Derivative of a Fourier transform
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From: Evgeny Turchyn on 14 Jun 2010 00:04 Hello All, let function f belong to $L_2(R)$. Let's consider its Fourier transform (as function from $L_2(R)$) $\hat f$. How do we define derivative $(\hat f)'(x)$ (taking into account that $\hat f$ is actually a class of functions that coincide a.e.)? Maybe someone can give a reference to a book where this notion is introduced? Thanks in advance. Best regards, Evgeny.
From: George Jefferson on 14 Jun 2010 00:45 "Evgeny Turchyn" <evgturchyn(a)gmail.com> wrote in message news:e8c14b34-535a-4efb-8e84-84ef801fe9e6(a)u26g2000yqu.googlegroups.com... > Hello All, > let function f belong to $L_2(R)$. Let's consider its Fourier > transform (as function from $L_2(R)$) $\hat f$. How do we define > derivative > $(\hat f)'(x)$ (taking into account that $\hat f$ is actually a class > of functions that coincide a.e.)? > Maybe someone can give a reference to a book where this notion is > introduced? > Thanks in advance. > Best regards, Evgeny. ? d/dw(f_hat(w)) <=> x*f(x) (up to some scaling constant)
From: David C. Ullrich on 14 Jun 2010 11:12 On Sun, 13 Jun 2010 21:04:13 -0700 (PDT), Evgeny Turchyn <evgturchyn(a)gmail.com> wrote: > Hello All, >let function f belong to $L_2(R)$. Let's consider its Fourier >transform (as function from $L_2(R)$) $\hat f$. How do we define >derivative >$(\hat f)'(x)$ (taking into account that $\hat f$ is actually a class >of functions that coincide a.e.)? That class of functions gives a _single_ distribution, and the distribution derivative is defined as usual, by duality. The function may or may not have an L^2 function for its derivative. If so then the function is continuous (ie one of the representatives is continuous) and the distribution derivative is almost everywhere the usual pointwise derivative. (There is an L^2 derivative if and only if x f(x) is in L^2.) > Maybe someone can give a reference to a book where this notion is >introduced? > Thanks in advance. > Best regards, Evgeny.
From: Simplane Simple Plane Simulate Plain Simple on 14 Jun 2010 13:19 On Jun 14, 8:12 am, David C. Ullrich <ullr...(a)math.okstate.edu> wrote: > On Sun, 13 Jun 2010 21:04:13 -0700 (PDT), Evgeny Turchyn > > <evgturc...(a)gmail.com> wrote: > >   Hello All, > >let function f belong to $L_2(R)$. Let's consider its Fourier > >transform (as function from $L_2(R)$)  $\hat f$. How do we define > >derivative > >$(\hat f)'(x)$ (taking into account that $\hat f$ is actually a class > >of functions that coincide  a.e.)? > > That class of functions gives a _single_ distribution, and the > distribution derivative is defined as usual, by duality. > > The function may or may not have an L^2 function for > its derivative. If so then the function is continuous (ie > one of the representatives is continuous) and the > distribution derivative is almost everywhere the usual > pointwise derivative. (There is an L^2 derivative > if and only if x f(x) is in L^2.) > > >   Maybe someone can give a reference to a book where this notion is > >introduced? > >   Thanks in advance. > >   Best regards, Evgeny. > > Results 1 - 10 for x f(x) is in L^2.). (0.21 seconds) [PDF] (R(y)f)(x) f(xy), f L2(G(Q)\G(A)), x,y G(A),File Format: PDF/ Adobe Acrobat - Quick View by J Arthur - 1997 - Related articles (R(y)f)(x) f(xy), f L2(G(Q)\G(A)), x,y G(A), on the Hilbert space. 1991 Mathematics Subject Classification. Primary: 22E55. ... www.claymath.org/cw/arthur/pdf/45.pdf x*=F(x) (l) codimK>l (2)by JS Il'jaÅ¡enko - 1972 - Related articles section of the line w =l/2 with the ball Q is a disc of smaller radius; ..... Proposition 2. Let X., F(x) and Y' be the same as in the corollary; codim Y' = ... www.turpion.org/php/full/infoFT.phtml?journal_id=sm&paper_id... conversions are not counted - Google Website Optimizer HelpFeb 22, 2009 ... (function(){var k='3499720701',d=document,l=d.location,c=d.cookie;function f(n) ... length+1,j<0?c.length:j)}}}var x=f('__utmx'),xx=f('__utmxx'),h=l.hash .... 2 replies. Original page and variations over multiple domains ... www.google.com/support/forum/p/websiteoptimizer/thread?tid...hl=en Iron-Based Layered Superconductor La[O1-xFx]FeAs (x = 0.05â0.12 ....by Y Kamihara - 2008 - Cited by 139 - Related articles Feb 23, 2008 ... Figure 2 (a) Electrical resistivity (Ï) versus temperature (T) for undoped and La[O1-xFx]FeAs: x = 0.04, 0.05, 0.11, and 0.12. ... dx.doi.org/10.1021/ja800073m Problems with Analytics and Optimizer recording visitors and ...Dec 10, 2009 ... (function(){var k='3781672910',d=document,l=d.location,c=d.cookie ... length+1,j<0? c.length:j)}}}var x=f('__utmx'),xx=f('__utmxx'),h=l.hash; ... here is for 2 versions of the sitewide header (the bit above the tabs). ... I (l-xf(\-XY(J\x)/(xa))dx,by S KUMAR - 1980 - Cited by 13 - Related articles (1 -xf(l +xY, we obtain the quadrature formula ... (i) /eHM , 0<//^l,. (ii) f'(x) exists and is continuous in a small neighbourhood of a, ... imamat.oxfordjournals.org/cgi/reprint/26/4/447.pdf Bounce Rate increases after implementing Website Optimizer ...Jan 7, 2010 ... (function(){var k='##########',d=document,l=d.location,c=d.cookie;function f(n){ .... length+1,j<0?c.length:j)}}}var x=f('__utmx'),xx=f('__utmxx'),h=l.hash; .... 1 of 2 people found this answer helpful. Did you? ... Lx(X) = f(X)x(X) +by W Koppelman - 1958 - Cited by 10 - Related articles of L has multiplicity <2. More general self-adjoint operators of the form. MX(X) = f(X)x(X) + I. I g(X )x( )d rti J M X can be shown differ from a direct ... www.pnas.org/content/44/12/1252.full.pdf Google Analytics works, but not Website Optimizer? - Google ...Apr 6, 2009 ... (function(){var k='1234567890',d=document,l=d.location,c=d.cookie ... length+1,j<0? c.length:j)}}}var x=f('__utmx'),xx=f('__utmxx'),h=l.hash; .... 2. In the website optimizer help it says to put this parameter into every ... Tf - (Tf)(x) = f(x - X).by LA Coburn - 1969 - Cited by 41 - Related articles the one-parameter semigroup = Tx:X:C R+} of isometries on L2(R+) defined by. (Tf)(x) = f(x - X). In this note we describe an analysis of the C*-algebra P ... www.pnas.org/content/62/4/1010.full.pdf 1 2 3 4 5 6 7 8 9 10 Next
From: Simplane Simple Plane Simulate Plain Simple on 14 Jun 2010 13:22
On Jun 14, 8:12 am, David C. Ullrich <ullr...(a)math.okstate.edu> wrote: > On Sun, 13 Jun 2010 21:04:13 -0700 (PDT), Evgeny Turchyn > > <evgturc...(a)gmail.com> wrote: > > Hello All, > >let function f belong to $L_2(R)$. Let's consider its Fourier > >transform (as function from $L_2(R)$) $\hat f$. How do we define > >derivative > >$(\hat f)'(x)$ (taking into account that $\hat f$ is actually a class > >of functions that coincide a.e.)? > > That class of functions gives a _single_ distribution, and the > distribution derivative is defined as usual, by duality. > > The function may or may not have an L^2 function for > its derivative. If so then the function is continuous (ie > one of the representatives is continuous) and the > distribution derivative is almost everywhere the usual > pointwise derivative. (There is an L^2 derivative > if and only if x f(x) is in L^2.) > > > Maybe someone can give a reference to a book where this notion is > >introduced? > > Thanks in advance. > > Best regards, Evgeny. > > This is the html version of the file http://www.claymath.org/cw/arthur/pdf/45.pdf. Google automatically generates html versions of documents as we crawl the web. Page 1 Centre de Recherches MathematiquesCRM Proceedings and Lecture NotesVolume 11, 1997The Problem of Classifying Automorphic Representations ofClassical GroupsJames ArthurIn this article we shall give an elementary introduction to an important problemin representation theory. The problem is to relate the automorphic representationsof classical groups to those of the general linear group. Thanks to the work of anumber of people over the past twenty- five years, the automorphic representationtheory of GL(n) is in pretty good shape. The theory for GL(n) now includesa good understanding of the analytic properties of Rankin-Selberg L-functions,the classification of the discrete spectrum, and cyclic base change. One wouldlike to establish similar things for classical groups. The goal would be an explicitcomparison between the automorphic spectra of classical groups and GL(n) throughthe appropriate trace formulas. There are still obstacles to be overcome. Howeverwith the progress of recent years, there is also reason to be optimistic.We shall not discuss the techniques here. Nor will we consider the possibleapplications. Our modest aim is to introduce the problem itself, in a form thatmight be accessible to a nonspecialist. In the process we shall review some ofthe basic constructions and conjectures of Langlands that underlie the theory ofautomorphic representations.1. We shall begin with a few of the basic concepts from the theory for thegeneral linear group. For the present, then, we take G = GL(n). The adeles of Qform a locally compact ringA = R x Q2 x Q3 x Q5 X ...in which Q embeds diagonally as a discrete subring. Consequently G(A) is a locallycompact group which contains G(Q) as a discrete subgroup. One can form theHilbert space L2(G(Q)\G(A)) of functions which are square integrable with respectto the right G(A)-invariant measure. The primary object of study is the regularrepresentation(R(y)f)(x) f(xy), f L2(G(Q)\G(A)), x,y G(A),on the Hilbert space.1991 Mathematics Subject Classification. Primary: 22E55.Supported in part by an NSERC Research Grant.This is the final form of the paper.© 1997 American Mathematical Society -------------------------------------------------------------------------------- Page 2 2JAMES ARTHURThe unitary representation R is highly reducible. For this discussion we shalldefine an automorphic representation informally as an irreducible unitary repre-sentation 7r of G(A) which occurs in the decomposition of R. This notion wouldbe precise certainly if 7r occurred as a discrete summand ofR. However, the irre- ducible constituents of R depend on several continuous parameters and one wantsto include all of these. The proper definition [14] in fact includes irreducible repre-sentations of G(A) which come from the analytic continuation of these parameters,but there is no need to consider such objects here. It is known [5] that any such 7rhas a decomposition7r = 7rR 0T2 ® 7r3 7r5 0 *as a restricted tensor product, with each irp being an irreducible unitary represen-tation of the group G(Qp).Anyone seeing these definitions for the first time could well ask why auto-morphic representations are interesting. To get a feeling for the situation, we fix aprime p and recall the construction of the unramified representations of G(Qp)-thesimplest family of irreducible representations {Trp} of this group.The representations in the family are determined by elements u = (u1,..., un)in Cn. Such an element defines a characterof the Borel subgroupB((Qp)= =abC G((Qp) [ \0bnnof GL(n,Qp) byXu(b) =Ib 1+(n-1)/2 Ib22U2+(n-3)/2 ..ln-(n-l/2Let 7r+ be the corresponding induced representation of G(Qp). It acts on a spaceof functions fp: G(Qp) - C which satisfyfp(bx) = Xu(b)fp(x), b E B(Qp), x E G(Qp),be right translation-(+u(y)fp)(z) = fp(xy), x,y C G(Qp).The vector (n - n 3,..-(n- 1)) comes from the usual Jacobian factor,and is included so that 7r+ will be unitary if u is purely imaginary. If u is purelyimaginary, 7r+ is known to be irreducible as well as unitary. In general, r+ canhave several irreducible constituents, but there is a canonical one-the irreducibleconstituent Trp,u of 7r+ which contains a G(Zp)-fixed vector. Thus, any u deter- mines an irreducible representation 7rp,u of G(Qp). Since the p-adic absolute valuesin the definition of Xu are powers of p, it is clear that 7rp,u remains the same if u istranslated by a vector in (2ri/ logp)Zn. In fact if u' is any other vector in Cn, itis known that 7rp,,' is equivalent to 7rp, if and only if(u ...,u )-=(U(1), .,u(n)) modzn)for some permutation a in Sn. -------------------------------------------------------------------------------- Page 3 THE PROBLEM OF CLASSIFYING AUTOMORPHIC REPRESENTATIONS3By definition, the unramified representations of G(Qp) are the ones in thefamily {7rp,u: u E Cn}. Setp-Ul0 \torp,J==\(0p-Un)regarded as a semisimple conjugacy class in GL(n, C). This is a special case of ageneral construction [13] of Langlands. In the present situation it gives a bijectionbetween the unramified representations of GL(n,Qp) and the semisimple conjugacyclasses in GL(n, C).Now suppose that ir is an automorphic representation of GL(n, A). It is knownthat the local components 7rp of 7r are unramified for almost all p. In other words,rr determines a familyt({r)={t(rrp):p ~ S}of semisimple conjugacy classes in GL(n, C). Here S = ST is a finite set of com- pletions of Q which includes the Archimedean place R. Returning to the originalquestion, automorphic representations are interesting because the correspondingfamilies t(7r) are believed to carry fundamental arithmetic information. What isimportant is not thefact that almost all 7rp are unramified-this would be true ofany irreducible representation of G(A) with some weak continuity hypothesis-butthat rr is automorphic. It is only then that the semisimple conjugacy classes {t(7rp)}will be related one to another ina way that is governed by fundamental arithmeticphenomena.In order to package the data t(7r) conveniently, one defines the local L-functionL(s,7rp) = det(I-t(7rp)p- S), E C, p E ,as the reciprocal (evaluatedat p-s) of the characteristic polynomial of the conjugacyclass t(7rp). One can then define a global L-functionLs(s, 7r)= JJ L(s,7rp)p~sas an Euler product which converges in some right half plane. It is known thatLs(s, 7r) has analytic continuation as a meromorphic function of s E C, and satisfiesa functional equation [9]. The basic proofis a generalization of the one used byTatefor GL(1). It exploits the embedding of GL(n) into the space of (n x n)-matrices.The proof entails defining local L- functions L(s, crp) for every p (including p = R).If one forms the productL(s, r) = IL(s, rrp)pover all p, the functional equation takes the formL(s, r) = e(s,7r)L(1l- ,-),where ir is the contragredient representation of xr, and the e-factor is a simplefunction of the forme(s, 7r) = a,(pr)s, ar E C, rr E Z.For an elementary example, take G = GL(1). Then G(A) = A* is the group ofideles, while G(Q)\G(A) = Q* \A* is the quotient group of idle classes. We shallconsider an automorphic representation r-= Ipr7rp with S = {RI,2}. Then if p d -------------------------------------------------------------------------------- Page 4 4JAMES ARTHURS,7rp is determined by an unramified character on the group B(Qp) = Q* = G(Qp).Any such prime is of course odd. Setp-^=l1, ifpp=1 (mod4),t(7rp) = P-U1 if p-3 (mod4)tp)-1, ifp 3 (mod 4).In other words,r()= IPP| =if p 3 (mod 4),where v(xp) C Z is the valuation of a point Xp E Qp. It is then easy to definecharacters 7rR and T2 on R* and Q* respectively so that 7r = ()p 7p is trivial on thesubgroup Q* of A*, and is hence an automorphic representation of GL(1). Observethat the definition of 7rp for p $S matches the splitting law of the prime p in theGaussian integers Z [v/--l;p is of the formp = (a + v-b) (a -b) = a2 + b2, a,b e Z,if and only ifp is congruent to 1 modulo 4. This is no co-incidence. The Kronecker-Weber theorem can be read as the construction ofan automorphic representation forany cyclic extension of Q in terms of how rational primes behave in the extension.The Artin reciprocity law gives a similar construction in the more general case thatQ is replaced byan arbitrarynumber field F. It can be regarded as a classification ofabelian extensions of F in terms of automorphic representations of GL(1) (relativeto F).This is a good point to recall Langlands' nonabelian generalization of the Artinreciprocity law. Suppose that0: Gal(Q/Q) - GL(n, C)is an n- dimensional representation of the Galois group of an algebraic closure of Qwhich is continuous, that is, which factors through a finite quotient Gal(E/Q) ofGal(Q/Q). Then b is unramified outside a finite set S = So of primes. For anyprime p ~ S, there is a Frobenius conjugacy class Frp in Gal(E/Q), and hence aconjugacy class O(Frp) in GL(n, C). Langlands conjectured that for any 0 there isan automorphic representation 7r of GL(n) such thatt(7rp) = q(Frp), p Sn S.This conjecture is very difficult, and has been established in only a limited numberof cases [15, 16, 4]. It is known, however, that there is at most one r with thisproperty [10].We recall also that there is an Artin L-function attached to 0 which is com-pletely parallel to the construction of an automorphic L-function. Itis defined byan Euler productL(s,q) = I L(s,qp)pwhich converges in a right half plane, with local factors given byL(s, p) = det(I- (Frp)p-)-s -------------------------------------------------------------------------------- Page 5 THE PROBLEM OF CLASSIFYING AUTOMORPHIC REPRESENTATIONS5if p does not belong to So. The function L(s, f) has analytic continuation andsatisfies a functional equationL(s, f) = E(s, 4)L(1 - s, ),with an e-factor of the form(s, ) = a,(pr)s a,C, r, E Z.Langlands' conjectural reciprocity law, which is actually a special case of his functo- riality principle, was formulated for all places p. It asserts that L(s, Crp) = L(s, ,p)for all p, or in global form, thatL(s, 7) =L(s, ).In other words, every Artin L-function is an automorphic L- function.2. Suppose now that G belongs to one of the other three families S0(2n+ 1),Sp(2n) and S0(2n) of classical groups. We shall assume that G is quasi-split. ThenG will actually be split if it is of the form S0(2n+ 1) or Sp(2n). In the remainingcase, G could be a nonsplit form of S0(2n) which splits over a quadratic extensionE of Q. (We exclude the exceptional quasi-split forms of S0(8).)With suitable modifications, the constructions of §1 all carry over to G. (Theywere introduced by Langlands for any reductive group over any global field F [13].)In particular, an automorphic representation of G(A) has a decomposition 7r =p 7rp, in which 7rp is an unramified representation of G(Qp) for all p outsidea finite set S = So. Each such 7rp is a constituent of a representation inducedfrom an unramified quasi- character of a Borel subgroup B(Qp) of G(Qp). Thereader unfamiliar with these things could try at this point to construct a semisimpleconjugacy class t(Trp), in analogy with GL(n). He/she will discover that such aconjugacy class exists, but that it occurs naturally in a complex group which isdual to G. If G is split, one can take the dual group G given by the tableGGS0(2n+ 1) Sp(2n, C)Sp(2n) SO(2n+ 1, C)S0(2n) SO(2n,C)If G is not split, one must take a semi- direct productG x Gal(E/Q),in which Gal(E/Q) acts on G = S0(2n, C) by conjugation through the isomorphismGal(E/Q) 0(2n,C)/SO(2n,C).The two cases are combined in Langlands' original construction of the L- groupLG= G > Gal(Q/Q),where Gal(Q/Q) acts trivially on G in case G is split, and acts on G through itsquotient Gal(E/Q) if G is not split. In the case of the general linear group, oneobviously takes L(GL(n)) to be thedirect product of GL(n, C) with Gal(Q/Q).Thus, to any automorphic representation 7r of G(A) there is associated a familyt(T) = {t(Trp) : p Sr} -------------------------------------------------------------------------------- Page 6 6JAMES ARTHURof semisimple conjugacy classes in the complex reductive group LG. We have toremind ourselves that the situation is more concrete than the final notation suggests;if G is split, for example, one can always replace LG bythe complex connected groupG. As with GL(n), the numerical data which determine these conjugacy classes arebelieved to carry fundamental arithmetic information. In fact, the data obtainedin this way ought to be a subset of the data obtained from general linear groups.This is the essence of the problem we shall presently discuss, and is also a specialcase of Langlands' functoriality principle. (For an introduction to the functorialityprinciple, see [1].)If the automorphic representations of classical groups are to be understood interms of GL(n), why study them at all? There are compelling reasons to do so.Suppose for example that G = Sp(2n). One can form the Siegel moduli spaceS(N) = r(N)\\,where 7- is the Siegel upper half space of genus n, and F(N) is the congruencesubgroup{? E Sp(2n, Z): y - I (mod N)}of Sp(2n, R). Then S(N) is a complex algebraic variety. The L2-cohomology ofS(N), H2) (S(N)), is a very interesting object which is directly related to certainautomorphic representations 7r of Sp(2n,A). For such rr, the conjugacy classest(Trp) are governed by the eigenvalues of Hecke operators acting on the cohomology.(See [3] for an introduction to these and related questions.) In this way one studiesquite different properties of 7r than one could get from the corresponding object ona general linear group.To attach an L-function to an automorphic representation 7r of G(A), one hasfirst to embed LG in a general linear group. Suppose thatLr: LG t GL(V)is a complex analytic, finite dimensional representation of LG. This determineslocal L-factorsL(s, p, Lr) = det (1 - Lr(t(7))p-)-PSIfor almost all p. One would like to be able to define L-factors for all p, and to showthat the Euler productL(s,iv, Lr)=)H L(s, P Lr)phas analytic continuation and functional equation. The case of G = GL(n) and Lrthe standard n- dimensional representation of GL(n, C) was discussed in §1. Despiteconsiderable progress [8], however, the general case is still far from solved.Finally, we recall that the Langlands reciprocity conjecture applies equally wellto L-homomorphismsOG :Gal(Q/Q) LGattached to G. (An L-homomorphism is one which is compatible with projectionsof the domain and co-domain onto Gal(Q/Q).) For each /G there should existan automorphic representation 7vG of G(A) with the property that for any LrLG -> GL(V), the Artin L-function L(s, LroCG) equals the automorphic L-functionL(s, TGr, Lr). -------------------------------------------------------------------------------- Page 7 THE PROBLEM OF CLASSIFYING AUTOMORPHIC REPRESENTATIONS7This completes our discussion of some of the general properties of automorphicrepresentations. We can now formulate the problem we set out to describe.Observe that for our classical group G there is a canonical embeddingrG : GGL(N, C),with N equal to either 2n or 2n + 1. This can beextended to an L-embeddingLrG : LG= G Gal(Q/Q) >- GL(N,C) x Gal(Q/Q) = L(GL(N)).By composing with LrG, we obtain a map QG - > ,Gal(Q/Q)LGCLrGL(GL(N))between L-homomorphisms into the two L- groups. We shall identify q with its pro-jection onto GL(N, C), that is, with an N-dimensional representation of Gal(Q/Q).As such it is self-contragredient. Conversely suppose that 0 is an arbitrary self- contragredient N-dimensional representation of Gal(Q/Q). We assume also that qis irreducible. Then q factors through an orthogonal or a symplectic group. Moreprecisely, there is a unique G, and an L- homomorphism qG for G, such that q isequivalent to LrG o 4G. (This is an easy consequence of the self-contragredience ofF--see for example §3 below.)The problem is to show that there is a similar mapping 7rG - * r betweenautomorphic representations. The mapping should reduce to qG --* f for the au-tomorphic representations attached (by Langlands' conjectural reciprocity law) toL-homomorphisms. As in this special case, the general mapping will be defined interms of the families t(7r) of conjugacy classes.Problem.(i) If7rG is an automorphic representation of the classical group G, show thatthere is an automorphic representation 7r ofGL(N, A) such thatLrG(tT(7G,p)) = t(7rp)for almost all p.(ii) Conversely, suppose that 7r is a self- contragredient automorphic representa-tion ofGL(N, A). If7r is cuspidal, show that 7r is the image of an automor-phic representation 7rG ofG(A), for a unique G as above.The problem is analogous to the base change problem, solved originally forGL(2) by Langlands [15]. That a similar question could be posed for the outerautomorphismX --+ =- tx-1, xGL(N),of GL(N) was I believe first noticed by Jacquet. However, there are some newphenomena here. The most obvious is the possibility of lifting representations frommore than one G to a given GL(N). If N = 2n is even, G could be either S0(2n, C)or Sp(2n, C); that is, G could be either S0(2n) or S0(2n + 1). It was pointed outby Shalika that one ought to be able to separate these two cases by looking at thesymmetric square and alternating square L-functions. -------------------------------------------------------------------------------- Page 8 8JAMES ARTHURLet S2 (respectively A2) be the finite dimensional representationg: XtgXg, g GL(2n,C),of GL(2n, C) on the space of symmetric (resp. skew-symmetric) (2nx 2n)-matrices.Consider a self- contragredient irreducible Galois representation0: Gal(Q/Q) - GL(2n, C).Then q factors through 0(2n, C) (resp. Sp(2n, C)) if and only if the representationS2 o ¢ (resp. A2 o q) of Gal(Q/Q) contains the trivial representation. This is thecase if and only if the Artin L- function L(s, S2 o I) (resp. L(s, A2 o I)) has a poleat s = 1. This suggests the following supplement to the problem.(iii) Suppose that 7r is a self-contragredient cuspidal automorphic representationofGL(2n). Show that Tr is the image ofan automorphic representation rTG ofS0(2n) (respectively SO(2n+ 1)) if and only ifthe automorphic L- functionL(s,I, S2) (resp. L(s, r, A2)) has a pole at s = 1.We shall state a second supplement to the problem that concerns automorphice- factors. Suppose that-:Gal(Q/Q) - GL(N, C)is an irreducible Galois representation. If we apply the functional equation of theArtin L- function L(s, g) twice, we obtainE(s,q)e(1 - s, ) = 1.Assume that q is self-contragredient. Setting s = 1, we see thatThe self- contragredience of ¢ means that it factors through an orthogonal or asymplectic group. If 0 factors through Sp(N, C), e (, ) can beeither 1 or -1;the actual value of this sign has interesting number theoretic implications [6]. If qfactors through O(N, C), however, e(1 b) is known to equal 1 [7]. One would liketo establish the automorphic version of this property.(iv) Suppose that rZis a self-contragredient cuspidal automorphic representationofGL(N). Ifir is the image ofan automorphic representation 7rG ofa groupG with G = SO(N, C), show that e (,7) = 1.3. It is known that an automorphic representation 7r of GL(N) is uniquelydetermined by the family t(7) of conjugacy classes. In other words, the map7r - t(w),from the automorphic representations of GL(N) to families of semisimple conjugacyclasses in GL(N, C), is injective. (The objects in the range are to be regardedas equivalence classes, two families being equivalent if they are equal at almostall p.) This is a theorem of Jacquet-Shalika [10], which is an extension of theearlier result for cuspidal automorphic representations. (Keep in mind that wehave adopted a restrictive definition of automorphic representation. What we arecalling an automorphic representation really includes an extra condition, that ofbeing globally tempered; it is only with this condition that the injectivity is valid.)The corresponding assertion for a classical group G is generally false. Ift= {tcGp : P S} -------------------------------------------------------------------------------- Page 9 THE PROBLEM OF CLASSIFYING AUTOMORPHIC REPRESENTATIONS9is a family of semisimple conjugacy classes in LG, the set of automorphic represen- tations TrG of G(A) such that t(TG) = tG could be an infinite packet. In particular,the mapping TrG -' 7r of our problem could have large fibres. An important part ofthe problem is to determine these fibres. There is a precise conjectural descriptionof the preimage of any 7r, based on the theory of endoscopy [12] and its extension tonontempered representations [2]. We shall not repeat it here. It suffices to say thatthe description is motivated by the case that 7r is attached to a self-contragredientGalois representation. We shall conclude this article with a few remarks on thestructure of such Galois representations.Consider an L-homomorphism':Gal(Q/Q)- L(GL(N)).We have agreed not to distinguish between such an object and the correspondingN-dimensional Galois representation. Thus, 4 has a decomposition-s= l*(D...**frqOrinto irreducible Galois representationsi : Gal(Q/Q) L(GL(Ni)),which occur with multiplicities fi. Suppose that 4 is self-contragredient. Thenthere is a permutation i -> i of period 2 on the set of indices such that qi = q, andi =- h,We are going to confine our attention to a special case. We assume that forevery i, fi = 1 and /i = /i. In particular, the irreducible representationa-iq(a) = t¢i(a)-1, a E Gal(Q/Q),is equivalent to 4i. It follows that for each i, there is a matrix Ai E GL(Ni, C) suchthat0i(o)-1 =AAii(a)A-1, a E Gal(Q/Q).Applying this equation twice, we see that tA-lAi is an intertwining operator forthe representation 4i. It follows from Schur's lemma that tAi = cAi for somec E C*. Applying this last identity twice, we find that c2 = 1, so that Ai is eitherskew-symmetric or symmetric. Therefore qi is either of symplectic or orthogonaltype. More precisely, if we replace qi by a suitable GL(Ni, C)-conjugate, we canassume that eitherImage(0i) C Sp(Ni,C) C GL(Ni, C)orImage(0i) C O(Ni, C) C GL(Ni, C).Separating the indices i into two disjoint sets I1 and 12 according to whether ¢i issymplectic or orthogonal, we obtain a decomposition0 = 0l E[ 0where01 =_ Oj: Gal(Q/Q)f Sp(Nj,C) C Sp(N1,C)jEI1jand¢2 = _Ok Gal(Q/ Q) > HO(Nk,C) C O(N2, ),kEI2kin which N1 = jNj and N2 =-ekNk. -------------------------------------------------------------------------------- Page 10 10JAMESARTHURThe maps )1 and 02 can be analyzed separately. For the first one, we notethat Sp(N1,C) is connected and equals (G1), where G1 = SO(N1 + 1). There isnothing more to say in this case. For the second case, observe that the mapGal(Q/Q) - O(N2,C)/SO(N2,C) = 2/2obtained from o2 by projection, determines a quadratic character 7r of Gal(Q/ Q).Suppose first that N2 is odd. Then O(N2,C) is the direct product of SO(N2,C)with Z/2Z. Setting G2 = Sp(N2 - 1), we use rT to define an embedding ofL(G2) = SO(N2,C) x Gal(Q/Q)intoL (GL(N2)) = GL(N2,C) x Gal(Q/Q)so that ¢2 factors through L(G2). Next suppose that N2 is even. Then O(N2, C)is a semi-direct product of SO(N2,C) with 2/22. Let G2 be the quasi-split formof SO(N2) obtained from r1 and the action of the nonidentity component of O(N2)on SO(N2). Again there is an embedding ofL(G2) = SO(N2,C) > Gal(Q/Q)77intoL(GL(N2)) = GL(N2,C) x Gal(Q/Q)such that )2 factors through L(G2).We have shown that the original Galois representation factors through LG, fora unique classical group G = G1 x G2. The groups obtained in this way (takentogether with the embeddings LG ,- L(GL(N))) are called the twisted endoscopicgroups for GL(N). (See [11]). They arise naturally from the twisted trace formulafor GL(N), which of course is where one would begin the study of our problem. Ifone is interested in the image and fibres of the maps 7rG --* 7, one should reallystate the problem in terms of these general endoscopic groups. However, for thestudy of classical groups, the primitive case that G equals G1 or G2 is obviouslywhat is important.The conjectural description of the contribution of 4 to the spectrum of G wehave alluded to (that is, the preimage in G of the automorphic representation 7r ofGL(N) attached to 0) is given in terms of a groupSo = So(G) = Cent(Image()), G),the centralizer in G of the image of 0 [2, Conjecture 8.1]. For example, ) shouldcontribute to the discrete spectrum of G if and only if So(G) is finite. It is clearthat for ) as above,So(GL(N)) = (C*).One also sees easily thatSG f(2/22)r, if each Ni is even,So(G)(Z/22)r-1, if some Ni is odd.Thus, ) contributes to the continuous spectrum of GL(N), but ought to contributeto the discrete spectrum of G. This property actually characterizes the specialcase we have been considering. If ei > 1 or 4i $ 4i for some i, and if 4 factorsthrough LG, the group S¢(G) will be infinite. Then ¢ should contribute only to -------------------------------------------------------------------------------- Page 11 THE PROBLEM OF CLASSIFYING AUTOMORPHIC REPRESENTATIONS11the continuous spectrum of G. In this more general situation, there could also beseveral different G such that q factors through LG.What is apparent is that one will need some analogue of the group SO(G) todetermine the fibres of the map 1rG -e 7r. It is no solution to use SO(G) itself- theLanglands reciprocity law is far from being established, and even if it were, it wouldnot be surjective. What we need instead is the construction of a group S,(G), forany self-contragredient automorphic representation 7r of GL(N), which reduces toSO(G) in case 7r comes from b. Now we can write 7r formally as7r = elTlT1 '''**er7Tr,where each 7ri is a (unitary) cuspidal automorphic representation of GL(Ni). Thenotation means that 7r is a representation induced from a parabolic subgroup withLevi componentGL(N)el x.. x GL(Nr)et,and embedded into L2(GL(N,Q)\ GL(N,A)) by an Eisenstein series. If we wouldhandle the cuspidal components7ri, we could copy the construction above; we wouldbe able to attach twisted endoscopic groups G = G1 x G2 to ir, and to define thegroups S (G). It is enough to treat the case that 7ri is self-contragredient. Onewould need to show that each such 7ri is attached to a unique endoscopic group Gifor GL(Ni), and that Gi is primitive in the sense that Gi equals either Sp(Ni, C)or SO(Ni, C). This is essentially part (ii) of the problem stated above.The remarks of this section have been concerned with setting up the definitions.One needs to define the group S,(G) in order even to state what the image andfibres of the maps lrG -- 7r should be. These groups are therefore at the heartof things. The required properties of the cuspidal components 7ri will have to beestablished as part of the full solution of the problem. One can foresee an elaborateinductive argument on the rank N of GL(N), which is based on the interplay ofthe stabilized twisted trace formula of GL(N), and the stabilized trace formulas ofthe endoscopic groups G.References1. J. Arthur, Automorphic representations and number theory, 1980 Seminar on Harmonic Anal-ysis (Montreal, Que., 1980), CMS Conf. Proc. 1, Amer. Math. Soc., Providence, R.I., 1981,pp. 3-51.2., Unipotent automorphic representations: Conjectures, Asterisque 171-172 (1989),13-71.3., L2- cohomology and automorphic representations, Advances in the MathematicalSciences-CRM's 25 years (Ed. L. Vinet), CRM Proceedings and Lecture Notes, AmericanMathematical Society, Providence, 1996, pp.???4. J. Arthur and L. Clozel, Simple algebras, base change and the advanced theory of the traceformula, Ann. of Math. Stud. 120, Princeton University Press, 1989.5. D. Flath, Decomposition of representations into tensor products, Automorphic Forms, Repre- sentations and L-Functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore.,1977), Part I, Proc. Sympos. Pure Math. XXXIII, Amer. Math. Soc., Providence, R.I., 1979,179-184.6. A. Frohlich, Galois module structure of algebraic integers, Ergeb. Math. Grenzgeb. (3) 1,Springer-Verlag, Berlin-New York, 1983.7. A. Fr6hlich and J. Queyrut, On the functional equations ofthe Artin L-function for charactersof real representations, Invent. Math. 20 (1973), 125-138.8. S. Gelbart and F. Shahidi, Analytic properties of automorphic L- functions, Perspect. Math.6, Academic Press, Inc., Boston, MA., 1988. -------------------------------------------------------------------------------- Page 12 12JAMES ARTHUR9. R. Godement and H. Jacquet, Zeta functions of simple algebras, Lecture Notes in Math., Vol.260, Springer-Verlag, Berlin-New York, 1972.10. H. Jacquet and J. Shalika, On Euler products and the classification of automorphic represen-tations. II, Amer. J. Math. 103 (1981), 777-815.11. R. Kottwitz and D. Shelstad, Twisted endoscopy 1: Definitions, norm mappings and transferfactors, preprint.12. J.-P. Labesse and R. Langlands, L-indistinguishability for SL(2), Canad. J. Math. 31 (1979),no. 4, 726-785.13. R. Langlands, Problems in the theory of automorphic forms, Lectures in modern analysis andapplications, III. Lecture Notes in Math., Vol. 170, Springer, Berlin,1970, 18-86.14., On the notion ofan automorphic representation, Automorphic forms, representationsand L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part1, Amer. Math. Soc., Providence, R.I., 1979, pp. 203-207.15., Base change for GL(2), Ann. of Math. Stud. 96. Princeton University Press, Prince-ton, N.J.; University of Tokyo Press, Tokyo, 1980.16. J. Tunnell, Artin's conjecture for representations of octahedral type, Bull. Amer. Math. Soc.(N.S.) 5 (1981), 173-175.DEPARTMENT OF MATHEMATICS, UNIVERSITY OF TORONTO, TORONTO, ONTARIO (CANADA)M5S 1A1E-mail address: arthurQmath. toronto. edu |