On the Spectral Theorem of Langlands Patrick Delorme April 22, 2021 A` Chantal Abstract We show that the Hilbert subspace of L2pGpF qzGpAqq generated by wave packets of Eisenstein series built from discrete series is the whole space. To- gether with the work of Lapid [18], it achieves a proof of the spectral theorem of R.P. Langlands based on the work of J. Bernstein and E. Lapid [7] on the meromorphic continuation of Eisenstein series from I use truncation on compact sets as J. Arthur did for the local trace formula in [2]. 1 Introduction We denote by E the isometry introduced by E. Lapid in [18], Theorem 2, whose proof involves the meromorphic continuation of of Eisenstein series built from discrete data. A dense subspace of its image is generated by wave packets of these Eisenstein series. We show that this image is equal to L2pGpF qzGpAqq. This is what Lapid calls the second half of the proof of the spectral theorem of Langlands. It achieves a new proof of the spectral theorem of R.P. Langlands ([17], [20]). Notice that the proof of Langlands describes the spectrum as residues of Eisenstein series built from cuspidal data. arXiv:2006.12893v3 [math.RT] 21 Apr 2021 One uses the notion of temperedness of automorphic forms introduced by J. Franke [15](cf. also [21] section 4.4). We show this notion of temperedness is weaker than the notion of temperedness introduced by Joseph Bernstein in [6]. We prove that for bounded sets of unitary parameters, the Eisenstein series are uniformly tempered when the parameter is unitary and bounded. One uses for this that the growth of an automorphic form is controlled by the exponents of its constant terms and that the constant terms of Eisenstein series are explicited. Some wave packets are in the Harish-Chandra Schwartz space (cf.[19]): this is due to J. Franke [15], section 5.3, Proposition 2 (2) but his proof rests on the work of Langlands. We give here a selfcontained proof (cf. Proposition 5.4) which uses a uniform bound for exponential polynomials of one variable in terms of its exponents (cf. [18]) together with the general scheme of Harish-Chandra’s study of wave packets in the Schwartz space in the real case (cf.[16], see also [5]). Then, one shows that if the image of E is not the full space, there would exist a tempered automorphic form orthogonal to these wave packets: the proof is similar to what we did for real symmetric spaces (cf. [11], [14]). One can compute an explicit asymptotic formula for the truncated inner product of this form with an Eisenstein series. Actually, here, we use ”true truncation” i.e. truncation on compact sets as in [2]. This uses partitions of GpF qzGpAq depending on the truncation parameter (cf. [1]). By a process of limit, as in [14], one computes the scalar product of this form with a wave packet of Eisenstein series. Then one shows that it implies that this form is zero. A contradiction which shows that E is onto. 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 .