-th row. In particular,

A_{k} = det ((ϕ_{i}(x_{j}))_{i,j=1}^{k−1})

which is nonzero by assumption. Since ϕ_{1}, …, ϕ_{m} are linearly independent, we can pick x_{k} such that the right hand side of (4.18) is nonzero. Therefore, by finite induction, there exist x_{1}, …, x_{m} such that (4.17) holds. Now for each i, 1 ≤ i ≤ m, we choose an approximation

{ψ_{i,ε}} ⊂ C_{c}^{0}(N_{P} (𝔸)M_{P} (ℚ)A_{P} (ℝ)^{0}\G(𝔸))

of the delta function at x_{i}, that is, for every continuous function ϕ on N_{P} (𝔸)M_{P} (ℚ)A_{P} (ℝ)^{0}\G(𝔸) we have

∫_{N_{P} (𝔸)M_{P} (ℚ)A_{P} (ℝ)^{0}\G(𝔸)} ψ_{i,ε}(x)ϕ(x) dx ⟶_{ε→0} ϕ(x_{i}).

Then we obtain

det (⟨ϕ_{i}, ψ_{j,ε}⟩_{i,j=1}^{m}) ⟶_{ε→0} det ((ϕ_{i}(x_{j}))_{i,j=1}^{m}) ≠ 0.

Hence, for ϵ ≪ 1, ψ_{i} = ψ_{i,ε}, 1 ≤ i ≤ m, is the desired set of compactly supported functions.

□

We can now establish Theorem 0.1. Proof of Theorem 0.1. Formula (4.15) expresses M_{P_{1}|P} (t, λ) as a quotient of entire functions and we know already that F (λ) is of order ≤ n + 2. Therefore, we are left with the estimation of the norm of M̂_{P_{1}|P} (t, λ). Let ϕ_{1}, …, ϕ_{m} be an orthonormal basis of A^{2}(P_{1}, χ, σ) and let ϕ ∈ A^{2}(P, χ, σ). Then it suffices to estimate ⟨M̂_{P_{1}|P} (t, λ)ϕ, ϕ_{j}⟩. Choose ψ_{1}, …, ψ_{m}∈ C_{c}^{0}(N_{P} (𝔸)M_{P} (ℚ)A_{P} (ℝ)^{0}\G(𝔸)) as in Lemma 4.2 such that det(⟨ϕ_{i}, ψ_{j}⟩_{i,j=1}^{m}) ≠ 0. Then

⟨M̂_{P_{1}|P} (t, λ)ϕ, ψ_{j}⟩ = ∑_{j=1}^{m}⟨M̂_{P_{1}|P} (t, λ)ϕ, ϕ_{i}⟩⟨ϕ_{i}, ψ_{j}⟩.

Let A = (⟨ϕ_{i}, ψ_{j}⟩_{i,j=1}^{m}). Then det A ≠ 0, and therefore, A is invertible. Let A^{−1} = (b_{ij}). Then

⟨M̂_{P_{1}|P} (t, λ)ϕ, ϕ_{i}⟩ = ∑_{j=1}^{m} b_{ij}⟨M̂_{P_{1}|P} (t, λ)ϕ, ψ_{j}⟩.  (4.19)

Since each ψ_{j}, 1 ≤ j ≤ m, has compact support, it follows from (4.16) that there exist C, c > 0 such that

|⟨M̂_{P_{1}|P} (t, λ)ϕ, ψ_{j}⟩| ≤ C ∥ ϕ ∥ e^{c∥λ∥^{n+2}}

for λ ∈ 𝔞_{P,ℂ}^{∗} and j = 1, …, m. Therefore, by (4.19) we get

∥ M̂_{P_{1}|P} (t, λ)ϕ ∥≤ C_{1} ∥ ϕ ∥ e^{c∥λ∥^{n+2}}, λ ∈ 𝔞_{P,ℂ}^{∗}.

This concludes the proof of Theorem 0.1.

□

Similar results hold for Eisenstein series. Suppose that Q ⊃ P are parabolic subgroups. Let ϕ ∈ A^{2}(P, χ, σ). Then by (4.3), ϕ equals a sum of iterated residues of cuspidal Eisenstein series which are determined by a collection {F_{B} | B ∈ P_{χ}, B ⊂ P } of meromorphic functions F_{B} from 𝔞_{B,ℂ}^{∗} to A^{2}(B, χ, σ) satisfying the assumptions above. Let λ ∈ 𝔞_{P,ℂ}^{∗} be a point in general position. Then it follows from Theorem 7.1 of [L1] that

\begin{matrix}E^{Q}(P, ϕ, λ, x) \\ = ∑_{\{B∈P_{χ}|B⊂P \}} ∑_{S∈S_{B}(P)} Res_{S,Λ→Λ_{S}}E^{Q}(B, F_{B}(Λ), Λ + λ, x). \end{matrix} (4.20)

We can now proceed in exactly the same way as above to prove Theorem 0.2.

Since the integral (1.6) is absolutely convergent for Re(λ) ∈ (𝔞_{P}^{∗})^{+}+ρ_{P}, each intertwining operator M_{P_{1}|P} (t, λ) is a holomorphic function on the tube domain {λ ∈ 𝔞_{P,ℂ}^{∗} | Re(λ) ∈ (𝔞_{P}^{∗})^{+} + ρ_{P} }. We shall now estimate ∥ M_{P_{1}|P} (t, λ) ∥ on this domain. Such an estimate is known to exist for the restriction of the intertwining operator to the cuspidal subspace [HC, Lemma 38]. We shall extend it to the noncuspidal case. In addition to that, we need estimates that are uniform with respect to χ and σ_{∞}. Recall that 𝔛 is the set of W_{0} conjugacy classes of pairs (M_{B}, r_{B}), where B is a standard parabolic subgroup and r_{B} is an irreducible cuspidal automorphic representation of M_{B}(𝔸)^{1}. Given any irreducible unitary representation π of M_{B}(𝔸)^{1}, let μ_{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. Given χ ∈ 𝔛 and (M_{B}, r_{B}) ∈ χ, the Casimir eigenvalue μ_{r_{B}} depends only on the class χ and we denote it by μ_{χ}. Let τ be a unitary irreducible representation of K_{∞}. Then we denote by λ_{τ} = τ(Ω_{K_{∞}}) the eigenvalue of the Casimir operator Ω_{K_{∞}} of K_{∞}. We shall now consider a slightly different space of automorphic forms. Let K_{f} be a compact open subgroup of G(𝔸_{f}) and let τ ∈ Π(K_{∞}). Let A^{2}(P, χ, τ)^{K_{f}} be the subspace of A^{2}(P, χ) of all K_{f}- invariant automorphic forms which transform under K_{∞} according to τ. The uniform estimate we are aiming at is based on the following lemma.

Lemma 4.3. Let P be a standard parabolic subgroup and let 𝔖_{P} be a Siegel domain with respect to P. Let K_{f} be an open compact subgroup of G(𝔸_{f}). Then there exist C, c > 0 and N ∈ ℕ, such that for all χ ∈ 𝔛, τ ∈ K̂_{∞}, and ϕ ∈ A^{2}(P, χ, τ)^{K_{f}},

|ϕ(x)| ≤ C(1 + μ_{χ}^{2} + λ_{τ}^{2})^{N} ∥ ϕ ∥ exp(cρ_{P_{0}}(H_{P_{0}}(x)), x ∈ 𝔖_{P} .

Proof. We observe that A^{2}(P, χ, τ)^{K_{f}} is also spanned by sums of iterated residues of the form (4.3). Consider one of the iterated residues in (4.3), say

ψ(x) = Res_{S,Λ→Λ_{S}}E^{P} (F_{B}(Λ), Λ, x),

where B ∈ P_{χ} and S ∈ S_{B}(P). Then for each x ∈ G(𝔸), the function ψ_{x}(m) = ψ(mx) of m ∈ M(ℝ) is an eigenfunction of the Casimir operator Ω_{M} of M(ℝ) with eigenvalue ∥ Λ_{S} ∥^{2} − ∥ ρ_{B} ∥^{2} +μ_{χ}. Furthermore, it follows from Langlands’ construction that ∥ Λ_{S} ∥≤ ||ρ_{B}||. Thus we can assume that ϕ ∈ A^{2}(P, χ, τ)^{K_{f}} is such that for any x ∈ G(𝔸),

Ω_{M}ϕ_{x} = λϕ_{x} and |λ − μ_{χ}| ≤∥ ρ_{B} ∥^{2} .

Set

K_{f}^{M} = M(𝔸) ∩ K_{f}.

Given k ∈ K_{∞}, let ϕ_{k} be the function defined by ϕ_{k}(x) = ϕ(xk), x ∈ G(𝔸). Then ϕ_{k} can be identified with a square integrable automorphic form on M(ℚ)A(ℝ)^{0}\M(𝔸) which is K_{f}^{M}-invariant. Recall from section 2 that there exists a finite subset C ⊂ M(𝔸_{f}) such that

M(ℚ)\M(𝔸)/K_{f}^{M} ≅ ⨆_{c∈C}(Γ_{c}\M(ℝ))

where Γ_{c} is an arithmetic subgroup of M(ℝ). Therefore, it suffices to consider ϕ ∈ L^{2}(Γ_{c}A(ℝ)^{0}\M(ℝ), τ) which satisfies

Ω_{M}ϕ = λϕ, |λ − μ_{χ}| ≤∥ ρ_{B} ∥^{2} .  (4.21)

Let Δ be the Laplacian of M(ℝ) with respect to the left-invariant metric on M(ℝ). Let Ω_{K} be the Casimir element of K_{∞}^{M}. Then

Δ = −Ω_{M} + 2Ω_{K}.

Let λ_{τ} = τ(Ω_{K}). Then by (4.21), we get

Δϕ = (−λ + 2λ_{τ})ϕ.  (4.22)

Now observe that the sectional curvature k_{M(ℝ)} of Γ_{c}A(ℝ)^{0}\M(ℝ) has a uniform bound |k_{M(ℝ)}| ≤ C_{1}. Given x ∈ Γ_{c}A(ℝ)^{0}\M(ℝ), let i(x) denote the injectivity radius at x. Set

ĩ(x) = min{C_{1}^{−1/2}, i(x)}.

Let p = dim(A(ℝ)^{0}\M(ℝ)) and put N = [p/4] + 1. Then by [CGT, (1.26)], we get for x ∈ Γ_{c}A(ℝ)^{0}\M(ℝ),

\begin{aligned}|ϕ(x)| & ≤ C(ĩ(x)^{−p/2} ∥ ϕ ∥_{B_{ĩ(x)}(x)} + ⋯ + ĩ(x)^{2N−p/2} ∥ Δ^{N}ϕ ∥_{B_{ĩ(x)}(x)}) \\ & ≤ C(ĩ(x)^{−p/2} ∥ ϕ ∥ + ⋯ + ĩ(x)^{2N−p/2} ∥ Δ^{N}ϕ ∥). \end{aligned}

Applying (4.22), it follows that

|ϕ(x)| ≤ Cĩ(x)^{−p/2}(1 + μ_{χ}^{2} + λ_{τ}^{2})^{p/2} ∥ ϕ ∥ .  (4.23)

Using the description of the metric on 𝔖_{P} [Bo2], it is easy to see that there exist C > 0 such that

Ce^{−2ρ_{P_{0}}(H_{P_{0}}(x))} ≤ ĩ(x), x ∈ 𝔖_{P} .

Combined with (4.23), the lemma follows.

□

We can now estimate the intertwining operator. Let P ∈ P(M), P_{1} ∈ P(M_{1}) and let t ∈ W (𝔞_{P} , 𝔞_{P_{1}}). Suppose that λ ∈ 𝔞_{P,ℂ}^{∗} is such that Re(λ) ∈ (𝔞_{P}^{∗})^{+} + ρ_{P}. Then for any ϕ ∈ A^{2}(P, χ, σ), M_{P_{1}|P} (t, λ)ϕ(x) is given by the absolutely convergent integral (1.6). Then we obtain

\begin{aligned}|M_{P_{1}|P} & (t, λ)ϕ(x)| ≤ e^{−(Re(tλ)+ρ_{P_{1}})(H_{P_{1}}(x))} \\ & ∫_{N_{1}(𝔸)∩w_{t}N(𝔸)w_{t}^{−1}\N_{1}(𝔸)} |ϕ(w_{t}^{−1}nx)|e^{(Re(λ)+ρ_{P})(H_{P} (w_{t}^{−1}nx))} dn. \end{aligned}

Since

Vol((N_{1}(ℚ) ∩ w_{t}N(ℚ)w_{t}^{−1})\(N_{1}(𝔸) ∩ w_{t}N(𝔸)w_{t}^{−1})) = 1,

we can also integrate over

(N_{1}(ℚ) ∩ w_{t}N(ℚ)w_{t}^{−1})\N_{1}(𝔸).

Now observe that

N_{1}(ℚ) ∩ w_{t}P (ℚ)w_{t}^{−1} = N_{1}(ℚ) ∩ w_{t}N(ℚ)w_{t}^{−1}

(see[MoWa, p.94]). Given λ ∈ 𝔞_{P,ℂ}^{∗}, put

ϕ_{λ}(x) = ϕ(x)e^{(λ+ρ_{P})(H_{P} (x))}.

Then, as in [MoWa, p.89], we get

\begin{matrix}∫_{N_{1}(ℚ)∩w_{t}P (ℚ)w_{t}^{−1}\N_{1}(𝔸)} | & ϕ_{λ}(w_{t}^{−1}nx)| dn ≤ \\ ≤ ∫_{N_{1}(ℚ)\N_{1}(𝔸)} ∑_{γ∈P (ℚ)\G(ℚ)}|ϕ_{λ}(γnx)| dn. \end{matrix}

Suppose that ϕ ∈ A^{2}(P, χ, τ)^{K_{f}}. Let 𝔖_{P} be a Siegel domain with respect to P. By Lemma 4.3 there exist C, c > 0 and N ∈ ℕ such that for all m ∈ 𝔖_{P}

|ϕ(mk)| ≤ C(1 + μ_{χ}^{2} + λ_{τ}^{2})^{N} ∥ ϕ ∥ exp(cρ_{P_{0}}(H_{P_{0}}(m))). (4.24)

Let P_{0} ∈ P(M_{0}) and suppose that P_{0} ⊂ P. We may assume that c ≥ 2. Let y: G(𝔸) → ℝ_{+}^{∗} be a function which satisfies

y(nmx) = e^{cρ_{P_{0}}(H_{P_{0}}(m))}y(x),  (4.25)

for all n ∈ N_{0}(𝔸), m ∈ M_{0}(𝔸), x ∈ G(𝔸), and

1 < y(k) for all k ∈ K.

For x ∈ G(𝔸), put

E^{M}(x) = ∑_{γ∈M(ℚ)∩P_{0}(ℚ)\M(ℚ)}y(γx).

This is an Eisenstein series for M(𝔸). By our choice of y, it is convergent and defines an automorphic form on N(𝔸)M(ℚ)\G(𝔸). Using (4.24) and (4.25), it follows as in [MoWa, p.88] that

|ϕ_{λ}(x)| ≤ C(1 + μ_{χ}^{2} + λ_{τ}^{2})^{N} ∥ ϕ ∥ e^{Re(λ)(H_{P} (x))}E^{M}(x)

for all x ∈ G(𝔸). Now observe that

∑_{γ∈P (ℚ)\G(ℚ)}e^{Re(λ)(H_{P} (γx))}E^{M}(γx)

= ∑_{γ∈P (ℚ)\G(ℚ)}e^{Re(λ)(H_{P} (γx))} ∑_{γ′∈P_{0}(ℚ)∩M(ℚ)\M(ℚ)}e^{cρ_{P_{0}}(H_{P_{0}}(γ′γx))}

= ∑_{γ∈P_{0}(ℚ)\G(ℚ)}e^{(Re(λ)+cρ_{P_{0}})(H_{P_{0}}(γx))}.

Let ω ⊂ N_{1}(𝔸) be a compact subset such that N_{1}(ℚ)ω = N_{1}(𝔸). Given m ∈ M(𝔸), let W ⊂ G(𝔸) be any compact subset such that ωmK ⊂ W. It follows as in [HC, p.45] that there exist C_{1} > 0 and H_{1} ∈ 𝔞_{0} such that

∑_{γ∈P_{0}(ℚ)\G(ℚ)}e^{(λ+ρ_{P_{0}})(H_{P_{0}}(γx))} ≤ C_{1} \frac{e^{(Re(λ)+ρ_{P_{0}})(H′_{1})}}{∏_{α∈Δ_{0}} |⟨Re(λ) + ρ_{P_{0}}, α⟩|}

for λ ∈ (𝔞_{0}^{∗})^{+} + ρ_{P_{0}}, x ∈ W. Since ∏_{α∈Δ_{0}} |⟨Re(λ) + ρ_{P_{0}}, α⟩| ≥ C_{2} for λ ∈ (𝔞_{0}^{∗})^{+} + ρ_{P_{0}}, we get

∫N_{1}(ℚ)\N_{1}(𝔸) ∑_{γ∈P_{0}(ℚ)\G(ℚ)}e^{(Re(λ)+cρ_{0})(H_{P_{0}}(γnmk))} dn

≤ sup_{x∈W} ∑_{γ∈P_{0}(ℚ)\G(ℚ)}e^{(Re(λ)+cρ_{P_{0}})(H_{P_{0}}(γx))}

≤ Ce^{(Re(λ)+ρ_{P_{0}})(H_{1})}.

Summarizing, we have proved the following lemma. Lemma 4.4. Let N = 2 dim(M(ℝ)/A(ℝ)^{0}). For every compact subset ω ⊂ M(𝔸), there exist C > 0 and H_{1} ∈ 𝔞_{0} such that for all χ ∈ 𝔛, τ ∈ K̂_{∞}, and ϕ ∈ A^{2}(P, χ, τ)^{K_{f}}, one has

\begin{matrix}|M_{P_{1}|P} (t & , λ)ϕ(mk)| \\ ≤ C ∥ ϕ ∥ (1 + μ_{χ}^{2} + λ_{τ}^{2})^{N}e^{Re(λ)(H_{1})}, \end{matrix}

for all λ ∈ 𝔞_{P,ℂ}^{∗} with Re(λ) ∈ (𝔞_{P}^{∗})^{+} + ρ_{P}, t ∈ W (𝔞_{P} , 𝔞_{P_{1}}), m ∈ ω, and k ∈ K. Let ϕ_{1}, …, ϕ_{l} be an orthonormal basis of A^{2}(P, χ, τ)^{K_{f}}. Choose functions ψ_{1}, …, ψ_{l} ∈ C_{c}^{0}(M(ℚ)\M(𝔸)^{1}K) as in Lemma 4.2 with respect to ϕ_{1}, …, ϕ_{l} such that

det(⟨ϕ_{i}, ψ_{j}⟩_{i,j=1}^{l}) ≠ 0.

Let (b_{rs}) be the inverse matrix to (⟨ϕ_{i}, ψ_{j}⟩_{i,j=1}^{l}). Then similarly to (4.19), we get

⟨M_{P_{1}|P} (t, λ)ϕ, ϕ_{i}⟩ = ∑_{j=1}^{l} b_{ij}⟨M_{P_{1}|P} (t, λ)ϕ, ψ_{j}⟩.

Applying Lemma 4.4, we obtain the following corollary. Corollary 4.5. Let the notation be as above. There exist C > 0 and H_{1} ∈ 𝔞_{0}, such that for all χ ∈ 𝔛, τ ∈ K̂_{∞}, ϕ ∈ A^{2}(P, χ, τ)^{K_{f}} and t ∈ W (𝔞_{P} , 𝔞_{P_{1}}),

∥ M_{P_{1}|P} (t, λ) ∥≤ C(1 + μ_{χ}^{2} + λ_{τ}^{2})^{N}e^{Re(λ)(H_{1})}

for all λ ∈ 𝔞_{P,ℂ}^{∗} with Re(λ) ∈ (𝔞_{P}^{∗})^{+} + ρ_{P}.

5. The rank one intertwining operators

We shall now investigate the rank one intertwining operators. Let P ∈ P(M) be of relative rank one and let P^{−} = N^{−}M be the opposite parabolic group to P. Let P_{1}, …, P_{r}, P_{1} = P, be a set of representatives for P(M)/W (𝔞_{P}). Then r = 1 or r = 2. Let P = P(M). Put

A^{2}(P, χ, σ) = ⨁_{j=1}^{r} A^{2}(P_{j}, χ, σ).  (5.1)

We define the intertwining operator

M(z): A^{2}(P, χ, σ) → A^{2}(P, χ, σ)  (5.2)

as a block matrix in the same way as in section 2, putting

M(z) = M_{P |P} (−1, zλ), z ∈ ℂ, if r = 1,  (5.3)

and

M(z) = (\begin{matrix}0 & M_{P_{1}|P_{2}}(1, −zλ) \\ M_{P_{2}|P_{1}}(1, zλ) & 0 \end{matrix}) , z ∈ ℂ,  (5.4)

if r = 2. Then M(z) is a meromorphic function of z ∈ ℂ which, as in section 2, satisfies

M(z)M(−z) = Id, M(z)^{∗} = M(\overline{z}), z ∈ ℂ.  (5.5)

Our main purpose is to study the meromorphic function det M(z). As an immediate consequence of Theorem 0.1, we get Theorem 5.1. There exist entire functions h_{1}(z) and h_{2}(z) of order ≤ n + 2 such that

det M(z) = \frac{h_{1}(z)}{h_{2}(z)}, z ∈ ℂ.

The following result about the poles of M(z) in the half-plane Re(z) ≥ 0 has recently been proved by J. Franke [Fr]. Theorem 5.2. The poles of M(z) in the half-plane Re(z) ≥ 0 are all real. Since M(z) is holomorphic for Re(z) > ||ρ_{P} ||, it follows that all poles of M(z) in the half-plane Re(z) ≥ 0 are contained in the interval (0, ||ρ_{P} ||]. In particular, the number of poles of M(z) in Re(z) ≥ 0 is finite. Remark. Recall that for the cuspidal intertwining operator M_{cusp}(z) a stronger result holds. The poles of M_{cusp}(z) in (0, ||ρ_{P} ||] are all simple. The same holds for any cuspidal Eisenstein series E(ϕ, z), ϕ ∈ A2_{cusp}(P, χ, σ): The poles of E(ϕ, z) in Re(z) ≥ 0 are all simple and contained in (0, ||ρ_{P} ||]. Furthermore, any residue of a cuspidal Eisenstein series E(ϕ, z) at a pole z_{0} ∈ (0, ||ρ_{P} ||] is square integrable. Each of these statements is false in the noncuspidal case. A counterexample is discussed by Franke in [Fr]. Our next goal is to express det M(z) as a Blaschke product. For the cuspidal rank one intertwining operators M_{cusp}(z), such a formula was derived in [Mu1, (6.8)]. To generalize this formula, we need the following auxiliary result. Proposition 5.3. Let h_{1} and h_{2} be two entire functions of order p ≥ 0. Put h = h_{1}/h_{2} and assume that h satisfies the following two conditions:

1) h(z) is holomorphic in Im(z) > 0.

2) |h(u)| = 1 for u ∈ ℝ. Then there exist C, c > 0 such that |h(z)| ≤ Ce^{c|z|^{p}} for all z with Im(z) ≥ 0.

Proof. To prove the proposition, we shall use some results from Nevanlinna theory. For details we refer to [Ne1, Chap. VI].

Put

ln^{+}x = max(ln x, 0), x ≥ 0.

Let f be a meromorphic function on ℂ. For any r ≥ 0, put

m(r, f) = \frac{1}{2π} ∫_{0}^{2π} ln^{+} |f(re^{iθ})| dθ.  (5.6)

Furthermore, let n(r, ∞) denote the number of poles of f in the closed disc D_{r} = {z ∈ ℂ | |z| ≤ r} where each pole is counted with its multiplicity. Let

N(r, f) = ∫_{0}^{r}\{n(x, ∞) − n(0, ∞)\}\frac{dx}{x} + n(0, ∞) ln r. (5.7)

Since the function x ⟼ n(x, ∞) is monoton increasing, it follows that

N(r, f) ≥ 0 for r ≥ 1.  (5.8)

Furthermore, one has

N(e^{u}, f) = ∫_{−∞}^{u}{n(e^{v}, ∞) − n(0, ∞)} dv + n(0, ∞)u

which shows that N(e^{u}, f) is a convex function of u. The characteristic function of f is then defined by

T (r, f) = m(r, f) + N(r, f), r ≥ 0.  (5.9)

Given a ∈ ℂ, put

m(r, a) = m (r, \frac{1}{f − a}) and N(r, a) = N (r, \frac{1}{f − a}) .

Then the first main theorem of Nevanlinna theory states that for all a ∈ ℂ,

m(r, a) + N(r, a) = T (r, f) + φ(r, a),  (5.10)

where φ(r, a) is bounded as r → ∞ [Ne1, p.166]. More precisely, let c be the first nonvanishing coefficient of the Laurent expansion of f(z)−a at z = 0. Then

|φ(r, a)| ≤ | ln |c|| + ln^{+} |a| + ln 2.

In particular, if we put a = 0 in (5.10), then we get

T (r, f^{−1}) = T (r, f) + φ(r),  (5.11)

and φ(r) is bounded as r → ∞. Furthermore, if f_{1} and f_{2} are two meromorphic functions, then one has

T (r, f_{1}f_{2}) ≤ T (r, f_{1}) + T (r, f_{2})  (5.12)

[Ne1, p.170]. Now let h_{1} and h_{2} be two entire functions of order p, which means that there exist C, c > 0 such that

|h_{i}(z)| ≤ Ce^{c|z|^{p}} for all z ∈ ℂ.  (5.13)

Put

h(z) = \frac{h_{1}(z)}{h_{2}(z)}, z ∈ ℂ.

If we apply (5.11) and (5.12) to h, we get

T (r, h) ≤ T (r, h_{1}) + T (r, h_{2}^{−1}) = T (r, h_{1}) + T (r, h_{2}) + φ(r), (5.14)

where φ(r) stays bounded as r → ∞. Since h_{1} and h_{2} are entire, it follows that

T (r, h_{j}) = m(r, h_{j}) = \frac{1}{2π} ∫_{0}^{2π} ln^{+} |h_{j}(re^{iθ})| dθ, j = 1, 2.

Now by (5.13) we have

ln^{+} |h_{j}(z)| ≤ C(1 + |z|^{p}).

Using (5.8) and (5.14), we obtain

m(r, h) ≤ T (r, h) ≤ Cr^{p}, r ≥ 1.  (5.15)

Now assume that h(z) is holomorphic in

ω_{R} = {z ∈ ℂ | |z| ≤ R, Im(z) ≥ 0}.

Let z = re^{iφ}, r ≥ 0. Then the following formula of Nevanlinna[Ne2, p. 29], [Di, p. 18] holds for any z in the interior of ω_{R}.

ln |h(z)| = \frac{1}{π} ∫_{−R}^{R} ln |h(t)| \{\frac{r sin φ}{|z − t|^{2}} − \frac{R^{2}r sin φ}{|R^{2} − zt|^{2}} \} dt

+ \frac{1}{2π} ∫_{0}^{π} ln |h(Re^{iθ})| \{\frac{R^{2} − r^{2}}{|Re^{iθ} − z|^{2}} − \frac{R^{2} − r^{2}}{|Re^{−iθ} − z|^{2}} \} dθ

− ∑_{\begin{matrix}|a_{m}|<R \\ Im(a_{m})>0 \end{matrix}} ln |\frac{R^{2} − \overline{a}_{m}z}{R(z − a_{m})} \frac{R(z − \overline{a}_{m})}{R^{2} − a_{m}z}| ,

(5.16)

where a_{m} runs over the zeros of h in ω_{R}. If |h(t)| = 1 for t ∈ ℝ, the first integral on the right hand side vanishes. Suppose that

|z| ≤ \frac{R}{2}.  (5.17)

Then the second integral on the right hand side of (5.16) can be estimated by

\frac{1}{2π} ∫_{0}^{π} |ln |h(Re^{iθ})|| dθ · 2\frac{R^{2}}{(R − r)^{2}}

≤ \frac{4}{π} ∫_{0}^{π} |ln |h(Re^{iθ})|| dθ

= \frac{4}{π} ∫_{0}^{π} ln^{+} |h(Re^{iθ})| dθ + \frac{4}{π} ∫_{0}^{π} ln^{+} |h(Re^{iθ})|^{−1} dθ.

By (5.15), the first integral can be estimated by CR^{p} for R ≥ 1. By symmetry, (5.15) holds also for h^{−1}. Hence, the second integral can also be estimated by C′R^{p}, R ≥ 1, for some constant C′ > 0. It remains to estimate the sum. Since Im(a_{m}) ≥ 0, we have

|\frac{z − \overline{a}_{m}}{z − a_{m}} | ≥ 1 for Im(z) ≥ 0.

This implies that

− ∑_{\begin{matrix}|a_{m}|<R \\ Im(a_{m})>0 \end{matrix}} ln |\frac{R(z − \overline{a}_{m})}{R(z − a_{m})}| ≤ 0.

Furthermore, by (5.17) we have |a_{m}z| ≤ R^{2}/2. Hence we get

\frac{1}{3} = \frac{R^{2} − R^{2}/2}{R^{2} + R^{2}/2} ≤ |\frac{R^{2} − \overline{a}_{m}z}{R^{2} − a_{m}z} | ≤ \frac{R^{2} + R^{2}/2}{R^{2} − R^{2}/2} = 3.

Thus

− ∑_{\begin{matrix}|a_{m}|<R \\ Im(a_{m})>0 \end{matrix}} ln |\frac{R^{2} − \overline{a}_{m}z}{R^{2} − a_{m}z} | ≤ ln 3 ∑_{\begin{matrix}|a_{m}|<R \\ Im(a_{m})>0 \end{matrix}} 1 = ln 3 n(R),

where n(R) denotes the number of zeros, counted with their multiplicity, of h(z) in |z| ≤ R. Since h is the quotient of the entire functions h_{1} and h_{2}, it is clear that n(R) is bounded from above by the number n_{1}(R) of zeros of h_{1}(z) in |z| ≤ R, where again each zero is counted with its multiplicity. To estimate n_{1}(R), we use Jensen’s formula [Ne1, p.164]. Assume that h_{1}(0) ≠ 0. Then

∫_{0}^{R} n_{1}(x)\frac{dx}{x} + ln |h_{1}(0)| = \frac{1}{2π} ∫_{0}^{2π} ln |h_{1}(Re^{iφ})| dφ.

Let

M(R) = max{|h_{1}(z)| | |z| = R}.

By (5.13) we have

ln M(R) ≤ C_{1}R^{p} for R ≥ 1.

Choose R so large that M(2R) > 1. Then we get

∫_{0}^{2R} n_{1}(x)\frac{dx}{x} ≤ ln M(2R) − ln |h_{1}(0)|.

Using that n_{1}(x) in monoton increasing, it follows that

n_{1}(R) ≤ \frac{1}{ln 2} ∫_{R}^{2R} n_{1}(x)\frac{dx}{x}

≤ \frac{1}{ln 2} (ln M(2R) − ln |h_{1}(0)|)

≤ C_{2}R^{p}

for R ≥ 1. Now we put R = 2r in (5.16), where R = |z|. Combining our estimates, it follows that there exists c > 0 such that

ln |h(z)| ≤ c|z|^{p}, |z| ≥ 2, Im(z) ≥ 0.

Hence there exists C > 0 such that |h(z)| ≤ Ce^{c|z|^{p}}, Im(z) ≥ 0.

□

Let K_{f} be an open compact subgroup of G(𝔸_{f}). Let Π(K, K_{f}) denote the subset of Π(K) consisting of all σ ∈ Π(K) such that σ_{f}|K_{f} = Id. Given σ ∈ Π(K, K_{f}), we denote by λ_{σ} the Casimir eigenvalue of σ_{∞}. Lemma 5.4. Given σ ∈ Π(K) and χ ∈ 𝔛, let M(z; χ, σ) be the intertwining operator acting in A^{2}(P, χ, σ) and let d_{χ,σ} = dim A^{2}(P, χ, σ). Set a(χ, σ) = (1 + log(1 + λ_{σ}^{2} + μ_{χ}^{2}))d_{χ,σ}. There exists c > 0 such that

e^{−ca(χ,σ) Re(z)}| det M(z; χ, σ)| ≤ 1  (5.18)

for all σ ∈ Π(K, K_{f}), χ ∈ 𝔛 and z with Re(z) ≥ ||ρ_{P} ||.

Proof. It follows from Corollary 4.5 that there exist C, c > 0 and N ∈ ℕ, such that for all σ ∈ Π(K, K_{f}), χ ∈ 𝔛 and ϕ, ψ ∈ A^{2}(P, χ, σ), one has

|⟨M(z; χ, σ)ϕ, ψ⟩| ≤ C ∥ ϕ ∥∥ ψ ∥ (1 + λ_{σ}^{2} + μ_{χ}^{2})^{N}e^{c Re(z)}

for Re(z) ≥ ||ρ_{P} ||. Hence we get

| det M(z; χ, σ)| ≤ e^{c_{1}(1+d_{χ,σ}) Re(z)}(1 + λ_{σ}^{2} + μ_{χ}^{2})^{Nd_{χ,σ}}
≤ e^{c_{2}((1+log(1+λ_{σ}^{2}+μ_{χ}^{2}))d_{χ,σ}) Re(z)}

for Re(z) ≥ ||ρ_{P} || and a certain constant c_{2} > 0 which is independent of χ.

□

By Theorem 5.2, det M(z) has only finitely many poles in Re(z) ≥ 0, which we denote by u_{1}, …, u_{l}. These poles are all real and contained in (0, ||ρ_{P} ||]. Thus u_{1}, …, u_{l} ∈ (0, ||ρ_{P} ||]. For each i, 1 ≤ i ≤ l, let m_{i} be the order of the pole u_{i}. Let q_{1} = exp(c(1 + log(1 + λ_{σ}^{2} + μ_{χ}^{2}))d_{χ,σ}) where c is the constant in (5.18). Put

ξ(z) = q_{1}^{−z} ∏_{i=1}^{l} (\frac{z − u_{i}}{z + u_{i}} )^{m_{i}} det M(z).  (5.19)

Lemma 5.5. The analytic function ξ(z) has the following properties: 1) ξ(z)ξ(−z) = 1 for all z ∈ ℂ.

2) \overline{ξ(z)} = ξ(\overline{z}) for all z ∈ ℂ.
3) |ξ(z)| = 1 for Re(z) = 0.

4) ξ(z) is holomorphic in the half-plane Re(z) > 0 and satisfies |ξ(z)| ≤ 1 for Re(z) ≥ 0.

Proof. By (5.5) we have

det M(z) det M(−z) = 1 and \overline{det M(z)} = det M(\overline{z}), z ∈ ℂ. (5.20)

Using this fact and the definition of ξ(z), we obtain 1), 2) and 3). The first part of 4) is clear from the definition of ξ(z). To prove the second part, consider the function ξ̃(z) = ξ(iz). By Theorem 5.1, there exist entire functions f_{1} and f_{2} of order ≤ n + 2 such that ξ̃ = f_{1}/f_{2}. By the above results, assumptions 1) and 2) of Proposition 5.3 are satisfied for ξ̃. Hence there exist C, c > 0 such that

|ξ(z)| ≤ Ce^{c|z|^{n+2}}, Re(z) ≥ 0.  (5.21)

By Lemma 5.4 we have

|ξ(z)| ≤ 1 for Re(z) ≥ ||ρ_{P} ||.  (5.22)

In particular, |ξ(z)| ≤ 1 for Re(z) = ||ρ_{P} ||. By 3), we have |ξ(z)| = 1 on Re(z) = 0. Since (5.21) holds for 0 ≤ Re(z) ≤ ||ρ_{P} ||, application of a Phragmen-Lindelöf-type theorem implies that |ξ(z)| ≤ 1 for 0 ≤ Re(z) ≤ |ρ_{P} |. Together with (5.22), we obtain 4).

□

By Lemma 5.5, ξ(z) is an analytic function in the half-plane Re(z) > 0 which is continuous and bounded on Re(z) ≥ 0. Hence we can apply Carleman′s theorem [T, Section 3.71] which implies the following lemma. Lemma 5.6. Let η run over all zeros of ξ(z) in the half-plane Re(z) > 0, where each zero is counted with its multiplicity. Then the series

∑_{η} \frac{Re(η)}{|η|^{2}}

converges. From the definition (5.19) of ξ(z) it is clear, that the same statement holds for the zeros of det M(z). By (5.20) we may as well take the sum over all poles of det M(z) and get a convergent series. By Theorem 5.1, there exist entire functions h_{1}(z) and h_{2}(z) of order ≤ n + 2 such that

ξ(z) = \frac{h_{1}(z)}{h_{2}(z)}, z ∈ ℂ.

Let η be a zero of ξ(z). Then it follows from Lemma 5.5, that \overline{η} is also a zero and −η, −\overline{η} are poles of ξ(z). Note that ξ(z) can have only finitely many zeros in (0, ∞). We denote these zeros by ρ_{1}, ..., ρ_{m}. For p ∈ ℕ let

E(u, p) = (1 − u) exp(u + \frac{u^{2}}{2} + ⋯ + \frac{u^{p}}{p}), u ∈ ℂ.

Then by Hadamard’s factorization theorem we get

ξ(z) = e^{P (z)} ∏_{i=1}^{m} \frac{z − ρ_{i}}{z + ρ_{i}} · \frac{∏_{η}E (\frac{z}{η} , n + 2) E (\frac{z}{\overline{η}} , n + 2)}{∏_{η}E (\frac{z}{−η} , n + 2) E (\frac{z}{−\overline{η}} , n + 2)},

where η runs over the zeros of ξ(z) in the quadrant Re(z) > 0, Im(z) > 0, and each zero is counted with its multiplicity. P (z) is a polynomial of order n + 2. Using Lemma 5.6, we can argue in exactly the same way as in [Mu1, pp. 517-518] to show that

ξ(z) = e^{Q(z)} ∏_{i=1}^{m} \frac{z − ρ_{i}}{z + ρ_{i}} · ∏_{η} \frac{(z − η)(z − \overline{η})}{(z + η)(z + \overline{η})},  (5.23)

where

Q(z) = az + πil, z ∈ ℂ,

for some a < 0 and l ∈ ℤ. Moreover, the infinite product equals

∏_{η} (1 − 4z \frac{Re(η)}{(z + η)(z + \overline{η})}) ,

which is absolutely convergent by Lemma 5.6. Thus we have proved the following theorem. Theorem 5.7. Let q = q_{1}e^{a} with q_{1} and a < 0 defined above. Let u_{1}, …, u_{l} ∈ (0, ||ρ_{P} ||] denote the poles of det M(z) in the half-plane Re(z) ≥ 0, and let η run over all poles, counted with their orders, of det M(z) in the half-plane Re(z) < 0. Then

det M(z) = det M(0)q^{z} ∏_{j=1}^{l}(\frac{z + u_{j}}{z − u_{j}} )^{m_{j}} ∏_{η} \frac{z + \overline{η}}{z − η} , z ∈ ℂ,(5.24)

and the infinite product is absolutely convergent.

One can use (5.24) to compute the logarithmic derivative of det M(z) which gives

\frac{d}{dz} log det M(z) = log q − ∑_{j=1}^{l} \frac{2m_{j}u_{j}}{z^{2} − u_{j}^{2}}

− ∑_{\begin{matrix}η \\ Re(η)<0 \end{matrix}} \frac{2 Re(η)}{(z − η)(z + \overline{η})}. (5.25)

Putting z = iλ, λ ∈ ℝ, we get

\frac{d}{dz} log det M(iλ) = log q + ∑_{j=1}^{l} \frac{2m_{j}u_{j}}{λ^{2} + u_{j}^{2}}

+ ∑_{\begin{matrix}η \\ Re(η)<0 \end{matrix}} \frac{2 Re(η)}{Re(η)^{2} + (λ − Im(η))^{2}} . (5.26)

6. Poles of intertwining operators

In this section we shall investigate the poles of the rank one intertwining operators. Our main purpose is to obtain estimates, which are uniform in χ and σ_{∞}, of the number of poles of any matrix coefficient of M(z) in a circle of radius R. In [Mu1] and [Mu2], we derived such estimates for cuspidal rank one intertwining operators. The proof can not be extended directly, but we will use the results of the previous sections to reduce the general case to the cuspidal one. Suppose that dim(𝔞_{M}/𝔞_{G}) = 1. Let P ∈ P(M). Let χ ∈ 𝔛 and σ ∈ Π(K). As in section 4, we denote by μ_{χ} the Casimir eigenvalue of χ and by λ_{σ} the Casimir eigenvalue of σ_{∞}. Throughout this section, we shall fix an open compact subgroup K_{f} of G(𝔸_{f}) and, as above, we denote by Π(K, K_{f}) the subset of Π(K) consisting of all σ ∈ Π(K) such that σ_{f}|K_{f} = Id. To begin with, we prove the following lemma.

Lemma 6.1. Let m = dim G. There exists C > 0 such that for all χ ∈ 𝔛 and σ ∈ Π(K, K_{f}), we have

dim A^{2}(P, χ, σ) ≤ C(1 + |μ_{χ}| + |λ_{σ}|)^{5m}.

Proof. Let K_{f}^{M} = M(𝔸_{f})∩K_{f}. Then K_{f}^{M} is an open compact subgroup of M(𝔸). As explained in section 2, there exist arithmetic subgroups Γ_{c} ⊂ M(ℝ), parametrized by a finite subset C ⊂ M(𝔸_{f}), such that

M(ℚ)\M(𝔸)/K_{f}^{M} ≅ ⨆_{c∈C} (Γ_{c}\M(ℝ))

(see (2.8)). Let K_{∞}^{M} = M(ℝ) ∩ K_{∞} and put

X_{M} = Z_{M}(ℝ)^{0}\M(ℝ)/K_{∞}^{M}.

Let σ_{∞}^{M} be the restriction of σ to K_{∞}^{M}. Then σ_{∞}^{M} defines a homogeneous vector bundle Ẽ_{σ} over X_{M} which can be pushed down to a locally homogeneous vector bundle E_{σ} over each Γ_{c}\X_{M}. Let Δ_{σ} = (∇^{σ})^{∗}∇^{σ} be the Bochner-Laplace operator of the canonical connection ∇^{σ} on E_{σ}. This is an elliptic operator acting in C^{∞}(Γ_{c}\X_{M}, E_{σ}), and with respect to the inner product induced by the metrics on Γ_{c}\X_{M} and E_{σ}, Δ_{σ} is formally self-adjoint. The point spectrum of Δ_{σ} consists of a sequence of eigenvalues 0 ≤ μ_{1} < μ_{2} < ⋯ with finite-dimensional eigenspaces E(μ_{i}). Given μ ∈ ℝ^{+}, let

L_{disc,μ}^{2}(Γ_{c}\X_{M}, E_{σ}) = ⨁_{μ_{i}≤μ} E(μ_{i}).

Now observe that there is a canonical isomorphism

C^{∞}(Ẽ_{σ}) ≅ (C^{∞}(Z_{M}(ℝ)^{0}\M(ℝ)) ⊗ V_{σ_{∞}})^{K_{∞}^{M}} .

With respect to this isomorphism, one has

Δ̃_{σ} = −R(Ω_{M}) ⊗ Id + λ_{σ}Id,  (6.1)

where R is the right regular representation (see [Mu2, Section3]). Let

ν = |μ_{χ}| + |λ_{σ}| + ||ρ_{P} ||.  (6.2)

Using the identifications described in section 2, together with (6.1) and (4.21), it follows that for each σ ∈ Π(K, K_{f}), A^{2}(P, χ, σ) can be identified with a subspace of

⨁_{c∈C} dim(σ_{∞})L_{disc,ν}^{2}(Γ_{c}\X_{M}, E_{σ}).

By Theorem 0.2 of [Mu2], we have

dim L_{disc,ν}^{2}(Γ_{c}\X_{M}, E_{σ}) ≤ C(1 + λ_{σ}^{4p} + ν^{4p}),  (6.3)

where p = dim M(ℝ). Furthermore, by Weyl′s dimension formula, there exists C_{1} > 0 and l ≤ 1/2 dim K_{∞} such that

dim(τ) ≤ C_{1}(1 + λ_{τ}^{l})  (6.4)

for all τ ∈ K̂_{∞}. Combining (6.2) - (6.4), we obtain the desired estimation of dim A^{2}(P, χ, σ).

□

First we deal with the cuspidal intertwining operators. Let M_{cusp}(z; χ, σ) be the restriction of M(z; χ, σ) to the space A2_{cusp}(P, χ, σ). We recall that for P ∈ P_{χ}, A2_{cusp}(P, χ, σ) = A^{2}(P, χ, σ) and A2_{cusp}(P, χ, σ) = {0} if P ∉ P_{χ}. Therefore,

M_{cusp}(z; χ, σ) = \{ \begin{matrix}M(z; χ, σ) ; & if P ∈ P_{χ}, \\ 0 ; & if P ∉ P_{χ}. \end{matrix}

Let η ∈ ℂ. If η is a pole of det M_{cusp}(z; χ, σ), let ν(η; χ, σ) denote the order of the pole η and if η is not a pole of det M_{cusp}(z; χ, σ), we put ν(η; χ, σ) = 0. Given R ≥ 0, let

N_{cusp}(R; χ, σ) = ∑_{\begin{matrix}η∈ℂ \\ |η|≤R \end{matrix}} ν(η; χ, σ).

Theorem 6.2. Let m = dim G. There exists C > 0 such that for all χ ∈ 𝔛, σ ∈ Π(K, K_{f}) and R ≥ 0, one has

N_{cusp}(R; χ, σ) ≤ C(1 + R^{2} + μ_{χ}^{2} + λ_{σ}^{2})^{3m}.

Proof. By Theorem 5.2, det M_{cusp}(z; χ, σ) has only finitely many poles in the half-plane Re(z) ≥ 0 which are all real. Let u_{1}, ..., u_{l} ∈ (0, ||ρ_{P} ||] denote these poles. Put

N +_{cusp}(χ, σ) = ∑_{i=1}^{l} ν(u_{i}; χ, σ).

By the adèlic version of Theorem 7.3 of [Mu2] there exists C > 0 such that

N +_{cusp}(χ, σ) ≤ C(1 + |μ_{χ}|^{3m} + |λ_{σ}|^{3m})  (6.5)

for all χ ∈ 𝔛 and σ ∈ Π(K, K_{f}). Let

N −_{cusp}(R; χ, σ) = N_{cusp}(R; χ, σ) − N +_{cusp}(χ, σ).

In order to estimate N −_{cusp}(R; χ, σ), we first observe that by the adèlic version of Theorem 7.1 of [Mu2] there exists C > 0 such that for all χ ∈ 𝔛, σ ∈ Π(K, K_{f}) and R ≥ 0, we have

| ∫_{−R}^{R} \frac{d}{dz} log detM_{cusp}(iu; χ, σ) du|
≤ C(1 + R^{2m} + μ_{χ}^{2m} + λ_{σ}^{2m}).

(6.6)

Next observe that for any R ≥ 0, we have

∑_{j=1}^{l} 2ν(u_{j}; χ, σ) ∫_{−R}^{R} \frac{u_{j}}{u_{j}^{2} + λ^{2}} dλ ≤ 2πN +_{cusp}(χ, σ).  (6.7)

Using (5.26), (6.5), (6.6) and (6.7), it follows that for all R ≥ 0, χ ∈ 𝔛 and σ ∈ Π(K, K_{f}),

|2R log q + ∫_{−R}^{R} ∑_{\begin{matrix}η \\ Re(η)<0 \end{matrix}} \frac{2 Re(η)}{Re(η)^{2} + (λ − Im(η))^{2}} dλ|

≤ 2πN +_{cusp}(χ, σ) + C(1 + R^{2} + μ_{χ}^{2} + λ_{σ}^{2})^{2m}

≤ C_{1}(1 + R^{2} + μ_{χ}^{2} + λ_{σ}^{2})^{2m},

(6.8)

where C_{1} > 0 is independent of χ and σ. We distinguish two cases: a) q ≥ 1. Then it follows from the definition of q that

0 ≤ log q ≤ c(1 + log(1 + λ_{σ}^{2} + μ_{χ}^{2})) d_{χ,σ},

where d_{χ,σ} = dim A^{2}(P, χ, σ) and c > 0 is independent of χ and σ ∈ Π(K, K_{f}). By Lemma 6.1, it follows that

| log q| ≤ C(1 + μ_{χ}^{2} + λ_{σ}^{2})^{3m}.

Hence, by (6.8) we get

∫_{−R}^{R} ∑_{\begin{matrix}η \\ Re(η)<0 \end{matrix}} \frac{| Re(η)|}{Re(η)^{2} + (λ − Im(η))^{2}} dλ

≤ C_{2}(1 + R^{2} + μ_{χ}^{2} + λ_{σ}^{2})^{3m}.

(6.9)

b) q < 1. Then log q < 0. Since all terms in the series on the left-hand side of (6.8) are negative, (6.9) holds in this case too.

Using (6.9) we obtain

N −_{cusp}(R; χ, σ) ∫_{−1}^{1} \frac{1}{1 + λ^{2}} dλ

≤ ∑_{\begin{matrix}|η|≤R \\ Re(η)<0 \end{matrix}} ∫_{−2R}^{2R} \frac{| Re(η)|}{Re(η)^{2} + (λ − Im(η))^{2}} dλ

≤ ∫_{−2R}^{2R} ∑_{\begin{matrix}η \\ Re(η)<0 \end{matrix}} \frac{| Re(η)|}{Re(η)^{2} + (λ − Im(η))^{2}} dλ

≤ C(1 + R^{2} + μ_{χ}^{2} + λ_{σ}^{2})^{3m}.

□

Let Φ_{1}, …, Φ_{d} be an orthonormal basis in A^{2}(P, χ, σ). If η is a pole of M_{cusp}(z; χ, σ), let κ_{ij}(η, χ, σ) be the order of ⟨M_{cusp}(z; χ, σ)Φ_{i}, Φ_{j}⟩ at η. Set κ(η; χ, σ) = max_{i,j} κ_{ij}(η; χ, σ). If η is not a pole of M_{cusp}(z; χ, σ), set κ(η; χ, σ) = 0. Corollary 6.3. Let m = dim G. There exists C > 0 such that for all χ ∈ 𝔛, σ ∈ Π(K, K_{f}) and R ≥ 0, one has

∑_{\begin{matrix}η∈ℂ \\ |η|≤R \end{matrix}} κ(η; χ, σ) ≤ C(1 + R^{2} + μ_{χ}^{2} + λ_{σ}^{2})^{4m}.

proof. We split the sum as follows

∑_{|η|≤R} κ(η; χ, σ) = ∑_{\begin{matrix}|η|≤R \\ Re(η)<0 \end{matrix}} κ(η; χ, σ) + ∑_{\begin{matrix}|η|≤R \\ Re(η)>0 \end{matrix}} κ(η; χ, σ).

Let η ∈ ℂ, Re(η) > 0. If η is a pole of M_{cusp}(z; χ, σ) then η is real and ν(η; χ, σ) = 1. Thus the second sum equals the number of poles of M_{cusp}(z; χ, σ) in Re(z) ≥ 0. Using the adèlic version of Corollary 6.2 of [Mu2], this number can be estimated by C(1 + μ_{χ}^{2} + λ_{σ}^{2})^{2m}, where m = dim G and C > 0 is a constant which is independent of χ and σ. It remains to estimate the first sum. Let η ∈ ℂ, Re(η) < 0, be a pole of M_{cusp}(z; χ, σ). We distinguish two cases.

a) M_{cusp}(z; χ, σ) is holomorphic in −η. Then it follows as in the proof of Corollary 7.6 of [Mu1] that

κ(η; χ, σ) ≤ ν(η; χ, σ)

b) −η is a pole of M_{cusp}(z; χ, σ). Arguing again as in the proof of Corollary 7.6 of [Mu1], we get

κ(η; χ, σ) ≤ ν(η; χ, σ) + 2 dim A^{2}(P, χ, σ).

Thus

∑_{\begin{matrix}|η|≤R \\ Re(η)<0 \end{matrix}} κ(η; χ, σ) ≤ N_{cusp}(R, χ, σ) + 2 dim A^{2}(P, χ, σ)N +_{cusp}(χ, σ).

Using Lemma 6.1, (6.5) and Theorem 6.2, the desired estimate follows.

□

Let P ∈ P(M), P_{1} ∈ P(M_{1}) and let t ∈ W (𝔞_{P} , 𝔞_{P_{1}}). Using (2.18) and (2.19) this result extends to all matrix coefficients of

M_{P_{1}|P} (t, λ) : A2_{cusp}(P, χ, σ) → A2_{cusp}(P_{1}, χ, σ).

i.e. we have Corollary 6.4. Let m = dim G. There exists C > 0 such that for

all χ ∈ 𝔛, σ ∈ Π(K, K_{f}), R ≥ 0, ϕ ∈ A2_{cusp}(P, χ, σ) and ψ ∈

A2_{cusp}(P_{1}, χ, σ), the number of poles, counted with their order, of the meromorphic function λ ∈ {Λ ∈ 𝔞_{M,ℂ}^{∗} |∥ Λ ∥≤ R} ↦ ⟨M_{P_{1}|P} (t, λ)ϕ, ψ⟩ in is bounded by C(1 + R^{2} + μ_{χ}^{2} + λ_{σ}^{2})^{4m}. We can now proceed to estimate the number of poles of the matrix coefficients of all rank one intertwining operators. As above, let P ∈ P(M), P_{1} ∈ P(M_{1}) and t ∈ W (𝔞_{P} , 𝔞_{P_{1}}). We recall that A^{2}(P, χ, σ) is spanned by sums of iterated residues of the form (4.3), and for any such ϕ, M_{P_{1}|P} (t, λ)ϕ is given by formula (4.7). Now consider any of the cuspidal intertwining operators

M_{B_{1}|B}(t_{B}, Λ) : A2_{cusp}(B, χ, σ) → A2_{cusp}(B_{1}, χ, σ)  (6.10)

occurring in (4.7). Given S ∈ S_{B}(P), let Σ(B, S) be the set of all points λ_{0} ∈ 𝔞_{P,ℂ}^{∗} such that there exist Φ ∈ A2_{cusp}(B, χ, σ), Ψ ∈ A2_{cusp}(B_{1}, χ, σ) and a singular hyperplane of ⟨M_{B_{1}|B}(t_{B}, Λ)Φ, Ψ⟩ which intersects 𝔱_{S,ℂ} = Λ_{S} +𝔞_{P,ℂ}^{∗} transversally in the point Λ_{S} +λ_{0}. Then the following Lemma is an immediate consequence of (4.6) and (4.7). Lemma 6.5. For each ϕ ∈ A^{2}(P, χ, σ) and ψ ∈ A^{2}(P_{1}, χ, σ), the set of poles of ⟨M_{P_{1}|P} (t, λ)ϕ, ψ⟩ is contained in the set

⋃_{{B∈P_{χ}|B⊂P }} ⋃_{S∈S_{B}(P)} Σ(B, S).

As explained in section 4, the points Λ_{S}, S ∈ S_{B}(P), are all contained in the compact set {Λ ∈ (𝔞_{B}^{P})^{∗} |∥ Λ ∥≤ ||ρ_{P} ||}. Using this fact and Lemma 6.5, the estimation of the number of poles, counted with their order, of ⟨M_{P_{1}|P} (t, λ)ϕ, ψ⟩ in the disk D_{R} = {λ ∈ 𝔞_{P,ℂ}^{∗} | ∥ λ ∥≤ R} can be reduced to the following problem: Consider any matrix coefficient

a_{Φ,Ψ}(Λ) = ⟨M_{B_{1}|B}(t_{B}, Λ)Φ, Ψ⟩

of the cuspidal intertwining operator (6.10), where Φ ∈ A2_{cusp}(B, χ, σ) and Ψ ∈ A2_{cusp}(B_{1}, χ, σ). Let 𝔥 = {H_{1}, …, H_{r}} be a set of singular hyperplanes of a_{Φ,Ψ}(Λ) satisfying

⋂_{k} H_{k} = {Λ_{𝔥}} and ∥ Λ_{𝔥} ∥≤ ||ρ_{P} || + R.  (6.11)

Let m(H_{k}) denote the order of the singularity of a_{Φ,Ψ}(Λ) along H_{k} and put

m(𝔥) = ∑_{k=1}^{r} m(H_{k}).

Put

N_{B}(R; χ, σ) = ∑_{𝔥}m(𝔥),

where the sum runs over all systems 𝔥 of singular hyperplanes of of a_{Φ,Ψ}(Λ) such that (6.11) holds. Then by Lemma 6.5, our problem of estimating the number of poles of ⟨M_{P_{1}|P} (t, λ)ϕ, ψ⟩ in D_{R} is reduced to the estimation of each of the numbers N_{B}(R; χ, σ). By (2.19) we can assume that B_{1} ∈ P(M_{B}). Furthermore, by (2.20) and the discussion following this equation, it suffices to consider M_{B_{1}|B}(1, Λ). Finally, as in (2.22), there is a sequence of adjacent chambers C_{i_{1}}, …, C_{i_{N}} in 𝔞_{B} with associated parabolic subgroups B_{i_{k}} ∈ P(M_{B}) such that B_{i_{1}} = B, B_{i_{N}} = B_{1}, and

\begin{aligned}M_{B_{1}|B} & (1, Λ) = M_{B_{i_{N}} |B_{i_{N−1}}}(1, Λ) ⋯ \\ & ⋯ M_{B_{i_{3}}|B_{i_{2}}}(1, Λ)M_{B_{i_{2}}|B_{i_{1}}}(1, Λ), Λ ∈ 𝔞_{B,ℂ}^{∗}. \end{aligned} (6.12)

By (2.25), each M_{B_{i_{k+1}}|B_{i_{k}}} (1, Λ) is induced from a rank one cuspidal intertwining operator

M_{B_{i_{k+1}}^{∗}|B_{i_{k}}^{∗}} (1, Λ^{∗}) : A^{2}(B_{i_{k}}^{∗}, χ, σ′) → A^{2}(B_{i_{k+1}}^{∗}, χ, σ′),

where Λ^{∗} is the projection of Λ to 𝔞_{B_{i_{k}}^{∗} ,ℂ}^{∗}. For each k, k = 1, ..., N − 1, we choose an orthonormal basis ϕ_{1,k}, ..., ϕ_{d_{k},k} in A^{2}(B_{i_{k}}^{∗}, χ, σ′). Given a pole η^{∗} of M_{B_{i_{k+1}}^{∗}|B_{i_{k}}^{∗}} (1, Λ^{∗}), we denote by κ_{ij,k}(η^{∗}; χ, σ′) the order of ⟨M_{B_{i_{k+1}}^{∗}|B_{i_{k}}^{∗}} (1, Λ^{∗})ϕ_{i,k}, ϕ_{j,k+1}⟩ in the possible pole η^{∗}. Put

κ_{k}(η^{∗}; χ, σ′) = max_{i,j} κ_{ij,k}(η^{∗}; χ, σ′).

Then it follows from (6.12) that there exists c > 0, independent of χ and σ, such that

N_{B}(R; χ, σ) ≤ ∏_{k=1}^{N−1} ( ∑_{∥η^{∗}∥≤c+R} κ_{k}(η^{∗}; χ, σ′)) .

The right hand side can be estimated by Corollary 6.4. Since N is bounded by the number of chambers in 𝔞_{B}, we get Proposition 6.6. Let m = dim G and let r be the number of chambers in 𝔞_{B} where B is any minimal parabolic subgroup of G. Then there exists C > 0 such that for all χ ∈ 𝔛, σ ∈ Π(K, K_{f}), R ≥ 0, ϕ ∈ A^{2}(P, χ, σ) and ψ ∈ A^{2}(P_{1}, χ, σ), the number of poles, counted with their order, of ⟨M_{P_{1}|P} (t, λ)ϕ, ψ⟩ in the disc of radius R is bounded by C(1 + R^{2} + μ_{χ}^{2} + λ_{σ}^{2})^{4rm}.

References

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

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