Harmonic Analysis of GL 2 and GL 3 Automorphic Forms

Transcription

1 Harmonic nalyi of GL 2 and GL 3 utomorphic Form. DeCelle Exerpt from oral paper (pril 2009) Document created 3/17/2010 Lat updated 3/17/2010 The tudy of pectral theory of automorphic form began with Rankin and Selberg in the late 1930, continued with Selberg and Roelcke in the 50, Gelfand, Fomin, and Graev in the 50 and 60, Harih-Chandra and Langland in the 60, and more recently Moeglin and Waldpurger in the utomorphic Form on SL 2 (Z)\SL 2 (R) Firt we treat the implet poible cae, automorphic form on SL 2 (Z)\SL 2 (R). Thi correpond to the familiar dicuion of SL 2 (Z)-invariant function on the upper half plane. From a modern point of view, conidering SL 2 (Z)\SL 2 (R) correpond to picking the archimedean place out of the more natural and coherent adelic verion of the tory, the harmonic analyi of automorphic form on Z GL 2 (Q)\GL 2 (). However, many of the difficultie that arie in the adelic dicuion are already preent in the archimedean cae, o we chooe to treat the archimedean cae firt. Here G SL 2. Recall the Iwaawa decompoition: G P K MNK where P i the tandard parabolic ubgroup (upper triangular matrice), K i the tandard maximal compact ubgroup (the orthogonal group SO(2)), M i the tandard Levi component of P (diagonal matrice), and N i the unipotent radical for P (upper triangular matrice with 1 on the diagonal.) Reduction theory and the theory of compact operator how that the pace L 2 cup(g Z \G R ) of quare-integrable cup form decompoe dicretely with finite multiplicity, i.e. for f in L 2 (G Z \G R ) atifying the Gelfand condition: f L2 F onb of cfm f, F F The orthogonal complement i panned by peudo-eientein erie Ψ ϕ (g) ϕ(γg) for ϕ Cc 0 (N R M Z \G R ) γ P Z \G Z The pace of peudo-eientein erie decompoe a the direct integral of Eientein erie E, parameterized by C, E (g) f (γg) γ P Z \G Z where f i a vector in an induced repreentation, coming from a character χ on M extended to P by left N-invariance. Thi i the ame a the claical preentation becaue and furthermore: N R M Z \G R /K M Z \M R Z R M Z \M R GL 1 (Z)\GL 1 (R) Z \R R + So if we aume that everything i pherical, with trivial central character, we can conider ϕ and χ a function on R +. So we can ue Mellin inverion to decompoe ϕ: ϕ Mϕ() y d ϕ, χ χ and o, after ome work, the peudo-eientein erie decompoe a: Ψ ϕ Ψ ϕ, E E 1

2 Note that we need Re() > 1 in order to do the manipulation required to obtain thi decompoition. To fold up the integral, we move the contour to the critical line, picking up a reidue. fter ome work, Ψ ϕ Ψ ϕ, E E + Ψ ϕ, 1 1 1, ir+ 2. Harmonic nalyi for utomorphic Form on Z GL 3 (k)\gl 3 () Having dicued the pectral theory for SL 2 (Z)\SL 2 (R) we now ue the ame framework to dicu the GL 3 cae. Here we work in an adelic etting, over an arbitrary number field k. we will ee, thi only alter the dicuion minimally. 2.1 Claifying/grouping utomorphic Form by Cupidal Support The goal i to decompoe the pace of quare-integrable automorphic form into irreducible ubrepreentation. in the cae of SL 2, we tart with the cupidal automorphic form. Given a parabolic P in G, and function f on Z G k \G, the contant term of f along P i c P f(g) f(ng) dn where N i the unipotent radical of P. n automorphic form atifie the Gelfand condition if, for all maximal parabolic P, the contant term along P i zero. If uch a function i alo z-finite (for example, it i an eigenfunction for Caimir) and K-finite (for example, it i pherical), it i called a cup form. Since the right action of G commute with taking contant term, the pace of function atifying the Gelfand condition i G-table, and o i a ubrepreentation. Gelfand and Pieteky-Shapiro howed that integral operator on thi pace are compact, o by pectral theory of compact operator, thi ubrepreentation decompoe into a direct um of irreducible, each with finite multiplicity. We will take thi for granted and decompoe the ret of L 2. Having ued the contant term to filter out the cup form, we now ue the map for further orting. a firt tep toward obtaining the L 2 decompoition of the non-cupidal automorphic form, we claify them according to their cupidal upport, i.e. the mallet parabolic on which they have a non-zero contant term. (Converely we can think of the larget parabolic on which it contant term i zero.) In GL 3, there are three conjugacy clae of proper parabolic ubgroup. In addition, the whole group may alo conidered to be parabolic in a trivial ene. We will conider the tandard parabolic ubgroup: P 3 GL 3, P 2,1 and P 1,2 the maximal parabolic, and P 1,1,1 the minimal parabolic, contained in both P 2,1 and P 1,2. Starting with the eaiet cae, we oberve that an automorphic form whoe contant term along P 3 GL 3 i zero i identically zero, and an automorphic form with cupidal upport P 3 i preciely a nonzero cup form. There i more to ay about automorphic form whoe cupidal upport i a maximal parabolic. Conider an automorphic form f with cupidal upport P 2,1 and let F c 2,1 f. Then F i a non-zero left N 2,1 -invariant function. So if it i pherical, it can be conidered a a GL 2 automorphic form. In fact it i a GL 2 cup form, ince the contant term of f along the minimal parabolic i zero. Latly, we have the automorphic form whoe cupidal upport i the minimal parabolic, i.e. thoe whoe contant term along P 1,1,1 i nonzero. While claifying automorphic form according to cupidal upport i helpful (becaue of certain adjointne relation, which allow u to prove the orthogonality of ubpace panned by them) it doe not give u a very concrete or explicit decription of the variou clae of automorphic form. Recall from the SL 2 cae that peudo-eientein erie provided an explicit decription of automorphic form with cupidal upport P, and the pace panned by peudo-eientein erie wa the orthogonal 2

3 complement to the pace of cup form. In GL 3 thing are more complicated, ince there are more parabolic ubgroup, but we will till ue peudo-eientein erie to decribe the orthogonal complement to the pace of cup form. Define peudo-eientein erie in a manner exactly analogou to the SL 2 cae: Ψ ϕ (g) ϕ(γ g) γ P k \G k where ϕ i a continuou, compactly upported function on Z N M k \G. In GL 3, there are three different kind of peudo-eientein erie, correponding to the three tandard parabolic ubgroup. It i relatively traightforward to check that the pace of all peudo-eientein erie i the orthogonal complement to the pace of cup form, but it will require more work to determine how many different kind of peudo-eientein erie we actually need in order to pan the complement. We tart with the following adjointne relation, the key to proving orthogonality. Claim. For any quare-integrable automorphic form f, and any peudo-eientein erie Ψ P ϕ, with P a parabolic ubgroup, f, Ψ P ϕ Z G k \G c P f, ϕ Z N P M k P \G Proof. Thi i a tandard winding/unwinding argument: f, Ψ P ϕ Z G k \G f(g)ψ P ϕ (g) dg Z G k \G f(g) ( ) ϕ(γg) dg Z G k \G γ P k \G k f(g)ϕ(g) dg Z P k \G f(g)ϕ(g) dg Z N k M k \G f(ng)ϕ(ng) dn dg Z N M k \G N k \N f(ng) dn ) ϕ(g) dg Z N M k \G ( c P f, ϕ Z N P M P k \G Note. Thi winding/unwinding i a pecific example of integrating over quotient. For a G a topological group and H a cloed ubgroup, f(g) dg f(hg) dh dg G H\G a long a the modular function are compatible. When the ubgroup i dicrete, we write the integral over the ubgroup a a um. From thi adjointne relation, it quickly follow that a (quare-integrable) automorphic form i a cup form if and only if it i orthogonal to all peudo-eientein erie, i.e. the orthogonal complement to the pace of cup form i panned by peudo-eientein erie. Further, we can ue thi adjointne relation to decompoe the pace panned by peudo-eientein erie into orthogonal ubpace. In particular, if f i in the pace panned by peudo-eientein erie, then it quickly follow from the adjointne relation that f ha cupidal upport P 2,1 or P 1,2 if and only if it i orthogonal to all P 1,1,1 peudo-eientein erie. So the orthogonal complement to cup form decompoe into two orthogonal ubpace: the pace panned by P 1,1,1 peudo-eientein erie, and the pace of automorphic form with cupidal upport P 2,1 or P 1,2. H 3

4 We need to determine which peudo-eientein erie are in the econd ubpace. lthough we might naively gue that all P 2,1 and P 1,2 peudo-eientein erie have cupidal upport P 2,1 or P 1,2, thi i not the cae. What i true i that a P 2,1 or P 1,2 peudo-eientein erie with cupidal data (i.e. one whoe data can be identified with a GL 2 cup form) ha cupidal upport P 2,1 or P 1,2. (To how thi we need to compute the contant term along P 1,1,1 of uch a peudo-eientein erie. Thi computation i not trivial to carry out, and it relie on the Bruhat decompoition of GL 3. See the appendix on contant term for further detail.) ny other P 2,1 or P 1,2 peudo-eientein erie (i.e. one with non-cupidal data) can be written a the um of a P 1,1,1 peudo-eientein erie and a P 2,1 or P 1,2 peudo-eientein erie with cupidal data. So the ubpace coniting of automorphic form with cupidal upport P 2,1 or P 1,2 i panned by P 2,1 and P 1,2 peudo-eientein erie with cupidal data. we will ee, the pace generated by P 1,2 peudo-eientein erie i actually the ame a the pace generated by P 2,1 peudo-eientein erie. Thi i an example of a more general phenomenon: peudo-eientein erie of aociate parabolic pan the ame pace. So we have the following decompoition of L 2 (Z G k \G ) into orthogonal ubpace: L 2 (Z G k \G ) (cfm) (pan of P 1,1,1 p-ei) (pan of P 2,1 p-ei, cpdl data) 2.2 Decompoing Peudo-Eientein Serie While we have a fairly nice decription of the non-cupidal automorphic form in L 2 (Z G k \G ) in term of peudo-eientein erie, we would prefer a decompoition in term of irreducible. Following the GL 2 cae, we will decompoe the peudo-eientein erie into genuine Eientein erie. (Since Eientein erie are image of principal erie, they are eigenfunction for Caimir and for the whole center of the univeral enveloping algebra. Typically principal erie are irreducible, o Eientein erie typically generate irreducible repreentation.) gain, due to the plurality of parabolic in GL 3, we have everal kind of Eientein erie in GL 3. The definition for GL 2 Eientein erie given above cale nicely to include all of thee: for a parabolic P, the P -Eientein erie i E χ f χ (γg) γ P k \G k where f χ i a (pherical) vector in a repreentation χ of M P, extended to a P -repreentation by left N-invariance, and induced up to G. The key to obtaining the pectral decompoition for GL 2 peudo-eientein erie i that the Levi component i a product of copie of GL 1, allowing u to reduce to the pectral theory for GL 1 (Mellin inverion). For GL 3 we are able to ue a imilar approach for minimal parabolic peudo-eientein erie, again becaue the Levi component i a product of copie of GL 1. The ame method will certainly not work for decompoing P 2,1 and P 1,2 peudo-eientein erie, becaue in thee cae the Levi component contain a copy of GL 2. So we turn our attention firt to the decompoition of the minimal parabolic peudo-eientein erie. We will need the functional equation of the Eientein erie. Note that becaue of the increae in dimenion, the ymmetry of the Eientein erie i more complex. The Eientein erie can no longer be parameterized by one complex number, ince the data f χ for the Eientein erie i on a product of three copie of GL 1. The ymmetrie of the Eientein erie can be decribed in term of the action of the Weyl group W on the tandard maximal toru (which, in thi cae, i the ame a the Levi component M) on it Lie algebra a, and the dual pace ia. For now, we give a minimal explanation of thi action, jut enough to decribe the contant term and the functional equation of the Eientein erie and ue them in the pectral decompoition. See the appendice for further detail on the computation of contant term and the derivation of the functional equation. For GL n the tandard maximal toru i the product of n copie of GL 1, and repreentation of are product of repreentation of GL 1 ; in the unramified cae, thee repreentation are jut y y i, for complex i. The Weyl group W i the group of permutation matrice in GL n. It act on by permuting the copie of GL 1, and it act on the dual in the canonical way, permuting the i, in the unramified cae. 4

5 We now decribe the contant term and the functional equation of the Eientein erie. The contant term of the Eientein erie (along the minimal parabolic) ha the form c P (E χ ) c w (χ) wχ where wχ i the image of χ under the action of w and c w (χ) i a contant depending on w and χ with c 1 (χ) 1. The Eientein erie ha functional equation c w (χ) E χ E wχ for all w W We tart the decompoition of Ψ ϕ by uing the pectral expanion of it data ϕ. Recall that ϕ i left N -invariant, o it i eentially a function on the Levi component, which i a product of copie of k \J. (By Fujiaki lemma, thi i the product of a ray with a compact abelian group. To implify the preent dicuion we will aume that the compact abelian group i trivial, a i the cae for number field with cla number one, e.g. k Q.) So pectrally decompoing ϕ i a higher-dimenional verion of Mellin inverion. ϕ ϕ, χ χ dχ Winding up, Ψ ϕ (g) ϕ, χ E χ (g) dχ ia Note that in order for thi to be valid, the parameter of χ mut have Re( i ) 1. However, in order to ue the ymmetrie of the functional equation, we need the parameter to be on the critical line. In moving the contour, we pick up ome reidue, which fortunately are contant. Breaking up the dual pace according to Weyl chamber and changing variable, Ψ ϕ (g) (reidue) ϕ, w χ E w χ (g) dχ 1t Weyl chamber Now uing the functional equation, Ψ ϕ (g) (reidue) ϕ, w χ c w (χ) E χ (g) dχ (1) ϕ, c w (χ) w χ E χ (g) dχ (1) We recognize the contant term of the Eientein erie, and apply the adjointne relation ϕ, c w (χ) w χ ϕ, c P E χ Ψ ϕ, E χ So we have, Ψ ϕ (g) Ψ ϕ, E χ E χ (g) dχ + (reidue) (1) Our next goal i to how that the remaining automorphic form, namely thoe with cupidal upport P 2,1 or P 1,2, can be written a uperpoition of genuine P 2,1 Eientein erie. To do thi it uffice to decompoe P 2,1 and P 1,2 peudo-eientein erie with cupidal upport. For thi dicuion we let P P 2,1 and Q P 1,2. We tart by looking more carefully at peudo-eientein erie with cupidal data. The data for a P peudo-eientein erie i mooth, compactly upported, and left Z Mk P N P -invariant. For now, we aume that the data i pherical, i.e. right K-invariant. Thi mean that thi function i determined by it behavior on Z Mk P \M P. In contrat to the minimal parabolic cae, thi i not a product of copie of GL 1, o we cannot imply ue the GL 1 pectral theory (Mellin inverion) to accomplih the decompoition. Intead, thi quotient i iomorphic to GL 2 (k)\gl 2 (), o we will ue the pectral 5

6 theory for GL 2. If η i the data for a P 2,1 peudo-eientein erie Ψ η, we can write η a a tenor product f ν on Z GL2()GL 2 (k)\gl 2 () Z GL2(k)\Z GL2() Saying that the data i cupidal mean that f i a cup form. Similarly the data ϕ ϕ F, for a P 2,1 -Eientein erie i the tenor product of a GL 2 cup form F and a character χ on GL 1. We how that Ψ f,ν i the uperpoition of Eientein erie E F, where F range over an orthonormal bai of cup form and i on a vertical line. Uing the pectral expanion of f and ν, ( ) ( ) η f ν f, F F ν, χ χ d cfm F cfm F η f,ν, ϕ F, ϕ F, d So the peudo-eientein erie can be re-expreed a a uperpoition of Eientein erie. Ψ f,ν (g) η f,ν (γg) γ P k \G k η f,ν, ϕ F, ϕ F, (γg) d γ P k \G k cfm F cfm F cfm F η f,ν, ϕ F, ϕ F, (γg) d γ P k \G k η f,ν, ϕ F, E F, (g) d In fact the coefficient η, ϕ GL2 i the ame a the pairing Ψ η, E ϕ GL3, ince Ψ η, E ϕ c P (Ψ η ), ϕ η, ϕ So the pectral expanion i Ψ f,ν cfm F Ψ f,ν, E F, E F, (g) d Notice that, o far, we have not had to hift the line of integration to the critical line ir. It now remain to how that peudo-eientein erie for the aociate parabolic, Q P 1,2, can alo be decompoed into uperpoition of P -Eientein erie. Notice that in the dicuion above, when we decompoed P -peudo-eientein erie into genuine P -Eientein erie, we did not ue the functional equation to fold up the integral, a in the cae of minimal parabolic peudo-eientein erie. For maximal parabolic Eientein erie, the functional equation doe not relate the Eientein erie to itelf, but rather to the Eientein erie of the aociate parabolic. We will ue thi functional equation to obtain the decompoition of aociate parabolic peudo-eientein erie. For a derivation of the functional equation, ee the appendix. For now, we tate the functional equation without proof: E Q F, b F, E P F,1 where b f, i a meromorphic function that appear in the computation of the contant term along P of the Q-Eientein erie. We conider a Q-peudo-Eientein erie Ψ Q f,ν with cupidal data. By the ame argument ued above to obtain the decompoition of P -peudo-eientein erie, we can decompoe Ψ Q f,ν into a uperpoition of Q-Eientein erie. Ψ Q f,ν (g) η f,ν, ϕ F, E Q F, (g) cfm F 6

7 Now uing the functional equation, Ψ Q f,ν (g) Ψ Q f,ν, b F, EF,1 P b F, EF,1 P cfm F cfm F Ψ Q f,ν, EP F,1 b F, 2 EF,1 P So we have a decompoition of Q-peudo-Eientein erie (with cupidal data) into P -Eientein erie (with cupidal data). In order to ue the functional equation we did have to move ome contour, but in thi cae there are no pole, o we do not pick up any reidue. We have decribed the pectral decompoition of L 2 (Z G k \G ) a the direct um/integral of irreducible. ny automorphic form ξ can be written a ξ ξ, f f + ξ, E 2,1 F, E2,1 F, + ξ, Eχ 1,1,1 Eχ 1,1,1 ξ, 1 dχ + 1, 1 GL 3 cfm f GL 2 cfm F Certainly thi expanion converge in L 2. To enure more convergence, for example uniform convergence on compact et, additional condition need to be impoed. The argument given above, proving the convergence of the SL 2 pectral expanion under ufficient differentiability condition, generalize to GL 3.. ppendice Here we include ome upplemental material, which may erve a a ueful addendum to the dicuion above. In the firt appendix, we provide the computation of ome contant term for GL 3 Eientein erie uing the Bruhat decompoition. The econd appendix include the derivation of the functional equation for GL 3 Eientein erie from their contant term..1 Contant Term of GL 3 Eientein Serie Since the contant term of GL 3 Eientein erie were ued repeatedly in the dicuion of the pectral decompoition of GL 3, we briefly dicu the way to obtain contant term uing the Bruhat decompoition. Recall the Bruhat decompoition of GL n G P wq P wq where W i the Weyl group and P and Q are parabolic. (1) w (W P )\W/(W Q) To compute the contant term along P of a Q-Eientein erie, c P (Eϕ Q )(g) ϕ(γβng) dn Nk P \N P γ Q k \G k /P k β Q k \Q k γp k ϕ(γβng) dn γ Q k \G k /P N P k k \N P β Q k \Q k γp k ϕ(wβng) dn w (W P )\W/(W Q) Nk P \N P β Q k \Q k wp k ϕ(wβng) dn w (W P )\W/(W Q) Nk P \N P β (w 1 Q k w P k )\P k Further computation i dependent on the choice of P and Q. We how the computation for everal of the contant term for GL 3 Eientein erie. Firt conider P Q P 1,1,1 the minimal parabolic. Then the contant term i of the form: c 1,1,1 (Eϕ 1,1,1 ) c w (χ) wχ where c 1 (χ) 1 7

8 when ϕ i in the principal erie I χ. We recall the computation that yield thi concluion. The double coet pace (W P )\W/(W P ) i the whole Weyl group W, and ince the Levi component i invariant under conjugation by element of W, P wp P wn for all w. So the contant term i c 1,1,1 (Eϕ 1,1,1 )(g) ϕ(wβng) dn β (w 1 P k w N k )\N k For w 1, ϕ(ng) dn vol( ) ϕ(g) ϕ(g) and for w w o, the long Weyl element, the interection wo 1 P k w o N k i trivial, o there i unwinding ϕ(w o γng) dn ϕ(w o ng) dn γ N N k and thi integral factor over prime becaue ϕ doe. The integral correponding to the four other element of the Weyl group have partial unwinding. Firt conider w σ, the element correponding to the reflection of the firt poitive imple root. Then the quotient (σ 1 N k σ N k )\N k i iomorphic to the GL 2 unipotent radical, here denoted N 1,1. So the integral i γ (σ 1 N k σ N k )\N k ϕ(σγng) dn N 1,1 1,1 k \N N 2,1 2,1 k \N N 1,1 N 2,1 2,1 k \N N 1,1 N 2,1 2,1 k \N γ N 1,1 k ϕ(σnug) dn du ϕ(uσng) du dn vol(n 2,1 2,1 k \N N ) ϕ(σng) dn 1,1 ϕ(σng) dn N 1,1 ϕ(σγnug) du dn Thi computation relie on the fact that σn 2,1 σ 1 N 2,1 and ϕ i N 2,1 -invariant. Thi lat integral factor over prime. We can compute the term correponding to the other Weyl element imilarly. For w τ, the element correponding to the reflection of the econd poitive imple root, N 1,1 ϕ(τng) dn For w τσ, Finally, for w στ, N 2,1 N 1,2 ϕ(τσng) dn ϕ(στng) dn Thee integral factor over prime, and the local integral are intertwining operator among principal erie: T w,χv : I χv I wχv. For example, conider the local integral for w σ. Uing right K v -invariance, T w,χv ϕ v (g) ϕ v (σng) dn N v ϕ v (σnn g m g ) dn N v ϕ v (σnm g ) dn N v 8

9 Changing variable n m g nm 1 g and uing the P -equivariance of ϕ v by χ v, T w,χv ϕ v (g) δ(m g ) ϕ v (σm g n) dn δ(m g ) N v χ v (σm g σ 1 )ϕ v (σn) dn N v Notice that thi i the action of W on χ v, o T w,χv ϕ v (g) δ(m g ) σχ v (m g ) ϕ v (σn) dn δ(m g ) σχ v (m g ) T σ,χv ϕ v (1) N v So the contant term i c 1,1,1 (Eϕ 1,1,1 )(g) ( v ) T w,χv ϕ v (1) δ(m g ) wχ(m g ) Defining c w (χ) to be the contant in front and renormalizing to eliminate the modular function, we obtain the deired expreion for the contant term: c 1,1,1 (Eϕ 1,1,1 )(g) c w (χ) wχ(g) Now we conider the cae where P i the minimal parabolic and Q i one of the maximal parabolic, ay P 2,1. We conider Q-Eientein erie with cupidal data. The contant term c 1,1,1 (Eϕ 2,1 ) i identically zero. To ee how thi can be computed, recall from above, c 1,1,1 (Eϕ 2,1 )(g) ϕ(wβng) dn w (W P )\W/(W Q) Nk P \N P β (w 1 Q k w P k )\P k in the previou cae, the Levi component of P i invariant under conjugation by W o QwP QwN, where N denote the unipotent radical of P. The quotient (W P )\W/(W Q) ha three ditinct coet, with repreentative w 1, τ, τσ. So, For w 1, the integral i c 1,1,1 (E 2,1 ϕ ) w1,τ,τσ ϕ(ng) dn N 1,1 1,1 k \N β (w 1 Q k w N k )\N k ϕ(wβng) dn vol(n 2,1 2,1 k \N N 2,1 2,1 k \N ) N 1,1 k ϕ(nug) du dn \N 1,1 ϕ(ng) dn which i zero becaue ϕ i cupidal. Similar computation how that the other two term are zero a well. Next we dicu the cae where P Q i a maximal parabolic, ay P 2,1. If the data ϕ for the Eientein erie i cupidal, the contant term i jut ϕ. From the initial computation, c 2,1 (Eϕ 2,1 )(g) ϕ(wβng) dn w (W P )\W/(W P ) β (w 1 P k w P k )\P k In thi cae, there are two double coet, with repreentative 1 and τ o, c 2,1 (Eϕ 2,1 )(m) ϕ(nm) dn + ϕ(τβnm) dn β (τ 1 P k τ P k )\P k Since ϕ i left N-invariant, the firt term i jut vol( ) ϕ(m). Showing that the econd term i zero take a little more work. The quotient (τ 1 P k τ P k )\P k i the emidirect product of quotient of M k and N k, o the um over β can be written a a double um over the M k part, a quotient iomorphic 9

10 to P GL2(k)\GL 2 (k), and the N k part, a quotient by the unipotent radical Uk 1 which i zero in the (2, 3) entry. ϕ(τβnm) dn ϕ(τµνng) dn β (τ 1 P k τ P k )\P k µ P GL2 (k)\gl 2(k) ν U 1 k \N k The um over µ come out of the integral, and the um over ν unwind yielding ϕ(τµnm) dn µ P GL2 (k)\gl 2(k) U 1 k \N Since M normalize N, a change of variable eliminate the µ, while introducing the modular function. Letting U 2 be the unipotent radical uch that N 2,1 U 1 U 2, the integral become δ(µ) ϕ(τu 1 u 2 g) du 1 du 2 µ P GL2 (k)\gl 2(k) U 2 U 1 k \U 1 Since τu 1 i a GL 2 unipotent radical, the inner integral i a GL 2 contant term, and thi i zero, becaue of the cupidality of ϕ. So the P 2,1 contant term for a P 2,1 -Eientein erie with cupidal data i jut equal to the term coming from the identity coet. c 2,1 (Ψ 2,1 ϕ ) vol(n 2,1 k 2,1 \N ) ϕ Finally conider the cae where P and Q are the aociate (maximal) parabolic, ay P P 2,1 and Q P 1,2. We decribe the contant term, but omit the computation. Let Eϕ P be a P -Eientein erie with cupidal data ϕ P on M P, ( ϕ P (m) ϕ P f, f() det 1) where f i a GL 2 cup form and i a complex number. ociated to thi P -Eientein erie i a Q-Eientein erie E Q ϕ with data ϕ Q on M Q, ( ϕ Q (m) ϕ Q 1 f, f() det ) Then the contant term along Q of the P -Eientein erie i of the form and imilarly, c P (Eϕ Q ) c 2,1 (Eϕ 1,2 ) a f, ϕ Q f,1 c Q (Eϕ P ) c 1,2 (Eϕ 2,1 ) b f, ϕ P f,1 The coefficient a f, and b f, are meromorphic function of..2 Functional Equation of GL 3 Eientein Serie Here we recall the derivation of the functional equation for GL 3 Eientein erie from their contant term. We et the dicuion in GL n, ince the ame argument work and are in fact clearer. The functional equation for minimal parabolic pherical Eientein erie are of the form E χ (χ, w) E wχ for all w W The exitence of uch equation follow from the functional equation of the GL 2 Eientein erie. The key i that a GL n minimal parabolic Eientein erie i the compoition of an Eientein erie for a next-to-minimal parabolic Q with omething iomorphic to a GL 2 Eientein erie. Since the Weyl group i generated by reflection, it uffice to conider intertwining operator given by a imple reflection: let σ be the Weyl element that flip the ith poitive root, and fixe all other 10

11 poitive root. The correponding next-to-minimal parabolic, Q i trictly upper triangular, except for the (i, i + 1) th entry. The quotient P k \G k i the direct um of Q k \G k and P k \Q k, which i a copy of the GL 2 quotient P GL2 (k)\gl 2 (k). So the Eientein erie i an iterated um: E χ (g) f(δγg) δ P k \Q k γ Q k \G k Conider the ubgroup H of GL n iomorphic to GL 2 that ha entrie in the two-by-two block tarting at the (i, i) th entry. For fixed g in G GL n, the map f g on H given by h f(hg) i in the GL 2 principal erie I χ (where χ i retricted to P H). So, for fixed g G the erie Ẽ χ,g (h) δ P k \Q k f g (δh) i a GL 2 Eientein erie. Parameterize χ by ( 1,..., n ) C n. Then the action of σ on χ interchange i and i+1. In the GL 2 cae, we uually take a quotient by the center, which enable u to parameterize χ by one C, and the action of σ i 1. The familiar functional equation of GL 2 Eientein erie can be retated a E χ (h) (χ, σ) E σχ (h) pplying thi to the iterated Eientein erie, we obtain the functional equation for GL n minimal-parabolic Eientein erie. E χ (g) Ẽ χ,γg (h) (χ, σ) Ẽ σχ,γg (h) (χ, σ) E σχ (g) γ Q k \G k γ Q k \G k Now we recall the way to obtain the contant (χ, w) from the contant term of E χ along P, c P (E χ ) σ W c σ (χ) σχ Taking the contant term of both ide of the functional equation yield: c P (E χ ) (χ, w) c P (E wχ ) c σ (χ) σχ (χ, w) c σ (wχ) σwχ σ W σ W c σw (χ) σwχ (χ, w) c σ (wχ) σwχ σ W σ W Since the wχ are linearly independent, c σw (χ) (χ, w) c σ (wχ) So, for all σ, w W, and, in particular, etting σ 1 give So the functional equation become (χ, w) c σw(χ) c σ (wχ) (χ, w) c w (χ) E χ c w (χ) E wχ Next we dicu the derivation of the functional equation for maximal parabolic Eientein erie from their contant term. Thi argument parallel the argument for GL 2, hinging on the fact that: (1) apart from their contant term, automorphic form are of rapid decay on Siegel et and (2) the 11

12 maximal parabolic Eientein erie have the ame Caimir eigenvalue. Uing the contant term decribed above, in Siegel et, E P f, ϕ P f, + a f, ϕ Q f,1 + (rapid decay) E Q f, ϕ Q f, + b f, ϕ P f,1 + (rapid decay) Manipulate the econd equation to obtain cancellation: end 1 and divide by b f,1. 1 b f,1 E Q f,1 ϕ P f, + 1 b f,1 ϕ Q f,1 + (rapid decay) Now ubtracting from E P f,, E P f, 1 b f,1 E Q f,1 ( a f, 1 b f,1 ) ϕ Q f,1 + (rapid decay) For Re(1 ) 0, thi difference i in L 2, and it i an eigenfunction for Caimir. By the elf-adjointne of Caimir, the eigenvalue mut be negative real. However, there i a continuum of eigenvalue for which thi i not true, and o the identity principle implie that the difference i identically zero, proving the functional equation. E P f, 1 b f,1 E Q f,1 12