CUBIC UNIPOTENT ARTHUR PARAMETERS AND MULTIPLICITIES OF SQUARE INTEGRABLE AUTOMORPHIC FORMS

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1 CUBIC UNIPOTNT ARTHUR PARAMTRS AND MULTIPLICITIS OF SQUAR INTGRABL AUTOMORPHIC FORMS W TCK GAN, NADYA GURVICH AND DIHUA JIANG 1. Introduction Let G be a connected simple linear algebraic group defined over a number field F. It is a fundamental problem in number theory and the theory of automorphic forms to describe the spectral decomposition of the unitary representation L 2 (G(F )\G(A)) of G(A). By abstract results of functional analysis, such a unitary representation possesses an orthogonal decompostion L 2 (G(F )\G(A)) = L 2 d (G(F )\G(A)) L2 cont(g(f )\G(A)) into the direct sum of its discrete spectrum and its continuous spectrum. The theory of isenstein series reduces the description of L 2 cont(g(f )\G(A)) to that of the discrete spectrum of certain reductive subgroups of G, and thus the basic question is the understanding of the discrete spectrum L 2 d (G(F )\G(A)). The discrete spectrum has a further orthogonal decomposition L 2 d (G(F )\G(A)) = L2 cusp L 2 res where L 2 cusp is the subspace of cusp forms, and L 2 res is the so-called residual spectrum. Let us write: L 2 cusp = ˆ π m cusp (π) π and L 2 res = ˆ π m res (π) π. It is known that m cusp (π) and m res (π) are finite. Two simple-minded questions can now be raised: Does there exist π such that m cusp (π) 0 and m res (π) 0? Can the collection of integers m cusp (π)} be unbounded? Here is a sample of some prior results on these questions: (i) When G = PGL n, the results of Jacquet-Shalika [JS] and the multiplicity one theorem imply that the answers are negative for both questions. (ii) When G = SL 2, it is a recent result of Ramakrishnan [R] that m cusp (π) 1 for any π. (iii) For a more general classical group G, it is known that the multiplicities in L 2 d or L2 cusp can be > 1. xamples of such failure of multiplicity one were constructed by Labesse- Langlands [LL] for the inner forms of SL 2, by Blasius [B] for SL n (with n 3) and by Li [L] for quaternionic unitary groups. However, in these examples, the multiplicities are bounded above for the given G and the given number field F. 1

2 2 W TCK GAN, NADYA GURVICH AND DIHUA JIANG In view of these results, it is commonly believed that the two phenomena described above do not arise when G is a classical group. In this paper, we show on the contrary that both phenomena do occur when G is the split exceptional group of type G 2. As an example, for each finite set S of places of F, with #S 2, we construct an irreducible unitary representation π S of G 2 (A) with m res (π S ) = 1, (1.1) m cusp (π S ) 1 6 (2#S + ( 1) #S 2) 1. We now briefly explain how one constructs the representation π S and demonstrates (1.1). Let S 3 be the finite algebraic group over F determined by the symmetric group on 3 letters. Then S 3 G 2 is a dual reductive pair in the disconnected linear algebraic group H = Spin 8 S 3. For each place v of F, the group H(F v ) has a distinguished representation Π v known as the minimal representation. Restricting the representation Π v to the subgroup S 3 (F v ) G 2 (F v ), we can write: Π v = η v π ηv. η v S 3 (F v) In the beautiful papers [HMS] and [V], Huang-Magaard-Savin (for non-archimedean v) and Vogan (for archimedean v) showed that each π ηv is a non-zero irreducible unitary representation and the π ηv s are mutually distinct. Moreover, if 1 denotes the trivial representation of S 3, then π 1 is unramified. Consider now the global situation. Let η = v η v be an irreducible representation of the compact group S 3 (A), so that η v is the trivial character for almost all v. Set π η = v π ηv. Using the global theta correspondence furnished by the dual pair S 3 G 2 in H, we show: Theorem 1.2. There is a natural embedding of vector spaces Hom S3 (A)(η, L 2 (S 3 (F )\ S 3 (A))) Hom G2 (A)(π η, L 2 d (G 2(F )\ G 2 (A))), and thus a natural G 2 (A)-equivariant embedding Hom S3 (A)(η, L 2 (S 3 (F )\ S 3 (A))) π η L 2 d (G 2(F )\ G 2 (A)). The above theorem is a special case of our main result (Theorem 8.2). To obtain the representation π S of (1.1), with S a given finite set of places of F, let r be the two-dimensional irreducible representation of the finite group S 3 and set: η S = ( v S r) ( v / S 1); π S = π ηs = ( v S π r) ( v / S π 1). The theorem implies that m res (π S ) + m cusp (π S ) m ηs := dim Hom S3 (A)(η S, L 2 (S 3 (F )\ S 3 (A))). The number m ηs is simply the multiplicity of η S in the space L 2 (S 3 (F )\ S 3 (A)) of automorphic forms on S 3. By the Peter-Weyl theorem, we see that m ηs = dim(η S 3(F ) S ),

3 CUBIC UNIPOTNT ARTHUR PARAMTRS 3 the multiplicity of the trivial representation in the restriction of η S to S 3 (F ). A simple character computation now gives: m ηs = 1 6 (2#S + ( 1) #S 2) if #S 1. On the other hand, H. Kim [K] and S. Zampera [Z] have determined the residual spectrum L 2 res completely. From their results, one sees that m res (π S ) = 1 if #S 1 and thus (1.1) follows immediately. The occurrence of π S in L 2 d (G 2(F )\ G 2 (A)) with multiplicity m ηs is predicted by the conjecture of Arthur on the discrete spectrum (cf. [A1] and [A2]), as we explain in Section 2 which serves as an extended introduction. We shall see that to every étale cubic F -algebra, one can naturally associate an Arthur parameter ψ for the group G 2 and our main result (Theorem 8.2) gives a construction of what one might hope is the global Arthur packet associated to ψ. Theorem 1.2 above is simply a special case of our main result, for the case = F F F. We now give an outline of the remainder of the paper. Section 3 introduces the relevant groups involved in this paper and Sections 4 and 5 are devoted to the properties of the local and global minimal representations respectively. There is nothing very illuminating here, but they are quite necessary and the proofs of various technical results there are deferred to the appendix at the end of the paper. The exception is the important Proposition 5.5, which is the main result of these preliminary sections. After recalling the local results of [HMS] and [V] in Section 6, we summarize in Section 7 the results of [K] and [Z] concerning the residual spectrum and give an interpretation of their results in terms of Arthur parameters. After this long preparation, the proof of our main global result (Theorem 8.2) is given in Section 8. The cuspidal representations constructed in the main theorem are non-tempered and have some very special properties. For one thing, these cusp forms are distinguished, in the sense that they have only one orbit of non-vanishing generic Fourier coefficients along the Heisenberg parabolic subgroup of G 2. Further, the cuspidal representations they afford are CAP with respect to the Borel subgroup (respectively, the non-heisenberg maximal parabolic subgroup) of G 2 if is Galois (respectively, non-galois) over F. Here, following Piatetski- Shapiro [PS], we say that a cuspidal representation π = v π v is CAP (cuspidal associated to parabolic) with respect to a parabolic P = M N if there is a cuspidal representation σ = v σ v of the Levi subgroup M such that π v is equivalent to a subquotient of Ind G(Fv) P (F σ v) v for almost all v. In [RS], Rallis and Schiffmann constructed CAP representations of G 2 with respect to the Borel subgroup and the Heisenberg parabolic. When is non-galois, the cuspidal representations constructed here seems to be the first examples of CAP representations with respect to the non-heisenberg maximal parabolic. It will be very interesting to see if these special properties characterize the cusp forms constructed in this paper. General Notations: If G is a connected reductive group over a number field F, we shall let A(G) denote the space of automorphic forms on G. This possesses a decomposition A(G) = A(G) cusp A(G) is, where A(G) cusp is the subspace of cusp forms and A(G) is is the subspace spanned by isenstein series. The definition of automorphic forms used here

4 4 W TCK GAN, NADYA GURVICH AND DIHUA JIANG differs from the usual definition (as given in [BJ]) in that we do not require an automorphic form to be K -finite, where K is a maximal compact subgroup of G(F ). In particular, A(G) is a representation of the group G(A). Similarly, instead of working with (g, K )- modules, we shall work with their canonical Casselman-Wallach globalizations (cf. [C] and [W]), which are representations of G(F ). For a smooth representation V of G(F ), we write V K for the associated (g, K )-module. 2. Cubic Unipotent Arthur Parameters In this section, we explain how the two questions raised in the introduction can be viewed in the framework of Arthur s conjecture on the discrete spectrum. Let F be a number field and let L F be the conjectural Langlands group of F. Let G be the split exceptional group over F of type G 2, so that the dual group of G is the complex Lie group G 2 (C). To each étale cubic algebra over F, we shall first attach a cubic unipotent Arthur parameter ψ : L F SL 2 (C) G 2 (C). The parameter ψ will factor through the Galois group of a finite extension of F (which is a quotient of L F ), and thus we shall simply construct a map ψ : Gal(F /F ) SL 2 (C) G 2 (C). To begin, observe that there is a natural bijection between the set of isomorphism classes of étale cubic algebras over F and the set of conjugacy classes of group homomorphisms ρ : Gal(F /F ) S 3. Here S 3 is the symmetric group on 3 letters and is the automorphism group of the split cubic algebra F F F. Henceforth, we fix the étale cubic algebra. Via ρ, we have an action of Gal(F /F ) on S 3 which allows one to define a twisted form S of the finite (constant) group scheme S 3. For any commutative F -algebra K, we have S (K) = Aut K ( K). Now let ι : SL 2 (C) G 2 (C) be a morphism furnished by the Jacobson-Morozov theorem, such that ( ) 1 1 ι 0 1 is a subregular unipotent element of G 2 (C). Note that this morphism is not injective; it factors through the quotient SO 3 (C) of SL 2 (C). It is known that the centralizer of ι(sl 2 (C)) is isomorphic to the finite group S 3, so that we have a subgroup We now set S 3 SO 3 (C) G 2 (C). ψ = ρ ι : Gal(F /F ) SL 2 (C) S 3 SO 3 (C) G 2 (C). This is the desired (global) cubic unipotent Arthur parameter. Observe that the centralizer in G 2 (C) of the image of ψ is a subgroup of S 3 isomorphic to S (F ).

5 CUBIC UNIPOTNT ARTHUR PARAMTRS 5 For each place v of F, we have a canonical conjugacy class of embeddings Gal(F v /F v ) Gal(F /F ) and thus we obtain from ψ a local Arthur parameter ψ,v : Gal(F v /F v ) SL 2 (C) G 2 (C). The centralizer of the image of ψ v is a subgroup of S 3 isomorphic to S (F v ). Remarks: According to Arthur [A2], a parameter ψ is called unipotent if it is trivial when restricted to L F. On the other hand, Moeglin [M] called a parameter ψ quadratic unipotent if its restriction to L F has image equal to a product of Z/2Z s, so that ψ LF factors through the Galois group of the compositum of finitely many quadratic extensions of F. We call the parameter ψ cubic unipotent since it factors through the automorphism group of the étale cubic algebra. We can now state Arthur s conjecture for the parameter ψ. Arthur s conjecture: For each place v and to each irreducible representation η v of the finite group S (F v ), we can attach a unitarizable admissible (possibly reducible, possibly zero) representation π ηv of G(F v ). The set A,v = π ηv : η v S (F v )} is called the local Arthur packet determined by v = F F v. It is also required that, for almost all v, π ηv is irreducible and unramified if η v is the trivial character. Further, one expects that the distribution,v = η v dim(η v ) Tr(π ηv ) is a stable distribution on G(F v ) and that certain identities involving transfer of distributions to endoscopic groups of G(F v ) should hold. We will not go into these important details here; the reader can consult [A2]. If η = v η v is an irreducible representation of S (A) (so that η v is the trivial representation for almost all v), let and π η = v π ηv m η := dim(η S (F ) ) = dim Hom S (A)(η, L 2 (S (F )\S (A))). Then there is a G(A)-equivariant embedding m η π η L 2 d (G(F )\G(A)). It is not difficult to see that the collection of numbers m η : η Ŝ(A)} is unbounded. For example, when = F F F, S = S 3 and we have seen in the introduction that for a non-empty finite set S of places of F, the representation η S of S (A) satisfies m ηs = 1 6 (2#S + ( 1) #S 2)

6 6 W TCK GAN, NADYA GURVICH AND DIHUA JIANG so that m ηs as #S. Hence, Arthur s conjecture predicts that the multiplicities of representations occurring in the discrete spectrum are unbounded when G = G 2. As the reader will undoubtedly realize by now, the key reason for this is the fact that the centralizer of the image of ψ v is non-abelian (equal to S 3 ) for infinitely many places v. This also (partially) explains why the two phenomena of the introduction do not arise in the case of classical groups: the component group of the centralizer of an Arthur parameter is always abelian there. Despite the above, Arthur s conjecture is not sufficient to imply the two phenomena highlighted in the introduction, since the conjecture does not distinguish between the cuspidal and residual spectrum (cf. the paper [M2] of Moeglin which addresses this issue conjecturally, at least for classical groups). However, the theory of isenstein series furnishes a systematic procedure for constructing the residual spectrum: L 2 res can be spanned by residues of isenstein series and can in principle be described explicitly as an abstract representation. As we mentioned earlier, for G = G 2, this has actually been carried out by H. Kim [K] and S. Zampera [Z]. Parts of the results of [K] and [Z] were established under the assumption that the symmetric cube L-function of GL 2 is entire for non-monomial cuspidal representations. This assumption has since been removed by Kim-Shahidi [KS]. As we mentioned in the introduction, a consequence of [K] and [Z] is that m res (π ηs ) = 1 if #S 1. Coupled with this, Arthur s conjecture gives affirmative answers to the two initial questions. It remains then to verify Arthur s conjectures for the parameter ψ. The results of Huang- Magaard-Savin [HMS, Prop. 6.1, Pg. 76] and Vogan [V, Thm , Pg. 788] can be viewed as a natural construction of what one might hope is the local Arthur packet A,v. Using the local packets furnished by [HMS] and [V], our main result (Theorem 8.2) can be viewed as a natural construction of what one might hope is the global Arthur packet associated to the parameter ψ. 3. Groups In this section, we set up some notations and introduce the basic objects of interest. Let F be any field of characteristic zero (for simplicity) and let be an étale cubic F -algebra, with norm map Nm. More explicitly, is one of the following: (3.1) F F F, F K, with K a quadratic extension of F, = a Galois field extension of F, a non-galois field extension of F. We shall say that is Galois in the first three cases, and non-galois otherwise. In any case, corresponds to a conjugacy class of group homomorphisms ρ : Gal(F /F ) S 3. This determines a twisted form S of the finite group scheme S 3, characterized by S (K) = Aut K ( K) for any F -algebra K.

7 CUBIC UNIPOTNT ARTHUR PARAMTRS 7 It also determines a simply-connected quasi-split group G of type D 4. Let us fix a Chevalley- Steinberg system of épinglage [BT, 4.1.3, Pg. 78] T, B, x α : G a (G ) α, α Φ} where T B is a maximal torus contained in a Borel subgroup (both defined over F ) and Φ denotes the roots of G F with respect to T F. We shall label the simple roots of G F according to the following diagram. α 1 α 2 α 3 α 4 The automorphism group Aut(G ) of G is a disconnected linear algebraic group and there is a split exact sequence: 1 G ad Aut(G ) S 1. The choice of épinglage defines a splitting of this exact sequence [Sp] and thus we can regard S as a subgroup of Aut(G ). This allows one to define the semi-direct product H = G S. The centralizer of S in H is easily seen to be a subgroup G G, isomorphic to the split exceptional group of type G 2. This gives a dual reductive pair S G H. Moreover, T = T G is a maximal split torus of G, contained in the Borel subgroup B = B G. If Φ G denotes the roots of G with respect to T, with short simple root α and long simple root β, then there is a natural restriction map Φ Φ G. Under this map, α 2 β, α i α, for i = 1, 3, 4. We now describe various parabolic subgroups of G and G. Let P = M N be the (standard) Heisenberg parabolic subgroup of G. Its Levi subgroup is (3.2) M = g Res/F GL 2 : det(g) G m } and is generated by the simple roots α 1, α 3 and α 4. We fix the isomorphism (3.2) so that the modulus character of P is given by δ P = det 5. The unipotent radical N is a 9-dimensional Heisenberg group, and thus Hom(N, G a ) is an 8-dimensional affine space on which M naturally acts. We let Ω denote the minimal non-trivial M -orbit in Hom(N, G a ). Let P = G P. Then P = M N is the (standard) Heisenberg parabolic subgroup of G. Its Levi subgroup is M = GL 2, and we pick the isomorphism so that the inclusion M M

8 8 W TCK GAN, NADYA GURVICH AND DIHUA JIANG is the natural inclusion GL 2 g Res /F GL 2 : det(g) G m }. The modulus character of P is thus δ P = det 3. The unipotent radical N is a 5-dimensional Heisenberg group, and thus Hom(N, G a ) is a 4- dimensional affine space on which M naturally acts. It is well-known that this representation of M is isomorphic to Sym 3 (F 2 ) det, and that the M(F )-orbits on Hom(N(F ), F ) are naturally parametrized by isomorphism classes of cubic F -algebras [HMS]. Moreover, the generic orbits (i.e. those of dimension 4) correspond to those cubic algebras which are étale. Given a cubic F -algebra, we shall denote by O the M(F )-orbit determined by. If is étale and x O, then the stabilizer M x (F ) of x in M(F ) is a finite subgroup isomorphic to S (F ). The inclusion N N now induces a natural projection and for x Hom(N(F ), F ), we let τ P : Hom(N, G a ) Hom(N, G a ) Ω (x) = y Ω (F ) : τ P (y) = x}. Now we have the following lemma whose proof is essentially contained in [HMS]: Lemma 3.3. Let x O (i) If, then Ω (x) is empty. be an element of a generic orbit. (ii) If =, then M x (F ) S (F ) acts naturally on Ω (x) with each group acting simply transitively. (iii) Ω (0) is empty. Now let Q = L U be the standard parabolic subgroup of G whose Levi subgroup L is generated by the simple root α 2. Then L = (SL2 Res /F G m )/ µ 2 and the isomorphism can be chosen so that the modulus character is δ Q = Nm 6. The unipotent radical U is a 3-step unipotent group and Hom(U, G a ) is a 6-dimensional affine space on which L acts. This representation of L is isomorphic to (F 2 ) and we let ω denote the set of non-zero v x such that x has rank 1. Let Q = G Q. Then Q = L U is the non-heisenberg maximal parabolic subgroup of G. Its Levi subgroup is L = GL 2, and this isomorphism is chosen so that the embedding L L is the natural one GL 2 = (SL2 G m )/ µ 2 (SL 2 Res /F G m )/ µ 2. The modulus character of Q is thus given by δ Q = det 5. Note that L and L have the same derived group SL 2. The unipotent radical is a 3-step unipotent group, and the inclusion U U induces a natural projection τ Q : Hom(U, G a ) Hom(U, G a ).

9 For x Hom(U(F ), F ), let Now we have the following simple lemma: Lemma 3.4. ω (0) is empty. CUBIC UNIPOTNT ARTHUR PARAMTRS 9 ω (x) = y ω (F ) : τ Q (y) = x}. We conclude this section with a description of the character group Hom F (L, G m ). This depends on the type of as described in (3.1). When = F F F, Hom F (L, G m ) = Zχ 1 Zχ 3 Zχ 4 where, for (g, a, b, c) SL 2 G 3 m, (3.5) On restriction to T, we have: (3.6) χ 1 : (g, a, b, c) bc, χ 3 : (g, a, b, c) ca, χ 4 : (g, a, b, c) ab. χ 1 = (α 1 + α 0 )/2, χ 3 = (α 3 + α 0 )/2, χ 4 = (α 4 + α 0 )/2, where α 0 = 2α 2 + α 1 + α 3 + α 4 is the highest root. Suppose now that = F K. Then we see by the above that Hom K (L, G m ) is generated by χ 1, χ 3 and χ 4. Assume without loss of generality that α 1 is a root defined over F, so that the action of Gal(K/F ) on Hom K (L, G m ) fixes χ 1 and exchanges χ 3 and χ 4. Hence Hom F (L, G m ) has basis χ 1, χ 3 + χ 4 }. For later purposes, it is convenient to define the following two elements of Hom F (L, G m ): µ 1 = 3χ 1 (χ 3 + χ 4 ), (3.7) µ 2 = 4χ 1 2(χ 3 + χ 4 ). These two elements generate a subgroup of index 2 in Hom F (L, G m ) and on restriction to T, we have µ 1 = α 2 + 2α 1 µ 2 = 2α 1 α 3 α 4. Finally, when is a field, and on restriction to T, Hom F (L, G m ) = Z (χ 1 + χ 3 + χ 4 ) χ 1 + χ 3 + χ 4 = 2α 0 α 2.

10 10 W TCK GAN, NADYA GURVICH AND DIHUA JIANG 4. Local Minimal Representation Let F be a local field (of characteristic zero) and let be an étale cubic F -algebra with associated homomorphism ρ : Gal(F /F ) S 3. Then G (F ) has a distinguished irreducible representation Π known as the minimal representation. This representation is trivial on the center of G (F ) and extends (non-canonically) to a representation of the adjoint group G ad (F ). In this section, we recall various properties of the minimal representation Π. A construction of the minimal representation Π was first given by Kazhdan [Ka]. However, at a crucial point in the construction, his argument is incomplete and the analytic difficulty encountered in fixing this gap is non-trivial. Fortunately, an independent treatment which bypasses this analtyic difficulty has now been given in [GaS]. In particular, all essential results of [Ka] are correct. Most of the properties of Π discussed below are contained in [Ka], [HMS] and [GaS]. Hence, we shall simply state the results and omit many of the proofs. Let r : S 3 GL 2 (C) denote the 2-dimensional irreducible representation of S 3. Then the composite r ρ is a 2-dimensional representation of Gal(F /F ). By Jacquet-Langlands [JL], this determines an irreducible admissible representation r of GL 2 (F ) given by the following table: r F F F σ(1, 1) F K σ(1, χ K ) Galois field σ(χ, χ 1 ) non-galois field monomial supercuspidal Here, χ K and χ denote the characters of F associated to K and by local class field theory, and σ(µ 1, µ 2 ) denotes the representation of GL 2 (F ) unitarily induced from the character µ 1 µ 2 of the diagonal torus. Let χ r denote the (quadratic) central character of r and define a representation of L ad (F ) = (GL 2 (F ) )/ F by: (4.1) σ = r (χ r Nm). Here, L ad is the image of L in G ad and is a Levi subgroup of the parabolic subgroup Qad (the image of Q in G ad ). For the description of Lad (F ) used above, see [HMS, Pg. 69]. Pulling back via the natural map L (F ) L ad (F ), we regard σ as a representation of L (F ). This representation of L (F ) is still irreducible (cf. [GaS, 9]). Consider the induced representation Ind G (F ) Q (F ) δ1/6 Q σ. Here and elsewhere, our induction is unnormalized. This has a unique irreducible submodule Π and it is shown in [GaS] that Π is the unique unitarizable minimal representation of G (F ). Moreover, it is self-contragredient. When F is non-archimedean, Π is spherical if and only if is unramified over F. When F is archimedean, Π is spherical if and only if = F F F.

11 CUBIC UNIPOTNT ARTHUR PARAMTRS 11 If is Galois, let ν denote the following character of M (F ): 1, if = F F F ; (4.2) ν = χ K det, if = F K; χ det, if is a field. From the above, one has: Proposition 4.3. If is Galois, then Π is the unique irreducible subrepresentation of Ind G (F ) P (F ) ν δ 1/5 P. Proof. Clearly, we have an embedding of representations of G (F ): Ind G (F ) P (F ) ν δ 1/5 P Ind G (F ) B (F ) (ν δ 1/5 P ) B (F ). On the other hand, by induction in stages through the parabolic subgroup Q, one sees that the latter representation is equal to Ind G (F ) Q (F ) δ1/6 Q σ. The proposition follows. Using this last proposition, one can extend Π to a representation of H (F ) when is Galois. More precisely, since the parabolic subgroup P and the characters ν and δ P are stable under the action of S (F ), we can define an action of S (F ) on Ind G (F ) P (F ) ν δ 1/5 P by: (s f)(g) = f(s 1 (g)), s S (F ). This action of S (F ) normalizes that of G (F ), thus preserving the unique submodule Π, and we obtain the desired extension of Π to a representation of H (F ). Of course, when is non-galois, there is no need to define an extension since H (F ) = G (F ). We now assume that F is non-archimedean. Fix a non-trivial additive (unitary) character ψ 0 of F. Composition with ψ 0 gives identifications Hom(N, G a ) = unitary characters of N (F )}, Hom(U, G a ) = unitary characters of U (F )}. The following two propositions describe the Jacquet modules of Π with respect to N and U. Proposition 4.4. (i) Let ψ be a non-trivial character of N (F ). Then 1, if ψ Ω ; dim(π ) N (F ),ψ = 0, otherwise. (ii) If = F F F, then as a representation of we have: M (F ) = (g 1, g 2, g 3 ) GL 2 (F ) 3 : det(g 1 ) = det(g 2 ) = det(g 3 )}, where after semi-simplification, 0 W (Π ) N δ 1 5 P 0 W = δ 1 5 P δ 1 5 P ((St 1 1) (1 St 1) (1 1 St)).

12 12 W TCK GAN, NADYA GURVICH AND DIHUA JIANG Here St denotes the Steinberg representation of GL 2 (F ) and St 1 1 is the representation St pr 1, where pr 1 : M (F ) GL 2 (F ) is given by pr 1 (g 1, g 2, g 3 ) = g 1. (iii) If = F K, then as a representation of we have: M (F ) = (g, h) GL 2 (F ) GL 2 (K) : det(g) = det(h)}, 0 W (Π ) N (F ) ν δ 1 5 P 0. Here, W is obtained by pulling back the representation σ( 1/2, χ K 3/2 ) of GL 2 (F ) via the projection pr : M (F ) GL 2 (F ). (iv) If is a Galois cubic field, then as a representation of M (F ), (Π ) N (F ) = δ 1 5 P (ν ν 1 ). (v) If is a non-galois field, then (Π ) N (F ) = 0. Proposition 4.5. (i) Let ψ be a non-trivial character of U (F ). Then 1, if ψ ω ; dim(π ) U (F ),ψ = 0, otherwise. (ii) If = F F F, then as a representation of L (F ), where the χ i s were defined in (3.5). (Π ) U = δ 1/6 Q σ χ 1 2 χ 3 2 χ 4 2, (iii) If = F K, then as a representation of L (F ), we have: where µ 1 and µ 2 were defined in (3.7). (iv) If is a field, then (Π ) U = δ 1/6 Q σ (χ K 2 µ 1 ) µ 2 1 (Π ) U = δ 1/6 Q σ. We now consider the case when F is archimedean. Proposition 4.6. (i) If = F F F, then where i 1, 3, 4}. (ii) If = F K, then dim Hom Q (F )(Π, χ i 2 ) 1 dim Hom Q (F )(Π, (χ K 2 µ 1 ) µ 2 1 ) 1.

13 CUBIC UNIPOTNT ARTHUR PARAMTRS 13 Proof. (i) This follows by Frobenius reciprocity and the fact that the induced representation Ind G (F ) Q (F ) χ i 2 contains exactly one spherical subquotient. (ii) This is similar to (i); one checks that the relevant induced representation contains the minimal K-type of Π with multiplicity one. 5. Global Minimal Representation Let F be a number field and an étale cubic F -algebra. In this section, we construct an automorphic realization θ : Π A(G ) L 2 (G (F )\G (A)) of the global minimal representation Π = v Π v, and study its Fourier coefficients along the two unipotent subgroups N and U. When is non-galois, the automorphic realization has been constructed by Kazhdan in [Ka]. Hence, we shall concentrate on the case when is Galois. When = F F F, the automorphic realization of Π was constructed by Ginzburg-Rallis-Soudry in [GRS] (starting from a different degenerate principal series representation). The proofs of many results in this section are standard exercises in the theory of isenstein series and we give the complete proofs for the case = F F F in the appendix at the end of the paper; the other cases are in fact easier. The only exception is Prop. 5.5 which is the main result of this section and whose proof is given at the end of the section. The 2-dimensional representation r ρ of Gal(F /F ) gives rise to an automorphic representation r = v r v of GL 2. When is Galois, r is a global principal series representation and thus can be embedded into A(GL 2 ) by the formation of isenstein series. When is non-galois, r is cuspidal automorphic. As in (4.1), we can define the automorphic representation σ = v σ v of L ad (A) and hence of L (A). Henceforth, we shall regard σ as a submodule of A(L ). Let ν be the character of F \A defined analogously as in (4.2), so that ν = v ν v. Since the minimal representation Π is self-contragredient, we deduce by Prop. 4.3 that when is Galois, dim Hom G (A)(Ind G (A) P (A) ν δ 4/5 P, Π ) = 1. For a standard K -finite section f s Ind G (A) P (A) χ δ s P, let (g, f s ) be the associated isenstein series. Then we have: Proposition 5.1. (i) Suppose that = F F F. For any standard K -finite section f s, (g, f s ) has at most a double pole at s = 4 5. This double pole is attained by the spherical section. Moreover, the space of automorphic forms spanned by the functions (s 4 5 )2 (, f s ) s=4/5 is an irreducible square-integrable automorphic representation isomorphic to Π. (ii) Suppose that = F K or is a Galois cubic field. For any standard K -finite section f s, (g, f s ) has at most a simple pole at s = 4 5. This pole is attained by some K - finite section, and the space of automorphic forms spanned by the residues of (g, f s ) at s = 4 5 is an irreducible square-integrable automorphic representation isomorphic to Π.

14 14 W TCK GAN, NADYA GURVICH AND DIHUA JIANG Proof. We give the proof of (i) in the appendix at the end of the paper. The other cases are similar. Together with [Ka, Pg ], Prop. 5.1 gives a (g, K ) G(A f )-equivariant embedding θ : (Π ) K A(G ) K L 2 (G (F )\G (A)) in all cases. By the automatic continuity theorem of Casselman-Wallach (cf. [C] and [W]), this extends to a G(A)-equivariant embedding θ : Π A(G ) L 2 (G (F )\G (A)). Moreover, we have defined in the last section an extension of Π to H (A) and the finite group S (F ) acts on A(G ) L 2 (G (F )\G (A)) by: (s f)(g) = f(s 1 (g)). It is then easy to see that the map θ is in fact S (F ) G (A)-equivariant. Let θ U : Π A(L ) be the L (A)-equivariant map defined by θ U (v)(l) = θ (v)(ul)du U (F )\U (A) for l L (A). Thus θ U is the constant term map along U. The following Proposition describes the image Im(θ U ) of this map and is the global analog of Prop It is proved in the course of proving Prop. 5.1 by examining the constant term along U of the relevant isenstein series at the point s = 4 5 ; we give the proof of (i) in the appendix. Proposition 5.2. (i) If = F F F, then Here the χ i s were defined in (3.5). (ii) If = F K, then Here the µ i s were defined in (3.7). (iii) If is a cubic field, then Similarly, we have the constant term map Im(θ U ) = δ 1/6 Q σ χ 1 2 χ 3 2 χ 4 2. Im(θ U ) = δ 1/6 Q σ (χ K 2 µ 1 ) µ 2 1. Im(θ U ) = δ 1/6 Q σ. θ N : Π A(M ), More generally, if ψ 0 : F \A S 1 is a non-trivial unitary character, then composition with ψ 0 gives an identification Hom(N, G a ) = unitary characters of N (A) trivial on N (F )}

15 CUBIC UNIPOTNT ARTHUR PARAMTRS 15 For any unitary character ψ of N (A) trivial on N (F ), we have the linear functional defined by: θ N,ψ (v) = θ N,ψ : Π C N (F )\N (A) θ (v)(n) ψ(n)dn. The following proposition describes the Fourier expansion of Π along N and is also proved in the course of proving Prop. 5.1; we give the proof of (ii) in the appendix. Proposition 5.3. (i) Suppose that ψ is non-trivial. The linear functional θ N,ψ if and only if ψ Ω (F ). is non-zero (ii) If = F F F, then as a representation of M (A), θ N (Π ) A(M ) is given by the exact sequence: 0 W θ N (Π ) δ 1/5 P 0. Here, W is defined as follows. Let V be the representation of GL 2 (A) defined by: Then there is an exact sequence 0 V σ( 1/2, 1/2 ) δ 1/5 P W δ 1/5 P ((V 1 1) (1 V 1) (1 1 V )) 0, where V 1 1 is regarded as a representation of M (A) via the first projection pr 1 : M GL 2 (as in Prop. 4.4(ii)) and so on. Moreover, W is realized as a subspace of A(M ) as follows. Let (g, f s ) be the isenstein series associated to a flat section f s σ( s, s ). Then (g, f s ) has a pole at s = 1 2 if f s is the spherical section. On the other hand, if f s is a flat section such that f 1 V, 2 then (g, f s ) is holomorphic and non-zero at s = 1 2. Thus the map f 1 (g, f 1 ) defines an 2 2 embedding of vector spaces: is : V A(GL 2 ), which is not GL 2 (A)-equivariant. Then W is spanned as a vector subspace of A(M ) by the constant functions on M (A) and the subspaces pr1 (is(v )) pr 2 (is(v )) pr 3 (is(v )). (iii) If = F K, then 0 W θ N (Π ) ν δ 1/5 P 0 where W is defined by pulling back the irreducible representation V = σ( 1/2, χ K 3/2 ) of GL 2 (A) via the projection pr : M GL 2. Moreover, if is : V A(GL 2 ) is the embedding defined by the theory of isenstein series, then W is the subspace of A(M ) consisting of the functions g is(v)(pr(g)) for v V. (iv) If is a Galois field extension of F, then θ N (Π ) = ν δ 1/5 P ν 1 δ1/5 P.

16 16 W TCK GAN, NADYA GURVICH AND DIHUA JIANG (v) If is non-galois, then θ N (Π ) = 0. The following corollary is immediate from Prop. 5.2 and Prop Corollary 5.4. Let v Π. When is Galois, θ U (v) L A(L) is, θ N (v) M A(M) is. When is non-galois, θ U (v) L A(L) cusp, (v) M = 0 A(M). θ N We come now to the main result of this section. Proposition 5.5. For any v Π, the restriction θ (v) G is a square-integrable automorphic form on G. Proof. Firstly, θ (v) G is certainly a smooth function of moderate growth, and is K -finite whenever v is a K -finite vector. Here K and K are the maximal compact subgroups of G(F ) and G (F ) respectively. Moreover, it was shown in [V] that, for v archimedean, Π v possesses an infinitesimal character when regarded as a representation of G(F v ). Hence θ (v) G is certainly Z(g v )-finite for any archimedean v. In other words, θ (v) G is an automorphic form on G. It remains to show that θ (v) G is square-integrable. As an automorphic form, θ (v) G has a decomposition [MW]: θ (v) G = f cusp + f P + f Q + f B where f cusp is a cusp from, f P has cuspidal support along P and so on. Consider the case when is Galois. By Lemmas 3.3(iii) and 3.4, as well as Prop. 5.3(i), we see that (θ (v) G ) U = θ U (v) L, (θ (v) G ) N = θ N (v) M, where (θ (v) G ) U denotes the constant term of the automorphic form θ (v) G along U. Indeed, for m M(A), we have (θ (v) G ) N (m) = θ (v)(nm)dn = N(F )\N(A) N(F )\N(A) = θ N (v)(m) + = θ N (v)(m). θ N (v)(nm) + ψ Ω (0) θ N,ψ (m) ψ Ω (F ) θ N,ψ (nm v) dn

17 CUBIC UNIPOTNT ARTHUR PARAMTRS 17 Thus Corollary 5.4 implies immediately that f P = f Q = 0. Hence it remains to show that f B is square-integrable. Since f B is concentrated along the Borel subgroup B, we need to examine the exponents of θ (v) G along B. By Prop. 5.2, one deduces that the exponents are given by: α β, 2α β, if = F F F ; α β, 2α β, if = F K; 2α β, if is a Galois field. Since all the exponents have negative coefficients, we deduce by Jacquet s square-integrability criterion [MW, Pg. 74] that θ (v) G is square-integrable. The case when is non-galois can be treated similarly; we omit the details. The proposition is proved completely. 6. Local Arthur Packets In this section, we return to the local situation and recall the results of Vogan [V, Thm , Pg. 788] and Huang-Magaard-Savin [HMS, Prop. 6.1, Pg. 76] which give a natural construction of what one might hope is the local Arthur packet A,v. Since the setting is entirely local, we shall suppress v from the notation. When σ is a tempered representation of GL 2 (F ) and Re(s) > 0, we let J P (σ, s) and J Q (σ, s) denote the Langlands quotients of the induced representations Ind G(F ) σ and Ind G(F ) Q(F ) δ 1 2 +s Q P (F ) δ 1 2 +s P σ. Let Π be the minimal representation of H (F ), as defined in Section 4. The following proposition describes the restriction of Π to the subgroup S (F ) G(F ). Proposition 6.1. As a representation of S (F ) G(F ), Π = η π η η Ŝ(F ) where the sum runs over irreducible characters η of S (F ) and each π η is a non-zero irreducible unitarizable non-generic representation of G(F ) and the π η s are mutually nonisomorphic. Remarks: How does one check that the π η s are mutually non-isomorphic? Recall that the unitary characters of N(F ) are parametrized by Hom(N(F ), F ) and the M(F )-orbits are parametrized by cubic F -algebras. Moreover, if ψ is a character of N(F ) in the M(F )-orbit O corresponding to, then the stabilizer of ψ in M(F ) is a finite subgroup isomorphic to S (F ). When F is non-archimedean, it was shown in [HMS, 1.10] that for any ψ O, (π η ) N,ψ = η as S (F )-modules. This proves that the π η s are mutually non-isomorphic without determining what each π η is. In [V, Thm ] and [HMS, 7], the irreducible representation π η is completely determined in many cases (and completely in the archimedean case). We summarize the result in the following proposition. Proposition 6.2. The following table determines π η in many instances.

18 18 W TCK GAN, NADYA GURVICH AND DIHUA JIANG η π η F F F 1 J Q (r, 1 5 ) r J P (St, 1 6 ) ɛ supercuspidal (if v finite) F K 1 J Q (r, 1 5 ) κ supercuspidal (if v finite) Galois field 1 J Q (r, 1 5 ) ν supercuspidal (if v finite) ν 2 supercuspidal (if v finite) non-galois field 1 J Q (r, 1 5 ) In the above table, ɛ is the sign character of S (F ) = S 3, κ is the non-trivial character of S (F ) = Gal(K/F ) and ν is a non-trivial (cubic) character of S (F ) = Gal(/F ). When F is archimedean, the special unipotent representations of G(F ) associated to the subregular unipotent orbit has been classified in [BV] and [V, Thm. 18.5] and are precisely the representations of G(F ) obtained in the restriction of Π. When F = R, let us describe the representations π η using the notations of [V, Thm. 18.5] and give their minimal K-types (with K = (SU 2 SU 2 )/ µ 2 ). Suppose first that = R R R. Then we have: Π = (1 J(ξ 1, (1, 0, 1))) (r J(H 2 ; (1, 1))) (ɛ J (H 2 ; (2, 0))). Here, π 1 = J(ξ 1, (1, 0, 1)) is spherical, π r = J(H 2 ; (1, 1)) has minimal K-type C 2 C 2 and π ɛ = J (H 2 ; (2, 0)) is a limit of discrete series with minimal K-type C C 3. On the other hand, when = R C, Π = (1 J(ξ x, (1, 0, 1))) (κ J(H 1 ; (1, 1)))

19 CUBIC UNIPOTNT ARTHUR PARAMTRS 19 where π 1 = J(ξ x, (1, 0, 1)) has minimal K-type C 3 C and π κ = J(H 1 ; (1, 1)) has minimal K-type C 5 C. In any case, observe that π η is completely determined when η is the trivial representation, and the description is uniform in. Also, if is unramified and η is the trivial representation, then π η is unramified. It is of course desirable to specify the representation π η completely in all cases, but for the global applications we have in mind, the results of the above Proposition are often sufficient. We shall let A = π η : η Ŝ(F )} and one can hope that A is the Arthur packet associated to the parameter ψ described in 2. How does one check that A is the right set? To do so, one would first need to check that the distribution = η dim(η) Tr(π η ) is a stable distribution. Further, one has to verify certain identities involving transfer of distributions to endoscopic groups of G. Both of these are important problems in local harmonic analysis and are beyond our means. However, there are reasons to believe that the set A is the right one. Besides the naturality of its construction, this belief should be justifiable when F = C using the results of [BV]. Moreover, there is a Langlands parameter φ ψ : L F G 2 (C) naturally associated to ψ [A2, Pg ] and defined by ( )) w φ ψ (w) = ψ (w, w. 1 2 If Σ is the L-packet associated to φ ψ, then one expects an injection Σ A [A2, Pg. 38], whose image is the singleton set containing the representation π 1 associated to the trivial character of the component group S (F ). This last expectation can actually be verified. Using Prop. 6.2, one can easily check that Σ is precisely the set π 1 } as predicted. This provides evidence for the claim that A is indeed the Arthur packet associated to ψ. 7. Arthur Parameters and Residual Spectrum In this section, we summarize the results of Kim and Zampera on the residual spectrum. It is most illuminating to formulate their results in the framework of Arthur s conjectures and hence we shall begin by describing (as much as possible) all the Arthur parameters of G = G 2. There are 5 conjugacy class of homomorphisms SL 2 (C) G 2 (C), corresponding to the 5 unipotent conjugacy classes in G 2 (C). These are: the trivial morphism, which corresponds to the identity element of G 2 (C); the long root SL 2, which corresponds to the unipotent conjugacy class of a non-trivial element in the root subgroup associated to a long root;

20 20 W TCK GAN, NADYA GURVICH AND DIHUA JIANG the short root SL 2, which corresponds to the unipotent conjugacy class of a non-trivial element in the root subgroup associated to a short root; the subregular SL 2, which corresponds to the subregular unipotent conjugacy class; the regular SL 2, which corresponds to the regular unipotent conjugacy class. The following table gives the centralizer in G 2 (C) of the image of these SL 2 s: SL 2 trivial long root short root subregular regular Centralizer G 2 (C) short root SL 2 long root SL 2 S 3 trivial Henceforth, let F be a number field and L F the conjectural Langlands group of F. We can now describe the possible Arthur parameters ψ : L F SL 2 (C) G 2 (C) and fantasize about the associated Arthur packet. For the purpose of describing the discrete spectrum, it suffices to look at those parameters ψ for which the centralizer of the image of ψ is finite. If ψ is such that ψ SL2 (C) is the trivial morphism, then ψ is a tempered parameter and for such, we have nothing intelligent to add. On the other hand, if ψ SL2 (C) is the regular morphism, then the resulting local and global Arthur packets are singletons consisting of the trivial representation of G(F v ) and G(A) respectively. We now consider the long and short root SL 2 s. From the above table, we see that there is a map ι : SL 2 (C) SL 2 (C) G 2 (C), where one copy of SL 2 is short root and the other is long root. This realizes (SL 2, SL 2 ) as a dual pair in G 2 and the kernel of ι is the diagonal µ 2. Hence the situations for the long and short root SL 2 s are entirely symmetrical. Given φ : L F SL 2 (C), set ψ = ι (φ id) : L F SL 2 (C) SL 2 (C) SL 2 (C) G 2 (C). The centralizer S ψ of the image of ψ is finite if and only if φ is an irreducible representation of L F, in which case S ψ = Z/2Z. Conjecturally, such a φ corresponds to a cuspidal representation σ φ of P GL 2 (A). We now consider the local parameter ψ v : L Fv L F G 2 (C), where L Fv is the Weil-Deligne group (resp. Weil group) of F v if v is finite (resp. archimedean), and determine the component group S ψv of the centralizer of image(ψ v ). There are two cases, depending on whether the local representation φ v : L Fv SL 2 (C) is irreducible or not; equivalently, whether the local component σ φ,v is a discrete series representation or not. Indeed, 1}, if φ v is reducible; S ψv = Z/2Z, if φ v is irreducible. Letting ɛ denote the non-trivial character of S ψv in the second case, we see that the local Arthur packet A ψv has the form A ψv = π 1 } or A ψv = π 1, π ɛ }

21 in the respective cases. CUBIC UNIPOTNT ARTHUR PARAMTRS 21 As explained at the end of the previous section, we can describe the representation π 1, at least in terms of σ φ,v. Indeed, π 1 has Langlands parameter φ ψv and is given by: J P (σ φv, 1 π 1 = the Langlands quotient 6 ), if ψ v SL2 (C) is the short root SL 2 ; J Q (σ φv, 1 10 ), if ψ v SL2 (C) is the long root SL 2. On the other hand, π ɛ is expected to be a supercuspidal representation (when v is finite), which remains to be determined. Remark: Since we do not know the Ramanujan conjecture for GL 2, the representation σ φv above is not known to be tempered. Hence, it may not be entirely accurate to call π 1 a Langlands quotient of the relevant induced representation. However, it is true that this induced representation has a unique irreducible quotient (this was checked in [Z]). In fact, the recent progress made by Kim and Shahidi towards Ramanujan conjecture shows that if σ φv is in the complementary series, it has the form σ(χ s, χ s ) for some quadratic character χ and more importantly, with s < 1 6 (this used to be an assumption in [K]). This implies that π 1 is a Langlands quotient, but its Langlands data is supported on the Borel subgroup, instead of P or Q. With the above description of the local packets, it is not difficult to write down the global Arthur packet which has 2 r(φ) elements, where r(φ) is the number of places v such that σ φ,v is a discrete series representation. Arthur s multiplicity formula says that only half of the elements in the global packet will appear (with multiplicity one) in the discrete spectrum, at least when r(φ) > 0. Of course, the only element of the global packet that we can specify is the representation π P,σφ := J P (σ φ, 1 π 1 = 6 ), if ψ v SL2 (C) is the short root SL 2 ; (7.1) π Q,σφ := J Q (σ φ, 1 10 ), if ψ v SL2 (C) is the long root SL 2, corresponding to the trivial character of v S ψ v. This is the only representation in the packet that can occur in L 2 res, since for example π ɛ is supercuspidal at a finite prime v. The other representations in this Arthur packet, if they occur in L 2 d at all, are cuspidal and nearly equivalent to π 1. Hence they are CAP with respect to P if ψ v SL2 (C) is the short root SL 2 ; Q if ψ v SL2 (C) is the long root SL 2. This concludes our discussion of the long and short root SL 2 s. We come now to the subregular parameters ψ which are parametrized by étale cubic F -algebras and have been treated in Section 2 in some detail. Fixing, we shall take the set A v constructed in the previous section as the Arthur packet. Then for each irreducible representation η = v η v of S (A), we have the global representation π η = v π ηv. Let A,res be the set of η given by the following table and such that m η 0. Here, the character ɛ and κ are as defined in Prop As an example, when = F F F, any η A,res is equal

22 22 W TCK GAN, NADYA GURVICH AND DIHUA JIANG to η S = ( r) ( 1) v S v / S for some finite set S of places of F and 1 m ηs = 6 (2#S + ( 1) #S 2), if S is non-empty; 1, if S is empty, is non-zero if and only if #S 1. η F F F no local component η v is the sign character ɛ F K no local component η v is ɛ or κ Galois field η is trivial non-galois field η is trivial We can now state the results of Kim and Zampera on the residual spectrum L 2 res. Let us write L 2 res = L 2 res(b) L 2 res(p ) L 2 res(q) where L 2 res(b) is the closed span of those automorphic forms whose cuspidal supports are along B and so on. Here is a reformulation of the results of [K] and [Z] in terms of the above picture: Proposition 7.2. L 2 res(b) = 1 L 2 res(q) = ( Galois non-galois η A,res π η. ) ( ) π 1 π Q,σ. Here the second sum is over all non-monomial cuspidal representations σ of PGL 2 for which L( 1 2, σ, Sym3 ) 0 σ

23 and π Q,σ is defined in (7.1). CUBIC UNIPOTNT ARTHUR PARAMTRS 23 L 2 res(p ) = σ π P,σ. Here the sum is over all cuspidal representation σ of P GL 2 such that and π P,σ is defined in (7.1). L( 1 2, σ) 0 The following consequence of Prop. 7.2 is what we need: Corollary 7.3. Let be an étale cubic F -algebra and η an irreducible representation of S (A). Then dim Hom G(A) (π η, L 2 1, if η A,res ; res) = 0, if η / A,res. Proof. Fixing the pair (, η), we need to show: (i) for any pair (, η ) (, η), π η π η ; (ii) for any cuspidal representation σ of P GL 2 (A), π η π P,σ and π η π Q,σ. For (i), assume first that the étale cubic algebras and are non-isomorphic. Then there are infinitely many places v of F such that η v = η v = 1 but v v. For such a place v, Prop. 6.2 gives: π ηv = JQ (r v, 1/5) and π η v = JQ (r v, 1/5). Since r v and r v are tempered and non-isomorphic, the representations π ηv and π η v have different Langlands data and are thus non-isomorphic. On the other hand, if = but η v η v for some place v, then π ηv π η v by Prop Now we come to (ii). To distinguish between π η and π P,σ, we consider the Jacquet modules of π ηv and π P,σv along P for a suitable finite place v. More precisely, there are infinitely many finite places v such that (a) v = F v F v F v and η v = 1; (b) σ v is infinite-dimensional and unramified. For such a place v, we see that as a representation of M(F v ) = GL 2 (F v ), (π ηv ) N = δ 1/3 P (1 1 St) after semi-simplification, whereas (π P,σv ) N contains the infinite-dimensional spherical representation δ 1/3 P σ v as a quotient. This shows that (π ηv ) N (π P,σv ) N and thus π ηv π P,σv. On the other hand, to distinguish between π η and π Q,σ, we take a finite place v satisfying (a) and (b) above and consider the Jacquet modules of the local representations along Q. In this case, we find that as a representation of L(F v ) = GL 2 (F v ), (π ηv ) U 3/10 = δ Q r v δ 2/5 Q

24 24 W TCK GAN, NADYA GURVICH AND DIHUA JIANG whereas (π Q,σv ) U contains the infinite-dimensional representation δ 2/5 Q σ v as a quotient. Since σ v and r v are both representations of P GL 2 (F v ), one sees that δ 2/5 Q σ v and δ 3/10 Q r v have different central characters and thus π ηv π Q,σv. The corollary is proved completely. Remarks: (i) The proposition provides further evidence that the set A v is the predicted local Arthur packet. (ii) In [K, Pg ], an attempt was made to interpret the constituents of L 2 res(b) in terms of Arthur parameters. However, the interpretation given there is not entirely accurate. To be precise, the interpretation given in [K, Pg. 1266, Case 1] and [K, Pg. 1267, Case 2], which concerns the cases = F K and a Galois field respectively, appears to be incorrect. (iii) When = F F F and η A,res, the embedding π η L 2 res was constructed by Moeglin-Waldspurger [MW, Appendix III], building on the work of Langlands [La, Appendix 3] who constructed the embedding of π 1. In this respect, let us take the opportunity to correct a minor inaccuracy in [A2, Pg. 56], where the example of Langlands was discussed. The discussion on [A2, Pg ] was meant to correct an error in [A1] concerning Langlands representation π 1. However, it was claimed incorrectly that a principal unipotent element of the subgroup SL 2 (C) SL 2 (C)/ µ 2 G 2 (C) is contained in a proper Levi subgroup of G 2 (C), when in fact such an element lies in the subregular orbit. In other words, the original description in [A1] was correct after all. The cuspidal representations in the global Arthur packet associated to ψ are nearly equivalent to π η for η A,res and are thus CAP with respect to B (resp. Q) if is Galois (resp. non-galois). In the next section, we give a construction of these cusp forms. 8. Global Arthur Packets We continue to assume that F is a number field and an étale cubic F -algebra. Given an automorphic representation η = v η v of S (A), the conjecture of Arthur predicts that π η = v π ηv occurs in the discrete spectrum of L 2 (G(F )\G(A)) with multiplicity m η. In this section, we verify this and give a construction of what one might hope is the global Arthur packet. Let Π = v Π v be the global minimal representation of H (A). As an abstract representation of S (A) G(A), we have: Π = η π η. η Ŝ(A) Hence, for each irreducible representation η of S (A), there is a unique (up to scaling) S (A) G(A)-equivariant embedding ι η : η π η Π. Now let θ : Π L 2 (G (F )\G (A))