Harmonic Analysis and Group Representations, Volume 47

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Harmonic Analysis and Group Representations James Arthur

armonic analysis can be interpreted Chandra. I have been motivated by the following broadly as a general principle that re- three considerations. lates geometric objects and spectral (i) Harish-Chandra’s monumental contributions objects. The two kinds of objects are to representation theory are the analytic foun- sometimes related by explicit formu- dation of the Langlands program. For many las,H and sometimes simply by parallel theories. people, they are the most serious obstacle to This principle runs throughout much of mathe- being able to work on the many problems that matics. The rather impressionistic table at the top arise from Langlands’s conjectures. of the opposite page provides illustrations from (ii) The view of harmonic analysis introduced different areas. above, at least insofar as it pertains to group The table gives me a pretext to say a word about representations, was a cornerstone of Harish- the Langlands program. In very general terms, the Chandra’s philosophy. Langlands program can be viewed as a series of far- (iii) It is more than fifteen years since the death reaching but quite precise conjectures, which de- of Harish-Chandra. As the creation of one of scribe relationships among two kinds of spectral the great mathematicians of our time, his work objects—motives and automorphic representa- deserves to be much better known. tions—at the end of the table. Wiles’s spectacular I shall spend most of the article discussing Har- work on the Shimura-Taniyama-Weil conjecture, ish-Chandra’s ultimate solution of what he long re- which established the proof of Fermat’s Last The- orem, can be regarded as confirmation of such a garded as the central problem of representation relationship in the case of elliptic curves. In gen- theory, the Plancherel formula for real groups. I eral, the arithmetic information wrapped up in shall then return briefly to the Langlands program, motives comes from solutions of polynomial equa- where I shall try to give a sense of the role played tions with rational coefficients. It would not seem by Harish-Chandra’s work. to be amenable to any sort of classification. The analytic information from automorphic represen- Representations tations, on the other hand, is backed up by the rigid A representation of a group G is a homomorphism structure of Lie theory. The Langlands program rep- R : G → GL(V) , resents a profound organizing scheme for fundamental arithmetic data in terms of highly struc- where V = VR is a complex vector space that one tured analytic data. often takes to be a Hilbert space. We take for I am going to devote most of this article to a granted the notions of irreducible, unitary, direct short introduction to the work of Harish- sum, and equivalence, all applied to representations of a fixed group G. Representations of a finite group G were studied by Frobenius, as a tool for James Arthur is professor of mathematics at the Univer- sity of Toronto. The author would like to thank the editor investigating G. More recently, it was the repre- for his interest in the article and for some very helpful sug- sentations themselves that became the primary gestions. objects of study. From this point of view, there are

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Geometric objects Spectral objects

linear algebra sum of diagonal entries sum of eigenvalues of the of a square matrix matrix

ﬁnite groups conjugacy classes irreducible characters

topology singular homology deRham cohomology

differential geometry lengths of closed geodesics eigenvalues of the Laplacian

physics particles (classical mechanics) waves (quantum mechanics)

number theory logarithms of powers of zeros of (s) prime numbers

algebraic geometry algebraic cycles motives

automorphic forms rational conjugacy classes automorphic representations

Table of Illustrations

always two general problems to consider, for any then provide an isomorphism from V to Vb that given G. makes R equivalent to Rb . Moreover, this isomor- 1. Classify the set (G) of equivalence classes of phism satisfies the Plancherel formula irreducible unitary representations of G. Z X 2. If R is some natural unitary representation of 2 b 2 |f (x)| dx = |fn| . G, decompose R explicitly into irreducible rep- R/Z n resentations; that is, find a G-equivariant iso- 2 → b Example 2. G = R, VR = L (R), and morphism VR VRb , where VRb = V is a space built explicitly out of irreducible representations, as a direct sum R(y)f (x)=f (x + y) ,y∈ G, f ∈ VR. M R n V ,n ∈{0, 1, 2,...,∞}, In this case (G) is parametrized by : ∈(G) ∈ (G) ⇐⇒ V = C , or possibly in some more general fashion. iy (y)v = e v, v∈ V ,∈ R. 2 Example 1. G = R/Z, VR = L (R/Z), and Here we define Vb = L2(R) and R(y)f (x)=f (x + y),y∈ G, f ∈ VR. Rb(y) ()=eiy(),, ∈ V,b ∈ R. This is the regular representation that underlies b classical Fourier analysis. The set (G) is param- Then V is a “continuous direct sum”, or direct in- etrized by Z as follows: tegral of irreducible representations. The Fourier ∈ ⇐⇒ C transform (G) V = , Z 2iny (y)v = e v, v ∈ V ,n∈ Z. → b ix ∈ ∞ R f f ()= f (x)e dx, f Cc ( ), R The space n o extends to an isomorphism from V to Vb that sat- b 2 2 V = L (Z)= c =(cn): |cn| < ∞ isfies the relevant Plancherel formula Z Z 1 supports a representation |f (x)|2 dx = |fb()|2 d. R 2 R b 2iny R(y)c = e cn , n These two examples were the starting point for of G that is a direct sum of all irreducible repre- a general theory of representations of locally com- sentations, each occurring with multiplicity one. pact abelian groups, which was established in the The Fourier coefficients earlier part of the twentieth century. Attention Z then turned to the study of general nonabelian lo- b 2inx cally compact groups. Representations of non- f → fn = f (x)e dx, R/Z abelian groups have the following new features.

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Z ∈ (i) Representations (G) are typically infinite k k2 | |2 f 2 = f (x) dx dimensional. G (ii) Decompositions of general representations R typically have both a discrete part (like Fourier equals the dual norm series) and a continuous part (like Fourier Z kbk2 kb k2 transforms). f 2 = f () 2 d , (G) Problem of the for any function f ∈ C∞(G). In the second inte- Plancherel Formula b c grand, kf ()k2 denotes the Hilbert-Schmidt norm Against the prevailing b b of the operator f (). The norm kf k then defines opinion of the time, Har- 2 a Hilbert space Vb, on which G G acts pointwise ish-Chandra realized early by left and right translation on the spaces of in his career that a rich Hilbert-Schmidt operators on {V }. The problem theory would require the was known to be well posed. A general theorem of study of a more restricted I. Segal from 1950, together with Harish-Chandra’s class of nonabelian groups. From the very be- proof in 1953 that G is of “type I”, ensures that ginning, he confined his the Plancherel measure d exists and is unique. attention to the class of The point is to calculate d explicitly. This in- semisimple Lie groups, or cludes the problem of giving a parametrization of slightly more generally, re- (G), at least up to a set of Plancherel measure ductive Lie groups. These zero. include the general linear The first person to consider the problem and groups GL(n, R), the spe- to make significant progress was I. M. Gelfand. He cial orthogonal groups established Plancherel formulas for a number of SO(p, q; R), the symplec- matrix groups, and laid foundations for much of tic groups Sp(2n, R), and the later work in representation theory and auto- the unitary groups morphic forms. However, some of the most severe U(p, q; C) . For the pur- difficulties arose in groups that he did not consider. Photograph by Herman Landshoff, courtesy of the Institute for Advanced Study. poses of the present arti- Harish-Chandra worked in the category of general Harish-Chandra. cle, the reader can in fact semisimple (or reductive) groups. His eventual take G to be one of these proof of the Plancherel formula for these groups familiar matrix groups. was the culmination of twenty-five years of work. Harish-Chandra’s long-term goal became that of It includes many beautiful papers, and many ideas finding an explicit Plancherel formula for any such and constructions that are of great importance in G . As in the two examples above, one takes their own right. I shall describe, in briefest terms, 2 VR = L (G), with respect to a fixed Haar measure a few of the main points of Harish-Chandra’s over- on G. One can then take R to be the 2-sided reg- all strategy, as it applies to the example ular representation G = GL(n, R). 1 R(y1,y2)f (x)=f (y1 xy2) , Geometric Objects In Harish-Chandra’s theory of the Plancherel for- y1,y2 ∈ G, f ∈ VR, mula, the geometric objects are parametrized by the of G G on V. The regular representation is spe- regular, semisimple conjugacy classes in G. In the cial among arbitrary representations in that it al- example G = GL(n, R) we are considering, these ready comes with a candidate for an isomorphism conjugacy classes are the ones that lie in the open with a direct integral. This is provided by the gen- dense subset eral Fourier transform ( ) Z the eigenvalues i ∈ C Greg = ∈ G : f → fb()= f (x)(x) dx, of are distinct G of G. They are classified by the characteristic poly- ∈ ∞ ∈ f Cc (G), (G), nomial as a disjoint union of orbits a ∞ which is defined on a dense subspace Cc (G) of TP,reg/WP , L2(G), and takes values in the vector space of fam- P { } ilies of operators on the spaces V . where P ranges over certain partitions The problem of the Plancherel formula is to n o compute the measure d on (G) such that the P =(1,... ,1, 2,... ,2): r1 +2r2 = n norm | {z } | {z } r1 r2

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of n. For a given P, these linear forms to the Schwartz space C(G) on     G (which he later defined), and their behavior as t 0   1   approaches the singular set in T . What is per-  .  P TP =  =  ..   , haps surprising at first glance is that the problems are not always amenable to direct attack. Harish- 0 tr1+r2 Chandra often established concrete inequalities where tk belongs to R if 1 k r1, and is of the by deep and remarkably indirect methods, that form ! fully exploited the duality between the geometric objects fG() and their corresponding spectral ana- k cos k k sin k logues. k sin k k cos k Spectral Objects if r1 +1 k r1 + r2, while TP,reg stands for the in- The spectral objects for Harish-Chandra were the tersection of TP with Greg. The group characters of representations in (G). Here, he Z Z r2 WP = Sr1 Sr2 ( /2 ) , was immediately faced with the problem that the space V is generally infinite dimensional, in which in which Sk denotes the symmetric group case the sum determining the trace of the unitary on k letters, acts in the obvious way by operators (x) on V can diverge. His answer was b permutation of the elements {tk :1 k r1 + r2} to prove that the average f () of these operators ∈ ∞ and by sign changes in the coordinates against a function f Cc (G) is in fact of trace {k : r1 +1 k r1 + r2}. class. He then defined the character of to be the The complement of Greg in G has Haar measure distribution 0. By calculating a Jacobian determinant, Harish- b ∈ ∞ Chandra decomposed the restriction of the Haar fG()=trf () ,fCc (G). measure to Greg into measures on the coordinates However, this was by no means sufficient for the defined by conjugacy classes. The resulting formula purposes he had in mind. is Z Differential equations play a central role in Har- f (x)dx = ish-Chandra’s analysis of both characters and orbital integrals. Let Z be the algebra of differential G X Z Z operators on G that commute with both left and 1 1 |WP | |D()| f (x x) dx d , right translation. One of Harish-Chandra’s earliest P theorems, for which he won the AMS Cole Prize in TP,reg TP \G 1954, was to describe the structure of Z as an al- n ∈ ∞ gebra over C. Let tP = C be the complexification for any f Cc (G). Here d is a Haar measure on T , and T \G represents the right cosets of T in of the Lie algebra of the Cartan subgroup TP of P P P R G, a set that can be identified with the conjugacy G = GL(n, ). By definition there is then a canonical isomorphism ∂ from the symmetric algebra class of any ∈ TP,reg . The function Y S(tP ) to the algebra of invariant differential oper- 2 ators on T . D()= (i j ) P 1i

TP \G unique up to the action of Sn. ∞ The equation (i) can be interpreted in terms of f ∈ C (G), c the traditional technique of separation of vari- for any element in TP,reg. This distribution is now ables. The relevant differential operators are of known as Harish-Chandra’s orbital integral, and is course the elements in Z, while the variables of sep- at the heart of much of his work. Harish-Chandra aration are defined by the coordinates of conjugacy needed to prove many deep theorems about orbital classes in Greg. The equation (ii) is a variant of integrals. The questions concern the extension of Schur’s lemma, which says that any operator

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commuting with the action of a finite group under integration formula, and the invariance of an irreducible representation is a scalar. The func- under conjugation by G, that Z tional comes from the characterization of ho- X Z→C 1 momorphisms that is given by the isomor- fG()= |WP | fG() () d , T ,reg phism z → hP (z). P P Harish-Chandra also used the separation of ∈ ∞ f Cc (G). variables technique to study the distribution on This is a particularly vivid illustration of the du- Greg obtained by restricting the character of any . It is easy to see that many of the differential ality between the geometric objects fG() and the f () operators ∂ hT (z) on a given TP are actually spectral objects G . The formula becomes more elliptic. This allowed him to apply the well known explicit if we substitute the expansion above for theorem that eigendistributions of elliptic opera- (). The resulting expression reduces the study tors are actually real analytic functions. In this of characters to the determination of the linear way, he was able to prove that functionals and the families of coefficients Z {cs }. ∈ ∞ fG()= f (x) (x) dx, f Cc (Greg), Greg Plancherel Formula and Discrete Series We can now state Harish-Chandra’s Plancherel for- for a real analytic function on G . The sepa- reg mula (for the group G = GL (R)) as follows. ration of variables that is part of his argument then n implies that for any TP, the function Theorem (Plancherel formula). For each character

1 c in the dual group ()=|D()| 2 (),∈ TP,reg, Tb = (R Z/2Z)r1 (Z R)r2 , satisfies the differential equations P D E there is an irreducible representation of G such ∈Z c ∂ hP (z) ()= hP (z), (),z. that Z X Z Thus, () is a simultaneous eigenfunction of a |f (x)|2dx = |W | 1 kfb( )k2m(c) dc , large family of invariant differential operators on P b c 2 G P TP the abelian group T . From this, it is not hard to P ∈ ∞ deduce, at least in the case that the coordinates of f Cc (G), n in C are distinct, that the restriction of () b for an explicit real analytic function m(c) on TP. to any connected component of TP,reg has a simple formula of the form Remarks. X (s )(H) 1. The Plancherel density m(c) actually vanishes if ()= cs e ,= exp H, Z R r2 Z s∈Sn the image of c in ( ) has any -component equal to zero. For any such c, the representation for complex coefficients {c }. s is not well defined by the formula, and can be We can see that the differential equations give c taken to be 0. detailed information about characters. To be able to apply this information to the study of the 2. The linear function attached to = c is Plancherel formula, however, Harish-Chandra re- equal to the differential of c. quired the following fundamental theorem. 3. Harish-Chandra actually stated the theorem in Theorem. The character of any representation the form of a Fourier inversion formula ∈ (G) is actually a function on G. In other X Z 1 words, extends to a locally integrable function f (1) = |WP | fG(c )m(c) dc , on G such that P b Z TP ∞ ∞ ∈ ∈ f Cc (G). fG()= f (x) (x) dx, f Cc (G). G To recover the Plancherel formula one needs only replace f by the function The proof of this theorem required many new Z ideas, which Harish-Chandra developed over the (f f )(x)= f (y)f (x1y) dy course of nine years. Atiyah and Schmid later gave G a different proof of the theorem, by combining some of Harish-Chandra’s techniques with methods from geometry. in the inversion formula. Note the duality with the The theorem provides a more concrete formula earlier integration formula, which can be written for the character of . It follows from the original in the form

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Z X Z 1 1 tan subgroup T that is compact. Moreover, he spec- f (x) dx = |W | f ()|D()| 2 d , P G ified the representations in the discrete series G P TP ,reg ∞ uniquely by a simple expression for their charac- f ∈ C (G). c ters on Treg that is a striking generalization of the Weyl character formula. This short introduction does not begin to con- The group GL(n, R) does not have a discrete se- vey a sense of the difficulties Harish-Chandra en- ries. The representations that appear in its countered, and was able to overcome. The most fa- Plancherel formula are all constructed from dis- mous is the construction of the discrete series, the crete series of SL(2, R) and characters of R. (For family of representations ∈ (G) to which the a given partition P, there are r copies of SL(2, R) Plancherel measure d attaches positive mass. 2 to consider; the representations are defined by We ought to say something about these objects, c “parabolic induction” from representations of the since they are really at the heart of the Plancherel subgroup of block diagonal matrices in GL(n, R) formula. of type P.) The example of GL(n, R) is therefore rel- It might be helpful first to recall Weyl’s classi- atively simple. Groups that have discrete series, fication of representations of compact groups, as such as Sp(2n, R) and U(p, q; C), are much more it applies to the special case of the unitary group difficult. What is remarkable is that the final state- G = U(n, C). By elementary linear algebra, any uni- ment of the general Plancherel formula, suitably tary matrix can be diagonalized, so there is only interpreted, is completely parallel to that of the one Cartan subgroup GL(n, R).     After he established the Plancherel formula for 1 0     real groups, Harish-Chandra worked almost ex-  .  | | T =  =  ..  : i =1 clusively on the representation theory of p-adic 0 n groups. This subject is extremely important for the analytic side of the Langlands program, but it has in G to consider. We can otherwise use notation a more arithmetic flavor. Harish-Chandra was able similar to that of GL(n, R). Weyl’s classification is to establish a version of the Plancherel formula for provided by a canonical bijection ↔ between p-adic groups. However, it is less explicit than his the irreducible representations ∈ (G) and the formula for real groups, for the reason that he did n subset of points =(1,... ,n) in Z such that not classify the discrete series. The problem of dis- i >i+1 for each i. This bijection is determined crete series for general G is still wide open, in uniquely by a simple formula Weyl established for fact, as is much of the theory for p-adic groups.1 the value of the character Nature of the Langlands Program ()=tr() , The analytic side of the Langlands program is con- ∈ C cerned with automorphic forms. The language of at any element Treg . (Since U(n, ) is compact, the general theory of automorphic forms, as op- is in fact finite dimensional.) The Weyl charac- posed to classical modular forms, is that of the rep- ter formula is the identity resentation theory of reductive groups. It is a lan- X 1 1 guage created largely by Harish-Chandra. s( ) ()= D() 2 sign(s) , Harish-Chandra’s influence on the theory of au- ∈ s Sn tomorphic forms is pervasive. It is not so much in the actual statement of his Plancherel formula, where for any ∈ Zn, denotes the product 1 but rather in the enormously powerful methods 1 n 2 1 n . (The denominatorQ D() is the canon- and constructions (including the discrete series) ical square root (i j ) of the discriminant. that he created in order to establish the Plancherel i

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group G = G(R) has been equipped with structure one specific example of the influence of Harish- necessary to define G(Z).) As above, one seeks in- Chandra’s work—that of the discrete series. formation about the decomposition of R into ir- Assume that G does have a compact Cartan reducible representations. In this case, however, subgroup. The Hecke operators {Tp,i()} associ- there is some interesting extra structure. The space ated to discrete series representations of G are L2( \G) comes with a family of semisimple oper- expected to be related to arithmetic objects at- ators {Tp,i}, the Hecke operators, which are pa- tached to algebraic varieties. In many cases, it is rametrized by a cofinite set {p : p 6∈ P } of prime known how to construct algebraic varieties for numbers, and a supplementary set of indices which this is so. Let K be a maximal compact sub- {i :1 i np} that depends on p and has order group of G, and assume that G is such that the bounded by the rank of G. These operators com- space of double cosets mute with R, and also with each other. If ∈ (G) S = \G/K is a representation that occurs discretely in R with multiplicity m(), the Hecke operators then has a complex structure.2 This is the case, for ex- provide a family ample, if G equals Sp(2n, R) or U(p, q; C). Then S is the set of complex points of an algebraic vari- {Tp,i(): p 6∈ P , 1 i np} ety. Moreover, it is known that this variety can be of mutually commuting m() m() -matrices. defined in a canonical way over some number field F (equipped with an embedding F C ). If It is the eigenvalues of these matrices that are R thought to carry the fundamental arithmetic in- G = SL(2, ), S is just a quotient of the upper half plane, and as varies, the varieties in this case de- formation. termine the modular elliptic curves of the Shimura- The most powerful tool available at present for Taniyama-Weil conjecture. The varieties in gen- the study of R (and the Hecke operators) is the eral were introduced and investigated extensively trace formula. The trace formula plays the role here by Shimura. Their serious study was later taken up of the Plancherel formula, and is the analogue of by Deligne, Langlands, Kottwitz, and others. the Poisson summation formula for the discrete It is a key problem to describe the cohomology subgroup Z of R. It is an explicit but quite com- H(S ) of the space S , and more generally, vari- plicated formula for the trace of the restriction of ous arithmetic objects associated with this coho- the operator Z mology. The discrete series representations are at the heart of the problem. There is a well defined ∈ ∞ R (f )= f (x)R (x) dx, f Cc (G), G procedure, based on differential forms, for pass- ing from the subspace of L2( \G) defined by (of and more generally, the composition of R (f ) with multiplicity m()) to a subspace, possibly 0, of 2 several Hecke operators, to the subspace of L ( \G) H (S ). Different correspond to orthogonal sub- that decomposes discretely. The formula is really spaces of H (S ), and as ranges over all repre- an identity of two expansions. One is a sum of sentations in the discrete series, these subspaces terms parametrized by rational conjugacy classes, span the part of the cohomology of H (S ) that is while the other is a sum of terms parametrized by primitive and is concentrated in the middle di- automorphic representations. The trace formula is mension. Moreover, the Hodge structure on this thus a clear justification of the last line of our part of the cohomology can be read off from the original table. It is also a typical (if elaborate) ex- parametrization of discrete series. Finally, there has ample of the kind of explicit formula that relates been much progress on the deeper problem of es- geometric and spectral objects on other lines of the tablishing reciprocity laws between the eigenval- table. ues of the Hecke operators Tp,i() and arithmetic I mention the trace formula mainly to point out data attached to the corresponding subspaces of its dependence on the work of Harish-Chandra. The H (S ). These are serious results, due to Lang- geometric side is composed of orbital integrals, to- lands and others, that I have not stated precisely, gether with some more general objects. The spec- or even quite correctly.3 The point is that the re- tral side includes the required trace, as well as sults provide answers to fundamental questions, some supplementary distributions. All of these 2 terms rely in one way or another on the work of By replacing with a subgroup of finite index, if neces- Harish-Chandra, for both their construction and sary, one also assumes that has no nontrivial elements of finite order. their analysis in future applications of the trace for- 3 mula. The axioms for a Shimura variety are somewhat more complicated than I have indicated. They require a slightly I shall say no more about the trace formula. It modified discussion, which applies to groups with non- is also not possible in the dwindling allotment of compact center. Moreover, if S is itself noncompact, space to give any kind of introduction to the Lang- H (S ) should really be replaced by the corresponding lands program. I shall instead comment briefly on L2-cohomology.

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which could not have been broached without Har- groups G(Qp), from which one can recover the 4 ish-Chandra’s classification of discrete series. complex numbers {p,i} and the algebras {Uq}. The discussion raises further questions. What An automorphic representation is said to be about the rest of the cohomology of S ? What tempered if its components R and p are all tem- about the representations ∈ (G) in the com- pered. I did not give the definition of tempered rep- plement of the discrete series? Harish-Chandra’s resentations for p-adic groups, but it is the same Plancherel formula included a classification of the as for real groups. representations that lie in the natural support of For the unramified the Plancherel measure (up to some questions of primes p 6∈ P , it is reducibility of induced representations, which were equivalent to a cer- later resolved by Knapp and Zuckerman). Such tain set of bounds representations are said to be tempered, because on the absolute their characters are actually tempered distribu- values of the com- tions on G—they extend to continuous linear forms plex numbers { } on the Schwartz space of G. Tempered represen- p,i . The validity tations that lie in the complement of the discrete of these bounds series are certainly interesting for automorphic for one particular forms, but they do not contribute to the coho- automorphic rep- mology of S. Nontempered representations, on the resentation of the other hand, have long been known to play an im- group G = SL(2) is portant role in cohomology. Can one classify the equivalent to a fa- nontempered representations ∈ (G) that occur mous conjecture of discretely in L2( \G)? Ramanujan, which To motivate the answers, let me go back to the was proved by last line of the original table. A conjugacy class in Deligne in 1973. G(Q) has a Jordan decomposition into a semisim- The conjectures ple part and a unipotent part. (Recall that an ele- of Langlands in- ment x ∈ GL(n, Q) is unipotent if some power of clude a general pa- the matrix x I equals 0. The Jordan decomposi- rametrization of Q tempered auto- Photograph © 1996 Randall Hagadorn, courtesy of the Institute for Advanced Study. tion for GL(n, ) is given by the elementary divi- Robert Langlands sor decomposition of linear algebra.) Since automorphic represen- morphic representations are dual in some sense tations. In the early 1980s, I gave a conjectural characterization of au- to rational conjugacy classes, it is not unreason- tomorphic representations that are nontempered. able to ask whether they too have some kind of Jor- Among other things, this characterization de- dan decomposition. scribes the failure of a representation to be tem- I can no longer avoid giving at least a provisional pered in terms of a certain unipotent conjugacy definition of an automorphic representation. As- class. It is not a conjugacy class in G(Q)—such ob- sume for simplicity that G(C) is simply connected. jects are only dual to automorphic representa- In general, one would like an object that combines tions—but rather in the complex dual group Gb of a representation ∈ (G) with any one of the G. Here Gb is the identity component of the L-group m() families { : p 6∈ P , 1 i n } of si- p,i p LG = Gb o Gal(Q/Q) that is at the center of Lang- multaneous eigenvalues of the Hecke operators. (It lands’s conjectures. In this way, one can construct is these complex numbers, after all, that are sup- a conjectural Jordan decomposition for automor- posed to carry arithmetic information.) It turns out phic representations that is dual to the Jordan de- that any such and any such family, as well as composition for conjugacy classes in G(Q). The some (noncommutative) algebras of operators { ∈P } conjectures for nontempered representations con- Uq : q obtained from the ramified primes, tain some character identities for the local com- can be packaged neatly together in the form of a A A A ponents R and p of representations of G( ). representation of the adelic group G( ). Here is R They also include a global formula for the multi- a certain locally compact ring that contains , and plicity of in L2 G(Q)\G(A) that implies quali- Q Q also the completions p of with respect to p-adic tative properties for the eigenvalues of Hecke op- absolute values. The rational field Q embeds di- erators. The local conjectures for R have been A agonally as a discrete subring of . Let us define established by Adams, Barbasch, and Vogan, by an automorphic representation restrictively as an very interesting methods from intersection irreducible representation of G(A) that occurs discretely in the decomposition of L2 G(Q)\G(A) . 4The proper definition of automorphic representation Any such determines a representation = R also includes representations that occur continuously in in (G) and a discrete subgroup G such that L2 G(Q)\G(A) , as well as analytic continuations of 2 \ occurs discretely in L ( G). It also determines such representations. If G(C) is not simply connected, irreducible representations {p} of the p-adic actually determines several discrete subgroups of G.

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homology. The remaining assertions are open. Finally, the most general conjecture relating However, the contribution of nontempered repre- motives with automorphic representations is stated sentations to cohomology is quite well understood. at the end of §2 of Langlands’s article The unipotent class that measures the failure of a R. P. Langlands, Automorphic repre- representation to be tempered turns out to be the sentations, Shimura varieties and mo- same as the unipotent class obtained from the ac- tives. Ein Märchen, Automorphic Forms, tion of a Lefschetz hyperplane section on coho- Representations and L-functions, Proc. mology. One can in fact read off the Lefschetz Sympos. Pure Math., vol. 33, Part 2, structure on H(S ), as well as the Hodge struc- Amer. Math. Soc., 1979, pp. 205–246. ture, from the parametrization of representations.

References For a reader who is able to invest the time, the best overall reference for Harish-Chandra’s work is still his collected papers. Harish-Chandra, Collected Papers, Vol- umes I–IV, Springer-Verlag, 1984. Harish-Chandra’s papers are very carefully written, and are not difficult to follow step by step. On the other hand, they are highly interdependent (even in their notation), and it is sometimes hard to see where they are leading. The excellent technical introduction of Varadarajan goes some way towards easing this difficulty. Weyl’s classification of representations of compact groups is proved concisely (in the special case of U(n, C)), in H. Weyl, The Theory of Groups and Quantum Mechanics, Dover Publica- tions, 1950, pp. 377–385. The following two articles are general intro- ductions to the Langlands program S. Gelbart, An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. N.S. 10 (1984), 177–219.

J. Arthur and S. Gelbart, Lectures on automorphic L-functions, Part I, L-functions and Arithmetic, London Math. Soc. Lecture Note Series, vol. 153, Cambridge Univ. Press, 1991, pp. 2–21. Other introductory articles on the Langlands program and on the work of Harish-Chandra are contained in the proceedings of the Edinburgh in- structional conference Representation Theory and Automor- phic Forms, Proc. Sympos. Pure Math., vol. 61, Amer. Math. Soc., 1996, edited by T. N. Bailey and A. W. Knapp. The general axioms for a Shimura variety are summarized in the Deligne’s Bourbaki lecture P. Deligne, Travaux de Shimura, Sémi- naire Bourbaki, 23ème Année (1970/71), Lecture Notes in Math., vol. 244, Springer-Verlag, 1971, pp. 123–165.

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