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Automorphic L-Functions Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 2, pp. 27-61 AUTOMORPHIC L-FUNCTIONS A. BOREL This paper is mainly devoted to the L-functions attached by Langlands [35] to an irreducible admissible automorphic representation re of a reductive group G over a global field k and to local and global problems pertaining to them. In the context of this Institute, it is meant to be complementary to various seminars, in particular to the GL2-seminars, and to stress the general case. We shall therefore start directly with the latter, and refer for background and motivation to other seminars, or to some expository articles on this topic in general [3] or on some aspects of it [7], [14], [15], [23]. The representation re is a tensor product re = @,re, over the places of k, where re, is an irreducible admissible representation of G(k,) [11]. Accordingly the L-func­ tions associated to re will be Euler products of local factors associated to the 7r:.'s. The definition of those uses the notion of the L-group LG of, or associated to, G. This is the subject matter of Chapter I, whose presentation has been much influ­ enced by a letter of Deligne to the author. The L-function will then be an Euler product L(s, re, r) assigned to re and to a finite dimensional representation r of LG. (If G = GL"' then the L-group is essentially GLn(C), and we may tacitly take for r the standard representation r n of GLn(C), so that the discussion of GLn can be carried out without any explicit mention of the L-group, as is done in the first six sections of [3].) The local L- ands-factors are defined at all places where G and 1T: are "unramified" in a suitable sense, a condition which excludes at most finitely many places. Chapter II is devoted to this case. The main point is to express the Satake isomorphism in terms of certain semisimple conjugacy classes in LG (7. l ). At this time, the definition of the local factors at the ramified places is not known in . general. For GLn and r no however, there is a direct definition [19], [25]. In the general case, the most ambitious scheme is to associate canonically to an irreducible admissible representation of a reductive group H over a local field E a representa­ tion of the Weil-Deligne group We of E into LH, and then use L- ands-factors associated to representations of We [60]. This problem is the main topic of Chapter III. The L-function L(s, re, r) associated to re and r as above is introduced in §13. In fact, it is defined in general as a product oflocal factors indexed by almost all places of k. It converges absolutely in some right half-plane (13.3; 14.2). Some of the main AMS (MOS) subject classifications (1970). Primary IOD20; Secondary 12A70, 12B30, 14GIO, 22E50. © 1979, American Mathematical Society 27 28 A. BOREL conjectural analytic properties (meromorphic continuation, functional equation), and the evidence known so far, are discussed in §14. From the point of view of[35], a great many problems on automorphic represen­ tations and their £-functions are special cases of one, the so-called lifting problem or problem of functoriality with respect to L-groups. It is discussed in Chapter V. It is closely connected with Artin's conjecture (see §17 and the base-change sem­ inar [17]). In §18 brief mention is made of some known or conjectured relations between automorphic £-functions and the Hasse-Weil zeta-function of certain varieties, to be discussed in more detail in the seminars on Shimura varieties [8], [40]. Thanks are due to H. Jacquet and R. P. Langlands for various very helpful com­ ments on an earlier draft of this paper. CONTENTS I. L-groups .................................................................................... 28 I. Classification ........................................................................... 29 2. Definition of the L-group ......................................................... 29 3. Parabolic subgroups .................................................................. 31 4. Remarks on induced groups ......................................................... 33 5. Restriction of scalars ............................................................... 34 II. Quasi-split groups; the unramified case ............................................. 35 6. Semisimple conjugacy classes in LG ............................................. 35 7. The Satake isomorphism and the L-group. Local factors in the unramified case ........................................................................ 38 III. Weil groups and representations. Local factors ..................................... 39 8. Definition of <P(G) ..................................................................... 39 9. The correspondence for tori ..................................................... .41 10. Desiderata .............................................................................. 43 11. Outline of the construction over R, C ............................................ .46 12. Local factors. ....................................................................... 48 IV. The L-function of an automorphic representation ................................. 49 13. The £-function of an irreducible admissible representation of GA- ....... .49 14. The £-function of an automorphic representation ........................... 52 V. Lifting problems ........................................................................... 54 15. L-homomorphisms of L-groups ................................................... 54 16. Local lifting ........................................................................... 55 17. Global lifting ........................................................................... 56 18. Relations with other types of £-functions ....................................... 58 References ....................................................................................... 59 CHAPTER I. L-GROUPS. k is afield, k an algebraic closure of k, k, the separable closure of kink, and rk the Galois group of k, over k. G is a connected reductive group, over kin I.I, 1.2, 2.1, 2.2, over k otherwise. §§1, 2 will be used throughout, §3 from Chapter III on. The reader willing to take on faith various statements about restriction of scalars need not read §§4, 5. AUTOMORPHIC £-FUNCTIONS 29 1. Classification. We recall first some facts discussed in [58]. 1.1. There is a canonical bijection between isomorphism classes of connected reductive k-groups and isomorphism classes of root systems. It is defined by as­ sociating to G the root datum cj;(G) = (X*(T), rp, X*(T), rpv) where Tis a maximal torus of G, X*(T) (X*(T)) the group of characters (I-parameter subgroups) of T and(/) ((/)v) the set ofroots (coroots) of G with respect to T. 1.2. The choice of a Borel subgroup B ::::> Tis equivalent to that of a basis L1 of W(G, T). The previous bijection yields one between isomorphism classes of triples (G, B, T) and isomorphism classes of based root data cj;0(G) = (X*(T), Ll, X*(T), LJv). There is a split exact sequence (1) (1)--------> Int G--------> Aut G--------> Aut cj;0(G)--------> (1). To get a splitting, we may choose Xa E Ga (a E Ll) and then have a canonical bijec­ tion (2) Two such splittings differ by an inner automorphism Int t (t ET). 1.3. Given r Erk there isgEG(ks) such thatg-rT-g-1 = T, g-TB·g-1 = B, whence an automorphism of cj;0(G), which depends only on r· We let µc: I'k---> Aut cj;0(G) be the homomorphism so defined. If G' is a k-group which is isomorphic to G over k (hence over ks), then µc = µ(;, <o> G, G' are inner forms of each other. 1.4. Let/: G ---> G' beak-morphism, whose image is a normal subgroup. Then f induces a map cj;(f): cj;(G)---> cj;(G') (contravariant (resp. covariant) in the first (last) two arguments). Given B, Tc Gas above, there exists a Borel subgroup B' (resp. a maximal torus T') of G' such that/(B) c B',f(T) c T', whence also a map cj;0(f): c/;o( G) ---> c/;o( G'). 2. Definition of the £-group. 2.1. The inverse system lff"6 to the based root datum If! 0 = ( M, Ll, M*, LJV) is cJ; "6 = (M*, LJV, M, Ll). To the k-group G we first associate the group LG 0 over C such that 0 0 0 ¢0(LG ) = cj;0(G)V. We let LT , LB be the maximal torus and Borel subgroup de­ fined by ¢'cf, and say they define the canonical splitting of LG 0 • Let/ be as in 1.4. Then/ also induces a map ¢"6(f) : cj; 0(G')V ---> ¢ 0(G)v. An alge­ 0 braic group morphism of LG' 0 into LG0 associated to it will be denoted Lf • Given one, any other is of the form Int f0Lf 00Int t' (t E LT0, t' E LT'0), and maps LT'0 0 (resp. LB'0) into LT0 (resp. LB ). 2.2. EXAMPLES. (1) Let G = GLn. Then LG 0 = GLn. In fact, let M = zn with {x;} its canonical basis. Let {e;} be the dual basis of M* = zn. Then lf!0(GLn) = (M, Ll, M*, LJV) with LI= {(x; - X;+1), 1 ~ i < n}, LJV = {(e; - ei+l), I ~ i < n}, hence c/;o = cJ; "6. (2) Let G be semisimple and lf!0(G) = (M, (/), M*, cpv). As usual, let P(W) c M ® Q be the lattice of weights of (/) and Q(W) the group generated by (/) in M. Define P((/)V) and Q(wv) similarly. As is known G is simply connected (resp. of adjoint type) if and only if P(¢) = M (resp. Q(</>) = M). Moreover P(W) = {A EM@ Qj(,l., cpv) c Z}, P((/)V) = {A E Mx @ Qj(,l., (/)) E Z}. 30 A. BOREL Therefore: G simply connected ¢>- LG 0 of adjoint type; G of adjoint type ¢>- LG 0 simply connected. (3) Let G be simple. Up to central isogeny, it is characterized by one of the types Am ... , G2 of the Killing-Cartan classification. It is well known that the map r/J0{G) -+ rjJ0{G)• permutes Bn and Cn and leaves all other types stable. Thus if G = SP2n (resp. G = PSpzn), then LG 0 = S02n+1 (resp. LG 0 = SpiDzn+1). In all other cases, G ,_ LG 0 preserves the type (but goes from simply connected group to adjoint group, and vice versa).
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