CAP automorphic representations of U E/F (4) I. Local A-packets

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1 CAP automorphic representations of U E/F (4 I. Local A-packets Takuya KONNO, Kazuko KONNO April 16, 003 Abstract This is the first part of our two papers devoted to an explicit description of the CAP automorphic representations of quasisplit unitary groups in four variables over number fields. We describe them along the lines of Arthur s conjecture [Art89]. In this first part, we calculate candidates of the local A-packets consisting of the local components of CAP automorphic representations, by means of the local θ- correspondence from unitary groups in two variables. For this, we need a version of the ε-dichotomy property of the local θ-correspondence for unitary groups in two variables, involving the Langlands-Shahidi local factors. We also verify that Hiraga s conjecture [Hir] on a relation between ZASS duality and A-packets is compatible with our candidates. Contents 1 Introduction 3 A-parameters of CAP type for unitary groups 8.1 Settings Global A-parameters for unitary groups Local A-parameters and induced representations Local A-parameters for G Review of the local Arthur conjecture Non-supercuspidal representations of G 4 (F Graduate School of Mathematics, Kyushu University, Hakozaki, Higashi-ku, Fukuoka, Japan takuya@math.kyushu-u.ac.jp URL: tkonno/ JSPS Post Doctoral Fellow at Graduate School of Mathematics, Kyoto University, Yoshida, Sakyo-ku, Kyoto, Japan kkonno@math.h.kyoto-u.ac.jp URL: kkonno/ Partially supported by the Grants-in-Aid for Scientific Research No (T.K and No (K.K, the Ministry of Education, Science, Sports and Culture, Japan 1

2 3.4 Induced representations over the real field Explicit Langlands classification for G, G The packets Π ψ (G, Restriction rule from GU( to U( via base change The groups Representations and test functions Norm map in the theory of base change Classification of σ-conjugacy classes σ-conjugacy classes in L and L Orbital integral transfers Definition of the base change for G Restriction from GU E/F ( to GL( Relation between the transfers for GU E/F ( and GL( Comparison of the base changes for GU E/F ( and GL( Restriction from GU E/F ( to U E/F ( Local theta correspondence for unitary groups in two variables Weil representations of unitary dual pairs Splitting of the metaplectic -cocycle Weil representations Local theta correspondence for non-archimedean U( The group GU( E B Dual pairs and Weil representations for similitude groups Review of the Shimizu-Jacquet-Langlands correspondence Local theta correspondence for GU( Correspondence for unitary groups in two variables The case F = R Candidates for local A-packets of U( Non-archimedean A-packets Induction principle in the non-archimedean case Local non-archimedean A-packets Relation with ZASS involution A-packets in the archimedean case Some tools in the Howe duality over R Candidates for real A-packets Non-elliptic parameters Split case A-parameters and representations Local θ-correspondence An incomplete multiplicity formula Multiplicity pairing The multiplicity inequality

3 1 Introduction This is the first one of two articles devoted to an explicit description of the CAP automorphic representations of the quasisplit unitary group in four variables U E/F (4 associated to a quadratic extension E/F of number fields. The term CAP is a short hand for the phrase Cuspidal but Associated to Parabolic subgroups. This is the name given by Piatetski-Shapiro [PS83] to those cuspidal automorphic representations which apparently contradict the generalized Ramanujan conjecture. More precisely, let G be a connected reductive group defined over a number field F and G be its quasisplit inner form. We write A = A F for the adele ring of F. An irreducible cuspidal representation π = v π v of G(A is a CAP form if there exists a residual discrete automorphic representation π = v π v of G (A such that, at all but finite number of v, π v and π v share the same absolute values of Hecke eigenvalues. It is a consequence of results of Jacquet-Shalika [JS81b], [JS81a] and Moeglin-Waldspurger [MW89] that there are no CAP forms on the general linear groups. For a central division algebra D of dimension n over F, however, the trivial representation of D (A is a CAP form, since it shares the same local component with the residual representation 1 GL(n,A at any place v where D is unramified. Also it is known that even the quasisplit rank one unitary group U E/F (3 of three variables has non-trivial CAP forms [GR90], [GR91]. But the first and most well-known examples of CAP forms are Howe-Piatetski-Shapiro s analogue of the θ 10 representation [Sou88] and the Saito-Kurokawa representations [PS83] of Sp( (for their local structures see also [Wal91]. In any case, local components of CAP forms at almost all places are non-trivial Langlands quotients by definition, and hence non-tempered. In order to describe such forms, J. Arthur proposed a series of conjectures [Art89]. The conjectural description is through the so-called A-parameters, homomorphisms ψ from the direct product of the hypothetical Langlands group L F of F with SL(, C to the L-group L G of G: ψ : L F SL(, C L G, considered modulo Ĝ-conjugation. We write Ψ(G for the set of Ĝ-conjugacy classes of A-parameters for G. By restriction, we obtain the local component ψ v : L Fv SL(, C L G v of ψ at each place v of F. Here L Fv is the local Langlands group defined in [Kot84, 1], and L G v is the L-group of the scalar extension G v = G F F v. The local conjecture, among other things, associates to ψ v a finite set Π ψv (G v of isomorphism classes of irreducible unitarizable representations of G(F v, called an A-packet. At all but finite number of v, Π ψv (G v is expected to contain a unique unramified element πv. 1 Using such elements, we can form the global A-packet associated to ψ: Π ψ (G := { v (i π π v Π ψv (G v, at any v; v (ii π v = πv, 1 at almost all v Arthur s global conjecture predicts the multiplicity of each element of Π ψ (G in the discrete spectrum of the right regular representation of G(A on L (G(F \G(A. Let us call an A-parameter ψ is of CAP type if 3 }.

4 (i ψ is elliptic. This is the condition for Π ψ (G to contain a discrete automorphic representation. (ii ψ SL(,C is non-trivial. According to the conjecture, the CAP automorphic representations of G(A is contained in the global A-packets associated to an A-parameter of CAP type. In this paper, we construct candidates for the local A-packets associated to the local components of A-parameters of CAP type for the quasisplit unitary group U E/F (4 in four variables. Although the description tells us nothing about the character relations conjectured in [Art89], it is explicit and fairly complete. In the subsequent one [Kon], we shall develop a theory of L-functions of U E/F (4 GL( E, and obtain a complete description of CAP automorphic representations of U E/F (4 in terms of the local A-packets constructed in this article. Since the main result of the present article is a case-by-case description of the A-packets (see 6, we present here an outline of our construction. Parameter considerations In, we classify the A-parameters of CAP type for G = G 4 := U E/F (4 together with their S-groups S ψ (G := π 0 (Cent(ψ, Ĝ/Z(Ĝ, which plays a central role in the conjectural multiplicity formula. Assuming the existence of the Langlands group, we first classify the elliptic A-parameters for general unitary group G n in n-variables (Prop... In practice, we need to replace the role of the Langlands group by the classification of automorphic representations of general linear groups. Thanks to Rogawski s result on the base change for G = U E/F (, this replacement is available for the A-parameters of CAP type for G (see Cor..3. Then we have a similar description for their local components in 3.1. At this point, we need to assume some local assertions of Arthur s conjecture, which we review in 3.. In particular, for each local component ψ of a CAP type A-parameter, we have the associated non-tempered Langlands parameter φ ψ. It is imposed that the L- packet Π φψ (G corresponding to φ ψ should be contained in Π ψ (G. In fact, we know from [Kon98], [Kon01] that Π φψ (G consist exactly of the local components of residual discrete representations. We have the S-group S ψ (G as in the global case. Let us postulate the following slightly stronger assertion than Arthur s conjecture, which might not be true in general. Assumption 1.1. There exists a bijection Π ψ (G π ( s s, π ψ Π(S ψ (G. Here Π(S ψ (G is the set of isomorphism classes of irreducible representations of S ψ (G. There are six types of elliptic A-parameters appearing as local components of the A- parameters of CAP type. According to the assumption, for three types of the six, Π ψ (G coincides with Π φψ (G so that no construction beyond [Kon98] is necessary. But for the rest three types, Π φψ (G contains only the half of the members of Π ψ (G. 4

5 Adams conjecture We use the local θ-correspondence to construct the set Π ψ (G \ Π φψ (G of missing members. Consider an m-dimensional hermitian space (V, (, and an n-dimensional skew-hermitian space (W,, over E. The unitary groups G(V and G(W for V and W form a type I dual reductive pair. For a pair ξ = (ξ, ξ of characters of E satisfying ξ F = ωe/f n, ξ F = ωe/f m, we have the Weil representation ω V,W,ξ = ω W,ξ ω V,ξ of G(V G(W 5.1 (cf. [HKS96]. We write R(G(V, ω W,ξ for the set of isomorphism classes of irreducible admissible representations of G(V which appear as quotients of ω W,ξ. For π V R(G(V, ω W,ξ, the maximal π V -isotypic quotient of ω V,W,ξ is of the form π V Θ ξ (π V, W for some smooth representation Θ ξ (π V, W of G(W. Similarly we have R(G(W, ω V,ξ and Θ ξ (π W, V for each π W R(G(W, ω V,ξ. The local Howe duality conjecture, which was proved by R. Howe himself if F is archimedean [How89] and by Waldspurger if F is a nonarchimedean local field of odd residual characteristic [Wal90], asserts the following: (i Θ ξ (π V, W (resp. Θ ξ (π W, V is an admissible representation of finite length of G(W (resp. G(V, so that it admits an irreducible quotient. (ii Moreover its irreducible quotient θ ξ (π V, W (resp. θ ξ (π W, V is unique. (iii π V θ ξ (π V, W, π W θ ξ (π W, V are bijections between R(G(V, ω W,ξ and R(G(W, ω V,ξ converse to each other. A link between local θ-correspondence and A-packets is given by the following conjectures of J. Adams [Ada89], which will be taken as a leading principle in our construction. Suppose n m. Then we have an L-embedding i V,W,ξ : L G(V L G(W given by ( ξ ξ 1 g (w w if w W E 1 n m i V,W,ξ (g w = ( g J n m 1 n m w σ if w = w σ, where w σ is a fixed element in W F \ W E and J n = diag(1, 1,..., ( 1 n 1. Let T : SL(, C Cent(i V,W,ξ, Ĝ(W be the homomorphism which corresponds to a regular unipotent element in Cent(i V,W,ξ, Ĝ(W GL(n m, C (the tail representation. Using this, we define the θ-lifting of A-parameters by θ V,W,ξ : Ψ(G(V ψ (i V,W,ξ ψ T Ψ(G(W. Conjecture 1. ([Ada89] Conj.A. The local θ-correspondence should be subordinated to the map of A-packets Ψ ψ (G(V Π θv,w,ξ (ψ(g(w. Here we have said subordinated because Π ψ (G(V R(G(V, ω W,ξ is often strictly smaller than Π ψ (G(V. When these two are assured to coincide, we can expect more: Conjecture 1.3 ([Ada89] Conj.B. For V, W in the stable range, that is, the Witt index of W is larger than m, we have Π θv,w,ξ (ψ(g(w = θ ξ (Π ψ (G(V, W. V ; dim E V =m 5

6 ε-dichotomy Consider the special case m = and W = V V. It turns out that the three types of A-parameters in question are exactly those of the form ψ = θ V,W,ξ (ψ 1 for some elliptic A-parameters ψ 1 for G. Thus we should be able to construct the desired candidates by calculating θ(τ V, W, τ V Π ψ 1 (G(V for various V. But for our global applications, we also need a parametrization of the members of Π ψ (G well-suited to the conjectural multiplicity formula. For this, we need one more ingredient. For the purpose of this introduction, it is best to restrict ourselves to the non-archimedean case. For the corresponding result in the case F = R, see 5.3. Theorem 1.4 (Th Suppose dim E V = and write W 1 for the hyperbolic skewhermitian space (E, ( Take an L-packet Π of G (F = G(W and τ Π [Rog90, Ch.11]. (i τ R(G(W, ω V,ξ if and only if ε(1/, Π ξξ 1, ψ F ω Π ( 1λ(E/F, ψ F = ω E/F ( det V. Here the ε-factor on the right hand side is the standard ε-factor for G twisted by ξξ 1. ω Π is the central character of the elements of Π, and λ(e/f, ψ F is the Langlands λ-factor [Lan70]. (ii If this is the case, θ ξ (τ, V = (ξ 1 ξ G(V τ V. Here (ξ 1 ξ G(V denotes the character G(V det U E/F (1, F z/σ(z ξ 1 ξ (z C. Recall the Jacquet-Langlands correspondence Π Π V between the (discrete if V is anisotropic L-packets of G (F and those of G(V [LL79]. Π τ τ V Π V is a certain bijection, which we cannot specify explicitly. This is a special case of the ε-dichotomy property of the local θ-correspondence for unitary groups over p-adic fields, which was proved for general unitary groups (at least for supercuspidal representations in [HKS96]. But since we need to combine this with our description of the residual spectrum [Kon98], we have to use the Langlands-Shahidi ε- factors [Sha90] instead of Piatetski-Shapiro-Rallis s doubling ε-factors used in that paper. By this reason, we deduce the theorem from the analogous result for unitary similitude groups [Har93] in 5.. The key is the following description of the base change lifting of representations of G (F to GL(, E, which is proved at length in 4. Proposition 1.5 (Cor Let π = ω π be an irreducible admissible representation of the unitary similitude group GU E/F ( E GL(, F / F, and write Π( π for the associated L-packet of G (F consisting of the irreducible components of π G (F. Then the standard base change of Π( π to GL(, E [Rog90, 11.4] is given by ω(detπ E, where π E is the base change lift of π to GL(, E [Lan80]. Candidates for the A-packets The candidates of the A-packets are constructed in 6. Here we explain the construction in the non-archimedean case ( 6.1, which is summarized in Fig. 1. Each A-parameter of our concern (or its restriction to L E SL(, C is of the form ψ = ψ 1 (ξ ξ 1 ρ, 6

7 π θ ξ (,W π + G(V π V θ ξ (,W J-L corr. G(V π V τ θ ξ (,V G(W = G 4 (F Witt tower G(W 1 = G (F Figure 1: Construction of A-packets where ψ 1 is an elliptic A-parameter for G. Take τ Π ψ1 (G and let (V, (, be the -dimensional hermitian space such that the condition of Th. 1.4 (i holds. If we write π V := θ ξ (τ, V (ξξ 1 G(V τv, then the induction principle in local θ-correspondence [Kud86] ( implies that π + := θ ξ (π V, W, (τ Π ψ1 (G form the local residual L-packet Π φψ (G 4. Next we take π V in the Jacquet-Langlands correspondent of the A- packet θ ξ (Π ψ1 (G, V, an irreducible representation of the unitary group G(V of the other (isometry class of -dimensional hermitian space V. Then π := θ ξ (π V, W is the so-called early lift or first occurrence. In particular, if π V is supercuspidal then so isπ. Following Conj. 1.3, we define Π ψ (G := {π ± τ Π ψ (G }. This gives the sufficiently many members of the packet as predicted by Assumption 1.1. Analogous construction in the archimedean case is given in 6.3. In 6., we show that our candidates of the A-packets are compatible with Hiraga s conjecture [Hir] on a relation between ZASS (Zelevinsky-Aubert-Schneider-Stühler duality and the A-packets. Although the global A-parameters of CAP type are elliptic, some of their local components are not. For later use, we also describe these non-elliptic A-packets in terms of local θ-correspondence in 6.4. Also the split case E F F, G GL(4 F is treated in 6.5. In the final section 7, we give an example of the half of the multiplicity formula in order to motivate the reader for the subsequent article. The result (Th. 7.1 is not satisfactory but it clarifies the role of global root numbers in the multiplicity formula, which is one of the main features in Arthur s conjecture. Parts of this work was done while the authors stayed at Johns-Hopkins University during the JAMI special period on Automorphic forms and Shimura varieties in 000. The authors heartily thank the staffs of Johns-Hopkins University, especially to Prof. S. Zucker, for their hospitality and encouragement. Also we are grateful to Prof. H. Yoshida for giving us a chance to participate the JAMI program. Notation For a subset S of a group G, we write Cent(S, G and Norm(S, G for the centralizer and normalizer of S in G, respectively. For a topological group or an algebraic group G, write G 0 for its connected component of the identity. diag(x 1,..., x r or 7

8 diag({x i } r i=1 stands for the matrix x 1 x..., x r where x i are square matrices. We write M n,m (R and M n (R for the R-modules of n m and n n matrices with entries in a ring R, respectively. A-parameters of CAP type for unitary groups We start with a classification of the elliptic A-parameters for unitary groups associated to a quadratic extension of number fields. The parameters which we shall treat in this paper are listed in Cor..3.1 Settings Throughout this section let E be a quadratic extension of a number field F. We write σ for the generator of the Galois group Γ E/F of this extension. We fix an algebraic closure F of F containing E. Γ and W F denote the Galois and Weil group of F over F, respectively. Also we need the hypothetical Langlands group L F of F. We use this only to describe A-parameters, and in practical considerations we use the classification of automorphic representations of GL(n instead. As for the theory of endoscopy and the notation related to it, we follow the book [KS99]. Writing 1 I n := 1.., ( 1 n 1 we introduce the outer automorphism θ n (g := Ad(I n t g 1 of GL(n. Let us realize the quasisplit unitary group G n associated to E/F in such a way that G n (R = {g GL(n, R F E θ n (σ(g = g} for any commutative F -algebra R. By the choice of θ n, the usual splitting of GL(n E G n F E gives a splitting spl Gn = (B n, T n, {X} stable under Γ E/F (an F -splitting. We fix the following set of L-group data (Ĝn, ρ Gn, η Gn for G n : Ĝ n = GL(n, C, { id if w W E, ρ Gn (w = otherwise, θ n and η Gn is the obvious identification between the dual of the based root datum for G n and the based root datum of Ĝn. The L-group of G n is the semidirect product L G n = Ĝ n ρgn W F. 8

9 Also we let H n := R E/F GL(n, where R E/F stands for Weil s restriction of scalars. The double copy of the standard splitting of GL(n defines a splitting spl Hn = (B H n, T H n, {X Hn } of H n F E GL(n E, which is actually an F -splitting of H n. The L-group L H n = Ĥ n ρhn W F is given by Ĥn = GL(n, C and { (h 1, h if w W E, ρ Hn (w(h 1, h = (h, h 1 otherwise, for (h 1, h GL(n, C. For the moment, consider a general connected reductive group G defined over F with the L-group L G. We define A-parameters for G as in the introduction [Art89]. In practice, we need to drop the boundedness of ψ(l F because of the lack of the generalized Ramanujan conjecture. We shall give an alternative condition which is accessible but more technical in 3.. Note that the classification results given below is not affected by this change. Two A-parameters are equivalent if they are Ĝ-conjugate. The set of equivalence classes of A-parameters for G is denoted by Ψ(G. For ψ Ψ(G, we write S ψ (G for the centralizer of the image of ψ in Ĝ, and set S ψ(g := S ψ (G/S ψ (G 0Z(ĜΓ. An A-parameter ψ is elliptic if S ψ (G 0 is contained in Z(ĜΓ, or equivalently, S ψ (G 0 ÂG with ÂG := (Z(ĜΓ 0. For G = G n, ÂG is trivial so that this condition is equivalent to the discreteness of S ψ (G. We write Ψ 0 (G for the subset elliptic elements in Ψ(G. Ψ CAP (G denotes the subset of elements of CAP type, defined in the introduction, of Ψ 0 (G. In what follows, we write A F := L F SL(, C for brevity. We fix w σ W F \ W E so that A F = A E w σ A E.. Global A-parameters for unitary groups Here we give a description of Ψ 0 (G n. This begins with an irreducible decomposition. Let ψ Ψ 0 (G n. Thanks to the semisimplicity of Langlands parameter, the representation ψ AE : A E GL(n, C admits an irreducible decomposition. Moreover looking at the action of A F /A E Γ E/F, this decomposition must be of the form r s ψ AE (τ j σ(τ j m j, (.1 i=1 τ m i i j=r+1 where σ(τ j := τ j Ad(w σ, and we have imposed { τ i σ(τ i for 1 i r, τ i τ j, if i j, τ j σ(τ j for r + 1 j s. Let V i be the τ i - (resp. τ i σ(τ i - isotypic subspace of ψ AE for 1 i r (resp. r + 1 i s. We may assume that s i=1 GL(V i is the standard Levi subgroup s i=1 GL(n i with n i := dim V i. Write ψ(w σ = ψ(w σ 0 w σ with ψ(w σ 0 Ĝn. For this to preserve each GL(n i -component, ψ(w σ 0 must be of the form ψ(w σ 0 = x s.. x 9 x 1, x i GL(n i, C.

10 Thus writing ˆn i := s j=i+1 n j, ň i := i 1 j=1 n j, we have ρ Gn (w σ ψ(w σ 0 = ( 1ˆn 1 I n1 = ( 1ˆn 1 θ n1 (x 1 ( 1ˆn I n.. I ns ( 1ň+ˆn θ n (x t (ψ(w σ I 1 n 1 ( 1ňs θ ns (x s ( 1ň I 1 n.. ( 1ňs In 1 s noting ň i + ˆn i = n n i, = ( 1 n n 1 θ n1 (x 1 ( 1 n n θ n (x.. ( 1 n ns θ ns (x s This gives diag({( 1 n n i x i θ ni (x i } s i=1 w σ = ψ(w σ 0 ρ Gn (w σ (ψ(w σ 0 w σ = ψ(w σ = ψ(w σ = diag({τ i (w σ m i } r i=1, {(τ i σ(τ i (w σ m i } s i=r+1 w σ. (. We write A and A E for the adele rings of F and E, respectively. ω E/F denotes the quadratic idele class character of F associated to E/F by the classfield theory. For each 1 i s, we shall fix a character ω i of A E /E whose restriction to A equals to ω n n i E/F. Define ψ i : A F L G ni, (1 i s by { ω 1 i τ m i i p WE for 1 i r, ψ i AE = ω 1 ψ i (τ i σ(τ i m i (w σ = x i w σ, i p WE for r + 1 i s, where p WE denotes the conjectural homomorphism L E W E. ω i are identified with characters of W E by the classfield theory. Lemma.1. (i ψ i is an well-defined A-parameter for G ni for each 1 i s. (ii ψ Ψ 0 (G n if and only if ψ i Ψ 0 (G ni for all 1 i s. Proof. (i follows immediately from (.. As for (ii, we have only to note that Schur s lemma implies s S ψ (G = Cent(ψ, Cent(ψ(A E, Ĝn = Cent(ψ, GL(n i, C = i=1 s S ψi (G ni. i=1 10

11 Thus we are reduced to study the ellipticity of each ψ i. First consider the case 1 i r. Writing V τi for the space of τ i, we have V i = V τi C m i and S ψi (G ni = Cent(ψ i (w σ, 1 Vτi GL(m i, C. Then ι := Ad(ψ i (w σ acts on GL(m i, C as an automorphism of order two, because ψ i (wσ = τ i (wσ 1 mi. For ψ i to be in Ψ 0 (G ni, it is necessary and sufficient that GL(m i, C ι is finite. Since Aut(GL(m i = Int(GL(m i θ mi, this forces m i = 1. Thus n i = dim τ i and ψ i (w σ acts as Ad(ψ i (w σ 0 θ ni on τ i. This also implies σ(τ i is isomorphic to τi, the contragredient of τ i. Next consider r + 1 i s. Similar argument as above shows m i = 1. We have an irreducible decomposition V i = V τi V σ(τi as an A E -module and S ψi (G ni = Cent(ψ i (w σ, C id Vτi C id Vσ(τi. Since Ad(ψ i (w σ ψ i AE = ψ i AE Ad(w σ, ψ i (w σ must interchange V τi and V σ(τi. These show that S ψi (G ni equals the diagonal subgroup C (id Vτi Ad(ψ i (w σ id Vτi. Hence such ψ i cannot be elliptic. Define Φ st 0 (G n to be the set of isomorphism classes of Langlands parameters ϕ : L F L G n such that Imϕ is bounded. ϕ AE is irreducible. For m N, let ρ m be the m-dimensional irreducible representation of SL(, C. Proposition.. Ψ 0 (G n is in bijection with the set of finite families of quadruples (d i, m i, ω i, ϕ i r i=1: (i d i, m i are positive integers satisfying n = r i=1 d im i. (ii ω i is an idele class character of E such that ω i A = ω n d i m i +1 E/F. (iii ϕ i Φ st 0 (G mi. such that ω i ϕ i ρ di ω j ϕ j ρ dj for 1 i j r. The bijection is given by the relation ψ AE = ω 1 (ϕ 1 LE ρ d1..., ψ(w σ = ϕ r (w σ Jm dr 1 r 1 dr ω r (ϕ r LE ρ dr ϕ 1 (w σ J d 1 1 m 1 1 d1.. w σ. Here J m = diag(1, 1,..., ( 1 m 1. 11

12 Proof. We retain the notation of the above argument. We may write ψ i AE ϕ i ρ di. Here ϕ i is an irreducible representation of L E with bounded image whose dimension we denote by m i. We obviously have n = r i=1 d im i. Since σ(ψ i AE (ψ i AE, we may choose an isomorphism ϕ i(w σ : θ mi ϕ i Ad(w σ. This must satisfy Ad(ϕ i(w σ θ mi (ϕ i(w σ ϕ i = ϕ i Ad(w σ = Ad(ϕ i(w σ ϕ i. The irreducibility of ϕ i implies ϕ i(w σ θ mi (ϕ i(w σ = zϕ i(wσ, for some z C. Moreover, applying Ad(ϕ i(w σ θ mi to this, we see zϕ i(w σ =ϕ i(w σ θ mi (ϕ i(w σ = Ad(ϕ i(w σ θ mi (ϕ i(w σ θ mi (ϕ i(w σ =z 1 Ad(ϕ i(w σ θ mi (ϕ i(w σ = z 1 ϕ i(ad(w σ w σ =z 1 ϕ i(w σ, so that z = ±1. Take an idele class character ω i of E satisfying ω i A ɛ i Z/Z is such that ( 1 ɛ i = z. Now one can easily check that { ω 1 i (wϕ ϕ i (w := i(w p WE (w if w L E, ϕ i(w σ w σ if w = w σ ϕ i = ω ɛ i E/F, where defines an well-defined element ϕ i Φ st 0 (G mi. We still have to determine ɛ i. For this, we note the tensor product decomposition θ ni = Ad(J d i 1 m i θ mi θ di. If we write ψ i (w σ = g i (ϕ 0 i (w σ J d i 1 m i 1 di w σ for some g i GL(n i, C, we have (ω i ϕ 0 i (w σ wwσ 1 ρ di (g = Ad(ψ i (w σ ψi 0 (w, g =Ad(g i (Ad(ϕ i (w σ (ω i ϕ 0 i (w ρ di (g =Ad(g i ((ω i ϕ 0 i (w σ wwσ 1 ρ di (g, for any w L E, g SL(, C. Thus g i must be a scalar matrix which, after a suitable conjugation in Ĝn i, we may assume to be 1 ni. We now use the equality between ψ 0 i (w σ =(ϕ i (w σ 0 J d i 1 m i =ϕ i (w σ 0 J d i 1 m i J d i 1 m i =( 1 (m i 1(d i 1 ϕ i (wσ 0 1 di 1 di θ ni (ϕ i (w σ 0 J d i 1 m i 1 di θ mi (ϕ i (w σ 0 θ mi (J d i 1 m i J 1 d i m i 1 di and ψi 0 (wσ = ω i (wσϕ 0 i (wσ 1 di to see that ɛ i (d i 1(m i 1 mod. We set ω i := ω iω i. We specialize this to the case n = 4. Φ 0 (H n denotes the set of equivalence (i.e. Ĥ n - conjugacy classes of elliptic Langlands parameters with bounded image. If we embed Ĥn diagonally into GL(n, C, Φ 0 (H n is exactly the set of induced representations ind L F L E ϕ E 1

13 where ϕ E runs over the set of isomorphism classes of irreducible n-dimensional representations of L E with bounded image. Then conjecturally, this is in bijection with the set of irreducible cuspidal representations of H n (A. Recall the standard base change map: ξ 1 : L G n g w (g, θ n (g w L H n. The map Φ st 0 (G n ϕ ξ 1 ϕ Φ 0 (H n is an well-defined injection. For n 3, its image consists of ϕ = ind L F L E ϕ E such that the corresponding irreducible cuspidal rerpesentation Π E of H n (A satisfies: (i σ(π E := Π E σ Π E. This forces that its central character ω Π E restricted to N E/F (A E is trivial. (ii ω ΠE is trivial on A. (iii The twisted tensor (generalized Asai-Oda L-function L Asai (s, Π E does not have a pole at s = 1 in the case n =. In fact, the stable cuspidal L-packets of G n (A lift exactly to these cuspidal representations by the base change lift corresponding to ξ 1 [Rog90, 11.5, Th ]. Thus, in practice, we can parametrize each element ϕ Π Φ st 0 (G n, (n 3 by the irreducible cuspidal representation Π E = ξ 1 (Π of H n (A satisfying the above conditions. We save η and µ to denote idele class characters of E whose restriction to A equal 1 and ω E/F, respectively. Corollary.3. The set Ψ CAP (G 4 consists of the following elements. (1 Stable parameters. Name ψ AE ψ(w σ (1.a ψ η (η ρ 4 p WE 1 4 w σ (1.b ψ Π,µ [µ(ϕ 0 Π L E ρ ] p WE [ϕ 0 Π (w σj 1 ] w σ Here Π runs over the set of stable L-packets of G (A containing a cuspidal representation. ( Endoscopic parameters. Name ψ AE ( ψ(w σ (.a ψ µ ((µ ρ 3 µ 1 p 3 WE w 1 σ ( (.b ψ Π,η ((η ρ ϕ 0 Π 1 L E p WE ϕ 0 Π (w w σ ( σ (.c ψ η ((η ρ (η 1 ρ p WE w 1 σ (.d ψ η,µ ((η ρ µ µ p WE w σ Here, in (.a µ = (µ, µ and µ can be µ, in (.b Π and ϕ Π are the same as in (1.b, in (.c η = (η, η with η η, and in (.d µ = (µ, µ with µ µ. 13

14 3 Local A-parameters and induced representations From now on, we turn to the local situation and study the local A-packets associated to the parameters listed in Cor..3. Thus let E be a quadratic extension of a local field F of characteristic zero. We write F for the module of F. We adopt analogous notation as in the global setting such as Γ E/F := Gal(E/F = σ, Γ = Gal( F /F, W F and w σ W F \ W E. The Langlands group of F is given by ([Kot84, 1] { W F if v is archimedean, L F := W F SU( if v is non-archimedean. 3.1 Local A-parameters for G 4 First we recall some general construction from [Art89]. Let G be a quasisplit connected reductive group over F and L G be its L-group. An A-parameter for G is a continuous homomorphism ψ : L F SL(, C L G such that (i ψ WF is semisimple and has bounded image. ψ (ii The composite W F L G pr W F is the identity. (iii ψ restricted to SL(, C or SU( SL(, C is analytic. Again we remark that the boundedness condition in (i might be too strong for our purpose, since the generalized Ramanujan conjecture is not yet known. An appropriate alternative condition will be given in 3. (Rem The notion of equivalence and ellipticity for A-parameters are defined in the same manner as in the global case. Also the groups S ψ (G and S ψ (G are defined. Although we consider only elliptic global parameters, the same is not always true for their local components. Thus we have to classify the whole Ψ(G in the local case. For this, we use the reduction argument of [Art89, 7]. Take a maximal torus A ψ in S ψ (G and set M ψ := Cent(A ψ, Ĝ. Recall that we fixed a splitting spl bg = (B, T, {X } of Ĝ in the description of L G. Taking a suitable Ĝ-conjugate of ψ, we may assume A ψ T and that M ψ is a standard Levi subgroup of Ĝ with respect to (B, T. Thus we have a standard parabolic subgroup P ψ = M ψ Û ψ of Ĝ. One can check that this is stable under the Γ-action ρ G. In fact, since ψ(w F commutes with A ψ, Ad(ψ(W F preserves the weight decomposition under Ad(A ψ so that Ad(ψ(w P ψ = P ψ, w W F. Writing ψ(w = ψ 0 (w w with ψ 0 (w Ĝ, this becomes ρ G (w( P ψ = Ad(ψ 0 (w 1 P ψ, w W F. Since ρ G (w( P ψ is again standard, we have ρ G (w( P ψ = P ψ and ψ 0 (w P ψ for any w W F, as claimed. Since G is quasisplit, we can fix its F -splitting spl G = (B, T, {X}. We now define P ψ = M ψ U ψ to be the standard parabolic subgroup of G with respect to (B, T having L P ψ = P ψ ρg W F as its L-group. The above discussion allows us to view ψ as an element of Ψ(M ψ. 14

15 Lemma 3.1. ψ Ψ 0 (M ψ. As a consequence, we have the disjoint decomposition Ψ(G = [P ] Ψ 0 (M/W (M. Here, [P ] runs over a system of representatives of the associated classes of parabolic subgroups of G, and W (M := Norm(A M, G/M is the Weyl group of A M in G. Proof. It follows from M ψ = Cent(A ψ, Ĝ that Aψ ÂM ψ. Since Aψ is a maximal torus in S ψ (G 0, we have S ψ (M ψ 0 = Cent(ψ, Cent(A ψ, G 0 = Cent(A ψ, S ψ (G 0 = A ψ ÂM ψ. In the G 4 case, a system of representatives of the associated classes of parabolic subgroups is given by the set of standard parabolic subgroups with respect to B 4. This consists of P i = M i U i, (i = 1, with a M 1 = m 1(a, g = g a H 1 θ 1 σ(a g G, 1 y y β y, y / U 1 = u 1 (y, β = 1 σ(y y = (y, y W 1 1 σ(y, β G a 1 { ( } a M = m (a = a H θ, σ(a { ( } 1 b b (R U = u (b = E/F M 1 ι, ( σ(b = b and G 4, B 4. Notice that our numbering of standard parabolics has been changed from [Kon98], [Kon01]. Here, for an algebraic group G over F, we write σ for the F -automorphism on R E/F G associated to σ by the F -structure of G. (W 1,, denotes the hyperbolic skew-hermitian space (R E/F G a with the form (x, x, (y, y = x σ(y x σ(y. ι denotes the main involution on M : ι ( a c d b = ( c d b a. Again as in the global case, we save η and µ for characters of E whose restriction to F are 1 and ω E/F, respectively. Here ω E/F is the sign character of F /N E/F (E. Any quasi-character ω of E is identified with a character of W E by the local classfield theory. For a connected reductive group G over F, we write Π(G(F Π unit (G(F Π temp (G(F Π disc (G(F for the set of isomorphism classes of irreducible admissible, unitarizable, tempered and square integrable representations (Harish-Chandra modules if v is archimedean. This abuse of terminology will be adopted throughout the paper. See of G(F, respectively. If F is non-archimedean, we write Π cusp (G(F for the subset of Π disc (G(F consisting of supercuspidal elements. Finally we write A F := L F SL(, C as in the global case. 15

16 Proposition 3.. The equivalence classes of A-parameters for G 4 with non-trivial restrictions to SL(, C are the followings. The reference to the global situation indicates the numbers in Cor..3. (G Ψ 0 (G consists of the following six types of elements as in the global case. (i Stable parameters. (a ψ η defined similarly as in (1.a (b ψ Π,µ defined similarly as in (1.b. In the local case, the stable elliptic L-packet Π has the base change lift Π E = ξ 1 (Π Π disc (H (F such that σ(π E Π E, ω ΠE F = 1 and L Asai (s, Π E is holomorphic at s = 0. (ii Endoscopic parameters. (a ψ µ defined similarly as in (.a, where µ = (µ, µ and µ can be µ. (b ψ Π,η defined similarly as in (.b, where Π is the same as in (G.1.b above. (c ψ η defined similarly as in (.c, where η = (η, η with η η. (d ψ η,µ defined similarly as in (.d, where µ = (µ, µ with µ µ. (M 1 Ψ 0 (M 1 consists of the parameters ψ M 1 ω,η, (ω Π unit (E : ψ M 1 ω,η AE = [(ω σ(ω 1 (η ρ ] p WE, ψ M 1 ω,η(w σ = [(ω(w σ 1 1 ] w σ, (M Ψ 0 (M consists of the parameters ψ M ω, (ω Π unit (E : ψ M ω AE = [(ω ρ (σ(ω 1 ρ ] p WE, ψ M ω (w σ = [(ω(w σ 1 1 ] w σ. Proof. The elliptic cases are similar to the global case. The only point is that a stable Langlands parameter ϕ Π for G is elliptic if and only if Π E Π disc (H (F (see for example [GL79]. The characterization of the image of the local base change in terms of the Asai-Oda L-factor is due to Goldberg [Gol93] (at least in the non-archimedean case. The parabolic cases are well known (cf. [Rog90]. 3. Review of the local Arthur conjecture To obtain the local A-packets associated to the parameters in Prop. 3., we postulate some local assertions of Arthur s conjecture. Let G be a connected reductive quasisplit group over F. We retain the notation of 3.1. For an F -parabolic subgroup P = MU of G with a Levi component M, we have the real vector spaces a M = Hom(X (M F, R and a M = X (M F R dual to each other. Here X (M F is the lattice of F -rational characters of M. We have the map H M : M(F a M defined by exp χ, H M (m = χ(m F, χ X (M F. Using this, we define the character e λ : M(F C associated to λ a M,C = a M R C by e λ (m = m λ := e λ,h M (m. If we write A M for the maximal F -split torus in the center 16

17 of M, the set Σ P of roots of A M in P can be viewed as a subset of a M. Similarly the set Σ P of corresponding coroots is a subset of a M. We write P and P for the subsets of simple roots and coroots in Σ P and Σ P, respectively. Write For ψ Ψ(G, we define ( t µ ψ : G m (C t a,+ P := {λ a M α (λ > 0, α P }. t 1 ψ SL(, C Ĝ. After a suitable Ĝ-conjugation of ψ, one may assume that µ ψ X (T and α (µ ψ 0, α (B, T = (B, T. Here (B, T denotes the set of simple roots of T in B. Then M ψ := Cent(µ ψ, Ĝ is a standard Levi subgroup of Ĝ, and we write P ψ = M ψ Û ψ for the corresponding standard parabolic subgroup. Since µ ψ is stable under Ad(ψ(W F Ad( M ψ ρ G (W F, we see that P ψ is ρ G (W F -stable. We obtain a standard parabolic subgroup P ψ = M ψ U ψ of G having L P ψ := P ψ ρg W F as its L-group. We can view ψ LF as an element of Ψ(M ψ. For an admissible representation (τ, V of M(F and λ a M,C, (IG P (τ λ, IP G(V λ denotes the parabolically induced representation ind G(F P (F [τ λ 1 U(F ] with (τ λ := e λ τ, V λ :=, we write J P G(τ λ for the Langlands quotient of IP G(τ λ. For any admissible representation π of finite length of G(F, JH(π denotes the set of isomorphism classes of its irreducible subquotients. Let χ be a non-degenerate character of U(F in the sense that its stabilizer in B(F is Z(G(F U(F. A χ-whittaker functional on an admissible representation (π, V of G(F is a (continuous if F is archimedean linear functional Λ : V C satisfying V. If τ Π temp (M(F and λ a,+ P Λ(π(uv = χ(uv, u U(F, v V. We say (π, V is χ-generic if it admits a non-zero χ-whittaker functional. When π is irreducible, it was shown by Shalika that the space of χ-whittaker functionals on V is at most one-dimensional [Sha74]. We write W = W G for the relative Weyl group Norm(A 0, G(F /T(F, where we have written A 0 := A T. It is a Coxeter group generated by the simple reflections r α associated to the simple roots α 0 of A 0 in B. In particular, we have the associated length function l B on W. We write w = w G for the longest element in W. In what follows we identify W with its fixed system of representatives in Norm(A 0, G(F. For a standard parabolic subgroup P = MU, we set w M = wm G := w (w M 1 G(F. We recall the following from [Sha81]. Proposition 3.3 ([Sha81], Prop. 3.1, 3.. Let χ and P = MU be as above. (a χ M := Ad(w 1 M χ U M (F is a non-degenerate character of U M (F M(F. (Note that this depends on the choice of the representative of w M! (b Let (τ, V be an admissible χ M -generic representation of M(F and Λ be a χ M - Whittaker functional on it. Then the integral IP G (Λ λ, φ := Λ(φ(w 1 w M (Ū(F M vχ(v dv, φ IG P (V λ 17

18 converges absolutely for any λ a M,C and defines a χ-whittaker functional on IG P (V λ. Moreover IP G(Λ λ, e λ φ is holomorphic in lambda for any φ IP G (V. We can now state the local Arthur conjecture which we shall need. Conjecture 3.4 ([Art89] 6, 7. (A For each ψ Ψ 0 (G there exists a finite subset Π ψ (G Π unit (G(F called the A-packet associated to ψ. Let us write Φ 0 (G for the subset of ϕ Ψ 0 (G such that ϕ SL(,C is trivial. Then we should have the disjoint decomposition Π disc (G(F = Π ϕ (G. (3.1 ϕ Φ 0 (G For general ψ Ψ(G, we have the A-packet Π ψ (M ψ attached to ψ Ψ 0 (M ψ. Then we define Π ψ (G := JH(I G P (π. ψ π Π ψ (M ψ Notice that I G (π is completely reducible since π is unitarizable. Writing Φ P ψ temp (G for the set of ϕ Φ(G with ϕ SL(,C = 1, we deduce from (3.1 and the Harish-Chandra s classification of tempered representation that Π temp (G(F = Π ϕ (G. ϕ Φ temp(g (B (1 Part (A asserts the existence of the packet Π ψ LF (M ψ Π unit (M ψ (F associated to ψ LF Φ temp (M ψ. The A-packet Π ψ (G should contain Π ψ(g := {J G P ψ (τ µψ τ Π ψ LF (M ψ }. ( Fix a non-degenerate character χ of U(F. For ϕ Φ 0 (G, Π ϕ (G should contain a unique χ-generic element δ χ (the generic packet conjecture. For ϕ Φ temp (G, we have Π ϕ (G = JH(IP G ϕ(δ. δ Π ϕ(m ϕ Applying Prop. 3.3, Π ϕ (G contains a unique χ-generic element τ χ JH(IP G ϕ(δ χ M ϕ. Finally for general ψ Ψ(G, Π ψ LF (M ψ contains a unique χ Mψ -generic element τ χmψ. The element π χ := JP G ψ (τ χmψ,µ ψ Π ψ(g Π ψ (G is called the χ-base point of Π ψ (G. (C (1 There should be a function, π χ ψ : S ψ (G Π ψ (G C satisfying the following conditions. (a For π Π ψ (G, S ψ (G s s, π π χ ψ 1, π π χ ψ R +. C is a class function such that (b If we write s ψ for the image of ψ(1 1 S ψ (G in S ψ (G, then there exists a sign character e ψ (, π π χ : S ψ (G {±1} such that s ψ s, π π χ ψ = e ψ ( s ψ, π π χ s, π π χ ψ, s S ψ (G. 18

19 ( If we define, π χ ψ : S ψ LF (M ψ Π ψ (G ( s, J G P ψ (τ µψ s, τ τ χmψ ψ LF C, then the following diagram commutes. Π ψ (G π, π π χ ψ Π(S ψ LF (M ψ inclusion inclusion Π ψ (G π, π π χ ψ Π(S ψ (G. Here Π(S ψ (G is the set of isomorphism classes of irreducible representations of S ψ (G, whose elements we identify with their characters. Remark 3.5. Since the generalized Ramanujan conjecture is not yet known, A-packets Π ψ (G for ψ with ψ(w F bounded might not be sufficient. Thus, in this paper, we replace the condition (i in the definition of local A-parameter by the following. (i ψ WF is semisimple. Each member of the L-packet Π ψ LF (M ψ appears as a local component of some cuspidal automorphic representation which is not of CAP type. Also the L-packet Π ψ (G is contained in Π unit(g(f. 3.3 Non-supercuspidal representations of G 4 (F In this subsection, we assume F is non-archimedean and calculate Π ψ (G 4 for the elliptic ψ in Prop. 3.. We first review the classification of Π unit (G 4 (F from [Kon01]. η defines a character η u : G 1 (F xσ(x 1 η(x C 1. We write η Gn for the composite G n (F det G 1 (F ηu C 1. Similarly for a character ω of E, we write ω Hm : H m (F h ω(det h C 1. We express irreducible representations of standard Levi subgroups of G(F = G 4 (F as follows. ω[λ] : T(F d(a 1, a ω 1 (a 1 a 1 λ 1/ E ω (a a λ / E C, ω[λ] π : M 1 (F m 1 (a, g ω(a a λ/ E π(g GL(V π, π E [λ] : M (F m (a det a λ/ E π E(a GL(V ΠE. Here ω = (ω 1, ω, ω i, ω Π(E, (π E, V πe Π(H (F, (π, V π Π(G (F, and E is the module of E. Also d(a 1, a = diag(a 1, a, σ(a 1, σ(a 1 1 T(F. Recall that the endoscopic liftings in the following three settings were established by Rogawski [Rog90, Ch. 11]. Standard base change for R E/F G This is the endoscopic lifting from the twisted endoscopic data (G, L G, 1, ξ η for (H, θ σ, 1 (see [KS99, Ch. II], where { ξ η : L (η(wg, η(w 1 θ (g w if w W E, G g w L H. (g, θ (g w σ if w = w σ Twisted base change for R E/F G This is the same as above except that the twisted endoscopic data (G, L G, 1, ξ µ is given by { ξ µ : L (µ(wg, µ(w 1 θ (g w if w W E, G g w L H. (g, θ (g w σ if w = w σ 19

20 Endoscopic lift for G This is the endoscopic lifting from the unique elliptic endoscopic data (G 1, L (G 1, s, λ µ 1 for G. Here ( z 1 µ(w w if w W E, λ µ 1 : L (G z µ(w 1 (z 1, z w ( L G. z 1 w σ if w = w σ z It follows from (the local analogue of Prop.. that the elliptic endoscopic parameters for G (F are of the form ( 0 1 ϕ µ LE = (µ µ pr WE, ϕ µ (w σ = w 1 0 σ. We recall from [Rog90] that the corresponding L-packet of G (F is This description is valid even if µ = µ. Π ϕµ (G = λ µ 1(1 (µ 1 µ G1. Lemma 3.6. The Langlands data (P ψ, Π ψ LF (M ψ, µ ψ for the elliptic local A-parameters in Prop. 3. (G are given by the following. Here in (.b, π is the unique member of Π. A-parameter P ψ µ ψ Π ψ LF (M ψ (1.a ψ η B (3, 1 η η (1.b ψ Π,µ P 1 µ H Π E (.a ψ µ P 1 µ Π ϕµ (G (.b ψ Π,η P 1 1 η π (.c ψ η P 1 I H B H η (.d ψ η,µ P 1 1 η Π ϕµ (G This is an immediate consequence of the description of parameters. Now we recall the results of [Kon01] on the composition series of IP G ψ (τ µψ, τ Π ψ LF (M ψ in the above lemma. We write δ0 H for the Steinberg representation of a connected reductive group H(F. We often drop the subscript 4 and write G = G 4. We fix a non-trivial character ψ F of F. This determines a non-degenerate character χ H n of U H n (F such that 1 x 1,... x 1,n (. χ H 1... n. = ψf ( 1.. x n 1,n Tr ( n 1 E/F x i,i+1, i=1 1 and its restriction to U n (F gives a non-degenerate character χ n. We also fix a system of representatives of W in Norm(A 0, G(F as follows. Write α i, (i = 1, for the simple roots of A 0 in G given by α 1 (d(a 1, a = a 1 a 1, α (d(a 1, a = a. 0

21 If we write r i W for the reflection associated to α i, W is the type C Weyl group generated by them. We choose a system of representatives so that r 1 = 1 1, r = , w = These choices assure that χ M = χ U M (F for any standard Levi subgroup M G. All the equalities are those in the Grothendieck group of admissible representations of finite length of G(F. (1.a For ψ η we have I G B(η[3] η[1] = η G δ G 0 + J G P 1 (η[3] η G δ G 0 + J G P (η H δ H 0 [] + η G. η G δ0 G Π disc (G(F, η G Π unit (G(F but the other two constituents are not unitarizable. (1.b For ψ Π,µ we have the following two possibilities. (i If Π is supercuspidal, we have I G P (µ H Π E [1] = δ G (µ H Π E + J G P (µ H Π E [1], where δ G (µ H Π E Π disc (G(F and J G P (µ H Π E [1] Π unit (G(F. (ii Π E = η H δ H 0. We have I G P ((µη H δ H 0 [1] = δ G 0 (ηµ + + δ G 0 (ηµ + J G P ((µη H δ H 0 [1], where δ G 0 (µη ± Π disc (G(F are labeled in such a way that δ G 0 (ηµ + is χ-generic. J G P ((µη H δ H 0 [1] Π unit (G(F. (.a For ψ µ, again we have two cases. (i If µ µ, then the L-packet Π ϕµ (G consists of two distinct supercuspidal representations π G (µ ± [Rog90, p. 17, (5], and only one of them, say, π G (µ + is χ -generic. We have I G P 1 (µ[] π G (µ ± = δ G (µ ± + J G P 1 (µ[] π G (µ ±. Here δ G (µ ± Π disc (G(F and δ G (µ + is χ-generic. J G P 1 (µ[] π G (µ ± Π unit (G(F. (ii If µ = µ, then Π ϕµ (G consists of the limit of discrete series representations τ G (µ ±, where only τ G (µ + is χ -generic [Rog90, p. 17, (6]. We have I G P 1 (µ[] τ G (µ ± = δ G 0 (µ ± + J G P 1 (µ[] τ G (µ ± + J G P (µ H δ H 0 [1], where δ G 0 (µ ± Π disc (G(F and only δ G 0 (µ + is χ-generic. The other constituents are also unitarizable. (.b The following three cases occur for ψ Π,η. 1

22 (i If Π E is supercuspidal, then so is π and I G P 1 (η[1] π = δ G 1 (η, π + J G P 1 (η[1] π, where δ G 1 (η, π Π disc (G(F and J G P 1 (η[1] π Π unit (G(F. (ii If Π E = η H δ H 0 with η η, then π = η G δ G 0. We have I G P 1 (η[1] η G δ G 0 = δ G 0 (η + J G P 1 (η[1] η G δ G 0. We have written η = (η, η and δ G 0 (η Π disc (G(F, J G P 1 (η[1] η G δ G 0 Π unit (G(F. (iii If π = η G δ G 0, then I G P 1 (η[1] η G δ G 0 = η G τ G 0 (δ G 0 + J G P 1 (η[1] η G δ G 0, where τ G 0 (δ G 0 Π temp (G(F and J G P 1 (η[1] η G δ G 0 Π unit (G(F. For later use, we also recall IP G 1 (η[1] η G = η G τ0 G (1 G + JP G (I H (η η[1], B H τ0 G (1 G Π temp (G(F, JP G (I H (η η[1] Π B H unit (G(F. (.c For ψ η, we have IP G (I H (η η [1] =δ B H 0 G (η + JP G 1 (η[1] η G δ G 0 + JP G 1 (η [1] η G δ G 0 + JP G (I H (η η [1], B H where δ G 0 (η is as in (.b.ii. The other constituents are also unitarizable. (.d For ψ η,µ, we have I G P 1 (η[1] π G (µ ± = δ G 1 (η, µ ± + J G P 1 (η[1] π G (µ ±, where δ1 G (η, µ ± Π disc (G(F and only δ1 G (η, µ + is χ-generic. JP G 1 (η[1] π G (µ ± Π unit (G(F, and π G (µ ± are as in (.a.i. We are now ready to determine the packets Π ψ (G associated to the parameters in Prop. 3.. Proposition 3.7. The L-packets Π ψ (G and their χ-base points for the elliptic ψ in Prop. 3. (G are given by the following. A-parameter S ψ (G Π ψ (G χ-base point (1.a ψ η trivial {η G } η G (1.b ψ Π,µ trivial {JP G (µ H Π E [1]} JP G (µ H Π E [1] (.a ψ µ Z/Z {JP G 1 (µ[] π ± π ± Π ϕµ (G } JP G 1 (µ[] π + (.b ψ Π,η Z/Z {JP G 1 (η[1] π} JP G 1 (η[1] π (.c ψ η Z/Z {JP G (I H (η η [1]} J B H P G (I H (η η [1] B H (.d ψ η,µ (Z/Z {JP G 1 (η[1] π G (µ ± } JP G 1 (η[1] π G (µ +

23 Here in (.a and (.d, the elements of Π ϕµ (G are labeled so that π + is χ -generic. Proof. The only thing that does not follow from the above list is the description of S ψ (G. Since the argument does not change, we consider the global case. Recall the description of ψ Ψ 0 (G n of Prop... It follows from Schur s lemma and the form of ψ(w σ that S ψ (G = (Z/Z r. Since Z(Ĝn Γ is the diagonal subgroup Z/Z of S ψ (G, we obtain S ψ (G (Z/Z r Induced representations over the real field In this subsection, we consider the case E v /F v C/R. Each of the characters η and µ of C are written explicitly as η a (z := (z/ z a/, µ b (z := (z/ z b/, a Z, b Z + 1, so that η a = (η a and µ b = (µ 1 b as notation suggests. Since Π disc (H n is {η a } a Z if n = 1 and empty otherwise, the Langlands parameters for G n are given by the following. (i Φ 0 (G n consists of the elements ϕ a, where a = (a 1,..., a n (Z+n+1 n satisfying a 1 > a > > a n : { diag(η a 1,..., η an if n is odd, ϕ a WC = diag(µ a 1 ϕ(w σ = I n w σ. (3.,..., µ an if n is even, (ii As for general parameters, we have the decomposition (Lem. 3.1 Φ(G n = [n/] r=0 Φ 0 (M (r /W (M (r, where P (r = M (r U (r is the standard parabolic subgroup with M (r H r 1 G n r. (iii Φ 0 (M (r consists of the elements ϕ b,ν,a, (b Z r, ν C r, a (Z + n + 1 n r given by ϕ b,ν,a WC = diag(ω b1,ν 1,..., ω br,ν r ; ϕ a WC ; ω br, ν r,..., ω b1, ν 1 id WC ϕ b,ν,a (w σ = diag(1 r ; I n r ; ( 1 br,..., ( 1 b 1. Here ω b,ν (z = (z/ z b/ z ν C, (z C Explicit Langlands classification for G, G 4 In this section we describe the L-packets associated to the parameters listed above for the unitary groups in two and four variables by means of Vogan s Langlands classification [KV95], [Vog84]. The groups to be considered are U(1, 1, U( and U(,. We first prepare some notation for general U(p, q, an inner form of G m. 3

24 Thus we take an R-form G p,q = U(p, q(r of G m : G p,q has the Lie algebra G p,q = {g H m (R t ḡi p,q g = I p,q }, I p,q = diag(1 p, 1 q. g p,q = u(p, q(r = {( a b tā } = a M p (C t b d t d = d Mq (C b M p,q(c, whose complexification is denoted by g = gl(m, C. As usual, we take the Cartan involution θ p,q = Ad(I p,q on g p,q, which determines the Cartan decomposition g = k p,q p p,q : {( a k p,q := d a M p (C d M q (C } {( 0p b, p p,q := c 0 q The corresponding maximal compact subgroup is {( k1 k K p,q = 1 U(p k U(q k }. b M p,q (C c M q,p (C We write T U(1 m for the diagonal fundamental Cartan subgroup of G p,q, and t for its Lie algebra. Writing X (T = m i=1 Ze i with the standard basis e i : T diag(t 1,..., t m t i G m, the set R(g, t of roots of t in g is given by R(g, t = R cpt R ncpt where { } 1 i j p or R cpt = R(k p,q, t = e i e j, p + 1 i j m { } 1 i p R ncpt = R(p p,q, t = ±e i e j p + 1 j m are the sets of compact and non-compact roots, respectively. We use the basis {e i } 1 i m to identify t with C m. In what follows, we call an admissible (g p,q, K p,q -module as an admissible representation of G p,q by abuse of terminology. Similarly, a (g p,q, K p,q -module is called a smooth representation of G p,q. This latter terminology is quite unusual, but it is harmless for our purpose and avails us to unify our descriptions in archimedean and non-archimedean cases. Discrete L-packets For each elliptic parameter ϕ a Φ 0 (G m, the corresponding L- packet Π ϕa (G p,q of G p,q contains m C p = m C q distinct discrete series representations. In fact, for each permutation (a i1,..., a ip ; a j1,..., a jq of the ordered family a satisfying a i1 > a i > > a ip, a j1 > a j > > a jq, 4 }.