CAP FORMS ON U E=F (4) TAKUYA KONNO. representations which Associated to Parabolic subgroups". More precisely, letg be a

Transcription

1 CAP FORMS ON U E=F (4) TAKUYA KONNO. What are CAP forms? The term CAP in the title is a short hand for the phrase Cuspidal automorphic representations which Associated to Parabolic subgroups". More precisely, letg be a connected reductie group defined oer a number field F, and G Λ be its quasisplit inner form. An irreducible automorphic representation ß = N ß of G(A )isacap form if (i) ß is cuspidal; (ii) There exists a residual discrete automorphic representation ß Λ = N ßΛ of G Λ (A ) such thatß ' ß Λ at all but finite number of. Assuming the suitable generalization of Ramanujan's conjecture, the CAP forms should be exactly the non-tempered cusp forms. Thus such forms should be beautifully described in terms of Arthur's conjecture. We write L G = b G oρg W F for the L-group of G where W F is the Weil group of μ F=F, μf being an algebraic closure of F. For the purpose of stating the conjecture, we introduce the hypothetical Langlands group L F of F.AnA-parameter for G is a continuous homomorphism ψ : L F SL(; C )! L G such that () The following diagram commutes: L F SL(; C ) proj??y ψ! L G??y proj W F () ψj L F is semisimple and ψ(l F ) is bounded; (3) ψj SL(;C) is analytic. We should also hae some releance condition (see Ex.. below). The set Ψ(G) of G- b conjugacy classes of A-parameters should parameterize the irreducible representations of G(A )whichappear in the L -automorphic spectrum of G. L The A-parameters ψ associated to CAP forms ß = ß can be characterized as follows. The cuspidality ofß implies, in particular, it appears in the discrete spectrum, so that ψ should be elliptic. That is, Imψ is not contained in the L-group of any proper F -parabolic subgroup of G. We write Ψ 0 (G) for the set of elliptic elements in Ψ(G). On the other hand the condition associated to parabolics" is equialent to the non-triiality of ψj SL(;C). Ψ CAP (G) denotes the subset of ψ Ψ(G) satisfying these two conditions. Let us look at some examples. Date: august, Mathematics Subject Classification. Primary F70; Secondary F85, E50. Talk at the conference Representations of p-adic Groups and Automorphic Forms", Takayama. The author is partially supported by the Grants-in-Aid for Scientific Research No , the Ministry of Education, Science, Sports and Culture, Japan. W F

2 TAKUYA KONNO Example.. Consider G = GL(n). L G is the direct product of b G = GL(n; C ) and WF, and an A-parameter is an n-dimensional representation of L F SL(; C ). ψ Ψ 0 (G) is equialent to its irreducibility. Thus each ψ Ψ(G) is of the form ψ = ' Ω ρ d ; where ' is an m-dimensional irreducible representation of L F, ρ d is the d-dimensional irreducible representation of SL(; C ) and n = dm. ' should corresponds to an irreducible cuspidal representation ß of GL(m; A ). Then it is a result of Moeglin-Waldspurger that the only automorphic representation attached to this is the global Langlands quotient Π ψ = O J GL(n) P (ß m d j det j (d )= Ω ß j det j (d 3)= Ω Ωß j det j ( d)= ); which appears in the residual spectrum. Here P m d denotes the standard parabolic subgroup of GL(n) with a Lei component isomorphic to GL(m) d. Thus there are no CAP forms on GL(n). Example.. Next look at the multiplicatie group G = D of a central diision algebra D of dimension n oer F. ψ Ψ CAP (G) is of the same form as in the preious example. But there should be areleance condition on ψ so that only the parameters of the form ψ = χ Ω ρ n ; dim χ = is releant. χ : L F! W F! C is identified with an idele class character of F. Certainly the corresponding representation Π ψ = χ ffi ν D=F is a CAP form, where ν D=F is the reduced norm of D. Example.3. Finally we describe Ψ CAP (G) for G = G n := U(n) E=F, the quasisplit unitary group in n-ariables attached to a quadratic extension E of F. We write ff for the generator of Gal(E=F), and define n by 0 n (g) :=Ad We realize G n so that ( ) n : : C A tg : G n (R) :=fg GL(n; R Ω F E) j n ( ff g)=gg for any commutatie F -algebra R. L G n = GL(n; C ) o ρgn W F, where ( id if w W E ; ρ Gn (w) = n otherwise. ψ is known to be determined by its restriction to L E SL(; C ). Now each ψ Ψ CAP (G) is of the form Here, ψj L E SL(;C) = rm i= (! i ' i ) Ω ρ di :

3 CAP FORMS ON UE=F (4) 3 ffl ' i is an irreducible m i -dimensional representation of L E, which corresponds to an irreducible cuspidal representation ß i of GL(m i ; A E ) such that ff ß ' ß _ and! ß ja, the central character of ß restricted toa, is triial. ffl! i is an idele class character of E whose restriction to A equals! n d i m i + E=F.! E=F is the quadratic character of A =F associated toe=f by the classfield theory. ffl n = P r i= d im i, and the! i ' i Ω ρ di are not equialent to each other. What are the CAP forms associated to the A-parameters in Ex..3? The cases n» is triial. When n = 3, it was proen by Gelbart-Rogawski that such representations are the theta liftings from G = U() E=F.Today we shall report the result of the joint work with Kazuko Konno in the case n = 4. There are 6 types of parameters for G 4 according to the following list of f(d i ;m i )g i : () Stable parameters: (.a) f(4; )g, (.b) f(; )g. () Endoscopic parameters: (.a) f(3; ); (; )g, (.b) f(; ); (; )g,(.c) f(; ); (; )g, (.d) f(; ); (; ); (; )g. In this talk, we shall concentrate to the interesting cases (.b) and (.c).. Local theory Let us write and 0 for characters of A E =E whose restriction to A is triial. Then the parameters to be considered are gien by (:b) ψ ;ß j L E SL(;C) =( Ω ρ ) Φ ' ß ; (:c) ψ j L E SL(;C) =( Ω ρ ) Φ ( 0 Ω ρ ); where =( ; 0 )andß is an irreducible cuspidal representation of GL(; A E ) satisfying the condition of Ex..3. At each place of F, these gie rise to the local parameters ψ : L F SL(; C )! L G. In this section, we shall construct the local A-packet Π ψ (G) attached to such ψ. Again for breity, we consider only those where E := E Ω F F is a quadratic extension of F and ß is (super) cuspidal in (.b) and 6= 0 in (.c). Also to keep in accordance with the theme of this conference, we restrict ourseles to non-archimedean. (The archimedean case is more explicit and needs some case-by-case treatment.).. Parabolic induction. Recall some assertion from Arthur's conjecture. For ψ Ψ 0 (G )we set S ψ (G) := Cent(ψ ; b G)=Z( b G), where = Gal( μ F =F ). In the case of G = G n,wehae S ψ (G) ' (Z=Z) r where r is as in Ex..3. Moreoer since F is non-archimedean, we may postulate that there exists a perfect duality Next set h ; i : S ψ (G) Π ψ (G)! C : t μ ψ : G m (C ) 3 t 7! SL(; C ) t ψ! b G: There is an F -parabolic subgroup P ψ = M ψ U ψ such thatc Mψ = Cent(μ ψ ; G)andμψ b is a bp ψ -dominant cocharacter of Mψ c. Associated to ψ j Ψ(M L F ψ) is the tempered L-packet Π (M ψjl F ψ). Then Arthur imposed Π 0 ψ (G) :=fj G P ψ (fi Ω e μ ψ ) j fi Π ψjl F (M ψ)g ρπ ψ (G): Here, again J G P (fi Ωe ) denotes the Langlands quotient ofi G P (fi Ωe ) = ind G(F ) P (F ) [(fi Ω e )Ω U (F) ].

4 4 TAKUYA KONNO We now examine these constructions in our G 4 -case. The standard parabolic subgroups of G 4 are G, P i = M i U i (i =, ) and the upper triangular Borel subgroup B = TU, where 8 < M = : m (a; g) = U = M = U = a 8 0 >< >: u (y;fi) = ρ m (a) = ρ u (b) = a g ( ff a) fi Afi fi y 00 y 0 fi hy;yi= ff y 0 ff y 00 ( ff a) fi fifi a H ff ; fi b fififi b (Res E=F M ) t ( ff b)= b 9 a Res E=F G m = g G ; ; fi C fi Afi fi ff : y =(y 00 ;y 0 ) W fi G a Here (W ; h ; i) is the hyperbolic skew-hermitian space (E ; ( )) oer E. Proposition.. The group S ψ (G) and Π 0 ψ (G) for parameters (.b), (.c) are gien as follows. A-parameter S ψ (G) Π 0 ψ (G) (.b) ψ ß; Z=Z fjp G ( jj = E Ω fi ) j fi T g (.c) ψ Z=Z fjp G (I( Ω 0 )j det j= E )g T in (.b) is the L-packet of G (F ) whose standardbase change to GL(;E ) is ß. I( Ω 0 ) in (.c) is the obious notation for the principal series representation of GL(;E )... Local theta correspondence. We still need to construct the members of Π ψ (G)n Π 0 ψ (G). For this we use the theta correspondence. We write G + := G and G for its anisotropic inner form. Fix a non-triial additie character ψ F of F. For a character of E haing the triial restriction to F,we can construct the Weil representation (! ;±;S) of the unitary dual pair G ± (F ) G n (F ). Again for simplicity,we assume that the residual characteristic of F is odd. Then for an irreducible admissible representation ß of G n (F ) (resp. fi of G ± (F )), we hae its (possibly zero) local Howe correspondent ± n; (ß) (resp. n (fi)), an irreducible representation of G ± (F ) (resp. G n (F )). Then we can proe the following. Proposition. (local -correspondence for U()). Consider the case n =. (i) ; ffl (ß) does not anish if and only if ffl(=;ß ;ψ F )! ß ( ) (E =F ;ψ F ) = ffl. (ii) If this is the case, we hae ( ffl ß _ if ffl =, ; (ß) = JL(ß _ ) otherwise. Here ß _ is the contragredient of ß and JL(ß _ ) is the Jacquet-Langlands correspondent" of ß _. Remark.3. This is the ffl-dichotomy property which is proed by Harris-Kudla-Sweet for general unitary dual pairs. But their result uses the ffl-factor defined by the doubling 9 >= >; ;

5 CAP FORMS ON UE=F (4) 5 method of Piatetskii-Shapiro-Rallis. The comparison conjecture between their ffl-factor and that defined by the Langlands-Shahidi method is not yet established in the present case. This is the reason why we rely on the construction of M. Harris using the Shimizu-Jacquet- Langlands correspondence. In general, we rarely hae a good understanding of the ramified local theory of theta correspondences. Except for few examples such as Waldspurger's study on the Shimura correspondence, all such examples put some strong restriction on the representations treated (e.g. triial in the example of Siegel-Weil formula, and one-dimensional in the study of Shalika-Tanaka). As was remarked aboe, we deduce the proposition from the similar result for GU() obtained by M. Harris. His method deduces the ramified local theory from the Shimizu- Jacquet-Langlands correspondence combined with the result of Waldspurger-Tunnel-Saito- Prasad on the restriction of a representation of GL(;F ) to elliptic Cartan subgroups. Our task is to deduce the unitary group case from the similitude group setting. This is achieed with the help of the following lemma. Let us identify the unitary similitude group e G = GU() with (Res E=F G m GL())=G m by (Res E=F G m GL())=G m 3 (z;g) 7! z g e G ρ Res E=F GL(): Thus each irreducible representation eß of e G (F ) is of the form!ωß, where! is a character of E and ß is an irreducible representation of GL(;F ) such that(!j F )! ß =. L Lemma.4. We hae eßj G (F ) = fi, where T is the unique L-packet of G fi T (F ) whose standard base change lift is!(det)ß E. ß E denotes the base change of ß to GL(;E ). We are now ready to complete the A-packets Π ψ. Consider the following diagram of local theta correspondences. G ffl (F ) 4! G 4 (F ) 4 JL l % j G ffl (F ) ψ ; ffl G (F ) The left ertical arrow is the Jacquet-Langlands correspondence", while the rightertical line indicates the Witt tower. For fi T in Prop.., write ffl := ffl(=;ß ;ψ F )! ß ( ) (E =F ;ψ F ) so that ffl ; (fi ) 6= 0. The tower property of theta correspondence implies 4 ffi ffl ; (fi ) ' ß + ψ; := J G P ( jj = E Ω fi ): On the other hand, we know from Prop.. that ffi JL ffi ffl ; (fi ) is zero. This again combined with the tower property says that ß ψ; := 4 ffi JL ffi ffl ; (fi ) is a cuspidal representation of G 4 (F ). We argue similarly in the case (.c) and obtain the following.

6 6 TAKUYA KONNO Theorem.5. Π ψ (G) =fß ± ψ; g,where ß± ψ; are defined aboe in the case (.b) and ß + = J G ψ; P (I( Ω 0 )j det j= E ); ß := ψ; 4 (( 0 ) G ); in the case (.c). Here ( 0 ) G : G (F )! det G (F ) 3 xff(x) 7! ( 0 )(x) C. Remark.6. (i) Similar results hold for other types of ß in the case (.b). In particular, when ß is the special representation, ß ψ; coincides with that in the case (.c). (ii) The archimedean case is treated in a similar way. In the case (.b) with T is endoscopic, jt j =while its Jacquet-Langlands correspondent is a singleton. But since ffl(=;fi ;ψr)! fi ( ) (C =R;ψR) =in this case, the theorem is still alid. 3. Global theory Wenow define the duality(multiplicity pairing) h ; i : S ψ (G) Π ψ (G)!f±g so that ß ψ; corresponds to the sign character of S ψ (G). The following erifies the multiplicity formula for the global A-packet Π ψ (G) = L Π ψ (G) conjectured by Arthur. L Theorem 3.. If we write m(ß) for the multiplicity of an irreducible representation ß = ß Π ψ (G) of G 4 (A ) in the L -automorphic spectrum, then X Y m(ß) = ffl ψ (μs) hμs; ß i: js ψ (G)j μssψ(g) Here ( sgn ffl ψ (s) = S in the case (.b) with ffl(=;ß ) =, ψ(g) otherwise. We show the inequality by the global theta correspondence. Also in some cases, this gies the exact equality. But in other cases including (.c), we need the analysis of the Fourier coefficients to hae the conerse inequality. Graduate School of Mathematics, Kyushu Uniersity, 8-80 Hakozaki, Higashi-ku, Fukuoka, Japan address: takuya@math.kyushu-u.ac.jp