GLOBAL JACQUET-LANGLANDS CORRESPONDENCE, MULTIPLICITY ONE AND CLASSIFICATION OF AUTOMORPHIC REPRESENTATIONS

Transcription

1 GLOBAL JACQUET-LANGLANDS CORRESPONDENCE, MULTIPLICITY ONE AND CLASSIFICATION OF AUTOMORPHIC REPRESENTATIONS by Aexandru Ioan BADULESCU 1 with an Appendix by Neven GRBAC Abstract: In this paper we generaize the oca Jacquet-Langands correspondence to a unitary irreducibe representations. We prove the goba Jacquet-Langands correspondence in characteristic zero. As consequences we obtain the mutipicity one and strong mutipicity one Theorems for inner forms of GL(n) as we as a cassification of the residua spectrum and automorphic representations in anaogy with resuts proved by Moegin-Wadspurger and Jacquet-Shaika for GL(n). Contents 1. Introduction. Basic facts and notation (oca) 5.1. Cassification of Irr n (resp. Irr n ) in terms of D (resp. D ), n 6.. Cassification of D n in terms of C, n 7.3. Loca Jacquet-Langands correspondence 7.4. Cassification of D n in terms of C, n. The invariant s(σ ) 8.5. Mutisegments, order reation, the function and rigid representations 8.6. The invoution 9.7. The extended correspondence 9.8. Unitary representations of G n Unitary representations of G n Hermitian representations and an irreducibiity trick 1 3. Loca resuts First resuts Transfer of u(σ, k) New formuas Transfer of unitary representations Transfer of oca components of goba discrete series 1 4. Basic facts and notation (goba) Discrete series Cuspida representations 1 Aexandru Ioan BADULESCU, Université de Poitiers, UFR Sciences SPMI, Département de Mathématiques, Tééport, Bouevard Marie et Pierre Curie, BP 30179, 8696 FUTUROSCOPE CHASSENEUIL CEDEX E-mai : baduesc@math.univ-poitiers.fr Neven GRBAC, University of Zagreb, Department of Mathematics, Unska 3, Zagreb, Croatia E-mai : neven.grbac@zpm.fer.hr 1

2 GLOBAL JACQUET-LANGLANDS 4.3. Automorphic representations Mutipicity one Theorems for G n The residua spectrum of G n Transfer of functions 4 5. Goba resuts Goba Jacquet-Langands, mutipicity one and strong mutipicity one for inner forms A cassification of discrete series and automorphic representations of G n Further comments 3 6. L-functions and ɛ -factors Bibiography 36 Appendix A. The Residua Spectrum of GL n over a Division Agebra 39 A.1. Introduction 39 A.. Normaization of intertwining operators 39 A.3. Poes of Eisenstein series 43 References Introduction The aim of this paper is to prove the goba Jacquet-Langands correspondence and its consequences for the theory of representations of the inner forms of GL n over a goba fied of characteristic zero. In order to define the goba Jacquet-Langands correspondence, it is not sufficient to transfer ony square integrabe representations as in the cassica oca theory ([JL], [FL], [Ro], [DKV]). It woud be necessary to transfer at east the oca components of goba discrete series. This resuts are aready necessary to the goba correspondence with a division agebra (which can be ocay any inner form). Here we prove, more generay, the transfer of a unitary representations. Then we prove the goba Jacquet-Langands correspondence, which is compatibe with this oca transfer. As consequences we obtain for inner forms of GL n the mutipicity one Theorem and strong mutipicity one Theorem, as we as a cassification of the residua spectrum à a Moegin-Wadspurger and unicity of the cuspida support à a Jacquet- Shaika. Using these cassifications we give counterexampes showing that the goba Jacquet- Langands correspondence for discrete series does not extend we to a unitary automorphic representations. We give here a ist of the most important resuts, starting with the oca study. We woud ike to point out that the oca resuts in this paper have aready been obtained by Tadić in [Ta6] in characteristic zero under the assumption that his conjecture U 0 hods. After we proved these resuts here independenty of his conjecture (and some of them in any characteristic), Sécherre announced the proof of the conjecture U 0 ([Se]). The approach is competey different and we insist on the fact that we do not prove the conjecture U 0 here but more particuar resuts which are enough to show the oca transfer necessary for the goba correspondence. Let F be a oca non-archimedean fied of characteristic zero and D a centra division agebra over F of dimension d. For n N set G n = GL n (F) and G n = GL n(d). Let ν genericay denote the character given by the absoute vaue of the reduced norm on groups ike G n or G n. Let σ be a square integrabe representation of G n. If σ is a cuspida representation, then it corresponds by the oca Jacquet-Langands correspondence to a square integrabe representation σ of G nd. We set s(σ ) = k, where k is the ength of the Zeevinsky segment of σ. If σ is not

3 GLOBAL JACQUET-LANGLANDS 3 cuspida, we set s(σ ) = s(ρ), where ρ is any cuspida representation in the cuspida support of σ, and this does not depend on the choice. We set then ν σ = ν s(σ ). For any k N we denote then by u (σ, k) the Langands quotient of the induced representation from k 1 i=0 σ σ ), and if α ]0, 1 [, we denote π (u (σ, k), α) the induced representation from νσ α u (σ, k) ν α σ u (σ, k). The representation π (u (σ, k), α) is irreducibe ([Ta]). Let U be the set of a representations of type u (σ, k) or π (u (σ, k), α) for a G n, n N. Tadić conjectured in [Ta] that (i) a the representations in U are unitary; (ii) an induced representation from a product of representations in U is aways irreducibe and unitary; (iii) every irreducibe unitary representation of G m, m N, is an induced representation from a product of representations in U. k 1 (ν i The fact that the u (σ, k) are unitary has been proved in [BR1] if the characteristic of the base fied is zero. In the third Section of this paper we compete the proof of the caim (i) (i.e. π (u (σ, k), α) are unitary; see Coroary 3.6) and prove (ii) (Proposition 3.9). We aso prove the Jacquet-Langands transfer for a irreducibe unitary representations of G nd. More precisey, et us write g g if g G nd, g G n and the characteristic poynomias of g and g are equa and have distinct roots in an agebraic cosure of F. Denote G nd,d the set of eements g G nd such that there exists g G n with g g. We denote χ π the function character of an admissibe representation π. We say a representation π of G nd is d-compatibe if there exists g G nd,d such that χ π (g) 0. We have (Proposition 3.9): Theorem. If u is a d-compatibe irreducibe unitary representation of G nd, then there exists a unique irreducibe unitary representation u of G n and a unique sign ε { 1, 1} such that for a g G nd,d and g g. χ u (g) = εχ u (g ) It is Tadić who first pointed out ([Ta6]) that this shoud hod if his conjecture U 0 were true. The sign ε and an expicit formua for u may be computed. See for instance Subsection 3.3. The fifth Section contains goba resuts. Let us use the Theorem above to define a map LJ : u u from the set of irreducibe unitary d-compatibe representations of G nd to the set of irreducibe unitary representations of G n. Let now F be a goba fied of characteristic zero and D a centra division agebra over F of dimension d. Let n N. Set A = M n (D). For each pace v of F et F v be the competion of F at v and set A v = A F v. For every pace v of F, A v M rv (D v ) for some positive integer r v and some centra division agebra D v of dimension d v over F v such that r v d v = nd. We wi fix once and for a an isomorphism and identify these two agebras. We say that M n (D) is spit at a pace v if d v = 1. The set V of paces where M n (D) is not spit is finite. We assume in the seque that V does not contain any infinite pace. Let G nd (A) be the group of adèes of GL nd (F), and G n(a) the group of adèes of GL n (D). We identify G nd (A) with M nd (A) and G n (A) with A(A). Let Z(A) be the center of G nd (A). If ω is a smooth unitary character of Z(A) trivia on Z(F), et L (Z(A)G nd (F)\G nd (A); ω) be the space of casses of functions f defined on G nd (A) with vaues in C such that f is eft invariant under G nd (F), f(zg) = ω(z)f(g) for a z Z(A) and amost a g G nd (A) and f is integrabe over Z(A)G nd (F)\G nd (A). The group G nd (A) acts by right transations on L (Z(A)G nd (F)\G nd (A); ω). We ca a discrete series of G nd (A) an

4 4 GLOBAL JACQUET-LANGLANDS irreducibe subrepresentation of such a representation (for any smooth unitary character ω of Z(A) trivia on Z(F)). We adopt the anaogous definition for the group G n(a). Denote DS nd (resp. DS n ) the set of discrete series of G nd(a) (resp. G n (A)). If π is a discrete series of G nd (A) or G n (A), and v is a pace of F, we denote π v the oca component of π at the pace v. We wi say that a discrete series π of G nd (A) is D-compatibe if π v is d v -compatibe for a paces v V. If v V, the Jacquet-Langands correspondence between d v -compatibe unitary representations of GL nd (F v ) and GL rv (D v ) wi be denoted LJ v. Reca that if v / V, we have identified the groups GL rv (D v ) and GL nd (F v ). We have the foowing (Theorem 5.1): Theorem. (a) There exists a unique injective map G : DS n DS nd such that, for a π DS n, we have G(π ) v = π v for every pace v / V. For every v V, G(π ) v is d v -compatibe and we have LJ v (G(π ) v ) = π v. The image of G is the set of D-compatibe eements of DS nd. (b) One has mutipicity one and strong mutipicity one Theorems for the discrete spectrum of G n(a). Since the origina work of [JL] (see aso [GeJ]), goba correspondences with division agebras under some conditions (on the division agebra or on the representation to be transferred) have aready been carried out (sometimes not expicity stated) at east in [F], [He], [Ro], [Vi], [DKV], [Fi] and [Ba4]. They were using simpe forms of the trace formua. For the genera resut obtained here these formuas are not sufficient. Our work is heaviy based on the comparison of the genera trace formuas for G n (A) and G nd(a) carried out in [AC]. The reader shoud not be mised by the fact that here we use directy the simpe formua Arthur and Coze obtained in their over 00 pages ong work. Their work overcomes big goba difficuties and together with methods from [JL] and [DKV] reduces the goba transfer of representations to oca probems. Let us expain now what are the main extra ingredients required for appication of the spectra identity of [AC] in the proof of the theorem. The spectra identity as stated in [AC] is roughy speaking (and after using the mutipicity one theorem for G nd (A)) of the type tr(σi )(f) + λ J tr(m J π J )(f) = m i tr(σ i )(f ) + λ j tr(m j π j )(f ) where λ J and λ j are certain coefficients, σ I (resp. σ i ) are discrete series of G nd(a) (resp. of G n (A) of mutipicity m i ), π J (resp. π j ) are representations of G nd(a) (resp. of G n (A)) which are induced from discrete series of proper Levi subgroups and M J and M j are certain intertwining operators. As for f and f, they are functions with matching orbita integras. The main step in proving the theorem is to choose a discrete series σ of G n (A) and to use the spectra identity to define G(σ ). The crucia resut is the oca transfer of unitary representations (Proposition 3.9.c of this paper) which aows to gobay transfer the representations from the eft side to the right side. This gives the correspondence when n = 1 as in [JL] or [Vi]. The troube when n > 1 is that we do not know much about the operators M j. We overcome this by induction over n. Then the Proposition 3.9.b shows that π j are irreducibe. This turns out to be enough to show that the contribution of σ to the equaity cannot be canceed by contributions from propery induced representations. In the seque of the fifth Section we give a cassification of representations of G n(a). We define the notion of a basic cuspida representation for groups of type G k (A) (see Proposition 5.5 and the seque). These basic cuspida representations are a cuspida. Neven Grbac wi show in his Appendix that these are actuay the ony cuspida representations. Then the residua discrete

5 GLOBAL JACQUET-LANGLANDS 5 series of G n (A) are obtained from cuspida representations in the same way the residua discrete series of GL n (A) are obtained from cuspida representations in [MW]. This cassification is obtained directy by transfer from the Moegin-Wadspurger cassification for G n. Moreover, for any (irreducibe) automorphic representation π of G n, we know that ([La]) there exists a coupe (P, ρ ) where P is a paraboic subgroup of G n containing the group of upper trianguar matrices and ρ is a cuspida representation of the Levi factor L of P twisted by a rea non ramified character such that π is a constituent (in the sense of [La]) of the induced representation from ρ to G n with respect to P. We prove (Proposition 5.7 (c)) that this coupe (ρ, L ) is unique up to conjugation. This resut is an anaogue for G n of Theorem 4.4 of [JS]. The ast Section is devoted to the computation of L-functions, ɛ -factors (in the sense of [GJ]) and their behavior under the oca transfer of irreducibe (especiay unitary) representations. The behavior of the ɛ-factors then foows. These cacuations are either we known or trivia, but we fee it is natura to give them expicity here. The L-functions and ɛ -factors in question are preserved under the correspondence for square integrabe representations. In genera, ɛ - factors (but not L-functions) are preserved under the correspondence for irreducibe unitary representations. In the Appendix Neven Grbac competes the cassification of the discrete spectrum by showing that a the representations except the basic cuspida ones are residua. His approach appies the Langands spectra theory. The essentia part of this work has been done at the Institute for Advanced Study, Princeton, during the year 004 and I woud ike to thank the Institute for the warm hospitaity and support. They were expounded in a preprint from the beginning of 006. The present paper contains exacty the same oca resuts as that preprint. Two major improvements obtained in 007 concern the goba resuts. The first one is the proof of the fact that any discrete series of the inner form transfers (based on a better understanding of the trace formua from [AC]). The second is a compete cassification of the residua spectrum thanks to the Appendix of Neven Grbac. The research at the IAS has been supported by the NSF feowship no. DMS I woud ike to thank Robert Langands and James Arthur for usefu discussions about goba representations; Marko Tadić and David Renard for usefu discussions on the oca unitary dua; Abderrazak Bouaziz who expained to me the intertwining operators. I woud ike to thank Guy Henniart and Coette Moegin for the interest they showed for this work and their invauabe advices. I thank Neven Grbac for his Appendix where he carries out the ast and important step of the cassification, and for his remarks on the manuscript. Discussions with Neven Grbac have been hed during our stay at the Erwin Schrödinger Institute in Vienna and I woud ike to thank here Joachim Schwermer for his invitation.. Basic facts and notation (oca) In the seque N wi denote the set of non negative integers and N the set of positive integers. A mutiset is a set with finite repetitions. If x R, then [x] wi denote the biggest integer inferior or equa to x. Let F be a non-archimedean oca fied and D a centra division agebra of a finite dimension over F. Then the dimension of D over F is a square d, d N. If n N, we set G n = GL n (F) and G n = GL n (D). From now on we identify a smooth representation of finite ength with its equivaence cass, so we wi consider two equivaent representations as being equa. By a character of G n we mean a smooth representation of dimension one of G n. In particuar a character is not unitary uness we specify it. Let σ be an irreducibe smooth representation of

6 6 GLOBAL JACQUET-LANGLANDS G n. We say σ is square integrabe if σ is unitary and has a non-zero matrix coefficient which is square integrabe moduo the center of G n. We say σ is essentiay square integrabe if σ is the twist of a square integrabe representation by a character of G n. We say σ is cuspida if σ has a non-zero matrix coefficient which has compact support moduo the center of G n. In particuar a cuspida representation is essentiay square integrabe. For a n N et us fix the foowing notation: Irr n is the set of smooth irreducibe representations of G n, D n is the subset of essentiay square integrabe representations in Irr n, C n is the subset of cuspida representations in D n, Irrn u (resp. Du n, Cu n ) is the subset of unitary representations in Irr n (resp. D n, C n ), R n is the Grothendieck group of admissibe representations of finite ength of G n, ν is the character of G n defined by the absoute vaue of the determinant (notation independent of n this wi ighten the notation and cause no ambiguity in the seque). For any σ D n, there is a unique coupe (e(σ), σ u ) such that e(σ) R, σ u Dn u and σ = ν e(σ) σ u. We wi systematicay identify π Irr n with its image in R n and consider Irr n as a subset of R n. Then Irr n is a Z-basis of the Z-modue R n. If n N and (n 1, n,..., n k ) is an ordered set of positive integers such that n = k n i then the subgroup L of G n consisting of bock diagona matrices with bocks of sizes n 1, n,..., n k in this order from the eft upper corner to the right ower corner is caed a standard Levi subgroup of G n. The group L is canonicay isomorphic with the product k G n i, and we wi identify these two groups. Then the notation Irr(L), D(L), C(L), D u (L), C u (L), R(L) extend in an obvious way to L. In particuar Irr(L) is canonicay isomorphic to k Irr n i and so on. We denote ind Gn L the normaized paraboic induction functor where it is understood that we induce with respect to the paraboic subgroup of G n containing L and the subgroup of upper trianguar matrices. Then ind Gn L extends to a group morphism ign L : R(L) R n. If π i R ni for i {1,,..., k} and n = k n i, we denote π 1 π... π k or abridged k π i the representation ind Gn k Gn i k σ i of G n. Let π be a smooth representation of finite ength of G n. If distinction between quotient, subrepresentation and subquotient of π is not reevant, we consider π as an eement of R n (identification with its cass) with no extra expanation. If g G n for some n, we say g is reguar semisimpe if the characteristic poynomia of g has distinct roots in an agebraic cosure of F. If π R n, then we et χ π denote the function character of π, as a ocay constant map, stabe under conjugation, defined on the set of reguar semisimpe eements of G n. We adopt the same notation adding a sign for G n: Irr n, D n, C n, Irr u n, D u n, C u n, R n. There is a standard way of defining the determinant and the characteristic poynomia for eements of G n, in spite of D being non commutative (see for exampe [Pi] Section 16). If g G n, then the characteristic poynomia of g has coefficients in F, it is monic and has degree nd. The definition of a reguar semisimpe eement of G n is then the same as for G n. If π R n, we et again χ π be the function character of π. As for G n, we wi denote ν the character of G n given by the absoute vaue of the determinant (there wi be no confusion with the one on G n )..1. Cassification of Irr n (resp. Irr n ) in terms of D (resp. D ), n. Let π Irr n. There exists a standard Levi subgroup L = k G n i of G n and, for a 1 i k, ρ i C ni,

7 GLOBAL JACQUET-LANGLANDS 7 such that π is a subquotient of k ρ i. The non-ordered mutiset of cuspida representations {ρ 1, ρ,...ρ k } is determined by π and is caed the cuspida support of π. We reca the Langands cassification which takes a particuary nice form on G n. Let L = k G n i be a standard Levi subgroup of G n and σ D(L) = k D n i. Let us write σ = k σ i with σ i D ni. For each i, write σ i = ν ei σi u, where e i R and σi u Du n i. Let p be a permutation of the set {1,,..., k} such that the sequence e p(i) is decreasing. Let L p = k G n p(i) and σ p = k σ p(i). Then ind Gn L p σ p has a unique irreducibe quotient π and π is independent of the choice of p under the condition that (e p(i) ) 1 i k is decreasing. So π is defined by the non ordered mutiset {σ 1, σ,..., σ k }. We write then π = Lg(σ). Every π Irr n is obtained in this way. If π Irr n and L = k G n i and L = k j=1 G n are two standard Levi subgroups of G n, if j σ = k σ i, with σ i D ni, and σ = k j=1 σ j, with σ j D n, are such that π = Lg(σ) = Lg(σ ), j then k = k and there exists a permutation p of {1,,..., k} such that n j = n p(i) and σ j = σ p(i). So the non ordered mutiset {σ 1, σ,..., σ k } is determined by π and it is caed the essentiay square integrabe support of π which we abridge as the esi-support of π. An eement S = i Gn L σ of R n, with σ D(L), is caed a standard representation of G n. We wi often write Lg(S) for Lg(σ). The set B n of standard representations of G n is a basis of R n and the map S Lg(S) is a bijection from B n onto Irr n. A these resuts are consequences of the Langands cassification (see [Ze] and [Rod]). We aso have the foowing resut: if for a π Irr n we write π = Lg(S) for some standard representation S and then for a π Irr n \{π} we set π < π if and ony if π is a subquotient of S, then we obtain a we defined partia order reation on Irr n. The same definitions and theory, incuding the order reation, hod for G n (see [Ta]). The set of standard representations of G n is denoted here by B n. For G n or G n we have the foowing Proposition, where σ 1 and σ are essentiay square integrabe representations: Proposition.1. (a) The representation Lg(σ 1 ) Lg(σ ) contains Lg(σ 1 σ ) as a subquotient with mutipicity 1. (b) If π is another irreducibe subquotient of Lg(σ 1 ) Lg(σ ), then π < Lg(σ 1 σ ). In particuar, if Lg(σ 1 ) Lg(σ ) is reducibe, it has at east two different subquotients. For G n, assertion (a) is proven in its dua form in [Ze] (Proposition 8.4). It is proven in its present form in [Ta] (Proposition.3) for the more genera case of G n. Assertion (b) is then obvious because of the definition (here) of the order reation, and since any irreducibe subquotient of Lg(σ 1 ) Lg(σ ) is aso an irreducibe subquotient of σ 1 σ... Cassification of D n in terms of C, n. Let k and be two positive integers and set n = k. Let ρ C. Then the representation k 1 i=0 νi ρ has a unique irreducibe quotient σ. σ is an essentiay square integrabe representation of G n. We write then σ = Z(ρ, k). Every σ D n is obtained in this way and, k and ρ are determined by σ. This may be found in [Ze]. In genera, a set X = {ρ, νρ, ν ρ,..., ν a 1 ρ}, ρ C b, a, b N, is caed a segment, a is the ength of the segment X and ν a 1 ρ is the ending of X..3. Loca Jacquet-Langands correspondence. Let n N. Let g G nd and g G n. We say that g corresponds to g if g and g are reguar semisimpe and have the same characteristic poynomia. We shorty write then g g.

8 8 GLOBAL JACQUET-LANGLANDS Theorem.. There is a unique bijection C : D nd D n such that for a π D nd we have χ π (g) = ( 1) nd n χ C(π) (g ) for a g G nd and g G n such that g g. For the proof, see [DKV] if the characteristic of the base fied F is zero and [Ba] for the non zero characteristic case. I shoud quote here aso the particuar cases [JL], [F] and [Ro] which contain some germs of the genera proof in [DKV]. We identify the centers of G nd and G n via the canonica isomorphism. Then the correspondence C preserves centra characters so in particuar σ Dnd u if and ony if C(σ) D u n. If L = k G n i is a standard Levi subgroup of G n we say that the standard Levi subgroup L = k G dn i of G nd corresponds to L. Then the Jacquet-Langands correspondence extends in an obvious way to a bijective correspondence D(L) to D (L ) with the same properties. We wi denote this correspondence by the same etter C. A standard Levi subgroup L of G n corresponds to a standard Levi subgroup or G r if and ony if it is defined by a sequence (n 1, n,..., n k ) such that each n i is divisibe by d. We then say that L transfers..4. Cassification of D n in terms of C, n. The invariant s(σ ). Let be a positive integer and ρ C. Then σ = C 1 (ρ ) is an essentiay square integrabe representation of G d. We may write σ = Z(ρ, p) for some p N and some ρ C d. Set then s(ρ ) = p and ν ρ = ν s(ρ ). p Let k and be two positive integers and set n = k. Let ρ C. Then the representation k 1 i=0 νi ρ ρ has a unique irreducibe quotient σ. σ is an essentiay square integrabe representation of G n. We write then σ = T(ρ, k). Every σ D n is obtained in this way and, k and ρ are determined by σ. We set then s(σ ) = s(ρ ). For this cassification see [Ta]. A set S = {ρ, ν ρ ρ, νρ ρ,..., ν a 1 ρ ρ }, ρ C b, a, b N, is caed a segment, a is the ength of S and ν a 1 ρ ρ is the ending of S..5. Mutisegments, order reation, the function and rigid representations. Here we wi give the definitions and resuts in terms of groups G n, but one may repace G n by G n. We have seen (Section. and.4) that to each σ D n one may associate a segment. A mutiset of segments is caed a mutisegment. If M is a mutisegment, the mutiset of endings of its eements (see Section. and.4 for the definition) is denoted E(M). If π G n, the mutiset of the segments of the eements of the esi-support of π is a mutisegment; we wi denote it by M π. M π determines π. The reunion with repetitions of the eements of M π is the cuspida support of π. Two segments S 1 and S are said to be inked if S 1 S is a segment different from S 1 and S. If S 1 and S are inked, we say they are adjacent if S 1 S = Ø. Let M be a mutisegment, and assume S 1 and S are two inked segments in M. Let M be the mutisegment defined by - M = (M {S 1 S } {S 1 S })\{S 1, S } if S 1 and S are not adjacent (i.e. S 1 S Ø), and - M = (M {S 1 S })\{S 1, S } if S 1 and S are adjacent (i.e. S 1 S = Ø). We say that we made an eementary operation on M to get M, or that M was obtained from M by an eementary operation. We then say M is inferior to M. It is easy to verify this extends by transitivity to a we defined partia order reation < on the set of mutisegments of G n. The foowing Proposition is a resut of [Ze] (Theorem 7.1) for G n and [Ta] (Theorem 5.3) for G n. Proposition.3. If π, π Irr n, then π < π if and ony if M π < M π.

9 GLOBAL JACQUET-LANGLANDS 9 If π < π, then the cuspida support of π equas the cuspida support of π. Define a function on the set of mutisegments as foows: if M is a mutisegment, then (M) is the maximum of the engths of the segments in M. If π Irr n, set (π) = (M π ). The foowing Lemma is obvious: Lemma.4. If M is obtained from M by an eementary operation then (M) (M ) and E(M ) E(M). As a function on Irr n, is decreasing. The next important Proposition is aso a resut from [Ze] and [Ta]: Proposition.5. Let π Irr k and π Irr. If for a S M π and S M π the segments S and S are not inked, then π π is irreducibe. There is an interesting consequence of this ast Proposition. Let N and ρ C. We wi ca the set X = {ν a ρ} a Z a ine, the ine generated by ρ. Of course X is aso the ine generated by νρ for exampe. If π Irr n, we say π is rigid if the set of eements of the cuspida support of π is incuded in a singe ine. As a consequence of the previous Proposition we have the Coroary.6. Let π Irr n. Let X be the set of the eements of the cuspida support of π. If {D 1, D,..., D m } is the set of a the ines with which X has a non empty intersection, then one may write in the unique (up to permutation) way π = π 1 π... π m with π i rigid irreducibe and the set of eements of the cuspida support of π i incuded in D i, 1 i m. We wi say π = π 1 π... π m is the standard decomposition of π in a product of rigid representations (this is ony the shortest decomposition of π in a product of rigid representations, but there might exist finer ones). The same hods for G n..6. The invoution. Aubert defined in [Au] an invoution (studied too by Schneider and Stuher in [ScS]) of the Grothendieck group of smooth representations of finite ength of a reductive group over a oca non-archimedean fied. The invoution sends an irreducibe representation to an irreducibe representation up to a sign. We speciaize this invoution to G n, resp. G n, and denote it i n, resp. i n. We wi write i and i when the index is not reevant or it is ceary understood. With this notation we have the reation i(π 1 ) i(π ) = i(π 1 π ), i.e. the invoution commutes with the paraboic induction. The same hods for i. The reader may find a these facts in [Au]. If π Irr n, then one and ony one among i(π) and i(π) is an irreducibe representation. We denote it by i(π). We denote i the invoution of Irr n defined by π i(π). The same facts and definitions hod for i. The agorithm conjectured by Zeevinsky for computing the esi-support of i(π) from the esi-support of π when π is rigid (and hence more generay for π Irr n, cf. Coroary.6) is proven in [MW1]. The same facts and agorithm hod for i as expained in [BR]..7. The extended correspondence. The correspondence C 1 may be extended in a natura way to a correspondence LJ between the Grothendieck groups. Let S = i G n L σ B n, where L is a standard Levi subgroup of G n and σ an essentiay square integrabe representation of L. Set M n (S ) = i G nd L C 1 (σ ), where L is the standard Levi subgroup of G nd corresponding to L. Then M n (S ) is a standard representation of G nd and M n reaizes an injective map from B n into B nd. Define Q n : Irr n Irr nd by Q n (Lg(S )) = Lg(M n (S )). If π 1 < π, then Q n (π 1 ) < Q n(π ). So Q n induces on Irr(G n), by transfer from G nd, an order reation << which is stronger than <.

10 10 GLOBAL JACQUET-LANGLANDS Let LJ n : R nd R n be the Z-morphism defined on B nd by setting LJ n (M n (S )) = S and LJ n (S) = 0 if S is not in the image of M n. Theorem.7. (a) For a n N, LJ n is the unique map from R nd to R n such that for a π R nd we have χ π (g) = ( 1) nd n χ LJn(π)(g ) for a g g. (b) The map LJ n is a surjective group morphism. (c) One has LJ n (Q n (π )) = π + where b j Z and π j Irr n. (d) One has π j <<π b j π j LJ n i nd = ( 1) nd n i n LJ n. See [Ba4]. We wi often drop the index and write ony Q, M and LJ. LJ may be extended in an obvious way to standard Levi subgroups. For a standard Levi subgroup L of G n which correspond to a standard Levi subgroup L of G nd we have LJ i G nd L = i G n L LJ. We wi say that π R nd is d-compatibe if LJ n (π) 0. This means that there exists a reguar semisimpe eement g of G nd which corresponds to an eement of G n and such that χ π (g) 0. A reguar semisimpe eement of G nd corresponds to an eement of G n if and ony if its characteristic poynomia decomposes into irreducibe factors with the degrees divisibe by d. So our definition depends ony on d, not on D. A product of representations is d-compatibe if and ony if each factor is d-compatibe..8. Unitary representations of G n. We are going to use the word unitary for unitarizabe. Let k, be positive integers and set k = n. Let ρ C and set σ = Z(ρ, k). Then σ is unitary if and ony if ν k 1 ρ is unitary. We set then ρ u = ν k 1 ρ C u and we write σ = Z u (ρ u, k). From now on, anytime we write σ = Z u (ρ, k), it is understood that σ and ρ are unitary. Now, if σ D u, we set k 1 u(σ, k) = Lg( ν k 1 i σ). The representation u(σ, k) is an irreducibe representation of G n. If α ]0, 1 [, we moreover set i=0 π(u(σ, k), α) = ν α u(σ, k) ν α u(σ, k). The representation π(u(σ, k), α) is an irreducibe representation of G n (by Proposition.5). Let us reca the Tadić cassification of unitary representations in [Ta1]. Let U be the set of a the representations u(σ, k) and π(u(σ, k), α) where k, range over N, σ C and α ]0, 1 [. Then any product of eements of U is irreducibe and unitary. Every irreducibe unitary representation π of some G n, n N, is such a product. The non ordered mutiset of the factors of the product are determined by π. The fact that a product of irreducibe unitary representations is irreducibe is due to Bernstein ([Be]). Tadić computed the decomposition of the representation u(σ, k) in the basis B n of R n.

11 GLOBAL JACQUET-LANGLANDS 11 Proposition.8. ([Ta4]) Let σ = Z(ρ, ) and k N. Let Wk be the set of permutations w of {1,,..., k} such that w(i) + i for a i {1,,..., k}. Then we have: k u(σ, k) = ν k+ ( ( 1) sgn(w) Z(ν i ρ, w(i) + i)). w W k One can aso compute the dua of u(σ, k). Proposition.9. Let σ = Z u (ρ u, ) and k N. If τ = Z u (ρ u, k), then i(u(σ, k)) = u(τ, ). This is the Theorem 7.1 iii) [Ta1], and aso a consequence of [MW1]..9. Unitary representations of G n. Let k, N and set n = k. Let ρ C and σ = T(ρ, k) D n. As for G n, one has σ D n u if and ony if ν k 1 ρ ρ is unitary; we set then ρ u = ν k 1 ρ and write σ = T u (ρ u, k). If now σ D u, we set and k 1 u (σ, k) = Lg( ν k 1 i σ σ ) i=0 k 1 ū(σ, k) = Lg( ν k 1 i σ ). The representations u (σ, k) and ū(σ, k) are irreducibe representations of G n. If moreover α ]0, 1 [, we set π(u (σ, k), α) = νσ α u (σ, k) ν α σ u (σ, k). The representation π(u (σ, k), α) is an irreducibe representation of G n (cf. [Ta]; a consequence of the (restated) Proposition.5 here). We have the formuas: (.1) ū(σ, ks(σ )) = and, for a integers 1 b s(σ ) 1, (.) ū(σ, ks(σ ) + b) = ( b s(σ ) i=0 ν i s(σ )+1 u (σ, k); s(σ ) b b+1 i ν u (σ, k + 1)) ( ν j s(σ ) b+1 u (σ, k)), with the convention that we ignore the second product if k = 0. The products are irreducibe, by Proposition.5, because the segments appearing in the esisupport of two different factors are never inked. The fact that the product is indeed ū(σ, ks(σ )) (and resp. ū(σ, ks(σ ) + b)) is then cear by Proposition.1. This kind of formuas has been used (at east) in [BR1] and [Ta6]. The representations u (σ, k) and ū(σ, k) are known to be unitary at east in zero characteristic ([Ba4] and [BR1]). j=1

12 1 GLOBAL JACQUET-LANGLANDS One has Proposition.10. Let σ = Z u (ρ u, ) and k N. If τ = Z u (ρ u, k), then (a) i (u (σ, k)) = u (τ, ) and (b) i (ū(σ, ks(σ ))) = ū(τ, s(σ )). Proof. The caim (a) is a direct consequence of [BR]. For the caim (b), it is enough to use the reation.1, the caim (a) here and the fact that i commutes with paraboic induction..10. Hermitian representations and an irreducibiity trick. If π Irr n, write h(π) for the compex conjugated representation of the contragredient of π. A representation π Irr n is caed hermitian if π = h(π) (we reca, to avoid confusion, that here we use = for the usua equivaent ). A unitary representation is aways hermitian. If A = {σ i } 1 i k is a mutiset of essentiay square integrabe representations of some G i, we define the mutiset h(a) by h(a) = {h(σ i )} 1 i k. If π Irr n and x R, then h(νx π) = ν x h(π), so if σ D and we write σ = ν e σ u with e R and σ u D u, then h(σ ) = ν e σ u D. An easy consequence of Proposition in [Ca] is the Proposition.11. If π Irr n, and A is the esi-support of π, then h(a) is the esi-support of h(π). In particuar, π is hermitian if and ony if the esi-support A of π satisfies h(a) = A. Let us give a Lemma. Lemma.1. Let π 1 Irr n 1 and π Irr n and assume h(π 1 ) π. Then there exists ε > 0 such that for a x ]0, ε[ the representation a x = ν x π 1 ν x π is irreducibe, but not hermitian. Proof. For a x R et A x be the esi-support of ν x π 1 and B x be the esi-support of ν x π. Then the set X of x R such that A x h(a x ) or B x h(b x ) is finite (it is enough to check the centra character of the representations in these mutisets). The set Y of x R such that the cuspida supports of A x and B x have a non empty intersection is finite too. Now, if x R\Y, a x is irreducibe by the Proposition.5. Assume moreover x / X. As a x is irreducibe, if it were hermitian one shoud have h(a x ) h(b x ) = A x B x (where the reunions are to be taken with mutipicities, as reunions of mutisets) by the Proposition.11. But if A x h(a x ) = and B x h(b x ) =, then this woud ead to h(a x ) = B x, and hence to h(π 1 ) = π which contradicts the hypothesis. We now state our irreducibiity trick. Proposition.13. Let u i Irr u n i, i {1,,..., k}. If, for a i {1,,..., k}, u i u i is irreducibe, then k u i is irreducibe. Proof. There exists ε > 0 such that for a i {1,,..., k} the cuspida supports of ν x u i and ν x u i are disjoint for a x ]0, ε[. Then, for a i {1,,..., k}, for a x ]0, ε[, the representation ν x u i ν x u i is irreducibe. As, by hypothesis, u i u i is irreducibe and unitary, the representation ν x u i ν x u i is aso unitary for a x ]0, ε[ (see for exampe [Ta3], Section (b)). So k ν x u i ν x u i is a sum of unitary representations. But we have (in the Grothendieck group) k (ν x u i ν x u i ) = k (νx u i ) k (ν x u i ). If k u i were reducibe, then it woud contain at east two different unitary subrepresentations π 1 and π (Proposition.1). But then, for some x ]0, ε[, (ν x k u i ) (ν x k u i )

13 GLOBAL JACQUET-LANGLANDS 13 contains an irreducibe, but not hermitian, subquotient of the form ν x π 1 ν x π (by Lemma.1). This subquotient woud be non-unitary which contradicts our assumption. 3. Loca resuts 3.1. First resuts. Let σ D n u and set σ = C 1 (σ ) Dnd u. Write σ = T u (ρ, ) for some N, n and ρ C u n. As C 1 (ρ ) D u nd we may write C 1 (ρ ) = Z u (ρ, s(σ )) for some. We set = s(σ ). Then we have σ = Z u (ρ, ) (means one can recover the ρ C u nd s(σ ) cuspida support of σ from the cuspida support of σ ; it is a consequence of the fact that the correspondence commutes with the Jacquet functor; the origina proof for square integrabe representations is [DKV], Theorem B..b). Let k be a positive integer and set k = ks(σ ). Let H be the group of permutations w of {1,,..., k } such that s(σ ) w(i) i for a i {1,,..., k }. For the meaning of Wk the foowing, see Proposition.8. This is Lemma 3.1 in [Ta5]: and W k Lemma 3.1. If w H, then for each j {1,,..., s(σ )}, the set of eements of {1,,..., k } equa to j mod s(σ ) is stabe under w, and w induces a permutation w j of {1,,..., k} defined by the fact that, if w(as(σ ) + j) = bs(σ ) + j then w j (a + 1) = b + 1. The map w (w 1, w,..., w s(σ )) is an isomorphism of groups from H to (S k ) s(σ ). One has w H Wk if and ony if for a j, w j Wk. Moreover, sgn(w) = s(σ ) j=1 sgn(w j). We have the foowing: Theorem 3.. (a) One has LJ(u(σ, k )) = ū(σ, k ). (b) The induced representation ū(σ, k ) ū(σ, k ) is irreducibe. (c) We have the character formua ū(σ, k ) = ν k + + s(σ ) 1 ( w H W k ( 1) sgn(w) k T(ν i ρ, w(i) i s(σ ) + )). Proof. (a) Let τ = T u (ρ, k) and set τ = C 1 (τ ). For the same reasons as expained for σ, we have τ = Z u (ρ, k ). We appy Theorem.7 (c) to ū(σ, k ) and ū(τ, ). We get and (3.1) LJ(u(σ, k )) = ū(σ, k ) + (3.) LJ(u(τ, )) = ū(τ, ) + π j <<ū(σ,k ) τ q <<ū(τ, ) We want to show that a the b j vanish. Let us write the dua equation to 3.1 (cf. Theorem.7 (d)). As i(u(σ, k )) = u(τ, ) (Proposition.9) and i (ū(σ, k )) = ū(τ, ) (Proposition.10), we obtain: b j π j c q τ q in

14 14 GLOBAL JACQUET-LANGLANDS (3.3) LJ(u(τ, )) = ε 1 ū(τ, ) + ε π j <<ū(σ,k ) b j i (π j ). for some signs ε 1, ε { 1, 1}. The equations 3. and 3.3 impy then the equaity: (3.4) ū(τ, ) + c q τ q = ε 1ū(τ, ) + ε ( b j i (π j )). τ q <<ū(τ, ) π j <<ū(σ,k ) First, observe that since π j ū(σ, k ) for a j, we aso have i (π j ) ū(τ, ) for a j. So by the inear independence of irreducibe representations in the Grothendieck group, ε 1 = 1 and the term ū(τ, ) cances. We wi now show that the remaining equaity τ q <<ū(τ, ) c q τ q = ε ( π j <<ū(σ,k ) b j i (π j )). impies that a the coefficients b j vanish. The argument is the inear independence of irreducibe representations and the Lemma: Lemma 3.3. If π j << ū(σ, k ), it is impossibe to have i (π j ) << ū(τ, ). Proof. The proof is compicated by the fact that we do not have in genera equaity < = << between the order reations. But this does not reay matter. Reca that π j << ū(σ, k ), means by definition Q(π j ) < Q(ū(σ, k )), i.e. there exists π j < u(σ, k ) such that the esi-support of π j corresponds to the esi-support of π j eement by eement by Jacquet-Langands. This impies the ony two properties we need: (*) the cuspida support of π j equas the cuspida support of ū(σ, k ) and (**) we have the incusion reation E(M π j ) E(Mū(σ,k )) (Lemma.4). The property (*) impies that, if π j = a 1 a... a x is a standard decomposition of π j in a product of rigid representations, then: - x = s(σ ), - we may assume that for 1 t s(σ ) the ine of a t is generated by ν t ρ and - the mutisegment M t of a t has at most k eements. So, if one uses the Zeevinsky-Moegin-Wadspurger agorithm to compute the esi-support M # t of i (a t ) (cf. [BR]), one finds that (M # t ) k, since each segment in M # t is constructed by picking up at most one cuspida representation from each segment in M t. This impies that ( i (a t ) ) k. As i (π j ) = i (a 1 ) i (a )... i (a x ) we eventuay have ( i (π j ) ) k. Assume now i (π j ) << ū(τ, ). We wi show that ( i (π j ) ) > k. Set Q( i (π j ) ) = γ and we know that γ < u(τ, ). We obviousy have in our particuar situation (γ) = s(σ )( i (π j ) ). So we want to prove (γ) > k. The mutisegment of γ is obtained by a sequence of eementary operation from the mutisegment of u(τ, ): at the first eementary operation on the mutisegment of u(τ, ) we get a mutisegment M such that (M ) > k and then we appy Lemma.4. We get, indeed, (γ) > k. So our assumption eads to a contradiction.

15 GLOBAL JACQUET-LANGLANDS 15 (b) The proof uses the caim (a) and is simiar to its proof. Let τ and τ be defined ike in (a). By the part (a) we know now that LJ(u(σ, k )) = ū(σ, k ) and LJ(u(τ, )) = ū(τ, ), so LJ(u(σ, k ) u(σ, k )) = ū(σ, k ) ū(σ, k ) and LJ(u(τ, ) u(τ, )) = ū(τ, ) ū(τ, ). Let us ca K 1 the Langands quotient of the esi-support of ū(σ, k ) ū(σ, k ) and K the Langands quotient of the esi-support of ū(τ, ) ū(τ, ). Using [BR] it is easy to see that i (K 1 ) = K. Then we may write, using Theorem.7 (c) and Proposition.1: (3.5) LJ(u(σ, k ) u(σ, k )) = K 1 + b j π j π j<<k 1 and (3.6) LJ(u(τ, ) u(τ, )) = K + ξ m <<K c m ξ m. We want to prove that a the b j vanish. Let us take the dua in the equation 3.5 (cf. Proposition.7 (d)): (3.7) LJ(i(u(σ, k ) u(σ, k ))) = ±(i (K 1 ) + π j<<k 1 b j i (π j )). We know that i(u(σ, k ) u(σ, k )) = u(τ, ) u(τ, ) because i commutes with the induction functor and we have i(u(σ, k )) = u(τ, ) by Proposition.9. As i (K 1 ) = K, we get from equations 3.6 and 3.7 after canceation of K (as in the equation 3.4): b j i (π j ) = ±( c m ξ m ). π j<<k 1 ξ m <<K To show that a the b j vanish, it is enough, by the inear independence of irreducibe representations, to show the foowing: Lemma 3.4. If π << K 1 it is impossibe to have i (π ) << K. Proof. The proof of Lemma 3.3 appies here with a minor modification. We write again π = a 1 a... a s(σ ) such that the ine of a t, 1 t s(σ ), is generated by ν t ρ. The difference here is that the mutisegment M of a t may have up to k eements. We wi prove though, that in this case again: Lemma 3.5. The mutisegment m # of i (a t ) verifies (m # ) k. This impies that (π ) k and the rest of the proof goes the same way as for (a). Proof. Let us denote m the mutisegment of a t (m and m # respect the notation in [MW1]). The mutisegment m # is obtained from m using the agorithm in [MW1] (cf. [BR]). As π << K 1, one has E(m) {ν k +1 ρ ρ, ν k + ρ ρ,..., ν +k ρ ρ } (it is the property (**) given at the beginning of the proof of Lemma 3.3). One constructs a the segments of m # ending with ν +k ρ ρ using ony cuspida representations in E(m) (Remark II.. in [MW1]). So the ength

16 16 GLOBAL JACQUET-LANGLANDS of the constructed segments is at most k. Let m be the mutisegment obtained from m after we dropped from each segment of m the cuspida representations used in this construction. We obviousy have then E(m ) {ν k ρ ρ, ν k +1 ρ ρ,..., ν +k 1 ρ ρ } which has again k eements. So going through the agorithm we wi find that a the segments of m # have ength at most k. (c) The caim (a) we have just proven aows us to transfer the formua of the Proposition.8 by LJ. We have LJ(u(σ, k )) = ν k + ( w W k ( 1) k sgn(w) LJ( Z(ν i ρ, w(i) + i))). The representations k Z(νi ρ, w(i)+ i) are standard. If w is such that, for some i, s(σ ) does not divide w(i) i, then LJ( k Z(νi ρ, w(i) + i)) = 0. If w satisfies s(σ ) w(i) i for a i, i.e. w H, then LJ( Z(ν i ρ, w(i) + i)) = T(ν i+ s(σ ) 1 ρ, w(i) i s(σ + ). ) k Hence the formua of (c). k Coroary 3.6. Let n, k N and σ D u n. (a) u (σ, k) u (σ, k) is irreducibe. π(u (σ, k), α) are unitary for α ]0, 1 [. (b) Write σ = T u (ρ, ) for some unitary cuspida representation ρ. Let W k be the set of permutation w of {1,,..., k} such that w(i)+ i for a i {1,,..., k}. Then we have: u (σ, k) = ν k+ σ ( w W k k ( 1) sgn(w) T(νσ i ρ, w(i) + i)) Proof. (a) It is cear that u (σ, k) u (σ, k) is irreducibe from the part (b) of Theorem 3. and the formua.1. The fact that this impies that a the π(u (σ, k), α) are unitary is expained in [Ta]. (b) We want to show that We use the equaity u (σ, k) = ν k+ σ ( w W k ū(σ, ks(σ )) = k ( 1) sgn(w) T(νσ i ρ, w(i) + i)). s(σ ) j=1 ν j s(σ )+1 u (σ, k) and the character formua for ū(σ, ks(σ )) obtained in Theorem 3. (c). Set U = ν k+ σ ( k ( 1) sgn(w) T(ν i σ ρ, w(i) + i)) R n. We have w W k

17 GLOBAL JACQUET-LANGLANDS 17 s(σ ) j=1 s(σ ) = ν k+ s(σ ) ν j s(σ )+1 ( j=1 s(σ ) = ν k + + s(σ ) 1 ( ν k + + s(σ ) 1 j=1 w W k w W k w 1,w,...,w s(σ ) Wk j=1 ν j s(σ )+1 U = k ( 1) sgn(w) T(νσ i ρ, w(i) + i)) = k ( 1) sgn(w) T(ν (i 1)s(σ )+j ρ, w(i) + i)) = s(σ ) k ( 1) sgn(wj) T(ν (i 1)s(σ )+j ρ, w j (i)+ i)). Using Lemma 3.1 we find that this ast formua is equa to the character formua of ū(σ, ks(σ )) (Theorem 3..c). As ū(σ, ks(σ )) is irreducibe, we wi show that so is U. The formua defining U is an aternated sum of Wk terms which are distinct eements of B n. The term k k+1 νi σ σ, corresponding to the trivia w, is maxima. To prove that, one may use Lemma.4 and the fact that one has ( k term t in the sum. The Langands quotient of this maxima term k appears then in the sum with coefficient 1. So we may write: m U = π 0 + b t π t t=1 k+1 νi σ σ ) =, whie (t) > for any other k+1 νi σ σ is u (σ, k) and where π 0 = u (σ, k), b t are non-zero integers, π t Irr n and the π t, 0 t m, are distinct, with the convention m = 0 if U = u (σ, k). The representations π t are rigid and supported on the same ine L (generated by ν k+ ρ ρ ). For different j in {1,,..., s(σ s(σ j )+1 )}, the ines ν L are different. So, as the π t are distinct (and have distinct esi-support), s(σ ) j=1 s(σ )+1 νj U is a inear combination of exacty (m + 1) s(σ ) irreducibe distinct representations each appearing with non-zero coefficient. As it is irreducibe, we have m = Transfer of u(σ, k). Let k,, q be positive integers, set n = kq and et ρ Cq u and σ = Z u (ρ, ) Dq u, τ = Zu (ρ, k) Dkq u. Let s be the smaest positive integer such that d sq. In the next Proposition we give the genera resut of the transfer of u(σ, k). The question has no meaning uness d n (i.e. s k) which we sha assume. Proposition 3.7. (a) If d q (i.e. s ), then σ = C(σ) is we defined; we have s = s(σ ) and LJ(u(σ, k)) = ū(σ, k). (b) If d kq (i.e. s k), then τ = C(τ) is we defined; we have s = s(τ ) and LJ(u(σ, k)) = ε i (ū(τ, )) where ε = 1 if s is odd and ε = ( 1) k s if s is even. (c) If d does not divide neither q, nor kq (i.e. s does not divide neither nor k), then LJ(u(σ, k)) = 0.

18 18 GLOBAL JACQUET-LANGLANDS Proof. (a) We have the formua for the decomposition of u(σ, k) in the standard basis B n (Proposition.8) so we may compute the formua for the decomposition of LJ(u(σ, k)) in the standard basis B n by transfer. On the other hand, we have the formua for the decomposition of ū(σ, k) in the standard basis B n using the formua. and the Coroary 3.6 (b). The equaity of the two decompositions in the basis B n eads again to the combinatoria Lemma 3.1 in [Ta5]. (b) Up to the sign ε, this is a consequence of the caim (a) and the dua transform, Theorem.7 (d), since i(u(τ, )) = u(σ, k). For the sign ε, see Proposition 4.1, b) in [Ba4]. (c) The proof is in [Ta6]. It is a consequence of Proposition.8 here, which is aso due to Tadić, and the foowing Lemma for which we give here a more straightforward proof. Lemma 3.8. Let k,, s N. Assume there is a permutation w of {1,,..., k} such that for a i {1,,..., k} one has s + w(i) i. Then s k or s. Proof. Let [x] denote the biggest integer ess than or equa to x. If y N, et N y denote the set {1,,..., y}. Assume s does not divide. Summing up a the k reations s + w(i) i we find that s k. So, if (s, ) = 1, then s k. Assume (s, ) = p. Then for a i {1,,..., k}, p w(i) i. Let w 0 be the natura permutation of N [ k p ] induced by the restriction of w to {p, p,..., [ k p ]p} and w 1 the k 1 natura permutation of N [ k 1 p ]+1 induced by the restriction of w to {1, p+1,..., [ p ]p+1}. Then for a i N [ k p ] one has s p p +w 0(i) i, and for a j N [ k 1 ]+1 one has s p p +w 1(j) j. As now ( s p, p ) = 1 we have aready seen that one has s p [ k p ] and s p and so p k. It foows s p k p, i.e. s k. p k 1 [ p ] + 1. This impies [ k p ] = [ k 1 p ] New formuas. The reader might have noticed that the dua of representations u(τ, ) and u (τ, ) are of the same type, whie the dua of representations ū(τ, ) are in genera more compicated. This is why the caim (b) of Proposition 3.7 ooks awkward. We coud not express i (ū(τ, )) in terms of σ = C(σ), and for the good reason that C(σ) is not defined since the group on which σ ives does not have the appropriate size (divisibe by d). Reca the hypothesis was s(σ ) k. We expain here that one can get a formua though, in terms of u (σ +, k s(σ ) ) and u (σ, k s(σ ) ), where σ + = C(σ +) and σ = C(σ ), and the representations σ + and σ are obtained from σ by stretching and shortening it to get an appropriate size for the transfer. The formuas we wi give here are required for the goba proofs. Let τ D n and = as(τ ) + b with a, b N, 1 b s(τ ) 1. We start with the formua.: ū(τ, ) = b s(τ ) b b+1 i ν u (τ, a + 1) ν j s(τ ) b+1 u (τ, a). So one may compute the dua of ū(τ, ) using Proposition.9; if τ = T u (ρ, k), we set σ + = T u (ρ, a + 1) and, if a 0, σ = T u (ρ, a); then (3.8) i (ū(τ, )) = b b+1 i ν j=1 s(τ ) b u (σ +, k) j=1 ν j s(τ ) b+1 u (σ, k) with the convention that if a = 0 we ignore the second product. In particuar the dua of a representation of type ū(σ, k) is of the same type (i.e. some ū(γ, p)) if and ony if s(σ ) k or σ is cuspida and k < s(σ ). One can see that comparing the formua 3.8 with the formua.1 and using the fact that a product of representations of the type ν α u (σ, k) determines its factors up to permutation ([Ta]).