2 then refer to this special case of Jacquet's relative trace formula as the \Fourier summation formula". In this note we study a general case of Jacq

Transcription

1 QUADRATIC CYCLES ON GL(2n) CUSP FORMS Yuval. Flicker Let E=F be a quadratic separable extension of global elds, with completions E v =F v, adeles A E =A, ideles A E =A. A central simple algebra over F is a matrix m m algebra M m (D d ) with entries in a division algebra D = D d central of degree (deg = p dim) d over F. We shall consider only D unsplit by E, by which we mean that at v where E v is a eld, the exponent (order in the Brauer group Br (F v )=Q=) ofd v =D(F v )is odd (when D v is split, namely is a matrix algebra, its exponent (= order in Br (F v )) is one). In this case exp D v is equal to exp Dv E (= order in Br (E v )), where Dv E = D v Fv E v (= D E (F v ), where D E = D F E = D Spec F Spec E). If D is not unsplit by E then exp Dv E may be half of exp D v, for some v where E v is a eld. Under our assumption that D is unsplit by E, the algebra D E is division, central over E of degree d = deg D. Let be a simple algebra of degree 2 central over F. The multiplicative group G of M m (D H ), where D H = D F H, is an algebraic F -group, which is an inner form of the split group G sp = GL(2n)=F, n = md. Put G = G(F ), G = G(A ); = (F ), = (A ), where is the center of G; and C = C(F ), C = C(A ), where C is the multiplicative group of M m (D E ). Then C is an inner form of the split group C sp = GL(n)=E. The group G can A B be realized as consisting of the invertible matrices of the form, A, B in C, bar "B A indicating the Gal (E=F)-action on the second factor in D E = D A A E. Here " is a xed element inf, outside the norm subgroup NE = N E=F E from E, unless H = GL(2)=F, in which case we take " =1. Indeed, H can be realized by such matrices with entries A, B in A E. The group C embeds in G via A 7! diag R (A; A). This note concerns the periods P () = CnC (h)dh of cusp forms in L2 (GnG ) over the cycle CnC. This cycle has nite volume and the convergence of the integral follows at once from the rapid decay of the cusp form on GnG. Cuspidal (automorphic) representations (= irreducible submodules of the G -module L 2 (GnG ) of cusp forms in the space L 2 (GnG ) of automorphic forms) which contain a form with a non-zero period are called here cyclic. The interest in such cyclic originates from studies of arithmetic cohomology, and liftings of automorphic forms. Such studies were initiated by Waldspurger [Wa] using the theory of the Weil representation, in the case of m = d = n =1. Jacquet [J1] introduced a new technique for the study of such cusp forms, which he named the \relative trace formula". It is based on integrating the kernel of the convolution operator K f (x; y) over x and y in two cycles C 1 nc 1 and C 2 nc 2. The case of the group C C and the subgroups C 1 = C 2 = C embedded diagonally, coincides with the standard trace formula. In general Jacquet's relative trace formula involves no traces; it is a summation formula, equating a geometric with a spectral sums. The case C 1 = C 2 considered in this note is called here the \bi-period summation formula". Another notable case is introduced in Jacquet [J2] (see also [F3]); there C 2 is a unipotent subgroup, and Fourier coecients of the cusp forms (in addition to cycles) are obtained. We Department of Mathematics, 231 W. 18th Ave., The Ohio State University, Columbus, OH ; icker@math.ohio-state.edu Typeset by AMS-TE 1

2 2 then refer to this special case of Jacquet's relative trace formula as the \Fourier summation formula". In this note we study a general case of Jacquet's bi-period summation formula, with arbitrary m; d; n. The general case { introduced here { poses several interesting questions, some of which are answered below, some are stated as \working hypothesis", and some are conjectured. The simple algebra M m (D d ) over F (m d = md = n) is more split than M m (D d ) if exp D v divides exp D v for all v. In particular, ord D v is odd when E v is a eld. The multiplicative group of M m (D H d ) is denoted by G. Similar terminology applies to the group, and locally: G splits more than G precisely when G v splits more than G v for all v. In particular, the split group G sp is more split than any of its inner forms. Denote by V the set of places where G v = G(F v ) is not isomorphic to G v = G (F v ). Theorem A. Suppose that G is more split than G, and that G (hence also G ) is unsplit by E. Let be a cuspidal G -module whose component u at some place u of F where G u ' G u, is supercuspidal. Denote by the cuspidal G -module which corresponds to (see [FK2]). Fix a place u 6= u, and put V = V [fu g. Assume that (WH1) and (WH2) hold for each component v of (v R 2 V ), and that u is bi-elliptic (denitions below). If is cyclic then so is (thus P ( )= C nc (h)dh 6= for some 2, where C = M m (D E d )). Moreover, the bi-character of v is not identically zero on the set of bi-regular elements of G v which come from G v, for all v. Accept (WH1) and (WH2) as valid for v (v 2 V ). If is cyclic, u is bi-elliptic, and the bi-character of v is not identically zero on the set of bi-regular elements of G v which come from G v (v 2 V ), then is cyclic. Two extreme cases where the Theorem applies are (1) when G = GL(2n)=F is split; (2) m = m, d = d, and exp D v = exp D v for all v (in this case the invariants of D v and D v can be dierent in Br (F v ) ' Q=, hence D v 6' D v for some v). The cuspidal G -module = v and the cuspidal G -module = v correspond if v ' v for almost all v (where G v ' G v). It is shown in [FK2] that the cuspidal G - modules with a supercuspidal component u at some place u 62 V occur with multiplicity one in L 2 (GnG ); that they satisfy the rigidity theorem: if 1 = 1v and 2 = 2v have supercuspidal components 1u ' 2u, and 1v ' 2v for almost all v, then 1 ' 2 ; and that the correspondence denes an embedding of the set of the cuspidal with a supercuspidal u into the set of the cuspidal with a supercuspidal u. The image consists of the whose local components v are obtained by the local correspondence of relevant representations of G v to relevant representations of G v, for all v. In particular, if corresponds to then v ' v for all v 62 V. In fact [FK2] sharpens the work of Bernstein-Deligne-Kazhdan-Vigneras [BDKV] and [F1; III] where the case of with a supercuspidal and in addition another square-integrable component, is dealt with. The global theorem requires in particular establishing the local correspondence not only for tempered local representations, but also for relevant local representations (since the generalized Ramanujan conjecture { asserting that all components of a cuspidal are tempered { is merely a conjecture). The notion of relevant representations (the representations which may be components of a cuspidal G -module) is introduced in [FK1] in a similar context (of an r-fold covering

3 of GL(n)), where they are shown to be irreducible and unitarizable. This notion was later used e.g. by Patterson and Piatetski-Shapiro [PPS]. Of course all the main ideas in the proof of the correspondence are due to Deligne and Kazhdan. Their proof in the case of m =1(d=n; i.e. G is anisotropic) { which is remarkably simple { is explained in [F2]. The proofs of [BDKV], [F2], [F1; III] and [FK1] are based on the \Deligne-Kazhdan" simple trace formula, and that of [FK2] on a sharper form of the simple trace formula, where regular, Iwahori-invariant functions, are used. The proof here does not involve any trace formula, yet we do use some of the ideas which play key roles in the development of the simple trace formula. Our main global tool is a new \bi-period summation formula", obtained on integrating over two copies of CnC the spectral and geometric expressions for the kernel of the convolution operator r(f) on L 2 (GnG ) for a test function f with a supercuspidal component f u. An observation of Kazhdan implies that r(f) factorizes through the natural projection to the space L 2 (GnG ) of cusp forms. On the spectral side of our formula we obtain the periods of the cyclic cusp forms. On the geometric side we obtain a new type of bi-orbital integrals. As in [BDKV], [F2], [F1; III], [FK1], we choose another component { say f u { of the test function f, and restrict its support to a certain set of \bi-elliptic bi-regular" elements in our bi-periodic sense. This choice of f u greatly simplies our study of the geometric side, indeed it makes our study possible. Yet the choice of f u restricts the applicability of our technique to and with a \bi-elliptic" (a notion presently to be dened) components at u. Our proof is based on two statements, (WH1) and (WH2), which we accept here as \working hypotheses". In Proposition we prove (WH1) in a special case. The (WH1) and (WH2) are analogues of similar statements for characters, whose proofs { we hope { are applicable (after some work) in our case too. As noted above, the present note can be viewed also as a motivation to study these hypotheses. Both hypotheses are local. They concern an irreducible admissible G v -module v (see [B]), where G v = G(F v ). Working hypothesis (WH1). Let v be an admissible irreducible G v -module. Then there exists at most one (up to a scalar multiple) C v -invariant linear form on v (thus there is a single form P v : v! C with P v ( v (h)) =P v ()for all h 2 C v and 2 v ). Alternatively put, dim Hom Cv ( v ; 1) 1, or: the restriction of v to C v has the trivial quotient with multiplicity at most one. A G v -module v with P v 6= is called cyclic. Each local component of a cyclic cuspidal is cyclic, but a cuspidal whose local components are all cyclic is not necessarily cyclic. Statements similar to (WH1) were established using techniques of Gelfand-Kazhdan [GK] (cf. [B], (5.16)-(5.17), (7.6)-(7.1), [R], [NPS]) to prove (existence in the case of GL(n) and) uniqueness of Whittaker models, the uniqueness of a GL(n; F v )-invariant linear form on an irreducible GL(n; E v )-module where E v =F v is a quadratic eld extension ([F3], p. 163), the uniqueness of a GL(2;F v )-invariant form on a GL(2;K v )-module where K v is a cubic extension of F v (Prasad [P], p. 1327), as well as in the cases of such pairs as (GL(n, 1);GL(n)), (O(n, 1);O(n)), (U(n, 1);U(n)) by Bernstein, Piatetski-Shapiro, Rallis. The case where E=F and D are split has recently been treated by Jacquet and Rallis (in fact, after this note was written). These techniques would eventually lead to a proof of (WH1). Let us verify (WH1) in a special case. 3

4 4. Proposition. Let D v be a division algebra of odd degree n central over F v, E v a quadratic eld extension of F v, and H v a central simple algebra of degree 2 over F v. Note that C v = (D v Fv E v ) embeds in G v = (D v Fv H v ). For any admissible irreducible G v -module v there exists at most one (up to a scalar multiple) C v -invariant linear form on v ; namely, Hom Cv ( v ; 1) has dimension 1. Proof. By a well-known criterion of Gelfand-Kazhdan [GK] (recorded also in [P], p. 1327; [F3], p. 163), it suces to nd a non-trivial involution g 7! g # on G v which preserves C v, and xes every C v -double coset in G v. We shall check that the involution g 7! g,1 has, A, B in A B this property. For that, realize G v as the group of invertible matrices "B A D v Fv E v, and embed C v in G v via A 7! diag (A; A). Here " =1if H v = GL(2;F v ), and,1 1 1 " = "(H v ) 2 F v,n Ev =F v E v if H v is a quaternion algebra. Since = " ",1, ",1,1 1 B 1,B 1,"BB and =, and there is some x in E "B 1,"B 1 1, "BB v with x =,x 6= so that complete. x x,1 A B "B A x x A,B =,"B A ; the proof is Let H v = Cc 1 ( v ng v ) denote the convolution algebra (a choice of a Haar measure is implicit) of compactly supported (modulo v ) smooth (= locally constant when v is nonarchimedean) complex-valued functions on G v which transform trivially under v. Fix an orthonormal basis f v g in the space of the irreducible admissible G v -module v. Introduce a bi-period distribution on H v by P v (f v )= v P v ( v (f v ) v )P v ( v ): The linear form P v lies in the dual v of v. It also denes an element { denoted P _ v { in the dual e v of the contragredient e v of v. Put hp v ; v i= P v ( v );hp _ v ; _ vi= P _ v ( _ v). Then P _ v decomposes as P _ v = P v hp _ v ; _ vi v, and hp v ; v (f v )P _ v i= v hp _ v ; _ vihp v ; v (f v ) v i is an alternative expression for P v (f v ). This P v (f v ) is clearly independent of the choice of the basis f v g of v. If 1v ;:::; kv are pairwise inequivalent, then P 1v ;:::;P kv are linearly independent. Since P v is independent of the choice of basis for v, it is bi-c v -invariant, namely its value at a fv(g) b =f v (a,1 gb), (a; b 2 C v ) is equal to its value at f v. In particular the distribution P v depends on f v only via the bi-period integral (; f v )= Cv=Cv\Cv,1 C v = v f v (hh )dh dh : The convergence of this bi-orbital integral is obvious when is bi-regular (see below). Note that without assuming (WH1), the bi-period distribution P v of v is not uniquely dened.

5 Working Hypothesis (WH2). Let v be a cyclic admissible irreducible G v -module. Then there exists a bi-c v -invariant complex valued function p(g; v ), which is smooth (= locally constant if v is non-archimedean) and not identically zero on a ariski open (hence dense) subset of G v (named bi-regular below), such that P v (f v )= v ng v f v (g)p(g; v )dg: In the archimedean case, this has been shown by Sekiguchi [S]. The function p(g; v ) is named here the bi-character of R v. It is analogous to the character (g; v )or v (g) of the trace distribution tr v (f v ) = f v (g)(g; v )dg, shown by Howe [H] and Harish-Chandra [HC2] to be locally constant on the regular set (which is ariski open), and moreover (see Harish-Chandra [HC3]) locally integrable on G R v. The proof of [HC2] shows that the restriction of P v to the space of functions f K v v (g) = K f v v (kgk,1 )dk (K v = good maximal compact subgroup of G v ) is represented by a smooth function on the regular set. Since tr v (f v ) = tr v (f K v v ), this establishes the result for the trace distribution. It would be interesting to extend this simple proof of [HC2] to apply in our case too. A similar question is dealt with in [FH], where it is shown, using Howe's orbit method as in [HC3], that the bi-character exists as a locally constant function on the relatively(=bi)- regular set (introduced there), in the case of GL(n; D v )-invariant distributions on GL(n; Dv)- modules, where D v is a division algebra central over F v, while Dv = D v Fv E v, where E v =F v is a quadratic eld extension. Avery recent work by Rader and Rallis extends this method to show that the bi-character is locally constant on the bi-regular set in the present case too. The case of a supercuspidal v is discussed in the Remark below. The local integrability ([HC3]) implies that the character is not identically zero on the regular set, in the case of the trace. The bi-character of [FH] is also locally integrable, hence not identically zero on the bi-regular set. This quadratic case is very close to that of Harish-Chandra's group case. But in general, p(g; v ) often fails to be locally integrable on G v. It may be supported on the closed proper subset of \bi-singular" elements. It will be interesting to determine which v satisfy (WH2). We expect all cyclic admissible G v -modules to satisfy (WH2) (in our case), in analogy with the archimedean case; see Sekiguchi [S]. This problem seems to be accessible to available techniques, but its solution would require a separate paper. The relation A A I B "B I A A,1 = I ABA,1 "ABA,1 I in C v ) shows that on the ariski open (dense) subset v = A B "B A 5 (I is the unit ; jaj 6=6=jBj of G v, a set of representatives for C v n v =C v is given by the matrices = () = I, " I where ranges over a set of representatives for the -conjugacy classes of -regular elements in C v, with j", Ij 6=. Here jaj = det A. Following Shintani [Sh] we dene the - conjugacy class of in C v to consist of faa,1 ; a in C v g. It is well-known and easy to see that the conjugacy class in C v of intersects Cv = GL(m; D v ). The map 7! yields an injection of the set of -conjugacy classes in C v into the set of conjugacy classes in Cv;

6 6 the image is easily described by means of the norm map from E v to F v, and the eigenvalues of. We shall say that g 2 G v is bi-regular if it is bi-conjugate (agb, for some a; b in C v, is equal) to = () with -regular (namely is regular: it has distinct eigenvalues, not equal to or ",1 ). A g 2 G v is called bi-elliptic if it is bi-conjugate to = () with a -elliptic (thus is elliptic, namely its conjugate lies in an elliptic torus of Cv). The ariski open dense set in (WH2) is the bi-regular set. A cyclic G v -module v is called bi-elliptic if its bi-character is not identically zero on the bi-elliptic bi-regular set of G v. Theorem A concerns with a bi-elliptic component u. Note that if g is bi-elliptic then lies in an elliptic torus of Cv, which is the multiplicative group Kv of a separable eld extension K v of degree n of F v. The image of the map 7! consists of the subgroup N Kv E v =K v (E v K v ) of Kv. A general bi-regular bi-conjugacy class can easily be described in terms of these -elliptic. Denote by p (z) = det(z, ) (the reduced norm is dened by the determinant on the group of points over a splitting eld, such as a separable closure of F v ) the characteristic polynomial of the conjugacy class in Cv of. In the case where Cv is the multiplicative group of a division algebra, the map 7! p is a bijection from the set of -regular (necessarily -elliptic) -conjugacy classes in C v, to the set of irreducible separable polynomials of degree d over F v whose eigenvalues lie in N Kv E v =K v (K v E v ), where K v is the separable extension of F v of degree d generated by the eigenvalue. The analogous statement holds globally with (F;D;:::) replacing (F v ;D v ;:::). In general, the map 7! p is an injection from the set of -regular (resp. -elliptic -regular) -conjugacy classes in C v, to the set of separable (resp. irreducible separable) polynomials of degree deg Cv over F v whose irreducible factors have degrees which are multiples of deg D v. The image is easily described on expressing as a product of elliptic regular factors (i.e. conjugating into the standard Levi factor of a minimal parabolic containing a conjugate of ). In particular, the set of bi-regular bi-conjugacy classes in G v embeds as a subset of the set of bi-regular bi-conjugacy classes in G v. A bi-regular bi-conjugacy class in G v so obtained is said here to come from G v. The set of bi-regular bi-elliptic bi-conjugacy classes in G v bijects with the set of bi-regular bi-elliptic bi-conjugacy classes in G v. With this denition, the statement oftheorem A is now complete. Remark. If v is cyclic and supercuspidal, then its bi-character is smooth on the bi-regular bi-elliptic set. Indeed, the linear form P v is the unique (up to a scalar multiple) non-zero bi-c v -invariant linear form on H v which vanishes on the orthogonal complement of the span of the space of matrix coecients of v. Hence P v (f v ) is a constant multiple of h v (f v ) v (h);e v (h ) _ idh dh C v = v C v = v = for any vector 6= in v. f v (g)h v (h gh); _ idg dh dh ; C v = v C v = v G v = v

7 7 If g is bi-regular bi-elliptic then it is of the form g = c ()c. Its bi-centralizer v (g) =f(h ;h)2 C v C v ;h gh = gg is equal to (,1! ) h = c t c,1 ; h = c,1 t c ; t 2 t v () ' v (); t where v () =ft2c v ;tt,1 = g is the -centralizer of in C v. Since is assumed to be -regular -elliptic, the -centralizer v () is an elliptic torus in Cv, isomorphic to the multiplicative group of the separable extension of F v of degree n(= deg Cv) generated by the elliptic regular elements. In particular the volume j v (g)= v j is nite, for such g. Now suppose that f v is supported on the bi-regular bi-elliptic set in G v. Then we may change the order of integration, obtaining (equality up to a scalar multiple depending on the choice of ): P v (f v )= G v = v f v (g)j v (g)= v j(g; c v )dg; where c v (g) = h v (g); _ i is a matrix coecient of v. In particular the bi-character p(g; v ) of a supercuspidal cyclic v is given on the bi-regular bi-elliptic set by p(g; v ) = j v (g)= v j(g; c v ). It is therefore smooth on the bi-regular bi-elliptic set. However, we have not veried that p(g; v ) is not identically zero on the bi-regular bielliptic set. In the classical case of characters, it is veried in [HC1] that the characters of the supercuspidal representations are locally integrable functions, and that their restrictions to the elliptic regular subset satisfy orthonormality relations. In particular the character of a supercuspidal representation is not identically zero on the elliptic regular set. It will be interesting to establish an analogue in our case. We obtain also purely local results. The following is a bi-analogue of Kazhdan's density theorem for characters (see [K; Appendix]). It does not rely on (WH2). Let E w be a commutative separable semi-simple algebra of dimension twoover F w (thus E w is a separable quadratic eld extension of F w, or E w = F w F w ). Assume that there exists a supercusp form f u on G u with (g; f u ) 6, or alternatively that there exists a supercuspidal cyclic u, in which case f u can be taken to be its matrix coecient. Of course, when n is odd and E u is a eld, we may take f u =1on G u =(D u Fu H u ),where D u is a division algebra of degree n central over F u. Theorem B. Assume that (WH1) holds for every irreducible admissible cyclic representation w of G w = GL(m w ;D H w), where D w is a division algebra central over F w of odd degree if F w is a eld, and H w is a simple algebra ofdegree two central over F w (D H w = D w Fw H w ). Then P w is dened. If f w 2 H w is a test function such that P w (f w )=for all cyclic w, then the bi-orbital integral (; f w ) is zero on the bi-regular set of in G w. A local analogue of Theorem A is stated next. Suppose that the bi-elliptic part of (WH2) holds for every admissible irreducible representation u of G u =(D u Fu H u ),where

8 8 D u is a division algebra central of degree n over F u, and H u is a quaternion algebra central over F u when E u is a eld, GL(2;F u ) if E u = F u F u (namely no assumption when n is odd). In other words, we assume that the bi-character p(g; u )ofany such u (not only supercuspidal as in the Remark following the statement of (WH2)), is locally constant on the bi-regular (necessarily bi-elliptic) set of G u. When D u Fu H u is a division algebra, each representation of its multiplicative group is supercuspidal, and its bi-character is clearly locally constant. Theorem C. Let u be a cyclic supercuspidal G u -module satisfying (WH1) and (WH2), where G u = GL(m u ;Du H ), D u an F u -central division algebra unsplit by E u. Then the corresponding square-integrable G u = GL(m u ;Du H )-module u is cyclic; here Du is an F u -central simple algebra of the same degree as D u, with exp Du dividing exp D u (Du is more split than D u ). Recall that the local correspondence is dened by means of character relations (see [F1; III]). The corresponding u is square-integrable, but not necessarily supercuspidal. A \split" analogue { where E = F F globally { of our work, is the subject matter of [F4]. Our local Theorems B and C specialize to the corresponding local theorems of [F4] when E u = F u F u. At least when n is odd the statements of our local theorems are preferable to those of [F4], since then the requirement at u (or w) is not present here. When n is odd our proofs of the local results can be considered better than the analogous proof in [F4] as we can work here with a global anisotropic group, for which the analysis simplies. Although the present note overlaps with [F4], we decided to separate the two in an attempt to make each of them readable independently of the other. We hope to compare our results here, in the quadratic case, with the results of [F4], in the split case, on another occasion. The global tool in our proofs is the following bi-period summation formula. 1. Proposition. Let f = f v be a test function on G (f v 2 H v for all v; f v = f v for almost all v) which has a supercuspidal component f u and a component f u supported on the bi-regular bi-elliptic set. Then P ((f))p () = fg j()=()j Y v (();f v ): Here ranges over the set of cuspidal cyclic G -modules with a supercuspidal component at u, ranges over an orthonormal basis of smooth vectors in the space of, and fg ranges over a set of representatives for the -elliptic -regular -conjugacy classes in G. Proof. Let K f (x; y) be the kernel of the convolution operator (r(f))(x)= R ng f(g)(xg)dg on L 2 (GnG ). Here f = f v is a product over all places v of F of f v 2 H v, such that f v is the unit element f v in the convolution algebra H v of spherical (bi-k v -invariant, K v being the standard maximal compact subgroup of G v ) function in H v, for almost all v. It is easy to see that (r(f))(x) = R ng K f (x; y)(y)dy, where K f (x; y) = P 2nG f(x,1 y). This is the geometric expansion of the kernel. We take the component f u of f to be a supercusp form. A well-known observation of Kazhdan (see [F1; III]) asserts that r(f) then factorizes through the natural projection

9 into the subspace L 2 (GnG ) of cusp forms in L 2 (GnG ). Then the kernel has the spectral expansion K f (x; y) = ((f))(x)(y): The rst sum ranges over the set of cuspidal G -modules (in fact with a supercuspidal component at u), and ranges over an orthonormal basis of smooth vectors in the space of. Note that it is { and not its equivalence class { which occurs here, by virtue of the multiplicity one theorem for such of [F1; III] and [FK2]. Our formula is obtained on integrating these two expressions for the kernel over x; y in CnC. The integral of the spectral expression is The integral over x; y in CnC C=C dx CnC 2G= P ((f))p (); P () = CnC (h)dh: of the geometric expression for the kernel is f(xy) dy = dx 2CnG=CC=C\C,1 nc f(xy) dy: We take the component f u of f at u to be supported on the bi-regular bi-elliptic set. Consequently the rational bi-c -orbits (the set of xy, with x; y in C, and in G) on which f is non-zero, are those of the bi-regular bi-elliptic, represented by = () = I, I where is a -elliptic -regular element of C. A complete set of representatives of these rational bi-orbits is given by (), as ranges over a set of representatives fg for the -conjugacy classes of -elliptic -regular elements in C, ji, j 6=. Note that a()a,1 = (AA,1 ), where a = diag (A; A). Since C \ C,1 = (), where () is the group of a in C such that AA,1 =, our double integral is equal to = fg j()=()j C=() dx nc f(x()y)dy: The double integral here can be expressed as a product, for f = f v, of local bi-orbital integrals. Thus we obtain 9 = fg j()=()j Y v (();f v ); where the sum is nite and the product is absolutely convergent, as required. The following is clear.

10 1 2. Lemma. Let f v 2 H v be a function on G v supported on the bi-regular set. If T v is a torus in Cv = GL(m v ;D v ), and Tv is its centralizer in C v = GL(m; Dv E ), denote by Tv =N (Tv) the quotient of Tv by the equivalence relation t t if t = t,1 ( 2 C v ), and by (Tv =N (Tv)) the set of (), 2 Tv =N (Tv). Then (; f v ) is a smooth function with compact support on the union of (Tv =N (Tv)) over a set of representatives ft v g of the conjugacy classes of F v -tori in Cv. Conversely, given a smooth compactly supported function () on the bi-regular subset of [ ftv g(tv =N (Tv)), there exists an f v 2 H v supported on the bi-regular set of G v, with () = (; f v ). Both statements hold with \bi-regular" replaced by \bi-regular and bielliptic" throughout, except that T v ranges then only over the elliptic conjugacy classes of F v -tori. Of course the discussion above holds not only for G but for any inner form of it, in particular for G. To establish the comparison of the Theorem, we compare the geometric sides of the bi-periodic summation formula for f = f v on G and for f = fv on G. For this comparison x a non degenerate dierential form of highest degree on G over F. It denes a Haar measure on G v and G v, hence on G and G, in a compatible way. These measures, dg v, dg, d g v and d g, are used to dene the bi-period orbital integrals (; f v ) and (; fv), as well as the distributions P () and P ( ). Denition. The functions f v 2 H v and f v 2 H v are called matching if ( ;f v) is zero on the bi-regular which do not come from G v, while if is a bi-regular element ofg vwhich comes from in G v, then ( ;f v)=(; f v ). For all v 62 V, where V is the nite set of places where G v is not isomorphic to G v, we take f v and f v to correspond to each other under this isomorphism. At the remaining nite number of places v in V, Lemma 2 guarantees the existence of f v matching any f v which is supported on the bi-regular set of G v. This f v can be taken to be supported on the bi-regular set of G v, in fact on the (open) set of such elements which come from G v. Conversely, given any f v whose bi-period orbital integrals are supported on the set of bi-regular elements of G v which come from G v, Lemma 2 guarantees the existence of an f v, supported on the bi-regular set of G v, matching f v. 3. Lemma. For any test functions f = f v on G and f = f v on G such that f v = f v for all v 62 V, f v = f v for almost all v, f u is a supercusp form and f u supported on the bi-regular bi-elliptic set of G u, u outside V ), and f v, f v matching for all v(2 V ), we have P ( (f ) )P ( )= P((f))P (): The sums range over the cuspidal G -modules and cuspidal G -modules, whose components at u are supercuspidal, and over orthonormal bases of smooth vectors in and in. Proof. Our choice of matching f and f, as well as matching measures, guarantees the equality of the geometric sides of the bi-period summation formulae for f on G and f on G of Proposition 1. Hence the spectral sides are equal.

11 4. Lemma. Let be a cuspidal G -module with a supercuspidal component u (u 62 V ), and the corresponding cuspidal G -module. Let S be a nite set of places of F containing V, u; u, and all archimedean places and those where v is not unramied. If f v 2 H v and fv 2 Hv are matching (v 2 V ), f u = fu is a supercusp form, and f u ;fu are supported on the bi-regular bi-elliptic sets of G u, G u, and f v = fv (v 2 S, V ), then Here K (S) = Q v62s 2 K (S) P ( S(f S) )P ( )= 2 K(S) P( S (f S ))P (): 11 K v (' K (S)), where K v is the standard maximal compact GL(2n; R E v ) of G v ' G v ' GL(2n; E v ); K(S) is the space of K (S)-xed vectors Q in ; ranges over an orthonormal basis of smooth vectors in K(S). Finally S (f S ) is v (f v ). Proof. We work with f and f whose components are spherical (K v -biinvariant) at each v 62 S. Note that v (f v ) acts as on unless is K v -invariant, in which case Q v (f v ) acts as multiplication by a scalar, denoted again by v (f v ). Putting S (f S ) = v (f v ), the identity of Lemma 3 can be written as 2 K (S) S (f S )P ( S(f S) )P ( )= v2s v62s 2 K(S) S (f S )P( S (f S ))P (): A standard argument { originally due to Langlands (in the case of GL(2)) { used in [F2], [F1], [FK1], [FK2],..., of \linear independence of characters", based on varying the spherical components of f at the v 62 S, using standard unitarity estimates, the Stone- Weierstrass theorem and the absolute convergence of the sums in Lemma 3, implies our claim. Of course, we use in the statement of the Lemma multiplicity one theorem for G and G ([F1;III], [FK2]), as well as rigidity theorem for G and G ([F1;III], [FK2]). 5. Proposition. Suppose that is a cuspidal cyclic G -module with a supercuspidal component u (u 62 V ) and a bi-elliptic component u (u 6= u). Suppose that (WH1) and (WH2) hold for v for all v 2 V and v = u. Then the corresponding cuspidal G -module is cyclic, its component at u is bi-elliptic, and the bi-character of v (v 2 V ) is not identically zero on the set of bi-regular elements of G v which come form G v. Proof. It suces to show that the side of in the identity displayed in Lemma 4 is non zero. Consider a smooth 1 in K(S) such that P ( 1 ) 6=. In this proof we regard K(S) as an abstract representation, rather than in N its automorphic realization. Denote by = S the preferred K (S)-xed vector in S = v, and x an orthonormal basis f v g of smooth v62s vectors in v. Then f ( N v2s v ); v 2f v g;v 2 Sg is an orthonormal basis of K(S). Any smooth vector in K(S) is a nite linear combination of such factorizable vectors. Expressing 1 as a linear combination of vectors including 1 = ( N v2s 1v ) etc., since P ( 1 ) 6= wemay assume that the restriction of P to 1 is non zero. At each v 2 S, V,

12 12 v 6= u, u, we choose f 1v 2 H v such that v (f 1v ) v = for all v 2 f v g, v 6= 1v, and v (f 1v ) 1v = 1v. Such achoice is possible since H v spans the algebra of endomorphisms of v. In fact this choice can be made also at the place u, where u is supercuspidal. Indeed, by the Schur orthogonality relations the matrix coecient f 1u (x) =( u (x) 1u ; 1u) _ acts as zero on any u orthogonal to 1u, and as a scalar multiple (we assume it is 1 on multiplying f 1u by a scalar) on 1u. Moreover, such a matrix coecient is a supercusp form (since u is supercuspidal), as required P to apply Lemma 4. With this choice of f v = f 1v (v 2 S, V; v 6= u ), our sum P ( S (f S ))P () ranges over the vectors whose component N outside V = V [fu gis V = ( 1v ). Put also f V N N =( f 1v )( fv ). v2s,v v2s,v v62s The side of in the identity of Lemma 4 can now be expressed as hp V ; V (f V )P _ V i; R where P V is the restriction of the linear form hp; i = P() = CnC (h)dh to V V ; P V lies in the dual V of V. The integral analogously denes a linear form P _ in the dual R e of the contragredient e of, which consists of the, 2. Namely hp _ ; i = CnC (h)dh. Denote by P _ V the restriction of P _ to ( V ) _ e V. Here f v _ g signies the dual basis of f v g, and v _ = v(e v ). Note that V (f V )P _ V 2 V. Hence hp V ; V (f V )P _ Vi is dened. It is equal to the side of in the identity of Lemma 4 as explained when P v (f v )was introduced, before (WH2) was stated. Note that V (f V ) is a cusp form for each cusp form. We shall now use (WH1) for v (v 2 V ). It asserts the uniqueness of the C v -invariant form P v on v, up to a scalar multiple. The existence of P v follows from the cyclicity of. Since the components of outside V [fu g are xed, there is a constant c(), depending on these components, such that P V = c() O v2v P v : Our sum then takes the form c() 2 Y v2v [fu g P v (f v ); P v (f v )=hp v ; v (f v )P _ v i: At the place u we use (WH2). We take f u which is supported on the bi-elliptic bi-regular set, such that P u (f u )= u ng u f u (g)p(g; u )dg is non-zero. The choice of such f u is clearly possible, since the bi-character p(g; u )of u is locally constant on the bi-regular set, and is assumed to be non-zero on the bi-regular bi-elliptic set. Similarly, at each v 2 V other than u, we can choose f v which is supported on the bi-regular set of G v, with P v (f v ) 6=, again using (WH2): the bi-character is smooth on

13 the bi-regular set, and is not identically zero there. As noted following Lemma 2, there are functions fv (v 2 V ) matching the f v. The matching fv will be supported on the set of bi-regular (also bi-elliptic when v = u ) elements of G v which come from G v. With this choice of f v (v 2 S), since is cyclic, the right side of the identity displayed in Lemma 4 is non-zero. Hence the left side is non-zero. This means that is cyclic, and Pv (f v) 6= (v 2 S) for the matching function fv. Since the matching function fv is supported R on the bi-regular (also bi-elliptic when v = u ) elements of G v which come from G v, and v ng fv(g)p(g; v)dg 6=,the bi-character p(g; v) is not identically zero on this v set, as asserted. 6. Proposition. Let be a cuspidal cyclic G -module which corresponds to a cuspidal G -module. Suppose that u is supercuspidal (u 62 V ), that u is bi-elliptic, and that for each v 2 V, the bi-character of v is not identically zero on the set of bi-regular elements which come from G v. Suppose also that (WH1) and (WH2) hold for v (v 2 V [fu g). Then is cyclic. 13 Proof. The discussion at the places v 2 S,V [fu g, including the case of the supercuspidal component atu, is as in Proposition 5. The assumptions at u and v 2 V permit producing matching functions f u and f v for functions f u and f v for which the left side of the identity displayed in Lemma 4 is non-zero. The proof then proceeds as that of Proposition 5. This completes our proof of Theorem A. Proof of Theorem B. Choose global elds E=F with completions E u =F u, E w =F w,aswell as a division algebra D over F unsplit by E, and a quaternion algebra H, such that the group of points over F u, F w of G = GL(m; D H )isg u ;G w. Assume that (g; f w ) is not identically zero on the bi-regular set of G w. We shall show that this leads to a contradiction. Since (; f u ), (; f w ) are locally constant on the bi-regular sets of G u, G w (Lemma 2), we can x a third place u, a bi-elliptic bi-regular global element in G, which is bi-elliptic in G u, and f u 2 H u which is supported on the bi-elliptic bi-regular set in G u, such that ( ;f v )6=(v=u; w; u ). Since 2 Kv E for almost all v, and fv, the integral ( ;fv) is non zero for all v outside some nite set S of places of F. At the remaining nite set of places we choose f v to be the characteristic function of a small neighborhood of in G v ; then ( ;f v )6=. It follows that ( ;f)6=, where f = f v, and that if is rational (in G) with (; f) 6=, then is bi-regular bi-elliptic (since it is such in G u ). Since f is compactly supported, such = () lies in a nite set of bi-orbits; indeed, the set of characteristic polynomials of the associated is both compact { depending on the support of f { and discrete (since is rational) in the set of polynomials of degree n over A (' A n+1 ). The totally disconnected topology on G u permits choosing an open closed neighborhood of the orbit of which does not intersect the orbits of the other rational with (; f) 6=. Replacing f u by its product with the characteristic function of this neighborhood, we obtain f such that (; f) 6=for a rational implies that is in the bi-orbit of. We now apply the bi-period summation formula of Proposition 1, to our function f on G. The requirements of this Proposition 1 are satised. Indeed, f u is supercuspidal, and f u is supported on the bi-elliptic bi-regular set. Our assumption that P w (f w )vanishes for

14 14 all w implies the vanishing of the spectral (left) side of the summation formula. Hence the geometric side is zero. But it contains a single term, indexed by. So ( ;f) =, a contradiction to the assumption that (g; f w ) is not identically zero on the bi-regular set of G w, as required. Remark. Theorem B and its proof remain valid if we do not assume (WH1), but instead we assume for all w that P w (f w ) =, where P w is dened by means of any C w -invariant linear form P w on the space of w. Proof of Theorem C. Suppose that exp G u = d u. We shall work with global elds E=F whose completions at the places u 1 = u;:::;u 2du are isomorphic to E u =F u, and at the places u 1 = u ;:::;u n they are E u =F u, and with a group G = GL(m; D H ) over F with G ui ' G u (1 i 2d u ) and G u i ' G u (1 i n). We shall carry out a comparison with the inner form G of G which is split at the places u du +1;:::;u 2du, but with G v ' G v for all other v. We compare the bi-period summation formulae for G and G of Proposition 1. At the places u du +1;:::;u 2du we use matrix coecients of u, while at the places u 1 ;:::;u uwe take the test functions to be supported on the bi-elliptic bi-regular set. At the places u 1 ;:::;u du we take the f ui and fu i to be matching and supported on the bi-regular set (of elements which come from G ui in the case of fu i ), as in Lemma 2. At all other places, f v = fv under G v ' G v. Since both f = f v and f = fv have supercuspidal components and components supported on the bi-elliptic bi-regular sets, Proposition 1 applies. Since f and f are matching the geometric parts of these formulae are equal. Note that f can be chosen so that the geometric side of the bi-period summation formula is non-zero. Indeed, since (g; f v ) is locally constant on the bi-regular set (Lemma 2), and is not identically zero there for v = u i or u i by our assumption on u and f u i, there is some rational bi-regular bi-elliptic element with ( ;f v ) 6= for such v. This relation clearly holds with f v = fv for almost all v. At the remaining nite set of places we choose f v supported on a small neighborhood of, and argue as in the proof of Theorem B that f can be chosen so that (; f) 6= for a rational implies that is in the bi-orbit of. Applying Proposition 1 with such an f we conclude that there exists a cuspidal cyclic G -module, in fact with the component v at v = u 1 ;:::;u 2du, and a bi-elliptic component at u 1 ;:::;u n. Since we have (WH1) and (WH2) for v (v = u i ) by assumption, and also at v = u i ((WH1) by Proposition, (WH2) on the bi-elliptic set by assumption), the proof of Proposition 5 implies that the corresponding cuspidal G -module is cyclic. In particular its components, including u, are cyclic, as required. References [B] [BDKV] [BJ] J. Bernstein, A. elevinskii, Representations of the group GL(n; F ) where F is a nonarchimedean local eld, Russian Math. Surveys 31 (1976), J. Bernstein, P. Deligne, D. Kazhdan, M.-F. Vigneras, Reprsentations des groupes reductifs sur un corps local, Hermann, Paris (1984). A. Borel, H. Jacquet, Automorphic forms and automorphic representations, Proc. Sympos. Pure Math. 33 (1979), I

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