International Congress of Mathematicians

124 J. W. Cogdell I. I. Piatetski-ShapiroStability of 7-factors. If ni :V and TT2, V are two irreducible admissible representationsofH(k v ) with the same central character, then for every sufficiently highlyramified character r] v of GL\(k v ) we have ^(s,ni^v X r) v ,ip v ) = 7(s,7r 2jt , x r h,'ipv)-Both of these facts are known for GL„, the multiplicativity being found in[9] and the stability in [10]. Multiplicativity in a fairly wide generality useful forapplications has been established by Shahidi [19]. Stability is in a more primitivestate at the moment, but Shahidi has begun to establish the necessary results in ageneral context in [20].To utilize these local results, what one now does is the following. At the placeswhere n v is ramified, choose n^ to be arbitrary, except that it should have the samecentral character as n v . This is both to guarantee that the central character of nis the same as that of n and hence automorphic and to guarantee that the stableforms of the 7-factors for n v and n^ agree. Now form II = 'II„. Choose ourcharacter r\ so that at the places v £ T we have that the F- and 7-factors forboth n v ® r) v and n^ ® r) v are in their stable form and agree. We then twist byT' = T ® T) for this fixed character r\. If n' £ T', then for v £ T, n' v is of theform n' v = Ind(\ \ Sl ® • • • ® | | Sm ) ® r) v . So at the places v £ T, applying bothmultilplcativity and stability, we have7(s,7T„ x n' v ,ip v ) = Jj7(s + Sj,7r„ ®i} v ,ip v )= Y[>y(s + Si,Yl v ®i} v ,ip v ) =7(s,II„ x n' v ,ip v )from which one deduces a similar equality for the F- and e-factors. From this itwill then follow that globally we will have L(s, n xn') = L(s, II x n') for all n' £ T'with similar equalities for the e-factors. This then completes Step 1.To complete our use of the highly ramified twist, we must return to the questionof whether L(s,n x n') can be made entire. In analysing F-functions via theLanglands-Shahidi method, the poles of the F-function are controlled by those of anEisenstein series. In general, the inducing data for the Eisenstein series must satisfya type of self-contragredience for there to be poles. The important observation ofKim is that one can use a highly ramified twist to destroy this self-contragredienceat one place, which suffices, and hence eliminate poles. The precise condition willdepend on the individual construction. A more detailed explanation of this can befound in Shahidi's article [21]. This completes Step 2 above.4. New examples of functorialityNow take k to be a number field. There has been much progress recently inutilizing the method described above to establish global liftings from split groupsH over k to an appropriate GL n . Among them are the following.1. Classical groups. Take H to be a split classical group over k, more specifically,the split form of either S0 2 „ + i, Sp 2 „, or S0 2 „. The the F-groups L H arethen Sp 2n (C), S0 2 „ + i(C), or S0 2 „(C) and there are natural embeddings into thegeneral linear group GL 2 „(C), GL 2 „+i(C), or GL 2 „(C) respectively. Associated toeach there should be a lifting of admissible or automorphic representations from