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Acknowledgment

I thank more than warmly Raphael Beuzart-Plessis for his constant help. I thank also warmly Jean-Loup Waldspurger for numerous comments on a previous version of this work. I thank also warmly Pascale Harinck for her careful reading and suggestions to improve the writing.

2 Notation

We introduce the notation for functions f, g defined on a set X with values in R`:

fpxq ăă gpxq, x P X if there exists C ą 0 such that fpxq ă Cgpxq, x P X. We will denote this also f ăă g. Let us denote fpxq„ gpxq, x P X if and only iff f ăă g and g ăă f. If moreover f and g take values greater or equal to 1, we write:

fpxq« gpxq if there exists N ą 0 such that

gpxq1{N ăă fpxq ăă gpxqN , x P X.

Let F be a number field and A its adele ring. If G is an algebraic group defined over F , we denote its unipotent radical by NG . Then the real vector space aG is 1 defined as usual, as well as the canonical morphism HG : GpAqÑ aG. Let GpAq be its kernel. From now on we assume that G is reductive. Let P0 be a parabolic subgroup of G

2 defined over F and minimal for this property. Let M0 be a Levi subgroup of P0. We have the notion of standard and semi-standard parabolic subgroup of G. Let K be a good maximal compact subgroup of GpAq in good position with respect to M0. If P is a semistandard parabolic subgroup of G we extend the map HP to a map

HP : GpAqÑ aP in such a way that HP ppkq“ HP ppq for p P P pAq,k P K. We have a Levi decomposition P “ MP NP . Let AP be the maximal split torus of the center of MP and A0 “ AM0 . Let GQ be the restriction of scalar from F to Q of G. We denote by AP pRq the 8 group of real points of the maximal split torus of the center of MP,Q and AP the neutral component of this real Lie group. The map HP induces an isomorphism 8 between the neutral component AP to aP . The inverse map will be denoted exp or expP . We define : rGsP “ MP pF qNP pAqzGpAq, rGs“rGsG.

The map HP goes down to a map rGsP Ñ aP . 1 1 8 The inverse image of 0 by this map is denoted rGsP and one has rGsP “rGsP AP . If P Ă Q are semistandard parabolic subgroups of G, we have the usual decompo- sition Q aP “ aP ‘ aQ. ˚ Q This allows to view elements of aQ as linear forms on aP which are zero on aP . Q Q,˚ The adjoint action of MP on the Lie algebra of MQ X NP determines ρP P aP . If

Q “ G we omit Q of the notation and we denote ρ for ρP0 . P If P is a standard parabolic subgroup of G, let ∆0 be the set of simple roots of A0 ˚ P in MP X P0 and ∆P Ă aP the set of restriction to aP of the elements of ∆P0 z∆0 . If Q Q P Q Ă P , one defines also ∆P as the set of restrictions to aP of elements of ∆0 z∆0 . ˇ Q Q One has also the set of simple coroots ∆P Ă aP . By duality we get simple weights ˆ Q Q ∆P denoted ̟α,α P ∆P . If Q “ G we omit Q of the notation. ` ` `` We denote by a0 the closed Weyl chamber and aP (resp. aP ) the set of X P aP such that αpXqě 0 (resp. ą 0 ) for α P ∆P . ˚ If P is a standard parabolic subgroup of G, we say that ν P aP,C is subunitary (resp. Reν X ,X a` Reν x α x strictly subunitary) if p qď 0 P P (resp. if “ αP∆P α where α ă 0 ˚ P for all α). If ν P aP , it is viewed as a linear form on a0 whichř is zero on a0 . Then ˚ ν P aP,C is subunitary if and only if one has:

` ReνpXqď 0,X P a0 (2.1)

` This follows from the fact α P ∆P is proportional to β Σ`,β α β where Σ0 is the P 0 |aP “ set of positive roots. In fact this sum is invariant byř the Weyl group generated by the reflections around roots which are 0 on aP . Let W be the Weyl group of pG, A0q. If P, Q are standard parabolic subgroups of

3 G, let W pQzG{P q be a set of representatives of QzG{P in W of minimal length. ´1 If s P W pQzG{P q the subgroup MP X s MQw is the Levi subgroup of a parabolic ´1 subgroup Ps contained in P and MQ X sMP s is the Levi subgroup of a parabolic subgroup Qw contained in Q. Let W pP, Qq be the set of w P W pQzG{P q such that ´1 wMP w Ă MQ. Let W pP |Qq be the set of w P W pQzG{P q such that wpMP q“ MQ. Hence: W pP |Qq“ W pP, QqX W pQ, P q´1.

By a Siegel domain sP for rGsP , we mean a subset of GpAq of the form:

P sP “ Ω0texpX|X P a0,αpX ` T qě 0,α P ∆0 uK (2.2)

1 where Ω0 is a compact of P0pAq , T P a0, such that GpAq“ MP pF qNP pAqsP . Let

`,P P a0 :“ tX P a0|αpXqě 0,α P ∆0 u.

Let us show

Any Siegel set s is contained in Ω texpX|X P a`,P uΩ where Ω is a P NP 0 (2.3) compact subset of GpAq and ΩNP is a compact subset of NP pAq.

8 There is a compact subset, in fact reduced to a single element, expT , Ω1 Ă A0 such that P `,P texpX|X P a0,αpX ` T qě 0,α P ∆0 u Ă texp X|X P a0 uΩ1.

The compact set Ω0 is a subset of ΩNP ΩP0XMP , where ΩNP (resp. ΩP0XMP ) is a compact subset of NP pAq (resp. pP0 X MP qpAq). Then the conjugate by exp ´ X, `,P `,P X P a0 of ΩP0XMP remains in a compact set, Ω2, when X varies in a0 . The compact subset Ω “ Ω2Ω1K satisfies the required property. Let P be a standard parabolic subgroup of G. If f is a function on GpAq with values ` in R , we denote frGsP the function on rGsP defined by

frGsP pgq“ infγPMP pF qNP pAqfpγgq,g P GpAq. (2.4)

We fix a height }.} on GpAq (cf. [20], I.2.2). From [8], Proposition A.1.1 (viii), one has }g}„}g}rGs,g P sG (2.5) Let us define: σpgq“ 1 ` logp}g}q,g P GpAq. We have

}mk}rGsP «}m}rMs. From this and (2.5), one deduces:

σrGsP pgq„ σpgq„ 1 `}H0pgq},g P sP . (2.6)

4 Let g P sP . From (2.3), one can write it g “ ω0expXω with ω0 P ΩNP ,ω P Ω,X P `,P a0 . Then taking into account

}g}«}expX} one has σpgq« σpexpXq and from (2.6): 1 `}H0pgq} „ 1 `}X},g P sP From this it follows:

σrGsP pgq„ 1 `}H0pgq} „ σpgq„ 1 `}X},g P sP . (2.7)

We normalize the measures as in [20], I.1.13. If X is a topological space, let CpXq be the space of complex valued continuous functions on X. Let Ω be a compact subset of GrAs and rΩs its image in rGs. Then, as Ω can be covered by a finite number of open sets on which the projection to rGs is injective, one has:

fpxqdx ăă fpGpF qgqdg,f P CprΩsq, f ě 0. (2.8) żrΩs żΩ

Let B be a symmetric bounded neighborhood of 1 in GpAq (a ball) and let ΞrGsP pxq“ ´1{2 pvolrGsP xBq . The equivalence class of the function does not depend of the choice of (2.9) B. We have (cf. [9] section 2.4): 2.1 Lemma. rGsP ρpH0pgqq Ξ pgq„ e ,g P sP .

Let G8 be the product of GpFvq where v describes the archimedean places and let Upg8q be the enveloping algebra of the Lie algebra g8 of this real Lie group. We have similar definition for subgroups of G defined over F . One has the Schwartz space CprGsq, denoted SprGsq in [19], corollary 2.6. From Lemma 2.1 and (2.7), it can be defined (see [19], section 1.1) as the space of functions 8 φ in C prGsq such that for all n P N and u P Upg8q:

´n rGs |pRuφqpxq| ăă σrGspxqΞ pxq, x PrGs.

5 3 Tempered automorphic forms

3.1 Definition of temperedness The space of automorphic forms on rGs “ GpF qzGpAq, ApGq is defined as in [20], I.2.17. In particular they are K-finite. If φ P ApGq, it has uniform moderate growth on GpAq (cf. lc. end of I.2.17, Lemma I.2.17 and Lemma I.2.5). It means that there exists r ą 0 such that for all u P Upg8q:

r |Ruφpgq| ăă }g}rGs,g P GpAq.

Let Ω be a compact subset of GpAq. Then one sees easily that this implies that for all u P Upg8q, one has

r |RuRωφpgq| ăă }g} ,g P GpAq,ω P Ω. (3.1)

If P is a standard parabolic subgroup of G, the space of automorphic forms on rGsP “ NP pAqMpF qzGpAq denoted ApNP pAqMpF qzGpAqq in l.c. will be denoted AP pGq. The constant term along P (cf. [20], I.2.6), φP , of an element φ of ApGq is an element of AP pGq. Similarly if Q is a standard parabolic subgroup contained in A A An P and φ P P pGq, φQ is a well defined element of QpGq. Let P pGq be the space of elements of φ P AP pGq such that:

ρP pXq φpexpXgq“ e φpgq,g P GpAq,X P aP .

A ˚ If φ P P pGq, and λ P aP,C we define

λpHP pgqq φλpgq“ e φpgq,g P GpAq.

˚ ˚ An We view SpaP q as the space of polynomial functions on aP and SpaP qb P pGq as a space of functions on GpAq by setting pp b φqpgq“ ppHP pgqqφpgq. If φ P AP pGq, one can write uniquely:

λpHP pgqq φpgq“ e pφ0,λqpgq, (3.2)

λPÿEP pφq

E ˚ ˚ An E where P pφqĂ aP,C, φ0,λ is a non zero element of SpaP qb P pGq. The set P pφq is A2 called the set of exponents of φ. We define also P pGq as the subspace of elements An of P pGq such that:

2 2 ´2ρP pHP pmqq }φ}P “ |φpmkq| e dmdk. ż ż 8 K AP MP pF qzMP pAq

6 We will need the following variant of [20], Lemma I.2.10:

a) The statement of [20], Lemma I.2.10 remains true if one changes P to a standard parabolic subgroup and, in the conclusion, one changes

P α to βP :“ infαP∆0z∆0 α. λ λ b) If one replaces mP0 pgq in the hypothesis by mP0 pgqp1 ` (3.3) n λ´tα }logpmP0 pgqq}q for some n, one can replace mP0 pgq by λ´tβP n mP0 pgq p1 `}logpmP0 pgqq}q in the conclusion. r r c) One can also replace in the statement }a} by p1`}loga}q for a P AG.

To prove a) one has simply to replace α by βP in (1) of the proof. n To prove b) one has to use that p1 `}plogmP0 pgqq}q is U0pAq invariant after (6) in the proof. (c) is obtained by replacing }a}r by p1 `}loga}qr in the proof.

3.1 Lemma. Let P be a standard parabolic subgroup of G. Let d ą 0. Let φ P AP pGq. The following conditions are equivalent: a) rGsP d |φpxq| ăă Ξ pxqσrGsP pxq , x PrGsP . b) For all Siegel domains sP , one has:

ρpH0pgqq d |φpgq| ăă e p1 `}H0pgq}q ,g P sP . c) For every compact subset Ω of GpAq, one has:

ρpXq d `,P |φpexpXωq| ăă e p1 `}X}q ,ω P Ω,X P a0 .

Proof. a) is equivalent to b) follows from Lemma 2.1 and (2.7). To prove c) implies b), we choose (cf. (2.3)), a compact subset Ω of GpAq and 8,`,P 8,`,P a compact subset, ΩNP of NP pAq such that sP Ă ΩNP A0 Ω, where A0 “ `,P texpX|X P a0 u. One has for g P sP :

`,P }X ´ H0pgq} ăă 1,g “ ωNP expXω,X P a0 ,ωNP P ΩNP ,ω P Ω. (3.4)

Then c) implies b) follows. Similarly b) implies c).

3.2 Definition. Let us define the space of tempered automorphic forms on rGsP , Atemp P pGq, as the space of automorphic forms satisfying, as well as its derivatives by elements of Upg8q, the equivalent properties a), b), c) of the preceding Lemma for some d.

3.3 Remark. This notion was introduced by J. Franke in [15] (cf. also [21], section 4.4 where the space of tempered form is denote AlogpGq).

7 3.4 Lemma. Let d ą 0. if φ P ApGq is such that all its derivatives by elements of 2 ´d Upg8q are in L prGs, σrGsdxq then φ is tempered. Proof. Let us use the notation of [6], Lemma-Definition 3.3. From the proof of this Lemma, one can take dmX pxq“ νpxqdx where dx is the GpAq-invariant measure on X “rGs and ν “pΞrGsq2, as it follows from the proof of l.c. Lemma-Definition 3.3 (ii). Let k ě dimG and f be a continuously k-times differentiable function on GpAq. Fix k a basis d1,...,dr of the space Upg8q of elements of Upg8q of degree ď k and define:

2 Qpfq“ |dif| . ÿi

k,J Let J be a compact open subgroup of GpAf q. Let CprGsq be the space of contin- uously k-times differentiable and fixed by J. One has from the Key Lemma of l.c., p. 686:

2 k,J |fpxq| ăă QpfqdmX “ Qpfqνpyqdy,x PrGs, f P CprGsq żxB żxB

d Let w “ σrGs. Then

|fpxq|2 ăă Qpfqνww´1dy,x PrGs. żxB We use now that w is a weight, in the sense of [6], Definition 3.1, as well as ν to get:

|fpxq|2 ăă νpxqwpxq Qpfqw´1dx ď νpxqwpxq Qpfqw´1dx, x PrGs. żxB żrGs

´1 As our hypothesis implies that rGs Qpfqw dx ă 8, this finishes the proof of the Lemma. ş

3.2 Characterization of temperedness and definition of the weak constant term A Atemp 3.5 Proposition. Let φ P pGq. It is in G if and only if the exponents of its constant term φP along any standard parabolic subgroup P of G are subunitary. Proof. Let us prove Let us show that the condition is necessary. Let φ P AtemppGq. Let X P aG. Then there exists d such that for all g PrGs:

|φpgexptXq| ăă p1 ` tqd, t ą 0.

It follows from [12], Proposition A.2.1, that the exponents of φ restricted to aG are `` ` unitary. Let P be a standard parabolic subgroup of G. Let X P aP XĂ a0 , where

8 `` ` aP is the interior of aP . Let g P GpAq, t P R. From the property c) of the definition of temperedness, applied to Ω “ g, one gets:

|φpexptXgq| ăă eρptXqp1 ` tqd, t ą 0. (3.5)

G Due to (3.1), one can apply (3.3) to the right translate by g of φ. Write X “ XG`X G G with XG P aG,X P a . Applying (3.3) b) and c) for the parameter t from [20], G 1 Lemma I.2.10 large and with g of l.c. equals to exptX P sG X GpAq , a equals to exptXG, one gets, for k ą 0,

ρptXq ´kt |φpexptXgq´ φP pexptXgq| ăă e e , t ą 0.

Together with (3.5) this implies that the exponential polynomial in t, φP pexptXgq satisfies: ρptXq d |φP pexptXgq| ăă e p1 ` tq , t ą 0. `` There is a dense open set O in aP such that different exponents of φP take different values on any element of O. We use the notation of (3.2). Then [12], Proposition A.2.1 gives: If φP,0,λpgq ­“ 0: ReλpXqď 0,λ P EP pφq,X P O.

As φP,0,λ is not identically zero, one has ReλpXq ď 0 for X P O, hence also for ` X P aP by density. This achieves the proof of (i). The sufficiency of the condition follows from [20], Lemma I.4.I.

3.6 Definition. Let φ P AtemppGq and let P be a standard parabolic subgroup of w G. We define the weak constant term of φ, denoted φP as the sum of the terms Atemp in (3.2) corresponding to unitary exponents. It is an element of P pGq from the preceding Proposition applied to MP (see below Lemma 3.7 for a detailed proof). Let ´ w E w E ´ w φP “ φP ´ φP and let us denote P pφq (resp. P pφq) the exponents of φP (resp. ´ φP ).

3.3 Transitivity of the weak constant term Let Q Ă P be standard parabolic subgroups of G. If φ is a function on GpAq and k P K, we define a function on MQpAq by:

k,MQ ´ρQpHQpmQqq φ pmQq“ e φpmQkq,k P K where ρQ P aQ is the restriction of ρ to aQ, which can be extended to a0 by zero on Q a0 . One has the following immediate properties, by coming back to the definitions: If φ P ApGq one has: (3.6) MQ,k MP ,k MQ,1 φQ “pφP qQ, pφQq “ rppφP q qQXMP s ,k P K

9 Notice that the function in bracket is a function on MP pAq, hence the upper index M ,1 ´ρ pH pmqq Q indicates that we multiply by e QXMP Q the restriction of this function to MQpAq. 3.7 Lemma. Let Q Ă P be as above. Atemp w (i) If φ P P pGq the exponents of φQ are subunitary and one can define φQ as the sum of the terms of φQ corresponding to unitary exponents. If φ P AtemppGq one has: w Atemp (ii) φP is in P pGq. (iii) w w w φQ “pφP qQ. (iv) w MQ,k w MP ,k w MQ,1 pφQq “ rppφP q qQXMP s ,k P K Proof. (i) is proved as Proposition 3.5 (i). (ii) From (3.6) and Proposition 3.5 applied to φ, one sees that the exponents w k,MP of pφP q are subunitary. Hence by this Proposition applied to MP , one sees: w,MP ,k Atemp φP P pMP q. Let ΩMP be a compact subset of MP pAq. Using K-finiteness, this gives that there exists d P N such that:

ρpXq d `,P |φP pexpXωkq| ăă e p1 `}X}q ,X P a0 ,ω P ΩMP ,k P K.

1 Every compact subset of GpAq is contained in a set of the form NP pAqΩ K where 1 Atemp Ω is a compact subset of MP pAq. Hence, recalling the definition of P pGq (cf. Definition 3.2), the preceding estimate achieves the proof of (ii). w ´ ´ (iii) Write φP “ φP ` φP . Then none of the exponents of pφP qQ is unitary. Hence as φQ “pφP qQ we get (iii). (iv) follows from the second assertion of (3.6).

3.4 A characterization of elements of square integrable au- tomorphic forms

Let us recall some facts from [6]. With the notation there, one can take dmX pxq“ pΞrGsqpxq2dx where dx is the GpAq-invariant measure on X “rGs, as it follows from the proof of l.c. Lemma 3.3(ii). Also rGs is of polynomial growth of rank dG (cf. the definition of this notion in l.c. p.689) equal to the split rank of G (cf. l.c. Example 1, p.698). Then taking into account what follows the definition of polynomial growth in l.c. and the criterion p.685, one gets:

´dG 2 p1 ` σrGspxqq Ξpxq dx ă8. (3.7) żrGs It follows from [20], Lemmas I.4.1 and I.4.11 that: A2pGqĂ CpGq.

10 From this and the definition of temperedness above, one has:

temp 2 For all φ P A pGq and ψ P A pGq, for all X P aG, the integral

φpg1expXqψppg1expXqdg1 żrGs1 (3.8) X X is absolutely convergent. It is denoted pφ, ψqG . Moreover X ÞÑ pφ, ψqG is an exponential polynomial in X. One defines similarly for φ P A2 Atemp P pGq, ψ P P pGq, an exponential polynomial, pP pφ, ψq on aP by X 1 pP pφ, ψqpXq“pφ, ψqP , X P aP , using integration on rGsP . We denote by Atemp,cpGq the space of ψ P AtemppGq such that for all φ P A2pGq, the Atemp,c polynomial pP pφ, ψq is zero. We define similarly P pGq. Then one has a direct Atemp,c A2 Atemp,c A2 sum: P pGq‘ P pGq and one can define, for φ “ φ1 ` φ2 P P pGq‘ P pGq Atemp and ψ P P pGq, an exponental polynomial denoted pP pφ, ψq equal to pP pφ2, ψq. With these definitions one has: Atemp 3.8 Lemma. (i) Let Q be a standard parabolic subgroup of G and let φ P Q pGqX An w QpGq such that φP “ 0 for any standard parabolic subgroup of G with P Ă Q, A2 Q ­“ P , P standard then φ P QpGq. Atemp w Atemp,c (ii) If φ P pGq and φQ P P pGq for all standard parabolic subgroup Q of G, then φ “ 0.

Proof. (i) We first prove the result for Q “ G. Let us show that for any standard parabolic subgroup of G, P ­“ G, the exponents of φP are strictly subunitary. Let ν be such an exponent. From the hypothesis it is subunitary but not unitary. If it is not strictly subunitary, there exists α P ∆P such that:

Reν “ xββ, xβ ď 0.

βP∆ÿP ztαu

Let Q be the maximal parabolic subgroup of G, containing P , such that ∆Q “ α|aQ . w Then ν|aQ is an exponent of φQ. But it is clear that it is unitary, hence φQ is non zero which contradicts our hypothesis. Hence ν is strictly subunitary. Then (i) for Q “ G follows from Lemma I.4.11 of [20]. Then the statement of (i) follows from Lemma 3.7 (iv) and what we have just proved for MQ instead of G. G Let us prove (ii) by induction on the dimension on a0 . If it is zero the claim is clear. G Suppose now dima0 ą 0. By applying the induction hypothesis to MP for a strict w standard parabolic subgroup P of G and Lemma 3.7, one sees that φP “ 0. Hence by (i), φ P A2pGq. As φ is in P Atemp,cpGq, one deduces from this that φ “ 0.

11 4 Uniform temperedness of Eisenstein series

4.1 Exponents of Eisenstein series A2 Let P be a standard parabolic subgroup of G. Let φ P P pGq. Let EP p., φ, λq be the Eisenstein series (cf. [7], (1.1)). Let P Ă Q be two standard parabolic A2 subgroups of G and w P W pP |Qq. One has the operators Mpw,λq : P pGq Ñ A2 ˚ QpGq meromorphivc in λ P aP,C ( cf. l.c. section 1.1, after (1.1). Q A One can define EP p., φ, λq which is, when defined, in QpGq and is characterized by: Q MQ,k MQ,k EP p., φ, λq “ EP XMQ p., φ ,λq,k P K. (4.1) They are analytic on the imaginary axis (cf. [7] Remark 1.3). We recall the formula for generic λ (cf. [18], Proposition 4, with the notation there)

Q pEP p., φ, λqqQ “ EQs p., Mps,λqφPs ,sλq. (4.2) sPW pÿQzG{P q

The exponents of EP p., φ, λqQ are given by [18], equation (13). Moreover they are subunitary by l.c. Lemma 6 for λ unitary. Hence by Proposition 3.5, EP p., φ, λq is A2 ˚ tempered for φ P P pGq,λ P iaP and the weak constant term is given by:

w Q pEP p., φ, λqqQ “ EQs p., Mps,λqφ,λq (4.3) sPWÿpP,Qq

˚ ´ which is holomorphic in a neighborhood of iaP . The same is true for EQ p., φ, λq“ w EQp., φ, λq´ EQp., φ, λq whose exponents are contained in E ´ E Q pλq“YsPW pQz G{P qzW pP,Qqtsp Ps pφq` λq|aQ u

˚ Hence, by analyticity, this inclusion holds for all λ in iaP . This implies: ˚ ´ For λ P iaP , X ÞÑ EQ pexpXgq,X P aQ is an exponential polynomial E˜´ E˜´ E ´ with exponents in a multiset Q pλq, where Q pλq is built from Q pλq by some repetitions, the multiplicities depending on the multiplicities (4.4) of the exponents of φ. Moreover the real parts of the exponents above ˚ do not depend on λ P iaP . 4.2 Uniform temperedness of Eisenstein series

˚ Let Λ be a compact subset of iaP . G,` Let µ P a0 , n P N. Let FΛ,µ,n be the space of functions F on GpAqˆΛ which satisfy for every compact subset Ω of GpAq and u P Upg8q:

n µpXq ` (4.5) |RuF pexpXω,λq| ăă p1 `}X}q e ,X P a0 ,ω P Ω,λ P Λ,

λpXq F pexpXg,λq“ e F pg,λq,X P aG,g P GpAq,λ P Λ.

12 4.1 Proposition. Let E : rGsˆ Λ Ñ C be defined by: Epg,λq“ EP pg,φ,λq. Then there exists n P N such that E P FΛ,ρ,n Proof. We will need the fact that lemma 1.4.1 of [20] holds uniformly for a set of automorphic forms which is bounded in a space of functions with given moderate growth and whose constant terms uniformly satisfy the assumption of that lemma. This is easy to see from the proof given in [20]. This applies to Eisenstein series from [7], Corollary 6.5, from the holomorphy of Eisenstein series on the imaginary axis and from (4.1), (4.2).

Let us show:

Atemp,c A2 ˚ If P ­“ G, EP p., φ, λq P pGq for all φ P P pGq,λ P iaP . (4.6)

Let X P aG and us look to

˚ Ipλq“ EpxexpX, φ, λqψpxexpXqdx,λ P iaP żrGs1 for ψ P A2pGq. From the uniform temperedness of Eisenstein series, it is a continous function in λ. Let z be an element of the center Zprg8, g8sq of the envelopping ˚ algebra of rg8, g8s. Let us assume that z is in the cofinite dimensional ideal of Zprg8, g8sq which annihilates ψ. One can assume that φ is Zpg8q eigen and let pz ˚ ˚ be the polynomial on iaP such that RzEP p., φ, λq“ pzpλqEP p., φ, λq,λ P iaP . Then on one hand:

˚ RzEpxexpX, φ, λqψpxexpXqdx,λ P iaP “ pzpλqIpλq żrGs1 and on the other hand, taking adjoint, this integral is zero. Moreover by the cofinite dimension of the annihilator in Zprg8, g8sq of ψ, there exists z as above such that pz in non identically zero. Then it follows, by continuity and density, that Ipλq is identically zero. This proves our claim.

5 Wave packets

5.1 Difference of a tempered automorphic form with its weak constant term Let Q be a parabolic subgroup of G. For δ ą 0, we define:

G,` G,` aQ,δ “ tX P aQ |αpXqě δ}X},α P ∆Qu.

13 5.1 Lemma. Let φ P AtemppGq. Let Ω be a compact subset of G. Let δ ą 0. Then there exists ε ą 0 such that:

w d ρpXq ρpY q´ε}Y } |pφ ´ φQq|pnQexpXexpY ωqăăp1 `}X}q e e ,

` G,` nQ P NQpAq,X P a0 ,Y P aQ,δ ,ω P Ω.

Proof. Using that conjugation by exp ´ X and exp ´ Y contracts NQpAq and as NQpF qzNQpAq is compact, possibly changing Ω, one is reduced to prove a similar claim, but without nQ. G,` Let SQ,δ be the intersection of the unit sphere of aQ with aQ,δ . It is compact. Let us look to the family of exponential polynomials in t P R:

w ` pY,X,ωptq :“ φQpexpXexptY ωq´ φQpexpXexptY ωq,X P a0 ,Y P SQ,δ,ω P Ω.

On one hand, from the definition of temperedness of φ and (3.3) b) and c), one gets that there exists d P N such that:

d ρpXq d tρpY q ` |φQpexpXexptY ωq| ăă p1 `}X}q e p1 ` tq e ,X P a0 ,Y P SQ,δ,ω P Ω, t ą 0.

w On the other hand the temperedness of φQ (cf. Lemma 3.7) and the definition of w w the temperedness of φQ implies a similar bound for φQpexpXexptY ωq. Hence by difference it follows that there exists d P N such that:

d ρpXq d tρpY q ` |pY,X,ωptq| ăă p1 `}X}q e p1 ` tq e ,X P a0 ,Y P SQ,δ,ω P Ω, t ą 0. (5.1)

Moreover the exponents of these exponential polynomials are equal to µpY q` ρpY q where µ is an exponent of φQ which is not imaginary. Hence its real part is equal to c α X Rec Rec αP∆Q α p q with α ď 0, with at least one α non zero. Hence there exists 1 1 εřą 0 such that for Y P SQ,δ, µpY qă´ε . 1 1 2 By applying (5.1) to Ω such that Ω contains texptY ||t|ă ε ,Y P SQ,δuΩ, one gets that the modulus of these polynomials restricted to the interval r´ε2,ε2s is bounded by a constant times p1 `}X}qdeρpXq. Applying Lemma 3 of [18] to the polynomials d ρpXq ´1 rp1 `}X}q e s pY,X,ω, one gets the required estimate.

Atemp G` G` 5.2 Lemma. Let φ P pGq. Let δ ą 0 and a0,Q,δ :“ tX P a0 |αpXqě δ}X},α P Q ∆P0 z∆P0 u. Let Ω be a compact subset of GpAq. Let φ P AtemppGq. There exists ε ą 0 such that

w ρpXq´ε}X} G` |φpnQexpXωq´ φQpnQexpXωq| ăă e , nQ P NQpAq,X P a0,Q,δ,ω P Ω.

G` G Proof. For X P a0,Q,δ, let Y be the element of aQ such that αpY q “ αpXq,α P Q G ∆P0 z∆P0 . Then, looking to coordinates in aQ, one sees that there exists δ1 ą 0 such ` G` 1 that δ1}Y } ď δ}X}. Hence Y P aQ,δ1 . Moreover as X P a0,Q,δ, X “ Y ´ X is in

14 G` 1 a0 . One gets the required estimate by using the preceding lemma with X instead of X as: }X1}ăă}X}`}Y }ăă}X} Q and if α P ∆P0 z∆P0 , δ}X}ď αpY qăă}Y }.

˚ A2 5.3 Lemma. Let Λ be a bounded subset of iaP and φ P P pGq. There exists ε ą 0 such that:

w ρpXq´ε}X} |EP pnQexpXω,φ,λq´ EP pnQexpXω,φ,λqQ| ăă e ,

G` nQ P NQpAq,X P a0,Q,δ,ω P Ω,λ P Λ. Proof. The proof is similar to the proof of the preceding lemma. One has to prove an analogous of Lemma 5.1 for Eisenstein series by using Proposition 4.1, that the real ˚ part of the exponents of EP p., φ, λqQ do not depend of λ P iaP and the expression of the weak constant term of Eisenstein series (cf. (4.3)).

5.2 Wave packets in the Schwartz space

˚ 5.4 Proposition. Let a be a smooth compactly supported function on iaP and φ P A2 P pGq. Then the wave packet

Ea :“ apλqEP p., φ, λqdλ ż ˚ iaP is in the Schwartz space CprGsq. 5.5 Remark. As already said in the introduction, this is due to Franke, [15], section 5.3, Proposition 2 (2). His proof rests on the main result of [3] for which Lapid in [18] has given a proof independent of [17]. We give below a more selfcontained proof.

G G Proof. We proceed by induction on the dimension of a0 . The case where dima0 “ 0 is immediate by classical Fourier analysis on aG: the classical Fourier transform of a compactly supported function on Rn is in the Schwartz space. G ` G Now we assume dima0 ą 0. Let S be the intersection of the unit sphere of a0 G,` ` with a0 . Let X0 in S . Let Q be the standard parabolic subgroup of G such that `` ` X0 P aQ . As X0 P S , Q is not equal to G. Let βQpXq :“ inf ∆ ∆Q αpXq,X P αP P0 z P0 ` a0. Then βQpX0qą 0. We choose a neighborhood S0 of X0 in S such that

βQpXqě βQpX0q{2,X P S0.

15 Let δ “ βQpX0q{2 . Then G,` S0 Ă a0,Q,δ. Let Λ be the support of a. We use the notation of Proposition 4.1. Let Ep., λq :“ w w EP p., φ, λq. Then Ep., λq is the sum of 2 terms: Ep., λq´ Ep., λqQ and Ep., λqQ. Let us show that, for all k P N one has:

´k ´k tρpXq | apλqF pexpXGexptXω, λqdλ|ăăp1 `}XG}q p1 ` tq e , ż ˚ (5.2) iaP t ą 0,XG P aG,X P S0,ω P Ω,λ P Λ,

w when F is any of these two families of functions. The case where F “ EQ follows from the induction hypothesis, using the formula for the weak constant term of Eisenstein series (cf. (4.3), (4.1)) and the fact that in this formula Mpw,λq is w analytic in λ (cf. [7], Remark 1.3). Let us treat the case where F “ E ´ EQ. One knows from Lemma 5.3 that there exists ε ą 0 such that:

w tρpXq´εt |EP pexpXGexptXω,φ,λq´ EP pexpXGexptXω,φ,λqQ| ăă e ,

XG P aG,X P S0,ω P Ω,λ P Λ. w By multiplying by a and integrating on iaP , we get (5.2) for F “ E ´ EQ and k “ 0. One applies this to successive partial derivatives of a with respect to elements of aG. w Then using that EQ transforms under aG by λ and applying integration by part one gets the result for all k. One can do the same for RuE, u P Upg8q. ` As a finite number of S0 covers S this achieves to prove the Proposition.

6 An isometry

We recall the statement of Theorem 2 of [18]. Let Pst be the set of standard parabolic subgroups of G. Let P be a standard parabolic subgroup of G. Let WP be the space of compactly supported smooth ˚ A2 functions on iaP taking values in a finite dimensional subspace of P . Write:

2 2 }φ}˚ “ }φpλq}P dλ. (6.1) ż ˚ iaP

For φ P WP , let

ΘP,φpgq“ EP pg,φpλq,λqdλ. ż ˚ iaP 2 8 Let LdiscpAM MpF qzMpAqq be the Hilbert sum of irreducible MpAq-subrepresentations 2 8 of L pAM MpF qzMpAqq. If P is a standard parabolic subgroup of G, let |PpMP q| be equal to the number of parabolic subgroups having MP as Levi subgroup. Consider the space L consisting

16 ˚ GpAq 2 8 of families of functions FP : iaP Ñ IndP pAqLdiscpAM MpF qzMpAqq where P describes the set of standard parabolic subgroups of G such that: 2 P ´1 2 }pFP q} “ | pMP q| }FP }˚ ă8 PÿPPst and ˚ FQpwλq“ Mpw,λqFP pλq,w P W pP |Qq,λ P iaP . (6.2) 1 Let L be the subspace of L consisting of those families such that FP P WP for all P . 6.1 Theorem. [18], Theorem 2 The map E from L1 to L2pGpF qzGpAqq

P ´1 pFP q ÞÑ | pMP q| ΘP,FP PÿPPst extends to an isometry E from L to L2pGpF qzGpAqq. 6.2 Lemma. We take the notation of Proposition 5.4. In particular P is fixed. Then Ea is in the image of E. Proof. For this one has to define a family in L1 whose image by E is a non zero ˚ A2 multiple of Ea. If ψ is a map on iaP with values in P pGq and s P W pP,P q, we define s.ψpλq :“ Mps,s´1λqψps´1λq. From the properties of composition of the intertwining operators, this defines an action of the group W pP |P q. Let

´1 ´1 FQpλq“ Mps,s λqψps λq. sPWÿpP |Qq

It is an easy consequence of the product formula for intertwining operators (cf. [7], Theorem 1.3 (4)) that FP is invariant by the action of W pP,P q and also that the 1 family pFQq satisfies (6.2). Moreover it is in L as the intertwining operators are analytic on the imaginary axis (cf. l.c. Remark 1.3). Then the functional equation for Eisenstein series (cf. lc. Theorem 1.3 (3)), implies that the image of pFQq by E is a non zero multiple of Ea.

7 Truncated inner product

If Q is a semistandard parabolic subgroup of G, let:

˚ θQpλq“ λpαˇq,λ P aQ,C. αźP∆Q

17 Q ˇ G Let LQ be the cocompact lattice of aG generated by ∆Q and let CQ “ volpaQ{LQq. 1 We fix a Siegel domain as in (2.2) associated to a compact set Ω0P0 pAq and to 0 0 0 1 1 T P a . We can choose Ω “ ΩN0 ΩM0 where ΩN0 (resp. ΩM0 ) is a compact subset 1 of N0pAq (resp.M0pAq ) such that

1 0 0 0 0 1 N pAq“ N pF qΩN0 , M pAq “ M pF qΩM0 . (7.1)

1 If C is a subset of a0 we define M0pCq “ tm P M0pAqX GpAq |H0pmq P Cu which 1 G is right invariant by M0pAq . We take T dominant and regular in a0 . We let ∆ dP0 pT q “ infαP P0 αpT q and if Q is a standard parabolic subgroup of G, TQ is the orthogonal projection of T on aQ. Q G If Q is a standard parabolic subgroup of G, we define the convex set CT of a0 by

Q G Q CT “ tX P a0 |αpX ´ T0qě 0,̟αpX ´ T qď 0,α P ∆0 , βpX ´ T qą 0, β P ∆Qu.

G Notice that CT is compact. MQ MQ G Let TMQ “ T ´ TQ, T0,MQ “ T0 ´ T0,G. Let us define C Ă a0 Ă a0 is defined TMQ G,`` G,`` with TMQ “ T ´TQ instead of T and T0,MQ instead of T0. Let aQ pT q“ TQ `aQ . We have: CQ “ C M ` aG,``pT q (7.2) T TMQ Q Q We define G G CT “ GpF qΩN0 M0pCT qK ĂrGs which is compact. Using (7.1), one has:

G G CT “ GpF qN0pAqM0pCT qK.

MQ Replacing N0 by N0 X MQ and G by MQ we define CT ĂrMQs by:

MQ MQ C “ MQpF qpN0 X MQqpAqM0pC qpK X MQpAqq. (7.3) TMQ T which is independent of the choice of Ω0. We define Q Q 1 CT “ QpF qN0pAqM0pCT qK Ă QpF qzGpAq . (7.4) Q Then CT is NQpAq invariant as

NQpAqQpF qN0pAq“ QpF qNQpAqN0pAq“ QpF qN0pAq.

As NQpAqN0pAq“ NQpAqpN0 X MQqpAq one has from (7.2):

Q G,`` MQ 1 C “ NQpAqexppa pT qqC K Ă QpF qzGpAq . (7.5) T Q TMQ

18 G We say that a strictly P0-dominant T P a0 is sufficiently regular if there exists a sufficiently large d ą 0 with dP0 pT qě d. We have the following result due to Arthur ([1], Lemma 6.4).

Let T be sufficiently regular. Q (i) For each standard parabolic subgroup Q of G, viewing CT as a subset of QpF qzGpAq, the projection to rGs is injective on this set. Its image (7.6) Q is still denoted CT . Q 1 (ii) The CT form a partition of rGs .

Q For a compactly supported function f on CT we have, using (7.5):

1 ´2ρQpXq 1 fpxqdx “ fpnQexpXmQkqe dnQdXdmQdk. ż Q ż G,`` MQ CT pNQpF qzNQpAqqˆaQ pT qˆCT ˆK MQ (7.7) as follows from the integration formula on GpAq related to the decomposition 1 G 1 1 1 GpAq “ NQpAqexpaQMQpAq K. Here dmQ is the measure on rMQs . Let G G ˆ a0,´ “ tX P a0 |ωpXqď 0,ω P ∆0u be the cone generated by the negative coroots and

G G ˆ G aQ,´pT q “ tX P aQ|ωpX ´ T qď 0,ω P ∆Qu“ TQ ` aQ X a0,´.

Let p be an exponential polynomial with unitary exponents on aQ and Z P aG. If ˚ µ P aQ,C has its real part strictly Q-dominant, the integral:

eµpX`ZqppXqdX ż G aQ,´pT q is convergent and has a meromorphic continuation in µ. When it is defined, its value ˚ in λ P iaQ is denoted: ˚ eλpXqppXqdX. ż G Z`aQ,´pT q A2 Atemp,c ˚ We use the notation following 3.8. We define for φ P QpGq‘ Q pGq, λ P aQ Atemp and Ψ P Q pGq, Z P aG:

˚ T Z λpXq rQpφλ, Ψq “ e pQpφ, ΨqpXqdX. (7.8) ż G Z`aQ,´pT q

Z Let Z P aG. If p is a polynomial on aQ, we define p the exponential polynomial on G aQ defined by: Z G p pXq“ ppX ` Zq,X P aQ

19 Z G,˚ and p pBq its Fourier transform viewed as a differential operator on aQ . Recall that CQ has been defined in the beginning of this section. One has:

T Z pλ´µqpTQ`Zq Z rQpφλ, Ψq “ CQ e rppφ, Ψ0,µq pBqθQspλ ´ µq, (7.9) µPEÿQpΨq

One can define rT pψ, Ψq where ψ is a linear combination of φλ. If Φ is a function 1 on GpAq and Z P aG, one defines a function on GpAq by:

ΦZ pg1q“ Φpg1expZq,g1 P GpAq1

Atemp A2 7.1 Theorem. Let Φ be an element of pGq and φ P P pGq. We denote by Ep., λq the function EP p., φ, λq. Let:

T Z Z Z ΩP0 pEpλq, Φq :“ Epx, λq Φ pxqdx, ż G CT

T Z T w w Z ωP0 pEpλq, Φq :“ rQpEpλqQ, ΦQq . QÿPPst Hc ˚ T (i) Let be the subset of λ P iaP where the summands of ωP0 pEpλ, Φq are analytic for all Z. From (7.9), this set contains the complementary set of a finite union of T Z Hc hyperplanes. The function ωP0 pEpλq, Φq on extends to an analytic function on ˚ iaP denoted in the same way. ˚ (ii) Let δ ą 0. Let Λ be a bounded set of iaP . There exists k P N, ε ą 0 such that the difference

T Z T Z T Z ∆P0 pEpλq, Φq :“ ΩP0 pEpλq, Φq ´ ωP0 pEpλq, Φq

´ε}T } k is an Ope p1 `}Z} q for λ P Λ, for T such that dP0 pT qě δ}T }, Z P aG.

G G The proof is by induction on dima0 . The statement is clear for dima0 “ 0. We 1 G1 G suppose that the Theorem is true for all groups G with dima0 ă dima0 .

c 7.2 Lemma. Let k0, δ ą 0. Then if Λ is a bounded subset of H , there exists C ą 0, k P N, ε ą 0 such that

T `S Z T Z ´ε}T } k |∆P0 pEpλq, Φq ´ ∆P0 pEpλq, Φq |ď Ce p1 `}Z}q , for λ P Λ,Z P aG, for T,S strictly P0-dominant such that dP0 pT q ě δ}T }, }S} ď k0}T }, }T0}ď}T }.

Proof. Let us define: Q G Q CT `S,T “ CT `S X CT . (7.10) and Q Q CT `S,T “ GpF qN0pAqM0pCT `S,T qK

20 G From (7.6), these subsets of rGs are disjoints. Moreover from (7.6), they cover CT `S. Let us show that, for T,S as in the Lemma, there exists δ1 ą 0 such that: Q Q αpXqě δ1}X},X P CT `S,T ,α P ∆0z∆0 (7.11)

Q Q Q Let X P CT `S,T and ∆0z∆0 . The definition of CT shows in particular that X “ 1 1 ` Q T ´ X ` Y where X “ Q d βˇ with d ą 0 and Y P a . Let α P ∆0z∆0 . Since βP∆0 β β Q Qř αpβˇqď 0 for each β P ∆0 , from the properties of simple roots, one has αpXqě αpT qě δ}T }. (7.12) Let us show: G }X ´ T0}ď}T ` S},X P CT `S. (7.13) G 1 1 As X P CT `S, X ´ T0 “ pT ` Sq ´ Y where Y is a linear combination with coefficients greater or equal to zero of coroots. Moreover T ` S and X ´ T0 are in ` a0 . Hence

pX ´ T0,X ´ T0qďpX ´ T0, T ` SqďpT ` S, T ` Sq which proves our claim. Hence

}X}ď}T }`}S}`}T0}ďp2 ` k0q}T } and αpXqě δ1}X} ´1 where δ1 “p2 ` k0q δ. This proves (7.11). From (7.12) one gets: }X}ąą}T }, if Q ­“ G. Hence from Lemma 5.2, one gets: Let Ω be a compact subset of GpAq. There exists ε ą 0 such that

w ρpXq´ε}T } Q |pΦ ´ ΦQqpnQexpXωq| ăă e , nQ P NQpAq,X P CT `S,T ,ω P Ω, (7.14)

and T,S as in the Lemma. Similarly one gets from Lemma 5.3:

w ρpXq´ε}T } |pEP pnQexpXω,φ,λq´ EP pnQexpXω,φ,λqQ| ăă e , (7.15) Q nQ P NQpAq,X P CT `S,T ,ω P Ω,λ P Λ. Using (2.8), the integration formula linked to the decomposition GpAq “ G N0pAqM0pAqK (cf. [20], I.1.13), and the fact that ρpXq ě ρpT0q for X P CT , one sees that:

G The volume of CT is bounded by a polynomial in }T }. (7.16)

21 Hence:

Q G The volume of CT `S,T Ă CT `S is bounded by a polynomial in }T }, (7.17) as }T ` S}ďp1 ` k0q}T }. Let us introduce:

Z Z w w,Z w,Z IQpT,λq :“ Epx, λq Φ pxqdx, IQ pT,λq :“ Epx, λqQ ΦQ pxqdx. ż Q ż Q CT `S,T CT `S,T Notice that:

w T T `S IGpT,λq“ IG pT,λq“ ΩP0 pEpλq, Φq, IQpT,λq“ ΩP0 pEpλq, Φq (7.18) QÿPPst For C ą 0,k P N let us consider the function of T and Z:

Ce´ε}T }p1 `}Z}qk (7.19)

It follows from (7.14), (7.15), as well as the tempered estimate for Φ and the uniform estimate for Eisenstein series (cf. Proposition 4.1) that: The difference of w |IQpT,λq´ IQ pT,λq| (7.20) is bounded for λ, T , S as in the Lemma, by a function of type (7.19). Let us define

G,`` G,`` G aQ pT ` S, T q :“ tTQ ` Y |Y P aQ ,̟αpY ´ Sqď 0,α P ∆QuĂ aQ. Let us show CQ “ aG,``pT ` S, T q` CMQ . (7.21) T `S,T Q TMQ

MQ G,`` Let X P CT and TQ ` Y P aQ pT ` S, T q. Let us show that X ` TQ ` Y is an Q element of CT `S,T . In view of (7.2), the only thing to prove is that it is an element G CT `S,T . One has:

X ` TQ ` Y ´ S ´ T “ Y ´ S ` X ´ TMQ .

Q Let α P ∆0z∆0 . Then ̟αpY ´S `X ´TMQq“ ̟αpY ´Sq which is less than or equal Q 0, by the definition of aQpT ` S, T q. Let α P ∆0 . The difference Y ´ SQ is a linear combination with coefficients less or equal to zero of elements of ∆ˇ Q hence of ∆ˇ 0. The MQ same is true for Y ´S “ Y ´SQ´SMQ . Hence ̟αpY ´Sqď 0. The definition of CT Q shows that ̟αpX´TMQ qď 0. Hence ̟αpY ´S`X´TMQ q“ ̟αpY ´Sqď 0,α P ∆0 . G This achieves to prove X ` TQ ` Y P CT `S,T as wanted. Hence

G,`` MQ Q aQ pT ` S, T q` CT Ă CT `S,T . (7.22)

22 Q The reciprocal inclusion follows easily from (7.2) and of the definition of CT `S,T . This achieves to prove (7.21). w w We use that ΦQ and EP px,φ,λqQ are left NQpAq-invariant and that the volume of NQpF qzNQpAq is equal to 1. If P is a parabolic subgroup of G with Levi subgroup MP and k P K, we have defined(cf. section 3.3) for any function on GpAq, the MP ,k function φ on MP pAq by:

MP ,k ´ρpHP pmqq φ pmq“ e φpmkq, m P MP pAq.

Thus, using (7.7), we get:

w w,Z w,Z IQ pT,λq“ Epx, λqQ ΦQ pxqdx “ ż Q CT `S,T

w,Z 1 w,Z 1 ´2ρQpXq 1 EpexpXmQk,λqQ ΦQ pexpXmQkqe dXdmQdk ż G,`` MQ aQ pT `S,T qˆCT

T w MQ,k w MQ,k X`Z 1 “ ΩP0XM prEpλqQs , rΦQs q dmQdXdk. ż G,`` MQ Q aQ pT `S,T qˆCT ˆK

Recall that by induction hypothesis, the Theorem 7.1 is true for MQ if Q ‰ G and one can apply this induction hypothesis. Taking into account (7.17) and the previous equality, one sees, using K-finiteness, that the difference of the preceding T ωT expression with the same expression, where ΩP0XMQ is replaced by P0XMQ , denoted JQpT,λq, is bounded by a function of type (7.19). One has:

T w MQ,k w MQ,k X`Z JQpT,λq“ ωP0XM prEpλqQs , rΦQs q dXdk ż G,`` Q aQ pT `S,T qˆK

T w MQ,k w MQ,k X`Z “ rR1 prEpλqQs , rΦQs q dXdk. żaG,``pT `S,T qˆK Q R1PPÿstpMQq

If R1 is a standard parabolic subgroup of MQ, let P1 be the standard parabolic subgroup of G contained in Q with P1 X MQ “ R1. Using Lemma 3.7 (iv), the definition (7.8), for MQ and R1, and integrating over K, one sees:

˚ J pT,λq“ pEpλqw , Φw qX`Z dX. Q M P1 P1 żaG,``pT `S,T q`a Q pT q P1PPstÿpGq,P1ĂQ Q P1XMQ,´

T We observe that JGpT,λq “ ωP0 pEpλq, Φq and one has seen that IGpT,λq “ T ΩP0 pEpλq, Φq. One writes:

T T T ΩP0 pEpλq, Φq“ ∆P0 pEpλq, Φq` ωP0 pEpλq, Φq.

23 Using what we have just proved and (7.18), and (7.20), we get:

The modulus of the difference

T `S T ΩP0 pEpλq, Φq´ ∆P0 pEpλq, Φq

T `S T `S T (7.23) “ ∆P0 pEpλq, Φq` ωP0 pEpλq, Φq´ ∆P0 pEpλq, Φq J T,λ J T,λ with p q “ QPPstpGq Qp q is bounded by a function of type (7.19). ř

Thus it is enough, to finish the proof of the Lemma, to prove:

T `S JpT,λq“ ωP0 pEpλq, Φq.

Using the expression of JQpT,λq above and interverting the sum over Q and P1, one sees that: ˚ 2 ´ ρP1 pXq w X`Z JpT,λq“ e pEpλq , Φ 1 q dX. M P1 P żaG,``pT `S,T q`a Q pT q`Z P1PPst,G,QÿPPstpGq,P1ĂQ Q P1XMQ,´

aMQ aQ aMQ aG,` T S, T Let P1XMQ,´´ be the interior in of P1XMQ,´ and let Q p ` q be the closure G,`` of aQ pT ` S, T q in aQ. Let us show:

The union

G,` MQ P a T S, T a T YQP st,P1ĂQ Q p ` q` P1XMQ,´´p MQ q (7.24)

G is disjoint and is a partition of aP1,´pT ` Sq.

G Let us consider the projection of aP1 on the closed convex cone aP1,´. By translating, G one sees, using e.g. [10] Corollary 1.4, that, if X P aP1,´pT ` Sq, there exists a 1 unique standard parabolic subgroup of G, Q with P1 Ă Q such that X “ X ` Y , X1 aMQ T ,Y aG,` T X aG T S Y aG,` T S, T P P1XMQ,´´p MQ q P Q p q. As P P1,´p ` q, one has P Q p ` q. G Hence the union in (7.24) contains aP1,´pT ` Sq and is disjoint. Reciprocally let us prove that for P1 Ă Q:

aMQ T aG,` T S, T aG T S . P1XMQ,´´p MQ q` Q p ` qĂ P1,´p ` q

X aMQ T ,Y To see this, by translation , it is enough to prove that if P P1XMQ,´´p MQ q P G G aQ,´ one has X ` Y P aP1,´ which is clear by convexity. This proves (7.24). Neglecting sets of measure zero, this implies that the sum JpT,λq is equal to T `S Z ωP0 pEpλq, Φq . This achieves to prove the Lemma.

24 We will give below a proof Theorem 7.1. It is done using first the argument of [2], Lemma 9.2 and second using wave packets as in [14] Lemma 3 and end of proof of Proposition 1 (see also the end of the proof of Theorem 1 in [13]).

One fixes δ ą 0 and one writes lim δ to describe the limit when }T } tends to T ÝÑ8 infinity verifying dpT qě δ}T }. One deduces from the preceding Lemma, as in ( [2], Lemma 9.2) that the limit

8 Z T Z ∆P0 pEpλq, Φq “ lim δ ∆P0 pEpλq, Φq T ÝÑ8 exists uniformly for λ in any compact subset of Hc and if Λ is a bounded set in Hc, there exists C,ε ą 0, k P N such that for λ P Λ and and T such that dpT qě δ}T }, Z P aG, one has:

8 Z T Z ´ε}T } k |∆P0 pEpλq, Φq ´ ∆P0 pEpλq, Φq |ď Ce p1 `}Z}q ,λ P Λ,Z P aG. (7.25) 8 Z Hc We prepare some Lemmas to prove that ∆P0 pEpλq, Φq is identically zero on . ˚ Using Proposition 5.4, we define a distribution TΦ,Z on iaP by:

Z Z 8 ˚ TΦ,Zpaq“ Eapxq Φ pxqdx, a P Cc piaP q, żrGs1 where Ea is the wave packet ia˚ apλqEpλqdλ. ş P 7.3 Lemma. The support S of TΦ,Z is a finite set.

˚ Proof. For λ P iaP , the center Zpg8q of Upg8q acts on Epλq by a character denoted χλ and Φ is annihilated by an ideal I of Zpg8q of finite codimension. Let us compute in two ways:

Z Z ˚ A :“ pzEapxqq Φ pxqdx, z P Zprg8, g8sq Ă Zpg8q, z P I, żrGs1 where z˚ is the adjoint of z. On one hand, looking to the action of z on Epλq and differentiating under the integral defining Ea we get: A “ TΦ,Zpppzqaq, where ppzqpλq“ χλpzq, which is a polynomial in λ. On the other hand:

Z Z ˚ A “ pEapxqq z Φ pxqdx “ 0 żrGs1

˚ From the equality above, if z P I the distribution ppzqTΦ,Z is equal to zero. Let ˚ ˚ ˚ ˚ I “ tz |z P Iu. As I is finite codimensional, the set of λ P iaP such that I Ă ker χλ is a finite set F. Hence if λ R F, there exists z P I˚ such that ppzqpλq ­“ 0. Hence TΦ,Z restricted to a neighborhood of λ is zero. Hence S Ă F.

25 8 ˚ S 7.4 Lemma. If a P Cc piaP q has its support in the complimentary set of , one has: T Z lim apλqΩP0 pEpλq, Φq dλ “ 0 δ ż ˚ T ÝÑ8 iaP Proof. From Fubini theorem and Lebesgue dominated convergence the limit is equal to TΦ,Zpaq, which is equal to zero by the preceding lemma.

8 ˚ Hc 7.5 Lemma. If a P Cc piaP q has its support in one has:

T Z lim apλqωP0 pEpλq, Φq dλ “ 0. δ ż ˚ T ÝÑ8 iaP

T Z Proof. This follows from the definition of ωP0 pEpλq, Φq , (7.9) and from the fact 8 n that the Fourier transform of a Cc function on R is rapidly decreasing.

Sc S ˚ 7.6 Lemma. If is the complimentary set of in iaP one has: 8 Z Hc Sc ∆P0 pEpλq, Φq “ 0,λ P X .

8 ˚ Proof. From the two preceding lemmas one has for all in Cc piaP q with support in the intersection Hc X Sc:

8 Z apλq∆P0 pEpλq, Φq dλ “ 0. ż ˚ iaP This implies the Lemma.

Proof. Let us finish the proof of the Theorem 7.1. The vanishing property of the preceding Lemma together with (7.25) shows that the bound of the theorem is true Hc Sc T Z for λ in a bounded subset of X . Recall that ΩP0 pEpλq, Φq is analytic in λ. ˚ T Z Hence, for any λ in iaP and any compact neighborhood of λ, V , ωP0 pEpλq, Φq is bounded on V X Hc X Sc. But this meromorphic function has only possible poles ˚ along hyperplanes. It follows that it is analytic on iaP . This proves the first part of the Theorem. The second part follows from (7.25) by continuity and density.

˚ Let Λ P aM be strictly P -dominant. If Q is a parabolic subgroup of G with Levi P Λ subgroup MQ one will denote by ψQ the characteristic function of :

Λ G CQ “ tX P aP |ωαpXqΛpαˇqą 0,α P ∆Qu

G Λ that we look as a tempered measure on aP by our choice of Haar measures. Let βQ be the number of elementsα ˇ of ∆ˇ Q such that Λpαˇqă 0. Then one has the following proposition, whose proof is analogous to Proposition 2 in [14], using (4.2) and (7.9).

26 7.7 Proposition. Using the notation of Theorem 7.1, the analytic function λ Ñ T Z ˚ ωP0 pEpλ, Φq is equal, as a distribution on iaP , to the sum: (a) on Q P Pst (b) on s P W pQ|P q E w (c) on µ P QpΦQq of:

1 1 Λ Λ ˆ 1 ´ w Z ´ βQs ´ C p M s ,λ φ, s ψ s λ s µ , Qrp Qp p q ΦQ,0,µq ˝ qpBqp´1q p Q ,TQs `Zq sp ´ q

s ´1 Λ where Q “ sQs and ψ s is the characteristic function of the translate of Q ,TQs`Z Λ Λ CQs , CQs ´TQs ´Z and the upper indexˆindicates that we take its Fourier transform. T ˚ Proof. First ωpλq :“ ωP0 pEpλq, Φq is analytic on iaP from Theorem 7.1 (i). Then, ωpλq is the limit when t to 0` of ωpλ ` tΛq in the sense of distributions. Then one uses [14], Lemma 11, for each term of the sum defining ωpλ ` tΛq. The Lemma follows.

The following theorem is the main result of this article. 7.8 Theorem. The image of the map E of Theorem 6.1 is equal to L2pGpF qzGpAqq. We start with a preliminary remark. Let us recall some result of Bernstein in [6]. We use the notation and terminology of l.c. The space rGs is of polynomial growth whose rank rkprGsq is equal to the split rank of G (cf. l.c., section 4.4, Example 1). Using the definition of a space of polynomial growth in l.c. p.689 and what follows d this definition, one gets that for d ą rkprGsq, p1 ` σrGsq is a summable weight. Moreover the automorphic forms which may contribute to the spectrum (see below 2 ´d for a precise meaning) are in some L pX, p1 ` σrGsq dxq, for some d ą 0, as well as their derivatives. Hence they are tempered, by Proposition 3.4.

7.9 Lemma. If the image of E is not equal to L2pGpF qzGpAqq, there exists a non zero tempered automorphic form Φ, transforming under a unitary character νG of aG and orthogonal to all the wave packets Ea of Proposition 5.4, when P , φ and a varies. Proof. The proof is similar to [11], Lemma 11. Let H be the orthogonal to the image of E, which is assumed to be non zero. One considers the decomposition of this representation of GpAq into an Hilbert integral of multiple of irreducible representations:

H “ Hπdµpπq. żGˆ The restriction ξ of the Dirac measure at the neutral element to the space H8 of C8 vectors, desintegrates:

ξ “ ξπdµpπq, żGˆ

27 H´8 GpF q H where ξπ Pp π q , i.e. is a GpF q-invariant distribution vector on π. Let 8 v “ vπdµpπq P H . żGˆ We assume that it is K-finite and non zero. Let pgnq be a dense sequence in GpAq. For p,q,n P N, let:

q Xp,q,n “ tπ P Gˆ||ă ξπ, πpgnqvπ ą|ď pΞpgnqp1 ` σrGspgnqq u

For all g P GpAq, the map π ÞÑă ξπ, πpgqvπ ą is µ-measurable. Hence all Xp,q,n are measurable as well as Xp,q “ XnPNXp,q,n. Moreover, from our preliminary remark, just after the statement of the Theorem, Yp,qPNXp,q is equal to Gˆ up to a set of µ- 0 measure 0. Let Xp,q be the set of elements π of Xp,q of π such that g ÞÑă ξπ, πpgqvπ ą 0 is non identically zero. As v is non zero, one can find p, q such that the set Xp,q is 0 of non zero measure. Let χ be the characteristic function of Xp,q. Then one has for any θ P L8pG,µˆ q, going back to the definition:

8 fθ :“ χpπqθpπqvπdµpπq P H . żGˆ

Hence by using the desintegration of ξ, viewing fθ as a function on GpAq, one has:

fθpgq“ χpπqθpπqă ξπ, πpgqvπ ą dµpπq,g P GpAq. żGˆ

Let us show that the map pπ,gq ÞÑă ξπ, πpgqvπ ą is measurable. Let g “ g8gf where g P GpA8q and gf P GpAf q. As v is smooth the map is locally constant in gf . One easily reduces to gf “ 1 and look to the dependence on pπ,g8q only. Then one uses the argument given in [11], p. 96 which uses step functions. Using (3.7), one can apply Fubini’s theorem to

Eapxqfθpxqdx “ θpπq Eapxqă ξπ, πpxqvπ ądxdµpπq. ż ż 0 ż rGs Xp,q rGs 0 This has to be zero for all θ. Hence for almost all π in Xp,q one has:

Eapxqă ξ, πpxqvπ ą dx “ 0 żrGs 0 for a given Ea. Using a separability argument, one finds an element π0 of Xp,q such that it is true for all Ea. One takes Φ “ă ξ, π0pxqvπ ą.

8 G ˚ 8 ˚ ˚ Let a “ a1 b a2 where a1 P Cc ppiaP q q and a2 P Cc piaGq ). Let νG P iaG which ˚ describes the action of aG on π0. Then, using Fourier inversion formula for iaG, one has:

EapxqΦpxqdx “ a2pνGq Ea1 pxqΦpxqdx żGpF qzGpAq żGpF qzGpAq1

28 where Ea1 “ G,˚ a1pλqEP px,φ,λqdλ. We want to compute iaP ş

I “ Ea1 pxqΦpxqdx żGpF qzGpAq1 using the preceding theorem. 7.10 Lemma. 0 I “ CP rppP pφ, ΦP,0,µq qpBqa1qspµ G q. |aP Ew Φ µPÿP p q Proof. We can compute I as limit. Using Lebesgue dominated convergence and Fubini theorems, one can write I as a limit. Let T be strictly P0-dominant. Then:

nT 0 I “ limnÑ`8 a1pλqΩP0 pEpλq, Φq dλ. ż G,˚ iaP From Theorem 7.1, one can replace Ω by ω. Then one uses Proposition 7.7. One sees easily that unless Qs “ P , the characteristic Λ function of CQs ´ nTQs tends to 0 in the sense of tempered distributions. But in this case Q is standard and conjugate to P . Hence Q “ P and s “ 1. Using Proposition 7.7, one computes easily the limit.

Now we can finish the proof of the Theorem. The hypothesis on Φ above shows that the right hand side of the equality of the Lemma is zero for all P , φ, a1, a2. One Atemp,c E w concludes, by varying a1, a2 and φ, that ΦP,0,µ P P pGq for all P and µ P P pΦq. Then, using Lemma 3.8 (ii), one concludes that Φ “ 0. A contradiction which finishes the proof.

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Patrick Delorme [email protected] Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France

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