International Congress of Mathematicians

660 F. ShahidiUsing this induction and local-global arguments (cf. Proposition 5.1 of [42]),it was proved in [42] thatTheorem 4.4. (Theorems 3.5 and 7.7 of [42]) a) For each i, 1 < i < m,and each v, there exist a local L-function L(s,ir v ,ri :V ), which is the inverse of apolynomial in q^swhose constant term is 1, if v < oo, and is the Artin L-functionattached to r t • ip' v , where tp' v : W' F —¥ L M V is the homomorphism of the Deligne-Weil group into L M V parametrizing n v , if either v = oo or n v has an Iwahori-fixedvector; and a root number e(s, / K v ,r i v ,ip v ) satisfying the same provisions, such thatifL(s,n,ri) = JJ_L(s,Tr v ,r i:V ) (4.1)andthen6) Lete(s,n,ri) = JJ_e(s,Tr v ,r i:V ,fi v ), (4.2)VL(s, / K,r i ) = e(s, / K,r i )L(l — s, 7r,Fj). (4.3)-y(s,ir v ,r i:V ,fi v ) = e(s,Tr v ,r i:V ,fi v )L(l - s,n v ,r iiV )/L(s,n v ,r iiV ). (4.4)Then each / y(s,ir v ,ri V ,fi v ) is multiplicative in the sense of equation (3.13) in Theorem3.5 of [42]. (See below.) Ifn v is tempered, then-f(s,ir v ,ri :V ,fi v ) determines thecorresponding root number and L-function uniquely and in fact that is how they aredefined. Suppose n v is non-tempered, then each L(s, Tr v ,r i:V ) is determined by meansof the analytic continuation of its quasi-tempered Langlands parameter and multiplicativityof corresponding 7 -functions. More precisely, if a v is the quasitemperedLanglands parameter that gives n v as a subrepresentation, thenL(s,TT v ,r i}V ) = JJ L(s,Wj(a v ),r' i{jhv ), (4.5)where the notation is as in part 3) of Theorem 3.5 of [42], provided that everyL-function on the right hand side is holomorphic for Re(s) > 0, whenever a v is(unitary) tempered (Conjecture 7.1 of [42], proved in many cases [3.6.42]). Theset Si,Wj and ifi.-. are defined as follows in which we drop the index v. Assumen C Lrad Me) (jv e) nM)tM (T ® 1J where M§(N^nM) is a parabolic subgroup o/M definedby a subset 9 C A, the set of simple roots of A 0 . Let 9' = WQ(9) C A and fix areduced decomposition WQ = w„-i • • • wi of WQ (Lemma 2.1.1 of [41])- For each j,there exists a unique root aj £ A such that Wj(ctj) < 0. For each j, 2 < j < n — 1,let Wj = Wj-i • • • wi- Set wi = 1. Let iij = 9j U {aj}, where 9i =9, 9 n = 9',and 9j + i = Wj(9j), 1 < j < n — 1. Then Mf^. contains M^.(N^. n MQ.) as amaximal parabolic subgroup andwj(a) is a representation of M$ i . The L-group L M$acts on the space of ri, but no longer necessarily irreducibly. Given an irreducibleconstituent of this action, there exists a unique j, 1 < j < n — 1, which under Wjis equivalent to an irreducible constituent of the action of L M$ j on the Lie algebraof the L-group of N^. n Mn r Let i(j) be the index of this subspace and denote by