ABSOLUTE CONVERGENCE OF THE SPECTRAL SIDE OF THE ARTHUR

TRACE FORMULA FOR GL_{n}

Werner Müller and Birgit Speh^{†} with Appendix by Erez M. Lapid

0. Introduction

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

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

between distributions on G(𝔸)^{1}. The left hand side is the geometric side and the right hand side the spectral side of the trace formula.

In this paper we are concerned with the spectral side of the trace formula. The distributions J_{χ} are initially defined in terms of truncated Eisenstein series. They are parametrized by the set of cuspidal data 𝔛 which consists of the Weyl group orbits of pairs (M_{B}, r_{B}), where M_{B} is the Levi component of a standard parabolic subgroup and r_{B} is an irreducible cuspidal automorphic representation of M_{B}(𝔸)^{1}. In the fine χ-expansion of the spectral side the inner products of truncated Eisenstein series are replaced by terms containing generalized logarithmic derivatives of intertwining operators. This leads to an integral-series that is only known to be conditionally convergent. It is an open problem to prove that the fine χ-expansion is absolutely convergent and the main purpose of this paper is to settle this problem for the group GL_{n}.

To explain our results in more detail, we need to introduce some notation. We fix a Levi component M_{0} of a minimal parabolic subgroup P_{0} of G. We assume that all parabolic subgroups considered in this paper contain M_{0}. Let P be a parabolic subgroup of G, defined over E, with unipotent radical N_{P}. Let M_{P} be the unique Levi component of P which contains M_{0}. We denote the split component of the center of M_{P} by A_{P} and its Lie algebra by 𝔞_{P}. For parabolic groups P ⊂ Q there is a natural surjective map 𝔞_{P} → 𝔞_{Q} whose kernel we will denote by 𝔞_{P}^{Q}. Let A^{2}(P) be the space of automorphic forms on N_{P} (𝔸)M_{P} (E)\G(𝔸) which are square-integrable modulo A_{P,ℚ}(ℝ)^{0}, where A_{P,ℚ} is the split component of the center of the group obtained from M_{P} by restricting scalars from E to ℚ. Let Q be another parabolic subgroup of G, defined over E, with Levi component M_{Q}, split component A_{Q} and corresponding Lie algebra 𝔞_{Q}. Let W (𝔞_{P} , 𝔞_{Q}) be the set of all linear isomorphisms from 𝔞_{P} to 𝔞_{Q} which are restrictions of elements of the Weyl group W (A_{0}). The theory of Eisenstein series associates to each s ∈ W (𝔞_{P} , 𝔞_{Q}) an intertwining operator

†partially supported by NSF grant DMS-0070561

Date: October 6, 2003.

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

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

M_{Q|P} (λ) := M_{Q|P} (1, λ).

Let Π(M_{P} (𝔸)^{1}) be the set of equivalence classes of irreducible unitary representations of M_{P} (𝔸)^{1}. Let χ ∈ 𝔛 and π ∈ Π(M_{P} (𝔸)^{1}). Then (χ, π) singles out a certain subspace A_{χ,π}^{2}(P) of A^{2}(P) [A3, p.1249]. Let \overline{A}_{χ,π}^{2}(P) be the Hilbert space completion of A_{χ,π}^{2}(P) with respect to the canonical inner product. For each λ ∈ 𝔞_{P,ℂ}^{∗} we have an induced representation ρ_{χ,π}(P, λ) of G(𝔸) in \overline{A}_{χ,π}^{2}(P).

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

𝔐_{L}(P, λ) =

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

where λ and Λ are constrained to lie in i𝔞_{L}^{∗}, and for each Q_{1} ∈ P(L), Q is a group in P(M_{P}) which is contained in Q_{1}. Then 𝔐_{L}(P, λ) is an unbounded operator which acts on the Hilbert space \overline{A}_{χ,π}^{2}(P). In the special case that L = M and dim 𝔞_{L}^{G} = 1, the operator 𝔐_{L}(P, λ) has a simple description. Let P be a parabolic subgroup with Levi component M. Let α be the unique simple root of (P, A_{P}) and let ω̃ be the element in (𝔞_{M}^{G})^{∗} such that ω̃(α^{∨}) = 1. Let \overline{P} be the opposite parabolic group of P. Then

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

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

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

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

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

So far, it is only known that ∑_{χ∈𝔛} |J_{χ}(f)| < ∞ and the goal is to show that the integral–sum obtained by summing (0.1) over χ ∈ 𝔛 and π ∈ Π(M(𝔸)^{1}) is absolutely convergent with respect to the trace norm. For a given Levi subgroup M let L(M) be the set of all Levi subgroups L with M ⊂ L. Put M(P, s) = M_{P |P} (s, 0). Denote by ∥ T ∥_{1} the trace norm of a trace class operator T. Let C^{1}(G(𝔸)^{1}) be the space of integrable rapidly decreasing functions on G(𝔸)^{1} (see [Mu4, §1.3] for its definition). Then our main result is the following theorem. Theorem 0.1. Let G = GL_{n}. Then the sum over all M ∈ L(M_{0}), L ∈ L(M), χ ∈ 𝔛, π ∈ Π(M(𝔸)^{1}), and s ∈ W^{L}(𝔞_{M})_{reg} of the product of

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

with

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

is convergent for all f ∈ C^{1}(G(𝔸)^{1}).

By Theorem 0.1, the spectral side for GL_{n} can now be rewritten in the following way. Denote by Π_{disc}(M(𝔸)^{1}) the set of all π ∈ Π(M(𝔸)^{1}) which are equivalent to an irreducible subrepresentation of the regular representation of M(𝔸)^{1} in L^{2}(M(E)\M(𝔸)^{1}). As in Section 7 of [A3], we shall identify any representation of M(𝔸)^{1} with a representation of M(𝔸) which is trivial on A_{M,ℚ}(ℝ)^{0}, where A_{M,ℚ} is the split component of the center of the group Res_{E/ℚ} GL_{n} obtained from G by restricting scalars from E to ℚ. For any parabolic group P, let A_{π}^{2}(P) = ⊕_{χ}A_{χ,π}^{2}(P) and for λ ∈ 𝔞_{P,ℂ}^{∗}, let ρ_{π}(P, λ) be the induced representation of G(𝔸) in \overline{A}_{π}^{2}(P), the Hilbert space completion of A_{π}^{2}(P). Given M ∈ L, L ∈ L(M), P ∈ P(M), s ∈ W^{L}(𝔞_{M})_{reg} and a function f ∈ C^{1}(G(𝔸)^{1}), let

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

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

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

Then for all functions f in C^{1}(G(𝔸)^{1}), the spectral side of the Arthur trace formula equals

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

Note that all sums in this expression are finite.

We shall now explain the main steps of the proof of Theorem 0.1. The proof relies on Theorem 0.1 of [Mu4]. In this theorem the absolute convergence of the spectral side of the trace formula has been reduced to a problem about local components of automorphic representations. So the main issue of the present paper is to verify that for GL_{n}, the assumptions of Theorem 0.1 of [Mu4] are satisfied.

Let M = GL_{n_{1}} × ⋯ × GL_{n_{r}} be a standard Levi subgroup and let P, Q ∈ P(M). We shall identify 𝔞_{M}^{∗} with ℝ^{r}. Given a place v of E and an irreducible unitary representation π_{v} = ⊗_{i=1}^{r}π_{v,i} of M(E_{v}), let J_{Q|P} (π_{v}, s), s ∈ ℂ^{r}, be the local intertwining operator between the induced representations I_{P}^{G}(π_{v}[s]) and I_{Q}^{G}(π_{v}[s]), where s = (s_{1}, ..., s_{r}) and π_{v}[s] = ⊗_{i}(π_{v,i}| det |^{s_{i}}). It follows from results of Shahidi [Sh5] that there exist normalizing factors r_{Q|P} (π_{v}, s), which are defined in terms of Rankin-Selberg L-functions, such that the normalized intertwining operators

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

satisfy the properties of Theorem 2.1 of [A7]. If v < ∞ and K_{v} is an open compact subgroup of G(E_{v}), denote by R_{Q|P} (π_{v}, s)_{K_{v}} the restriction of R_{Q|P} (π_{v}, s) to the subspace H_{P} (π_{v})^{K_{v}} of K_{v}-invariant vectors in the Hilbert space H_{P} (π_{v}) of the induced representation. If v|∞, let K_{v} ⊂ G(E_{v}) be the standard maximal compact subgroup. For every σ_{v} ∈ Π(K_{v}) we denote by ∥ σ_{v} ∥ the norm of the highest weight of σ_{v}. Given π_{v} ∈ Π(G(E_{v})) and σ_{v} ∈ Π(K_{v}), let R_{Q|P} (π_{v}, s)_{σ_{v}} be the restriction of R_{Q|P} (π_{v}, s) to the σ_{v}-isotypical subspace of H_{P} (π_{v}). Finally for any place v, let Π_{disc}(M(E_{v})) be the subspace of all π_{v} in Π(M(E_{v})) such that there exists an automorphic representation π in the discrete spectrum of M(𝔸) whose local component at v is equivalent to π_{v}. Then the main result that we need to prove Theorem 0.1 is the following proposition. Proposition 0.2. Let v be a place of E. For all M ∈ L and P, Q ∈ P(M) the following holds.

1) If v < ∞, then for every open compact subgroup K_{v} of GL_{n}(E_{v}) and every multi-index α ∈ ℕ^{r} there exists C > 0 such that

(0.2)  ∥ D_{u}^{α}R_{Q|P} (π_{v}, iu)_{K_{v}} ∥≤ C

for all π_{v} ∈ Π_{disc}(M(E_{v})) and u ∈ ℝ^{r}.

2) If v|∞, then for every multi-index α ∈ ℕ^{r} there exist C > 0 and N ∈ ℕ such that

(0.3)  ∥ D_{u}^{α}R_{Q|P} (π_{v}, iu)_{σ_{v}} ∥≤ C(1+ ∥ σ_{v} ∥)^{N}

for all u ∈ ℝ^{r}, σ_{v} ∈ Π(K_{v}) and π_{v} ∈ Π_{disc}(M(E_{v})).

The normalization used in [Mu4] differs slightly from the normalization by L-functions. However, it is easy to compare the two normalizations and it follows that Proposition 0.2 holds also with respect to the normalization used in [Mu4]. Together with Theorem 0.1 of [Mu4], this implies Theorem 0.1. Actually in [Mu4] we considered only reductive algebraic groups G defined over ℚ. However, passing to the group G′ = Res_{E/ℚ} G which is obtained from G by restriction of scalars, it follows immediately that the results of [Mu4] can also be applied to reductive algebraic groups defined over a number field.

The main analytic ingredients in the proof of Proposition 0.2 are a non-trivial uniform bound toward the Ramanujan hypothesis on the Langlands parameters of local components of cuspidal automorphic representations [LRS] and the determination of the residual spectrum [MW]. Furthermore Corollary A.3 is important for the proof of (0.3).

Let us explain this in more detail. First note that any local component π_{v} of a cuspidal automorphic representation π of GL_{m}(𝔸) is generic [Sk]. This implies that π_{v} is equivalent to a fully induced representation [JS3], i.e.,

π_{v} ≅ I_{P (E_{v})}^{G(E_{v})}(τ_{1}[t_{1}], ..., τ_{r}[t_{r}]),

where P is a standard parabolic subgroup of type (n_{1}, ..., n_{r}), τ_{i} are tempered representations of GL_{n_{i}}(E_{v}) and the t_{i}’s are real numbers satisfying

t_{1} > t_{2} > ⋯ > t_{r}.

Here τ_{i}[t_{i}] is the representation g ↦ τ_{i}(g)| det(g)|^{t_{i}}. For a unitary generic representation π_{v} the parameters t_{i} satisfy |t_{i}| < 1/2. In [LRS], Luo, Rudnick and Sarnak proved that for an unramified π_{v} which is the local component of a cuspidal automorphic representation of GL_{m}(𝔸), one has

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

First we extend this result of Luo, Rudnick and Sarnak to all local components of cuspidal automorphic representations of GL_{m}(𝔸). Then we use the description of the residual spectrum of GL_{m}(𝔸), given by Mœglin and Waldspurger [MW], to prove similar bounds for the local components of all automorphic representations in the residual spectrum of GL_{m}(𝔸) (cf. Proposition 3.5 for the precise statement). As a consequence, it follows that for every local component π_{v} of an automorphic representation π in the discrete spectrum of M(𝔸)^{1} the normalized intertwining operator R_{Q|P} (π_{v}, s) is holomorphic in the domain Re(s_{i} − s_{j}) > 2/(n^{2} + 1), 1 ≤ i < j ≤ r. This is the key result which is needed to prove Proposition 0.2. Combined with Corollary A.3 it immediately implies (0.3). For a finite place v we use that by Theorem 2.1 of [A7] any matrix coefficient of R_{Q|P} (π_{v}, s) is a rational function of q_{v}^{s_{i}−s_{j}}, i < j. Together with the above result, this implies (0.2).

In an earlier version of this paper, the first two authors were only able to establish (0.3) for a fixed K_{v}-type, so that Theorem 0.1 could only be proved for K-finite functions f ∈ C^{1}(G(𝔸)^{1}). With the help of the appendix which was kindly provided by E. Lapid, the K-finiteness assumption could be lifted.

To extend the results of this paper to other reductive groups G one would need, in particular, the existence of non-trivial uniform bounds on the local components of cuspidal automorphic representations of G(𝔸). For a discussion of this problem we refer to [Sa]. Also note that [CL] is a step in this direction.

The paper is organized as follows. In section 2 we compare the two different normalizations of intertwining operators and we prove some estimate for conductors. In section 3 we estimate the (continuous) Langlands parameters of local components of cuspidal automorphic representations of GL_{m} which generalizes results of Luo, Rudnick and Sarnak [LRS] to the case of ramified representations. Then we use the description of the residual spectrum of GL_{m} by Mœglin and Waldspurger [MW] to obtain estimations for the Langlands parameters of all local components of automorphic representations in the discrete spectrum of GL_{m}. We use these results in section 4 to prove Proposition 0.2 and Theorem 0.1. In the appendix, the normalized intertwining operators for real Lie groups are studied. The main results is Corollary A.3 which proves estimations for derivatives of matrix coefficients of intertwining operators along the imaginary axis, under the assumption that the intertwining operators are holomorphic in a fixed strip containing the imaginary axis.

Acknowledgment. The first two authors thank E. Lapid for many comments and suggestions which helped to improve the paper considerably and for providing the appendix by which the K-finiteness assumption in an earlier version of the paper could be lifted.

1. Preliminaries

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

P = N_{P} M_{P} ,

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

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

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

𝔞_{M} = Hom(X(M)_{E}, ℝ).

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

𝔞_{M}^{∗} = X(M)_{E} ⊗ ℝ.

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

M = GL_{n_{1}} × ⋯ × GL_{n_{r}} .

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

1.2. Let H be a reductive algebraic group defined over ℚ, let F be a local field of characteristic 0 and let K be an open compact subgroup of H(F). We shall denote by Π(H(𝔸)) (resp. Π(H(F)), Π(K), etc.) the set of equivalence classes of irreducible unitary representations of H(𝔸) (resp. H(F), K, etc.).

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

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

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

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

Let

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

be the induced representation and denote by H_{P} (π) the Hilbert space of the representation I_{P}^{G}(π, s). Sometimes we will denote I_{P}^{G}(π, s) by I_{P}^{G}(π_{1}[s_{1}], ..., π_{r}[s_{r}]).

2. Normalizing factors for local intertwining operators

Let F be a local field of characteristic 0. If F is non-Archimedean, let O be the ring of integers of F and let 𝔓 be the unique maximal ideal of O. Let q be the number of elements of the residue field O/𝔓. Let K = GL_{n}(O). If F is Archimedean, let K be the standard maximal compact subgroup of GL_{n}(F), i.e., K = O(n), if F = ℝ, and K = U(n), if F = ℂ.

Let M = GL_{n_{1}} × ⋯ × GL_{n_{r}} be a standard Levi subgroup. We identify 𝔞_{M} with ℝ^{r}. Let P_{1}, P_{2} ∈ P(M). Given π ∈ Π(M(F)), let

J_{P_{2}|P_{1}}(π, s), s ∈ ℂ^{r},

be the intertwining operator which intertwines the induced representations I_{P_{1}}^{G}(π, s) and I_{P_{2}}^{G}(π, s). The intertwining operator J_{P_{2}|P_{1}}(π, s) is defined by an integral over N_{P_{1}}(F)∩N_{\overline{P}_{2}}(F) which converges for Re(s) in a certain chamber of 𝔞_{M}^{∗}. It follows from [A7] and [CLL, §15] that the intertwining operators can be normalized in a suitable way. This means that there exist scalar valued meromorphic functions r_{P_{2}|P_{1}}(π, s) of s ∈ ℂ^{r} such that the normalized intertwining operators

R_{P_{2}|P_{1}}(π, s) = r_{P_{2}|P_{1}}(π, s)^{−1}J_{P_{2}|P_{1}}(π, s)

satisfy the properties of Theorem 2.1 of [A7]. The method used in [A7] works for every reductive group G. For GL_{n}, however, it follows from results of Shahidi [Sh1], [Sh5] that local intertwining operators can be normalized by L-functions.

The normalizing factors defined by the Rankin-Selberg L-functions can be described as follows. Fix a nontrivial continuous character ψ of the additive group F^{+} of F and equip F with the Haar measure which is selfdual with respect to ψ. First assume that P is a standard maximal parabolic subgroup with Levi component M = GL_{n_{1}} × GL_{n_{2}}. Let π_{i} ∈ Π(GL_{n_{i}}(F)), i = 1, 2. If F is non-Archimedean, let L(s, π_{1} × π_{2}) and ϵ(s, π_{1} × π_{2}, ψ) be the Rankin-Selberg L-function and the ϵ-factor, respectively, as defined in [JPS]. If F is Archimedean, let the L-function and the ϵ-factor be defined by using the Langlands parametrization (cf. [A7], [Sh2]). Then the normalizing factor can be regarded as a function r_{\overline{P} |P} (π_{1} ⊗ π_{2}, s) of one complex variable which is given by

(2.1)  r_{\overline{P} |P} (π_{1} ⊗ π_{2}, s) = \frac{L(s, π_{1} × π̃_{2})}{L(1 + s, π_{1} × π̃_{2})ϵ(s, π_{1} × π̃_{2}, ψ)}.

For arbitrary rank, the normalizing factors are products of normalizing factors associated to rank one groups in M. Let e_{i}, i = 1, …, r, denote the standard basis of (ℝ^{r})^{∗}. Then there exist σ_{1}, σ_{2} ∈ S_{r} such that the set of roots of (P_{1}, A_{M}) and (P_{2}, A_{M}), respectively, are given by

(2.2)  Σ_{P_{k}} = {e_{i} − e_{j} | 1 ≤ i, j ≤ r, σ_{k}(i) < σ_{k}(j)}, k = 1, 2.

Put

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

Then

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

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

\begin{aligned}r & _{P_{2}|P_{1}}(π, s) := \\ & ∏_{(i,j)∈I(σ_{1},σ_{2})} \frac{L(s_{i} − s_{j}, π_{i} × π̃_{j})}{L(1 + s_{i} − s_{j}, π_{i} × π̃_{j})ϵ(s_{i} − s_{j}, π_{i} × π̃_{j}, ψ)}. \end{aligned} ; (2.3)

Since the Rankin-Selberg L-factors are meromorphic functions, it follows that r_{P_{2}|P_{1}}(π, s) are meromorphic functions of s ∈ ℂ^{r} and as explained in [A7, §4] and [AC, p.87], they satisfy all properties that are requested for normalizing factors.

In order to be able to apply the results of [Mu4] we have to compare the normalizing factors r_{Q|P} (π, s) with those used in [Mu4] which we denote by r̃_{Q|P} (π, s). If F is Archimedean, the normalizing factors r̃_{Q|P} (π, s) are defined as the Artin L-factors and therefore, coincide with the r_{Q|P} (π, s). Assume that F is non-Archimedean. By the construction of the normalizing factors it suffices to consider the case where P is maximal, Q = \overline{P} and π is square integrable. Let P be a standard maximal parabolic subgroup of GL_{m} with Levi component

M = GL_{m_{1}} × GL_{m_{2}} .

Then the normalizing factor may be regarded as a function r̃_{\overline{P} |P} (π, s) of one complex variable s. We recall the construction of r̃_{\overline{P} |P} (π, s) for square integrable representations π [CLL]. It follows from [Si1], [Si2] that for every π ∈ Π_{2}(M(F)) there exists a rational function U_{P} (π, z) such that the Plancherel measure μ(π, s) is given by

μ(π, s) = U_{P} (π, q^{−s}).

The rational function U_{P} (π, z) is of the form

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

where |α_{i}| ≤ 1, |β_{i}| ≤ 1, i = 1, ..., r, and a ∈ ℂ is a constant such that

a ∏_{i=1}^{r} \frac{α_{i}}{β_{i}} > 0.

Let b ∈ ℂ be such that

|b|^{2} ∏_{i=1}^{r} \begin{matrix}\overline{α} & i \\ \overline{β} & i \end{matrix} = a

and set

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

Then the normalizing factor r̃_{\overline{P} |P} (π, s) is defined by

r̃_{\overline{P} |P} (π, s) = V_{P} (π, q^{−s})^{−1}.

By definition we have

μ_{P} (π, s) = (\overline{r̃_{\overline{P} |P} (π, −\overline{s})}r̃_{\overline{P} |P} (π, s))^{−1} ,

which is one of the main conditions that the normalizing factors have to satisfy.

Let π_{1} and π_{2} be tempered representations of GL_{m_{1}}(F) and GL_{m_{2}}(F), respectively. By Corollary 6.1.2 of [Sh1] the Plancherel measure is given by

μ(π_{1} ⊗ π_{2}, s) = q^{f(π̃_{1}×π_{2})} \frac{L(1 + s, π_{1} × π̃_{2})}{L(s, π_{1} × π̃_{2})} \frac{L(1 − s, π̃_{1} × π_{2})}{L(−s, π̃_{1} × π_{2})},

where f(π̃_{1} ×π_{2}) ∈ ℤ is the conductor of π̃_{1} ×π_{2}. Using the description of the Rankin-Selberg L-functions for tempered representations [JPS] (see also section 3), it follows that

(2.5)  L(s, π_{1} × π̃_{2}) = ∏_{i=1}^{m_{1}} ∏_{j=1}^{m_{2}} (1 − a_{ij}q^{−s})^{−1} ,

with complex numbers a_{ij} satisfying |a_{ij}| < 1. Furthermore by [JPS], the ϵ-factor ϵ(s, π_{1} × π_{2}, ψ) has the following form

(2.6)  ϵ(s, π_{1} × π_{2}, ψ) = c(π_{1} × π_{2}, ψ)q^{−f(π_{1}×π_{2},ψ)s},

with c(π_{1} × π_{2}, ψ) ∈ ℂ − {0} and f(π_{1} × π_{2}, ψ) ∈ ℤ. Let

c(ψ) = max{r | 𝔓^{−r} ⊂ ker ψ}

Then

(2.7)  f(π_{1} × π_{2}, ψ) = n_{1}n_{2}c(ψ) + f(π_{1} × π_{2}),

with f(π_{1} × π_{2}) ∈ ℤ independent of ψ. For simplicity assume that c(ψ) = 0. By Lemma 6.1 of [Sh1] we have

(2.8)  |c(π_{1} × π_{2})| = q^{f(π̃_{1}×π_{2})/2}.

Thus the ϵ-factor can be written as

ϵ(s, π_{1} × π_{2}, ψ) = W (π_{1} × π_{2})q^{(1/2−s)f(π_{1}×π_{2})},

where the root number W (π_{1} × π_{2}) satisfies |W (π_{1} × π_{2})| = 1. Finally, observe that f(π_{1} × π̃_{2}) = f(π̃_{1} × π_{2}). Using (2.5), (2.6) and (2.8) it follows that the constant b in (2.4) can be chosen to be ϵ(0, π_{1} × π̃_{2}, ψ) and

(2.9)  r̃_{\overline{P} |P} (π_{1} ⊗ π_{2}, s) = \frac{L(s, π_{1} × π̃_{2})}{L(1 + s, π_{1} × π̃_{2})ϵ(0, π_{1} × π̃_{2}, ψ)}.

Comparing (2.1) and (2.9), it follows that

(2.10)  r_{\overline{P} |P} (π_{1} ⊗ π_{2}, s) = \frac{ϵ(0, π_{1} × π̃_{2}, ψ)}{ϵ(s, π_{1} × π̃_{2}, ψ)}r̃_{\overline{P} |P} (π_{1} ⊗ π_{2}, s).

This can be extended to parabolic groups of arbitrary rank in the usual way. Let M be a standard Levi subgroup of GL_{n} of type (n_{1}, ..., n_{r}) and let P_{1}, P_{2} ∈ P(M). Using the product formula for r̃_{P_{2}|P_{1}}(π, s) [A7, p.29] and the corresponding product formula (2.3) for r_{P_{2}|P_{1}}(π, s), we extend (2.10) to all tempered representations π of M(F). Finally, if π is any irreducible unitary representation of M(F), it can be written as a Langlands quotient π = J_{R}^{M}(τ, μ), where R is a parabolic subgroup of M, τ is a tempered representation of M_{R}(F) and μ is a point in the chamber of 𝔞_{R}^{∗}/𝔞_{M}^{∗} attached to R. Then

r_{P_{2}|P_{1}}(π, s) = r_{P_{2}(R)|P_{1}(R)}(τ, s + μ)

and a similar formula holds for r̃_{P_{2}|P_{1}}(π, s) [A7, (2.3)]. Let σ_{1}, σ_{2} ∈ S_{r} be attached to P_{1}, P_{2} such that Σ_{P_{i}} is given by (2.2). Then we get Lemma 2.1. For all irreducible unitary representation π = ⊗_{i=1}^{r}π_{i} of M(F) we have

r_{P_{2}|P_{1}}(π, s) = ∏_{(i,j)∈I(σ_{1},σ_{2})} \frac{ϵ(0, π_{i} × π̃_{j}, ψ)}{ϵ(s_{i} − s_{j}, π_{i} × π̃_{j}, ψ)}r̃_{P_{2}|P_{1}}(π, s), s ∈ ℂ^{r}.

Since we will be concerned with logarithmic derivatives of normalizing factors, we need estimates for f(π_{1} ×π_{2}). Let f(π_{i}) be the conductor of π_{i}, i = 1, 2. Then by Theorem 1 of [BH] and Corollary (6.5) of [BHK] we have

(2.11) 0 ≤ f(π_{1} × π_{2}) ≤ n_{1}f(π_{1}) + n_{2}f(π_{2}) − inf{f(π_{1}), f(π_{2})}

for all admissible smooth representations π_{i} of GL_{n_{i}}(F), i = 1, 2. Furthermore, by Corollary (6.5) of [BHK], we have f(π_{1} × π_{2}) = 0 if and only if there exists a quasicharacter χ of F^{×} such that both π_{1} ⊗χ∘det and π_{2} ⊗ χ^{−1} ∘ det are unramified principal series representations.

By (2.11) it suffices to estimate the conductors f(π_{i}), i = 1, 2.

Given an open compact subgroup K ⊂ GL_{m}(F), let

Π(GL_{m}(F); K) = {π ∈ Π(GL_{m}(F)) | π^{K} ≠ {0}}.

Lemma 2.2. For every open compact subgroup K of GL_{m}(F) there exists C > 0 such that f(π) ≤ C for all π ∈ Π(GL_{m}(F); K).

Proof. In the first step we reduce the proof to the case of squareintegrable representations. Let π ∈ Π(GL_{m}(F)). Then there exist a parabolic subgroup P of GL_{m} of type (m_{1}, ..., m_{r}), tempered representations τ_{j} of GL_{m_{j}}(F) and real numbers t_{i} with t_{1} > ⋯ > t_{r} such that π is isomorphic to the Langlands quotient J_{P}^{GL_{m}}(τ_{1}[t_{1}], ..., τ_{r}[t_{r}]). By Theorem 3.4 of [J] it follows that

f(π) = f (I_{P}^{GL_{m}}(τ_{1}[t_{1}], ..., τ_{r}[t_{r}])) = ∑_{j} f(τ_{j}).

Furthermore a tempered representation τ of GL_{d}(F) is full induced: τ = I_{Q}^{GL_{d}}(σ_{1}, ..., σ_{l}), where Q is a parabolic subgroup of GL_{d} of type (d_{1}, ..., d_{l}) and σ_{i} is a square-integrable representation of GL_{d_{i}}(F), i = 1, ..., l. Then by (3.2.3) of [J] we get

f(τ) = ∑_{j} f(σ_{j}).

Next we relate the K-invariant subspaces. We may assume that K ⊂ GL_{m}(O_{F}) is a congruence subgroup. Suppose that π is a subquotient of an induced representation I_{P}^{GL_{m}}(σ), where P is a parabolic subgroup of GL_{m} of type (m_{1}, ..., m_{h}), σ = ⊗_{i}ρ_{i} and ρ_{i} is an admissible representation of GL_{m_{i}}(F). If π^{K} ≠ {0}, then I_{P}^{GL_{m}}(σ)^{K} ≠ {0}. Furthermore we have

I_{P}^{GL_{m}}(σ)^{K} = (I_{GL_{m}(O_{F})∩P}^{GL_{m}(O_{F})} (σ))^{K}

↪ ⨁_{GL_{m}(O_{F})/K} I_{K∩P}^{K} (σ)^{K}

≅ ⨁_{GL_{m}(O_{F})/K} σ^{K∩P} .

Now observe that

K ∩ P = ∏_{i=1}^{h} K_{i},

where K_{i} ⊂ GL_{m_{i}}(O_{F}) are congruence subgroups. Then

σ^{K∩P} ≅ ⊗_{i=1}^{h}ρ_{i}^{K_{i}}.

Thus if π^{K} ≠ {0}, then ρ_{i}^{K_{i}} ≠ {0} for all i, 1 ≤ i ≤ h. Combined with the above relations of the conductors, we reduce to the case of square-integrable representations.

Let 1 denote the trivial representation of K. By [HC2, Theorem 10] the set Π_{2}(GL_{m}(F), K) of square-integrable representations π of GL_{m}(F) with [π|_{K} : 1] ≥ 1 is a compact subset of the space Π_{2}(GL_{m}(F)) of square-integrable representations of GL_{m}(F). By the definition of the topology in Π_{2}(GL_{m}(F)) [HC2, §2], the set Π_{2}(GL_{m}(F), K) decomposes into a finite number of orbits under the canonical action of iℝ on Π_{2}(GL_{m}(F)) given by π ↦ π[it]. Since the conductor remains unchanged under twists by unramified characters, the lemma follows. □

3. Estimation of the Langlands parameters

For the unramified places, the Langlands parameters of local components of cuspidal automorphic representations of GL_{n}(𝔸) have been estimated by Luo, Rudnick and Sarnak [LRS]. The main purpose of this section is to extend the estimations of [LRS] first to ramified places and then to local components of automorphic representations in the discrete spectrum of GL_{n}(𝔸) in general. To deal with cuspidal automorphic representations we follow the method of [LRS] which uses properties of the Rankin-Selberg L-functions. First note that any local component of a cuspidal automorphic representation of GL_{n} is generic [Sk]. Let F be a local field. By [JS3], any irreducible generic representation π of GL_{n}(F) is equivalent to a fully induced representation

π = I_{P}^{G}(σ, s),

where P is a standard parabolic subgroup of type (n_{1}, ..., n_{r}), s = (s_{1}, ..., s_{r}) ∈ ℝ^{r} satisfies s_{1} ≥ s_{2} ≥ ⋯ ≥ s_{r} and σ is a square integrable representation of M_{P} (F). We shall refer to s as the (continuous) Langlands parameters of π. We also note that an irreducible induced representation I_{P}^{G}(σ, s), σ square-integrable and s ∈ ℝ^{r}, is unitary, only if it is equivalent to its hermitian dual representation IG ; P̄(σ, −s). By [KZ, Theorem 7] this implies that there exists a w ∈ W (a_{M}) of order 2 so that

(σ_{1}[s_{1}] ⊗ ⋯ σ_{r}[s_{r}])^{w} = σ_{1}[−s_{1}] ⊗ ⋯ ⊗ σ_{r}[−s_{r}].

Hence we have

(3.1)  {σ_{j}[s_{j}]} = {σ_{k}[−s_{k}]}.

Moreover by the classification of the unitary dual of GL_{n}(F) it follows that

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

The key result that we need about the local Rankin-Selberg L-functions is the following lemma. Lemma 3.1. Let π be an irreducible unitary generic representation of GL_{n}(F), and let (s_{1}, ..., s_{r}) ∈ ℝ^{r} be the Langlands parameters of π. Then L(s, π × π̃) has a pole at the point

s_{0} = 2 max_{j} |s_{j}|.

Proof. For the proof we need to describe the L-factors in more detail. Let π ≅ I_{P}^{G}(σ, s) be a fully induced representation with σ = ⊗_{i}σ_{i} for discrete series representations σ_{i} of GL_{n_{i}}(F) and Langlands parameters s = (s_{1}, ..., s_{r}) satisfying the above conditions. Then by the multiplicativity of the local L-factors [Sh6] we get

(3.2)  L(s, π × π̃) = ∏_{i,j=1}^{r} L(s + s_{i} − s_{j}, σ_{i} × σ̃_{j}).

If F is non-Archimedean, then this is Proposition 9.4 of [JPS]. If F is Archimedean, (3.2) follows from the Langlands classification (see §2 of [Sh6]). This reduces the description of the L-factors to the case of square-integrable representations. We distinguish three cases according to the type of the field F.

1. F = ℝ.

The Rankin-Selberg local L-factors are defined in terms of L-factors attached to semisimple representations of the Weil group W_{ℝ} by means of the Langlands correspondence [L3]. If τ is a semisimple representation of W_{ℝ} of degree n and π(τ) is the associated irreducible admissible representation of GL_{n}(ℝ), then

L(s, π(τ)) = L(s, τ).

Furthermore, if

τ = ⨁_{1≤j≤m} τ_{j}

is the decomposition into irreducible representations of W_{ℝ}, then

L(s, τ) = ∏_{j} L(s, τ_{j}).

If τ′ is another semisimple representation of W_{ℝ} of degree n′ and π(τ′) is the associated irreducible admissible representation of GL_{n′}(ℝ), then the Rankin-Selberg local L-factor is given by

L(s, π(τ) × π(τ′)) = L(s, τ ⊗ τ′).

This reduces the computation of the L-factors to the case of irreducible representations of the Weil group.

The irreducible representations of the Weil group W_{ℝ} of ℝ are either 1 or 2 dimensional. The associated representations of GL_{m}(ℝ), m = 1, 2, are square-integrable and all square-integrable representations are obtained in this way. Note that GL_{m}(ℝ) does not have square-integrable representations if m ≥ 3. To describe the L-factors, we define Gamma factors by

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

Suppose that τ is a two-dimensional irreducible representation of W_{ℝ}. Then

τ = ind_{ℂ^{∗}}^{W_{ℝ}}θ,

where θ is a (not necessarily unitary) character of W_{ℂ} = ℂ^{∗}. Thus there exist t ∈ ℂ and k ∈ ℤ such that

θ(z) = |z|^{t}(z/\overline{z})^{k/2}, z ∈ ℂ^{∗}.

Then the L-factor is defined as

L(s, τ) = Γ_{ℂ}(s + t + |k|/2).

The one-dimensional irreducible representations of W_{ℝ} are of the form

ψ_{ϵ,t} : (z, σ) ∈ W_{ℝ} → sign^{ϵ}(σ)|z|^{t}

where ϵ = 0, 1 and ”sign” is the sign character of the Galois group. The L-factor of ψ_{ϵ,t} is given by

L(s, ψ_{ϵ,t}) = Γ_{ℝ}(s + t + ϵ).

Next we have to consider the tensor products of irreducible representations.

If τ = ind_{ℂ^{∗}}^{W_{ℝ}}θ and τ′ = ind_{ℂ^{∗}}^{W_{ℝ}}θ′ are two-dimensional representations of W_{ℝ}, then we have

τ ⊗ τ′ = ind_{ℂ^{∗}}^{W_{ℝ}}(θ ⊗ ind_{ℂ^{∗}}^{W_{ℝ}}θ′|_{ℂ^{∗}})
= ind_{ℂ^{∗}}^{W_{ℝ}}(θ ⊗ θ′) ⊕ ind_{ℂ^{∗}}^{W_{ℝ}}(θ ⊗ θ′^{σ}),

where σ is the nontrivial element of the Galois group. Suppose that θ′(z) = |z|^{t′}(z/\overline{z})^{k′/2}. Then we get

L(s, τ ⊗ τ′)
= Γ_{ℂ}(s + t + t′ + |k + k′|/2)Γ_{ℂ}(s + t + t′ + |k − k′|/2).

Similarly, if ψ = ψ_{ϵ,t′} is a one-dimensional representation, then we have

τ ⊗ ψ = ind_{ℂ^{∗}}^{W_{ℝ}}(θ ⊗ ψ|_{ℂ^{∗}}),

and therefore, we get

L(s, τ ⊗ ψ) = Γ_{ℂ}(s + t + t′ + |k|/2).

Finally, if ψ = ψ_{ϵ,t} and ψ_{ϵ′,t′} are two one-dimensional representations of W_{ℝ}, then

L(s, ψ ⊗ ψ′) = Γ_{ℝ}(s + t + t′ + ϵ̃),

where 0 ≤ ϵ̃ ≤ 1 and ϵ̃ = ϵ + ϵ′ mod 2.

For k ∈ ℤ let D_{k} be the k-th discrete series representation of GL_{2}(ℝ) with the same infinitesimal character as the k-dimensional representation. Then D_{k} is associated with the two-dimensional representation τ_{k} = ind_{ℂ^{∗}}^{W_{ℝ}}(θ_{k}) of W_{ℝ} where the character θ_{k} of ℂ^{∗} is defined by θ_{k}(z) = (z/\overline{z})^{k/2}. For ϵ ∈ \{0, 1\} let ψ_{ϵ} the character of ℝ^{×}, defined by ψ_{ϵ}(r) = (r/|r|)^{ϵ}. It corresponds to the character ψ_{ϵ,0} of W_{ℝ}. Using the above description of the L-factors, we get

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

where 0 ≤ ϵ_{1,2} ≤ 1 with ϵ_{1,2} ≡ ϵ_{1}+ϵ_{2} mod 2. Up to twists by unramified characters this exhausts all possibilities for the L-factors in the squareintegrable case.

2. F = ℂ.

As in the real case, the local L-factors are defined in terms of the Lfactors attached to representations of the Weil group W_{ℂ} by means of the Langlands correspondence. The Weil group W_{ℂ} is equal to ℂ^{∗}.

Furthermore we note that GL_{m}(ℂ) has square-integrable representations only if m = 1. For r ∈ ℤ let χ_{r} be the character of ℂ^{∗} defined by χ_{r}(z) = (z/\overline{z})^{r}, z ∈ ℂ^{∗}. Then it follows that

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

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

3. F non-Archimedean.

Let π be a square-integrable representation of GL_{m}(F). By [BZ] there is a divisor d|m, a standard parabolic subgroup P of GL_{m}(F) of type (d, …, d), and an irreducible supercuspidal representation ρ of GL_{d}(F) so that π is the unique quasi-square-integrable component of the induced representation I_{P}^{G}(ρ_{1}, …, ρ_{r}), where r = m/d and ρ_{j} = ρ ⊗ | det |^{j−(r+1)/2}, j = 1, ..., r. We will write π = Δ(r, ρ). The representation π is unitary (or equivalently square-integrable) if only if ρ is unitary. Moreover the contragredient of Δ(r, ρ) is given by Δ̃(r, ρ) = Δ(r, ρ̃). Let σ_{1} = Δ(r_{1}, ρ_{1}) and σ_{2} = Δ(r_{2}, ρ_{2}) be square integrable representation of GL_{m_{1}}(F) and GL_{m_{2}}(F), respectively. Then by Theorem 8.2 of [JPS] we have

L(s, σ_{1} × σ̃_{2}) = ∏_{j=1}^{min(m_{1},m_{2})} L(s + (m_{1} + m_{2})/2 − j, ρ_{1} × ρ̃_{2}).

Thus the description of the Rankin-Selberg L-functions is reduced to the case of two supercuspidal representations. Let ρ_{i}, i = 1, 2, be supercuspidal representations of GL_{k_{i}}(F). By Proposition (8.1) of [JPS] we have L(s, ρ_{1} × ρ_{2}) = 1 if k_{1} > k_{2}. Since L(s, ρ_{1} × ρ_{2}) = L(s, ρ_{2} × ρ_{1}), the same holds for k_{1} < k_{2}. Let k_{1} = k_{2}. Then by Proposition (8.1) of [JPS], L(s, ρ_{1} × ρ_{2}) = 1, unless ρ_{1} and ρ_{2} are in the same twist class, i.e., there exists t ∈ ℂ such that ρ_{2} ≅ ρ_{1}[t]. In this case we have

L(s, ρ_{1} × ρ̃_{1}[t]) = L(s + t, ρ_{1} × ρ̃_{1}) = (1 − q^{−a(t+s)})^{−1} ,

where a|k_{1} is the order of the cyclic group of unramified characters χ = | det |^{u} such that ρ_{1} ⊗ χ ≅ ρ_{1}. Let σ = Δ(r, ρ). Then we get

\begin{aligned}L(s, σ × σ̃) & = ∏_{j=1}^{r} L(s + r − j, ρ × ρ̃) \\ & = ∏_{j=1}^{r} (1 − q^{−a(s+r−j)})^{−1} , \end{aligned}

where a is the order of the cyclic group of unramified characters χ so that ρ ⊗ χ is isomorphic to ρ.

From the above description of the local L-factors we conclude that they have the following two properties. Let π be a square-integrable representation of GL_{m}(F). Then L(s, π × π̃) has a pole at s = 0. Furthermore, if π_{1} and π_{2} are square-integrable representations of GL_{m_{1}}(F) and GL_{m_{2}}(F), respectively, then L(s, π_{1} × π_{2}) has no zeros.

Now we are ready to prove the lemma. Let s = (s_{1}, ..., s_{r}) be the Langlands parameters of π. Let 1 ≤ i ≤ r. Then it follows from (3.1) and (3.2) that L(s, π × π̃) contains the factor L(s − 2s_{i}, σ_{i} × σ̃_{i}). Using the above properties of the L-factors in the square-integrable case, it follows that L(s, π × π̃) has a pole at 2s_{i}. By (3.1), −s_{i} occurs also in s. Hence L(s, π × π̃) has a pole at 2|s_{i}|. In particular, L(s, π × π̃) has a pole at 2 max_{i} |s_{i}|. □

Next we recall some facts about ray class characters. Let E be a number field. Let 𝔮 be a nonzero integral ideal of E and denote by C(𝔮) the wide ray class group of E modulo 𝔮. We note that the term ”wide” means that no positivity condition has been imposed at the real places of E. Then a character of C(𝔮) is unramified at all infinite places. Now recall that any character χ of C(𝔮) can be identified with a character of the idele class group C_{E} = I_{E}/E^{×} which is trivial on the congruence subgroup C_{E}^{𝔮} = I_{E}^{𝔮}E^{×}/E^{×} [Neu]. Then χ = ⊗_{v}χ_{v} where χ_{v} is a character of E_{v}^{×} which is unramified at all places v|∞ and all finite places 𝔭 ∤ 𝔮. Furthermore for a finite place 𝔭 ∤ 𝔮, the character χ_{𝔭} is given by

(3.3)  χ_{𝔭}(α) = χ(𝔭)^{v_{𝔭}(α)}, α ∈ E_{𝔭}^{×},

where v_{𝔭} : E_{𝔭}^{×} → ℤ is the 𝔭-adic valuation.

Let S be any finite set of finite places of E. For an integral ideal 𝔮 of E let X_{𝔮} denote the set of all wide ray class characters of conductor 𝔮 such that χ(𝔭) = 1 for all 𝔭 ∈ S. Therefore we have (𝔭, 𝔮) = 1 for all 𝔭 ∈ S. By (3.3) it follows that

(3.4)  χ_{v} = 1 for all v ∈ S ∪ S_{∞}.

Let X_{𝔮}^{∗} be the subset of X_{𝔮} consisting of all primitive characters. Set X = ∪_{𝔮}X_{𝔮} and X^{∗} = ∪_{𝔮}X_{𝔮}^{∗}. For χ ∈ X_{𝔮}^{∗} and a cuspidal automorphic representation π of GL_{n}(𝔸), the partial Rankin-Selberg L-function L_{S}(s, (π ⊗ χ) × π̃) is defined to be

L_{S}(s, (π ⊗ χ) × π̃) = ∏_{v∉S} L(s, (π_{v} ⊗ χ_{v}) × π̃_{v}).

We shall use the following result of Luo, Rudnick and Sarnak which is the main result of [LRS]. Theorem 3.2. Given n, π, S as the above and any β > 1 − 2/(n^{2} + 1), there are infinitely many χ ∈ X^{∗} such that

L_{S}(β, (π ⊗ χ) × π̃) ≠ 0.

Now we can establish our extension of Theorem 2 of [LRS]. Proposition 3.3. Suppose that π = ⊗_{v}π_{v} is a cuspidal automorphic representation of GL_{n}(𝔸) . Let s_{v} = (s_{1,v}, …, s_{k,v}) ∈ ℝ^{k} be the Langlands parameters of the representation π_{v}. Then we have

max_{j} |s_{j,v}| < \frac{1}{2} − \frac{1}{n^{2} + 1}.

Proof. We follow the proof of Theorem 2 in [LRS]. Let v be a place of E and set S = {v}. Let χ ∈ X^{∗}, where X^{∗} is the set of ray class characters with respect to S which we defined above. The Rankin-Selberg L-function

L(s, (π ⊗ χ) × π̃) = L(s, (π_{v} ⊗ χ_{v}) × π̃_{v})L_{S}(s, (π ⊗ χ) × π̃).

is holomorphic in the whole complex plane except for simple poles at s = 1 and s = 0 if π ⊗ χ ≅ π. This follows from the work of Jacquet, Piateski-Shapiro, Shalika, Shahidi, Mœglin and Waldspurger [JPS], [JS1], [JS2], [MW], [Sh2]. Choosing the conductor of χ sufficiently large, we have π ⊗ χ ≅ π̸. Thus by Theorem 3.2 we may choose χ ∈ X^{∗} such that L(s, (π ⊗ χ) × π̃) is an entire function and by (3.4) we have χ_{v} = 1. Suppose that s_{0} > 0 is a pole of

L(s, π_{v} × π̃_{v}) = L(s, (π_{v} ⊗ χ_{v}) × π̃_{v}).

Then s_{0} must be a zero of L_{S}(s, (π ⊗ χ) × π̃). Assume that s_{0} > 1 − 2(1 + n^{2})^{−1}. Then by Theorem 3.2 there exists χ ∈ X^{∗} with L_{S}(s_{0}, (π ⊗ χ) × π̃) ≠ 0, χ_{v} = 1, and L(s, (π ⊗ χ) × π̃) entire. Hence it follows that s_{0} < 1 − 2(1 + n^{2})^{−1}. Together with Lemma 3.1 the proposition follows. □

We shall now establish a similar result for the local components of residual automorphic representations of GL_{n}(𝔸). First we recall some facts about representations of GL_{n} over a local field F. Any irreducible unitary representation π of GL_{n}(F) is equivalent to a Langlands quotient J_{R}^{G}(τ, μ). This is the unique irreducible quotient of an induced representation I_{R}^{G}(τ, μ) where τ is a tempered representation of M_{R}(F) and μ is a point in the positive chamber attached to R. A slight variant of this description is as follows. Recall that a tempered representation τ of GL_{m}(F) can be described as follows. There exist a standard parabolic subgroup Q of type (m_{1}, ..., m_{p}) and square-integrable representations δ_{j} of GL_{m_{j}}(F) so that τ is isomorphic to the full induced representation I_{P}^{GL_{m}}(δ_{1}, ..., δ_{p}). Hence by induction in stages, there exist a standard parabolic subgroup P of G of type (n_{1}, ..., n_{r}), discrete series representations δ_{i} of GL_{m_{i}}(F) and real numbers s_{1} ≥ s_{2} ≥ ⋯ ≥ s_{r} such that π is equivalent to the unique irreducible quotient

J_{P}^{G}(δ_{1}[s_{1}] ⊗ ⋯ ⊗ δ_{r}[s_{r}]),

of the induced representation I_{P}^{G}(δ_{1}[s_{1}] ⊗ ⋯ ⊗ δ_{r}[s_{r}]) [MW, I.2]. We call s_{1}, ..., s_{r} the (continuous) Langlands parameters of π.

The residual spectrum for GL_{n}(𝔸) has been determined by Mœglin and Waldspurger [MW]. Let π = ⊗_{v}π_{v} be an irreducible automorphic representation in the residual spectrum of GL_{n}(𝔸). By [MW], there is a divisor k|n, a standard parabolic subgroup P of type (d, ..., d), and a cuspidal automorphic representation ξ of GL_{d}(𝔸), d = n/k, so that the representation π is a quotient of the induced representation

I_{P (𝔸)}^{G(𝔸)} (ξ[(k − 1)/2] ⊗ ⋯ ⊗ ξ[−(k − 1)/2]) .

Lemma 3.4. Let π and ξ be as above. Let v be a place of E and let s_{1} ≥ ⋯ ≥ s_{r}, be the Langlands parameters of ξ_{v}. Then the Langlands parameters of π_{v} are given by

(\frac{k − 1}{2} + s_{1}, ..., \frac{k − 1}{2} + s_{r}, ..., −\frac{k − 1}{2} + s_{1}, ..., −\frac{k − 1}{2} + s_{r}) .

Proof. Since ξ_{v} is a local component of a cuspidal automorphic representation ξ of GL_{d}(𝔸), it is generic. Using induction in stages, it follows that there exist a standard parabolic subgroup R of GL_{d} of type (n_{1}, ..., n_{r}), discrete series representations δ_{i,v} of GL_{n_{i}}(E_{v}) and real numbers (s_{1}, ..., s_{r}) satisfying

(3.5)  s_{1} ≥ s_{2} ≥ ⋯ ≥ s_{r}; |s_{j}| < 1/2, j = 1, ..., r.

such that ξ_{v} is isomorphic to the full induced representation

ξ_{v} ≅ I_{R}^{GL_{d}}(δ_{1,v}[s_{1}] ⊗ ⋯ ⊗ δ_{r,v}[s_{r}]).

Let Q = M_{Q}N_{Q} be the standard parabolic subgroup of GL_{n} whose Levi component M_{Q} is a product of k copies of M_{R} and let

δ_{v} = (δ_{1,v} ⊗ ⋯ ⊗ δ_{r,v}) ⊗ ⋯ ⊗ (δ_{1,v} ⊗ ⋯ ⊗ δ_{r,v}),

where (δ_{1,v} ⊗ ⋯ ⊗ δ_{r,v}) occurs k times. Define

μ(k, s) = (\frac{k − 1}{2} + s_{1}, …, \frac{k − 1}{2} + s_{r}, \frac{k − 3}{2} + s_{1}, …, −\frac{k − 1}{2} + s_{r}) .

By induction in stages we have

I_{P}^{G}(ξ_{v}[(k − 1)/2] ⊗ ⋯ ⊗ ξ_{v}[−(k − 1)/2]) = I_{Q}^{G}(δ_{v}, μ(k, s)).

Furthermore, by (3.5) the coordinates of μ(k, s) are decreasing. Thus the induced representation I_{Q}^{G}(δ_{v}, μ(k, s)) has a unique irreducible quotient which must be isomorphic to π_{v}. □

Next we recall a different method to parametrize irreducible unitary representations of GL_{n}(F). Let d|n and k = n/d. Let P be the standard parabolic subgroup of type (d, ..., d). Let δ be a discrete series representation of GL_{d}(F) and let a, b ∈ ℝ be such that b − a ∈ ℕ. Then the induced representation

I_{P}^{G}(δ[b] ⊗ δ[b − 1] ⊗ ⋯ ⊗ δ[a])

has a unique irreducible quotient which we denote by J(δ, a, b). Especially, if a = −(k − 1)/2 and b = (k − 1)/2, then we put

(3.6)  J(δ, k) := J(δ, a, b).

By Theorem D of [Ta] and [Vo], for every irreducible unitary representation of GL_{n}(F) there exist a standard parabolic subgroup P of type (n_{1}, ..., n_{r}), k_{i}|n_{i}, discrete series representations δ_{i} of GL_{d_{i}}(F), d_{i} = n_{i}/k_{i}, and real numbers s_{1}, ..., s_{r} with |s_{i}| < 1/2, i = 1, ..., r, such that π is isomorphic to the fully induced representation:

(3.7)  π ≅ I_{P}^{G}(J(δ_{1}, k_{1})[s_{1}] ⊗ ⋯ ⊗ J(δ_{r}, k_{r})[s_{r}]).

Using this parametrization, we get the following analogue to Proposition 3.3 for local components of automorphic representations in the residual spectrum of GL_{n}(𝔸). Proposition 3.5. Let π_{v} be a local component of an automorphic representation in the residual spectrum of GL_{n}(𝔸). There exist k|n, a parabolic subgroup P of type (kn_{1}, …, kn_{r}), discrete series representations δ_{i,v} of GL_{n_{i}}(E_{v}) and real numbers s_{1}, ..., s_{r} satisfying

s_{1} ≥ s_{2} ≥ ⋯ ≥ s_{r}, |s_{i}| < 1/2 − (1 + n^{2})^{−1}, i = 1, ..., r,

such that

π_{v} ≅ I_{P}^{G}(J(δ_{1,v}, k)[s_{1}] ⊗ ⋯ ⊗ J(δ_{r,v}, k)[s_{r}]).

Proof. By the proof of Lemma 3.4, π_{v} is equivalent to a Langlands quotient of the form

π_{v} ≅ J_{Q}^{G}(δ_{v}, μ(k, s)),

where the parameters s satisfy (3.5). Set

b_{i} = \frac{k − 1}{2} + s_{i}, a_{i} = −\frac{k − 1}{2} + s_{i}, i = 1, ..., r.

Suppose there exist 1 ≤ i < j ≤ r such that the triples (δ_{i,v}, a_{i}, b_{i}) and (δ_{j,v}, a_{j}, b_{j}) are linked in the sense of I.6.3 or I.7 in [MW]. Suppose that s_{i} ≥ s_{j}. Then it follows from (2) and (3)(i) on p.622 or from (1) and (2) on p.624 of [MW] that a_{i} ≥ a_{j} + 1 and b_{i} ≥ b_{j} + 1. This implies 1 ≤ |s_{i} − s_{j}| which contradicts (3.5). Hence the triples (δ_{i,v}, a_{i}, b_{i}) are pairwise not linked. Now observe that

J(δ_{i,v}, a_{i}, b_{i}) = J(δ_{i,v}, k)[s_{i}].

Let P be the standard parabolic subgroup with Levi component

GL_{kn_{1}} × ⋯ × GL_{kn_{r}} .

Let δ̃_{v} = ⊗_{i=1}^{r}δ_{i,v} and set

J(δ̃_{v}, k) := ⊗_{i=1}^{r}J(δ_{i,v}, k).

Then it follows from Proposition I.9 of [MW] that the induced representation

I_{P}^{G}(J(δ̃_{v}, k), s) := I_{P}^{G}(J(δ_{1,v}, k)[s_{1}] ⊗ ⋯ ⊗ J(δ_{r,v}, k)[s_{r}])

is irreducible. By Lemma I.8, (ii), of [MW], I_{P}^{G}(J(δ̃_{v}, k), s) is a quotient of the induced representation I_{Q}^{G}(δ_{v}, μ(k, s)). Since I_{P}^{G}(J(δ̃_{v}, k), s) is irreducible, this quotient must be the Langlands quotient. Thus

π_{v} ≅ J_{Q}^{G}(δ_{v}, μ(k, s)) ≅ I_{P}^{G}(J(δ_{1,v}, k)[s_{1}] ⊗ ⋯ ⊗ J(δ_{r,v}, k)[s_{r}]).

By construction, the s_{i}’s are the Langlands parameters of a local component of a cuspidal automorphic representation. Therefore it follows from Proposition 3.3 that they satisfy

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

□

This result has an important consequence for the location of the poles of normalized intertwining operators (see Proposition 4.2).

4. Proof of the main results

In this section we prove Proposition 0.2 and Theorem 0.1. To this end we need some preparation.

Let M be a standard Levi subgroup of type (n_{1}, ..., n_{r}) and let P, P′ ∈ P(M). Given a place v of E, let Π_{disc}(M(E_{v})) denote the set of all π_{v} ∈ Π(M(E_{v})) which are local components of some automorphic representation π in the discrete spectrum of M(𝔸). Without loss of generality, we may assume that P is a standard parabolic subgroup. Let π_{v} ∈ Π_{disc}(M(E_{v})). Then π_{v} = ⊗_{i}π_{i,v} with π_{i,v} ∈ Π_{disc}(GL_{n_{i}}(E_{v})), i = 1, ..., r. By Proposition 3.5 there exist a standard parabolic subgroup R_{i} of GL_{n_{i}} with

M_{R_{i}} = GL_{n_{i1}} × ⋯ × GL_{n_{im_{i}}},

k_{ij}|n_{ij}, discrete series representations δ_{ij} of GL_{d_{ij}}(E_{v}), d_{ij} = n_{ij}/k_{ij}, and s_{ij} ∈ ℝ satisfying

(4.1)  s_{i1} ≥ ⋯ ≥ s_{im_{i}}, |s_{ij}| < \frac{1}{2} − \frac{1}{n^{2} + 1},

such that

π_{i,v} ≅ I_{R_{i}}^{GL_{n_{i}}} (J(δ_{i1}, k_{i1})[s_{i1}] ⊗ ⋯ ⊗ J(δ_{im_{i}}, k_{im_{i}})[s_{im_{i}}]) .

Set R = ∏_{i} R_{i}. Then R is a standard parabolic subgroup of M. Put m = m_{1} + ⋯ + m_{r}. We identify {(i, j) | i = 1, ..., r, j = 1, ..., m_{i}} with {1, ..., m} by

(i, j) ↦ ∑_{k<i} m_{k} + j.

For 1 ≤ l ≤ m let (i, j) be the pair that corresponds to l. Put

δ_{l} = δ_{ij}, k_{l} = k_{ij}, s_{l} = s_{ij}.

Set

(4.2)  J_{π_{v}} = ⊗_{i=1}^{m}J(δ_{i}, k_{i}), s_{π_{v}} = (s_{1}, ..., s_{m}).

Combing the above equivalences, we get

π_{v} ≅ I_{R}^{M}(J_{π_{v}}, s_{π_{v}}).

Let P (R) and P′(R) be the parabolic subgroups of GL_{n} with P (R) ⊂ P, P′(R) ⊂ P′, P (R)∩M = R and P′(R)∩M = R. Then by induction in stages, the induced representation I_{P}^{G}(π_{v}, s) is unitarily equivalent to the induced representation I_{P (R)}^{G}(J_{π_{v}}, s + s_{π_{v}}). The unitary map μ which provides this equivalence, is given by the evaluation map as in [KS, p.31]. Here we identify s ∈ ℂ^{r} with an element in ℂ^{m} by

(s_{1}, ..., s_{r}) ↦ (s_{1}, ..., s_{1}, s_{2}, ..., s_{2}, ..., s_{r}, ..., s_{r}),

where s_{i} is repeated m_{i} times. [KS, p.31]. Lemma 4.1. Let μ be the unitary equivalence of the induced representations as above. Then we have

(4.3)  μ ∘ R_{P′|P} (π_{v}, s) = R_{P′(R)|P (R)}(J_{π_{v}}, s + s_{π_{v}}) ∘ μ.

Proof. First consider the unnormalized intertwining operators. Recall that the unnormalized intertwining operators are defined by integrals which are absolutely convergent in a certain shifted chamber [KS, Theorem 6.6], [Sh2]. If we compare these integrals in their range of convergence as in [KS, p.31], it follows immediately that (4.3) holds for the unnormalized intertwining operators. So it remains to consider the normalizing factors. If we apply Lemma 3.4 to each component π_{i,v} of π_{v}, it follows that π_{v} is equivalent to a Langlands quotient of the form J_{Q}^{M}(δ_{v}, μ(k, s_{π_{v}})), where δ_{v} = ⊗_{i=1}^{m}δ_{i} and μ(k, s_{π_{v}}) = (μ(k_{1}, s_{π_{1,v}}), ..., μ(k_{r}, s_{π_{r,v}})). Hence by (2.3) in [A7] we get

r_{P′|P} (π_{v}, s) = r_{P′(Q)|P (Q)}(δ_{v}, s + μ(k, s_{π_{v}})).

Let S_{i} be the standard parabolic subgroup of M_{R_{i}} with Levi component

(GL_{d_{i1}} × ⋯ × GL_{d_{i1}}) × ⋯ × (GL_{d_{ip_{i}}} × ⋯ × GL_{d_{ip_{i}}}),

where each factor GL_{d_{ij}} occurs k_{i} times. Let S = ∏_{i} S_{i}. Then M_{S} and M_{Q} are conjugate. Let w ∈ S_{m} be the element which conjugates M_{S} into M_{Q}. Referring again to (2.3) in [A7], we get

r_{P′(R)|P (R)}(J_{π_{v}}, s + s_{π_{v}}) = r_{P′(R(S))|P (R(S))}(wδ_{v}, s + wμ(k, s_{π_{v}})).

By (r.6) of [A8, p.172] it follows that

r_{P′(R)|P (R)}(J_{π_{v}}, s + s_{π_{v}}) = r_{P′(R(S)^{w})|P (R(S)^{w})}(δ_{v}, s + μ(k, s_{π_{v}})),

where R(S)^{w} = w^{−1}R(S)w. Finally note that P (Q), P′(Q), P (R(S)^{w}) and P′(R(S)^{w}) have the same Levi component and the reduced roots satisfy

Σr ; \overline{P′(R(S)^{w})} ∩ Σ_{P (R(S)^{w})}^{r} = Σr ; \overline{P′(Q)} ∩ Σ_{P (Q)}^{r}.

Using the product formula (r.1) in [A8, p.171], it follows that

r_{P′(R(S)^{w})|P (R(S)^{w})}(δ_{v}, s + μ(k, s_{π_{v}})) = r_{P′(Q)|P (Q)}(δ_{v}, s + μ(k, s_{π_{v}})).

Combining the above equations, we get

r_{P′|P} (π_{v}, s) = r_{P′(R)|P (R)}(J_{π_{v}}, s + s_{π_{v}}),

and this finishes the proof of the lemma. □

We say that R_{Q|P} (π_{v}, s) has a pole at s_{0} ∈ ℂ^{r}, if R_{Q|P} (π_{v}, s) has a matrix coefficient with a pole at s_{0}. Otherwise, R_{Q|P} (π_{v}, s) is called holomorphic in s_{0}. Proposition 4.2. Let M = GL_{n_{1}} × ⋯ × GL_{n_{r}} be a standard Levi subgroup of GL_{n} and let P, P′ ∈ P(M). For every place v of E and for all π_{v} ∈ Π_{disc}(M(E_{v})), the normalized intertwining operator R_{P′|P} (π_{v}, s) is holomorphic in the domain

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

Proof. Using Lemma 4.1 we immediately reduce to the consideration of the corresponding problem for R_{P′(R)|P (R)}(J_{π_{v}}, s + s_{π}), regarded as function of s ∈ ℂ^{r}. By Proposition I.10 of [MW], the intertwining operator R_{P′(R)|P (R)}(J_{π_{v}}, s) is holomorphic in the domain of all s ∈ ℂ^{m} satisfying Re(s_{i} − s_{j}) > −1, 1 ≤ i < j ≤ m. Furthermore by (4.1) the absolute value of all components of s_{π} is bounded by 1/2 − (1 + n^{2})^{−1}. Combining these observations the claimed result follows.

□

Proof of Proposition 0.2:

Let M be a standard Levi subgroup of type (n_{1}, ..., n_{r}) and let P, Q ∈ P(M). We distinguish between the Archimedean and non-Archimedean case.

Case 1: v|∞.

Let K_{v} be the standard maximal compact subgroup of GL_{n}(E_{v}). Given π_{v} ∈ Π(M(E_{v})) and σ_{v} ∈ Π(K_{v}), denote by R_{Q|P} (π_{v}, s)_{σ_{v}} the restriction of R_{Q|P} (π_{v}, s) to the σ_{v}-isotypical subspace H_{P} (π_{v})_{σ_{v}} of the Hilbert space H_{P} (π_{v}) of the induced representation. If π_{v} belongs to Π_{disc}(M(E_{v})), then by Proposition 4.2, R_{Q|P} (π_{v}, s)_{σ_{v}} has no poles in the domain of all s ∈ ℂ^{r} satisfying | Re(s_{i})| < (1 + n^{2})^{−1}, i = 1, ..., r. Using the factorization of normalized intertwining operators and Corollary A.3 it follows that there exist constants C > 0 and k ∈ ℕ such that

∥ D_{u}R_{Q|P} (π_{v}, iu)_{σ} ∥≤ C (1+ ∥ σ_{v} ∥)^{k}

for all π_{v} ∈ Π(M(E_{v})), σ_{v} ∈ Π(K_{v}) and u ∈ ℝ^{r}. This proves (0.3).

Case 2: v < ∞.

Using the properties of the normalized intertwining operators [A7, Theorem2.1], one can factorize R_{P′|P} (π_{v}, s) in a product of normalized intertwining operators associated to maximal parabolic subgroups. Thus we immediately reduce to the case where P is maximal and P′ = \overline{P}.

Consider a matrix coefficient (R_{\overline{P} |P} (π_{v}, s)v_{1}, v_{2}), where ∥ v_{1} ∥=∥ v_{2} ∥= 1. By Theorem 2.1 of [A7], there is a rational function f(z) of one complex variable z such that

(4.4)  f(q^{−s}) = (R_{\overline{P} |P} (π_{v}, s)v_{1}, v_{2}), s ∈ ℂ.

We shall now investigate the properties of the rational function f. By Proposition I.10 of [MW] we know that (R_{\overline{P} |P} (π_{v}, s)v_{1}, v_{2}) is holomorphic in the half-plane Re(s) > 0. Hence f(z) is holomorphic in the punctured disc 0 < |z| < 1. Moreover by unitarity of R_{\overline{P} |P} (π_{v}, it), t ∈ ℝ, we have |(R_{\overline{P} |P} (π_{v}, it)v_{1}, v_{2})| ≤ 1, t ∈ ℝ, and hence |f(z)| ≤ 1 for |z| = 1. To determine the behaviour of f at z = 0 we observe that the unnormalized intertwining operator J_{\overline{P} |P} (π_{v}, s) is defined by an integral which is absolutely and uniformly convergent in some half-plane Re(s) ≥ c. Especially, J_{\overline{P} |P} (π_{v}, s) is uniformly bounded for Re(s) ≫ 0. The normalizing factor r_{\overline{P} |P} (π_{v}, s) is given by (2.1). It follows from the expressions (2.5) and (2.6) for the L-factors and the epsilon factor, that there exist polynomials P (z) and Q(z) with P (0) = Q(0) = 1, a constant a ∈ ℂ and m ∈ ℤ such that

r_{\overline{P} |P} (π_{v}, s) = aq^{ms} \frac{P (q^{−s})}{Q(q^{−s})}, s ∈ ℂ.

The integer m is given by (2.7) and it follows from (2.11) that there exists c ≥ 0, which depends on the choice of a nontrivial continuous character of E_{v}^{+}, such that −c ≤ m for all π_{v} ∈ Π(M(E_{v})). Thus by the maximum principle it follows that for 0 < |z| ≤ 1 we have

(4.5)  |f(z)| ≤ \begin{cases}1 & : m ≥ 0 \\ |z|^{m} & : m < 0. \end{cases}

Now assume that π_{v} ∈ Π_{disc}(M(E_{v})). Then by Proposition 4.2, f(z) is actually holomorphic for |z| < q^{2/(1+n^{2})}. Set

δ = min{2, q^{2/(1+n^{2})}}.

Note that δ > 1. Let ρ_{1}, ..., ρ_{r} be the poles of f, where each pole is counted with its multiplicity. Let −l be the order of f at infinity. Set

g(z) = f(z)z^{−l} ∏_{j} \frac{z − ρ_{j}}{1 − \overline{ρ}_{j}z} .

Since |ρ_{j}| > δ, j = 1, ..., r, the rational function g(z) is holomorphic for |z| > 1, bounded for |z| ≥ 1 and satisfies |g(z)| = |f(z)| ≤ 1 for |z| = 1. Thus |g(z)| ≤ 1 for |z| ≥ 1 and hence,

|f(z)| ≤ |z|^{l} ∏_{j=1}^{r} |\frac{1 − \overline{ρ}_{j}z}{z − ρ_{j}} | = |z|^{l} ∏_{j=1}^{r} |\frac{\overline{z} − 1/ρ_{j}}{1 − z/ρ_{j}} |, |z| ≥ 1.

For 1 ≤ |z| < (1+δ)/2 the right hand side is bounded by (\frac{1+δ}{2})^{l} (\frac{7}{δ−1})^{r}. Together with (4.5) it follows that there exists C > 0, which is independent of π_{v}, such that in the annulus 2/(1 + δ) < |z| < (1 + δ)/2 we have

|f(z)| ≤ C (\frac{1 + δ}{2} )^{l} (\frac{7}{δ − 1})^{r} .

Using Cauchy’s formula we obtain a similar bound for any derivative of f. By (4.4) this leads to a bound for any derivative of (R_{\overline{P} |P} (π_{v}, s)v_{1}, v_{2}) in a strip | Re(s)| < ε for some ε > 0. To complete the proof we need to verify that for a given open compact subgroup K_{v} of GL_{n}(E_{v}), the numbers r and l are bounded independently of π_{v} if π_{v}^{K_{v}∩M(E_{v})} ≠ 0.

First consider r. By Theorem 2.2.2 of [Sh2, p.323] there exists a polynomial p(z) with p(0) = 1 such that p(q^{−s})J_{\overline{P} |P} (π_{v}, s) is holomorphic on ℂ. Moreover the degree of p is bounded independently of π_{v}. Using the definition of the normalizing factors (2.1), it follows immediately that there exists a polynomial p̃(z) whose degree is bounded independently of π_{v} such that p̃(q^{−s})R_{\overline{P} |P} (π_{v}, s) is holomorphic on ℂ. This proves that r is bounded independently of π_{v}.

To estimate l, we fix an open compact subgroup K_{v} of GL_{n}(E_{v}). Our goal is now to estimate the order at ∞ of any matrix coefficient of R_{\overline{P} |P} (π_{v}, s)_{K_{v}}, regarded as a function of z = q^{−s}. Write π_{v} as Langlands quotient π_{v} = J_{R}^{M}(δ_{v}, μ) where R is a parabolic subgroup of M, δ_{v} a square-integrable representation of M_{R}(E_{v}) and μ ∈ (𝔞_{R}^{∗}/𝔞_{M}^{∗})_{ℂ} with Re(μ) in the chamber attached to R.Then

R_{\overline{P} |P} (π_{v}, s) = R_{\overline{P} (R)|P (R)}(δ_{v}, s + μ)

with respect to the identifications described in [A7, p.30]. Here s is identified with a point in (𝔞_{R}^{∗}/𝔞_{G}^{∗})_{ℂ} with repsect to the canonical embedding 𝔞_{M}^{∗} ⊂ 𝔞_{R}^{∗}. Using again the factorization of normalized intertwining operators we reduce to the case of a square-integrable representation. Let 1 denote the trivial representation of K_{v}. By the same reasoning as in the proof of Lemma 2.2 we get

[I_{P}^{G}(δ_{v}, s)|_{K_{v}} : 1] ≤ #(GL_{n}(O_{v})/K_{v})[δ_{v}|_{K_{v}∩M(E_{v})} : 1].

By [HC2, Theorem 10] the set Π_{2}(M(E_{v}), K_{v}) of square-integrable representations of M(E_{v}) with [δ|_{K_{v}∩M(E_{v})} : 1] ≥ 1 is a compact subset of the space Π_{2}(M(E_{v})) of square-integrable representations of M(E_{v}). Under the canonical action of i𝔞_{M}^{∗}, the set Π_{2}(M(E_{v}), K_{v}) decomposes into a finite number of orbits. In this way our problem is finally reduced to the consideration of the matrix coefficients of R_{\overline{P} |P} (π_{v}, s)_{K_{v}} for a finite number of representations π_{v}. This implies the claimed bound for l. □

Proof of Theorem 0.1:

Recall from §2 that at finite places the normalization of the local intertwining operators differs from the normalization used in [Mu4]. Let M = GL_{n_{1}} × ⋯ × GL_{n_{r}}, Q, P ∈ P(M) and v a finite place of E. Let r̃_{Q|P} (π_{v}, s), s ∈ ℂ^{r}, be the normalizing factor used in [Mu4] and let

R̃_{Q|P} (π_{v}, s) = r̃_{Q|P} (π_{v}, s)^{−1}J_{Q|P} (π_{v}, s)

be the corresponding local normalized intertwining operator. Then it follows from Lemma 2.1 together with (2.11) and Lemma 2.2 that for every multi-index α ∈ ℕ_{0}^{r} there exists C > 0 such that

∥ D_{u}^{α}R̃_{Q|P} (π_{v}, u)_{K_{v}} ∥≤ C ∑_{|β|≤|α|} ∥ D_{u}^{β}R_{Q|P} (π_{v}, u)_{K_{v}} ∥

for all u ∈ ℝ^{r} and all π_{v} ∈ Π_{disc}(M(E_{v})). Hence Proposition 0.2 holds also with respect to R̃_{Q|P} (π_{v}, s). Together with Theorem 0.1 of [Mu4] we obtain Theorem 0.1 of the present paper. □

Remark. As the proof of Proposition 0.2 shows, the estimations (0.2) and (0.3) hold for all generic representations π_{v} of M(E_{v}) whose Langlands parameters s_{1}, ..., s_{r} satisfy a non-trivial bound of the form |s_{i}| < 1/2 − ε, where ε > 0 is independent of π_{v}. We note that this assumption is really necessary and can not be removed in general. Especially, as the following example shows, the estimations can not be expected to be uniform in all π_{v} ∈ Π(M(E_{v})).

Example.

Let G = GL_{4}(ℝ) and P the standard parabolic subgroup with M_{P} = GL_{2} × GL_{2}. Consider the representation I_{P}^{G}(σ × σ, s) where σ is the spherical principal series representation induced from the character μ = (μ, −μ), μ real and 0 ≤ μ ≤ 1/2 of the Borel subgroup of GL_{2}(ℝ). We may assume that s = (s, −s) with s real. Then

I_{P}^{G}(σ × σ, s) = I_{B}^{G}(μ + s, −μ + s, μ − s, μ − s).

For fixed 0 ≤ μ < 1/2 this representation is irreducible for 0 ≤ |s| < 1/2 − μ and reducible for |s| = 1/2 − μ [Sp]. The intertwining operator R_{P̄|P} (σ ×σ, s) is therefore well defined on the interval 0 ≤ |s| < 1/2−μ and has a pole for −s = 1/2 − μ. □

The example shows that the poles of the normalized intertwining operator can be arbitrary close to the imaginary axis. Thus, we can not expect to have uniform bounds of the derivatives of the normalized intertwining operators along the imaginary axis for all unitary π.

Appendix A

by Erez M. Lapid

Let G be the real points of a connected reductive group defined over ℝ. Let K be a maximal compact subgroup of G and let P = M_{P} N_{P} be a parabolic subgroup of G with its Levi decomposition. Write M = M_{P} = ^{0}MA_{M} in the usual way. Let σ be an irreducible unitary representation of ^{0}M acting on a Hilbert space H_{σ} and let H_{σ}^{∞} be its smooth part. We denote by I_{σ}^{∞} the space of smooth functions f : K → H_{σ}^{∞} such that f(mk) = σ(m)f(k) for any m ∈ K_{M} = M ∩ K and k ∈ K with the inner product

⟨f_{1}, f_{2}⟩ = ∫_{K}(f_{1}(k), f_{2}(k))_{H_{σ}} dk

We denote the Lie algebra of A_{M} by 𝔞_{M}. Let P′ be another parabolic subgroup of G containing M as its Levi part. For any ν ∈ 𝔞_{M,ℂ}^{∗}, let J_{P′|P} (ν) be the usual intertwining operator on I_{σ}^{∞} ([Wal2, Chapter 10]) and let R_{P′|P} (σ, ν) = r_{P′|P} (σ, ν)^{−1}J_{P′|P} (ν) be the normalized intertwining operator (cf. [A7]). Finally, for any irreducible representation γ of K we denote by I_{σ}^{G}(γ) = I_{σ}(γ) the γ-isotypic part of I_{σ}^{∞}. We also denote by ∥γ∥ the norm of the highest weight of γ.

The purpose of this appendix is to give a bound for the matrix coefficients of the operator R_{P′|P} (σ, ν) on any K-type near the unitary axis. By factoring R_{P′|P} (σ, ν) it is enough to consider the “basic” case where P, P′ are adjacent – say along the root α. In this case the operator J_{P′|P} (σ, ν) depends only on (ν, α) and will be written as J_{P′|P} (s) for (ν, α) = 4s(ρ_{P} , α). Similarly for R_{P′|P} (σ, s).

It follows from [Wal2, Lemma 10.1.11, Theorem 10.1.6, 10.1.13] that the poles of J_{P′|P} (s) (counted with multiplicities) are contained in ⋃_{i=1}^{r}(ρ_{i} − ℕ) for some complex numbers ρ_{1}, …, ρ_{r}. By the nature of the normalization factors we may enlarge the set {ρ_{i}} to assume that the same holds for R_{P′|P} (σ, s) as well. Let

M_{σ}^{+} = max{0, Re ρ : ρ is a pole of R_{P′|P} (σ, s)}

and for any γ ∈ K̂ set

M_{σ,γ}^{−} = max{0, − Re(ρ) : ρ is a pole of R_{P′|P} (σ, s)|_{I_{σ}(γ)}}.

Finally, let

δ = min\{\frac{1}{2}, |Re(ρ)| : ρ is a pole of R_{P′|P} (σ, s)\}.

Lemma A.1. For any γ ∈ K̂ and any unit vectors φ_{1}, φ_{2} ∈ I_{σ}(γ) and any ϵ > 0 we have

|(R_{P′|P} (σ, s)φ_{1}, φ_{2})| ≤ [(M_{σ}^{+} + M_{σ,γ}^{−} + 1)/ϵ]^{r}

in the strip |Re(s)| < δ − ϵ.

Proof. Let f(s) = (R_{P′|P} (σ, s)φ_{1}, φ_{2}). It is a rational function of s ([A7]). We also have |f(s)| ≤ 1 for s ∈ iℝ since R_{P′|P} (σ, s) is unitary there. Define

g(s) = f(s) × ∏_{i=1}^{r} ∏_{j=⌈Re ρ_{i}+δ⌉}^{⌊Re ρ_{i}+M_{σ,γ}^{−}⌋} \frac{s − ρ_{i} + j}{s + \overline{ρ_{i}} − j} .

Then g(s) is holomorphic (and rational) for Re(s) ≤ 0 and |g(s)| = |f(s)| ≤ 1 on iℝ. Thus |g(s)| ≤ 1 for Re(s) ≤ 0. It follows that in this region

|f(s)| ≤ ∏_{i=1}^{r} ∏_{j=⌈Re ρ_{i}+δ⌉}^{⌊Re ρ_{i}+M_{σ,γ}^{−}⌋} |\frac{s + \overline{ρ_{i}} − j}{s − ρ_{i} + j} | ≤ ∏_{i=1}^{r} ∏_{j=⌈Re ρ_{i}+δ⌉}^{⌊Re ρ_{i}+M_{σ,γ}^{−}⌋} |\frac{Re(s + \overline{ρ_{i}} − j)}{Re(s − ρ_{i} + j)}| .

Then for 0 ≥ Re s ≥ −δ + ϵ each factor is bounded by

∏_{j=⌈Re ρ_{i}+δ⌉}^{⌊Re ρ_{i}+M_{σ,γ}^{−}⌋} \frac{− Re s − Re ρ_{i} + j}{Re s − Re ρ_{i} + j} < ∏_{j} \frac{Re s − Re ρ_{i} + j + 1}{Re s − Re ρ_{i} + j}

< \frac{M_{σ,γ}^{−} + 1}{ϵ}.

Similarly, one shows that for 0 ≤ Re s < δ − ϵ

|f(s)| < [\frac{M_{σ}^{+} + 1}{ϵ}]^{r} .

□

The following Proposition will be proved below. Proposition A.2. There exists a constant c depending only on G such that

(1.1)  M_{σ}^{+} ≤ c, r ≤ c M_{σ,γ}^{−} ≤ c(1 + ∥γ∥)

for all unitary σ.

By Cauchy’s formula, Lemma A.1 and Proposition A.2 will imply the following. Corollary A.3. For any differential operator D(s) with constants coefficients there exist constants c′, k′ (depending only on G) such that

(1.2)  ∥D(s)R_{P′|P} (σ, s)_{I_{σ}(γ)}∥ ≤ c′ (\frac{1 + ∥γ∥}{δ})^{k′}

for all γ ∈ K̂ and s ∈ iℝ. Remark A.4. The example in §4 emphasizes that the dependence on δ is essential if σ is not tempered. This is already important in order to lift the K-finiteness assumption in the absolute convergence of the contribution of an individual cuspidal datum. This point was overlooked in [A4] (cf., p. 1329]). More precisely, the property [A5, (7.6)] holds only for tempered representations. We mention that it follows from [KS, Theorem 16.2] that for all σ tempered we have δ > δ_{0} where δ_{0} > 0 depends only on G.

We will now prove Proposition A.2. We first deal with the first part of (1.1). More precisely, we have Lemma A.5. There exists s_{0} ∈ ℝ, depending only on G, such that J_{P′|P} (σ, s) converges and r_{P′|P} (σ, s) is holomorphic and non-zero for all Re(s) > s_{0}. In particular, M_{σ}^{+} ≤ s_{0}.

Proof. Let (Q, τ, λ) be Langlands data for σ, i.e., Q is a parabolic subgroup of M with Levi subgroup L, τ is a tempered representation of L and λ is a real parameter in the positive Weyl chamber of 𝔞_{Q}^{∗} and σ is the irreducible quotient of the standard module defined by Q and λ. By [Wal1, 5.5.2, 5.5.3], or [BW, Ch. XI, Theorem 3.3] ∥λ∥ is bounded in terms of G only. Moreover identifying I_{σ}^{∞} with a quotient of I_{τ}^{∞}, we may identify J_{P′|P} (σ, s) with J_{QN_{P′}|QN_{P}} (τ, s + λ) on the quotient space (cf. [A7, p. 30] or §4). Moreover, we have r_{P′|P} (σ, s) = r_{QN_{P′}|QN_{P}} (τ, s + λ). By factoring J_{QN_{P′}|QN_{P}} (τ, s + λ) and r_{QN_{P′}|QN_{P}} (τ, s + λ) the Lemma easily reduces to the tempered case. Similarly, we reduce to the squareintegrable case. For σ square-integrable we can take s_{0} = 0 ([A7]). □

The same argument reduces the second statement of (1.1) to the squareintegrable case. This case follows from [KS, Theorem 16.2] and the compatibility of the normalization factors with Artin’s factors ([A7]).

To continue the proof of Proposition A.2 we suppress for the moment the assumption that P, P′ are adjacent and set Σ(P′|P) = Σ(P)∩Σ(\overline{P′}) where \overline{P′} is the parabolic opposite to P′ and Σ(P) = Σ(P, A_{M}) be the set of reduced roots of A_{M} in P.

The main assertion is the following. Lemma A.6. There exists a constant d (depending only on G) such that for any γ ∈ K̂ J_{P′|P} (ν) is holomorphic and injective on I_{σ}(γ) in the domain

{ν ∈ 𝔞_{M,ℂ}^{∗} : Re(ν, α) > d(1 + ∥γ∥) for all α ∈ Σ(P′|P)}.

The last inequality of (1.1) then follows from Lemma A.5, Lemma A.6 and the relation

R_{P |P′}(σ, −s)R_{P′|P} (σ, s) = id .

It remains to prove Lemma A.6. Clearly we may assume, by passing to the derived group, that G is semisimple. We first need some more notation. Let P_{0} = ^{0}M_{0}A_{0}N_{0} be a minimal parabolic subgroup of G, contained in P, so that ^{0}M_{0} is compact. Let 𝔱 be a maximal abelian subalgebra of ^{0}𝔪 and let 𝔥 = 𝔱 ⊕ 𝔞_{0} (a direct sum with respect to the Killing form). Then 𝔥_{ℂ} is a Cartan subalgebra of 𝔤_{ℂ} and the real vector space 𝔥_{R} spanned by the co-roots is i𝔱 + 𝔞_{0} ([Wal1, 2.2.5]). The Weyl group W = W (𝔤_{ℂ}, 𝔥_{ℂ}) acts on 𝔥_{R} as well as on 𝔥_{ℂ}^{∗}. We identify the characters of the center of the universal enveloping algebra of 𝔤_{ℂ} as W - orbits of 𝔥_{ℂ}^{∗} via the Harish-Chandra isomorphism. A similar discussion applies to M. We denote by χ_{σ} the infinitesimal character of σ.

For any (finite dimensional) irreducible representation σ′ of ^{0}M_{0} and μ ∈ 𝔞_{0,ℂ}^{∗} we denote by π_{σ′,μ} = π_{σ′,μ}^{G} the corresponding principal series representation on G. Its infinitesimal character is (the W -orbit) of χ_{σ′} +μ where χ_{σ′} ∈ i𝔱^{∗} (the infinitesimal character of σ′) is the translate of the highest weight of σ′ by the half-sum of positive roots in 𝔪_{0}. Lemma A.7. There exists a constant c depending only on G such that any unitary representation σ of M can be embedded (infinitesimally) as a subrepresentation of a (non-unitary) π_{σ′,μ}^{M} and ∥Re μ∥ ≤ c(1 + ∥γ∥) whenever Hom_{K_{M}} (γ, σ) ≠ 0.

Proof. Suppose first that σ is square-integrable. Using the Casselman subrepresentation Theorem (e.g. [Wal1, Ch. 4] or [Kn, Theorem 8.37]) we may embed σ in some principal series π_{σ′,μ}^{M}. By comparing infinitesimal characters we infer that μ ∈ 𝔞_{0}^{∗} and

∥χ_{σ}∥^{2} = ∥μ∥^{2} + ∥χ_{σ′}∥^{2} ≥ ∥μ∥^{2}.

On the other hand by [Wa2, p.398], (cf. [Wal2, p. 258]) the square of the norm of any K-type of σ is bounded below, up to a fixed additive constant, by ∥χ_{σ}∥^{2}. The lemma follows in this case.

To treat the general case we use the Langlands classification Theorem to imbed σ in S(τ, λ) where Q is a parabolic subgroup with Levi subgroup L, τ is a square- integrable representation of L, λ is in the closed negative Weyl chamber of 𝔞_{Q}^{∗} and S(τ, λ) is the corresponding induced representation. As in the proof of Lemma A.5 we have ∥λ∥ < C independently of σ. All K-types of S(τ, λ) (and hence, of σ) contain a K_{L}-type of τ in their restriction to K_{L}. Hence, by induction in stages, the Lemma reduces to the square integrable case. □

We will now reduce Lemma A.6 to the case where P is a minimal parabolic of G.

Imbed σ in π_{σ′,μ}^{M} as in the Lemma and suppose that Re(μ + ν, β) > 0 for all β ∈ Σ(P_{0}N_{P′}|P_{0}N_{P}). Then J_{P′|P} (π_{σ′,μ}, ν) can be identified with J_{P_{0}N_{P′}|P_{0}N_{P}} (σ′, μ + ν) and it is given by an absolutely convergent integral. Its restriction to I_{σ}^{∞} is J_{P′|P} (σ, ν). Thus, in that region the injectivity of J_{P′|P} (σ, ν) on I_{σ}(γ) follows from that of J_{P_{0}N_{P′}|P_{0}N_{P}} (σ′, μ+ ν). We note that the restriction to A_{M} defines a bijection α ↔ α′ between Σ(P_{0}N_{P′}|P_{0}N_{P}) and Σ(P′|P), and we have (ν, α) = (ν, α′). The reduction follows.

By factoring J_{P′|P} as a product of “basic” intertwining operators we may also assume that P′ is adjacent to P. Let Q = LV be the parabolic subgroup generated by P and P′. Then L has rank one and it follows from the argument of [Wal2, 10.4.5] that J_{P′|P} (σ, γ) is injective on I_{σ}(γ) if and only if J_{P′∩L|P ∩L}^{L}(σ, ν^{L}) is injective on I_{σ}^{L}(γ′) for all γ′ ∈ K̂_{L} which occur in the restriction of γ. We observe that ∥γ′∥ ≤ ∥γ∥ for such γ′. Hence, we reduce to the case where G is of rank one, P is minimal and P′ = P̄. Once again we can assume that G is semisimple as well. From now on we assume that this is the case.

For Re(s) > 0 the representation π_{σ,sα} is of finite length and its Langlands quotient is given by the image of J_{P′|P} (σ, sα). Thus, J_{P′|P} (σ, sα) is not injective on I_{σ}(γ) if and only if γ occurs in one of the subquotients of π_{σ,sα} other than the Langlands quotient. Assume that this is the case and let π′ be any such subquotient. Then by [Wal1, Corollary 5.5.3] the Langlands parameter of π′ is smaller than that of π. Thus, either π′ is square-integrable or π′ can be imbedded in π_{σ′,s′} with 0 ≤ Re(s′) < Re(s). In the first case, the infinitesimal character of π′ is in 𝔥_{R}^{∗}, i.e., s ∈ ℝ, and by ([Wa2, p. 398])

C + ∥γ∥^{2} ≥ ∥χ_{π′}∥^{2} = ∥χ_{σ}∥^{2} + s^{2}∥α∥^{2} ≥ s^{2}∥α∥^{2}

for a certain constant C. It follows that s is bounded by a constant multiple of ∥γ∥. In the second case, we have

(1.3)  χ_{σ′} + s′α = w(χ_{σ} + sα)

for some w ∈ W. Write wα = ξα + β with ξ ∈ ℝ and β ∈ i𝔱^{∗}. If β = 0 then ξ = ±1, w stabilizes i𝔱^{∗} and we obtain s = ±s′ – a contradiction. Thus, β ≠ 0. Projecting (1.3) onto i𝔱^{∗} we obtain

χ_{σ′} = (wχ_{σ})_{i𝔱^{∗}} + sβ.

On the other hand, since γ occurs π_{σ,sα}, σ occurs in the restriction of γ to ^{0}M = K_{M} and hence ∥σ∥ ≤ ∥γ∥. Similarly, ∥σ′∥ ≤ ∥γ∥. Once again, it follows that |s| is bounded by a constant multiple of 1 + ∥γ∥. This concludes the proof of Lemma A.6.

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

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

Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, NY 14853-4201, U.S.A.

E-mail address: speh@math.cornell.edu

Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel

E-mail address: erezla@math.huji.ac.il