International Congress of Mathematicians

Converse Theorems, Functoriality, and Applications 125H (A) to the appropriate GLjv(A). The first lifting that resulted from the combinationof the Converse Theorem and the Langlands-Shahidi method of controllingautomorphic F-functions was the weak lift for generic cuspidal representations fromS0 2 „ + i to GL 2 „ over a number field k obtained with Kim and Shahidi [2]. We cannow extend this to the following result.Theorem. [2, 3] Let H be a split classical group over k as above and n aglobally generic cuspidal representation o/H(A). Then there exists an automorphicrepresentation n of GLJV (A) for the appropriate N such that n^ is the local Langlandslift of n v for all archimedean places v and almost all non-archimedean placesv where n v is unramified.In these examples the local Langlands correspondence is not understood at theplaces v where n v is ramified and so we must use the technique of multiplicativityand stability of the local 7-factors as outlined in Section 3. Multiplicativity hasbeen established in generality by Shahidi [19] and in our first paper [2] we reliedon the stability of 7-factors for S0 2 „ + i from [5]. Recently Shahidi has establishedan expression for his local coefficients as Mellin transforms of Bessel functions insome generality, and in particular in the cases at hand one can combine this withthe results of [5] to obtain the necessary stability in the other cases, leading to theextension of the lifting to the other split classical groups [3].2. Tensor products. Let H = GL ro x GL„. Then L H = GL ro (C) x GL„(C).Then there is a natural simple tensor product map from GL TO (C) x GL n (C) toGL TOn (C). The associated functoriality from GL„ x GL TO to GL TO „ is the tensorproduct lifting. Now the associated local lifting is understood in principle since thelocal Langlands conjecture for GL„ has been solved. The question of global functorialityhas been recently solved in the cases of GL 2 x GL 2 to GL 4 by Ramakrishnan[17] and GL 2 x GL 3 to GL 6 by Kim and Shahidi [15, 16].Theorem. [17, 15] Let m be a cuspidal representation o/GL 2 (A) and 7r 2 acuspidal representation o/GL 2 (A) (respectively GL 3 (A)J. Then there is an automorphicrepresentation II o/GL 4 (A) (respectively GL 6 (A)) such that Il v is the localtensor product lift of m, v x 7r 2jt , at all places v.In both cases the authors are able to characterize when the lift is cuspidal.In the case of Ramakrishnan [17] n = m x 7r 2 with each 7r, cuspidal representationof GL 2 (A) and II is to be an automorphic representation of GL 4 (A). Toapply the Converse Theorem Ramakrishnan needs to control the analytic propertiesof L(s,U x n') for n' cuspidal representations of GLi(A) and GL 2 (A), that is,the Rankin triple product F-functions L(s,II x n') = L(s,ni x 7r 2 x n'). This hewas able to do using a combination of results on the integral representation for thisF-function due to Garrett, Rallis and Piatetski-Shapiro, and Ikeda and the work ofShahidi on the Langlands-Shahidi method.In the case of Kim and Shahidi [15, 16] 7r 2 is a cuspidal representation ofGL 3 (A). Since the lifted representation II is to be an automorphic representationof GL 6 (A), to apply the Converse Theorem they must control the analytic propertiesof L(s, II x n') = L(s, 7Ti x 7T 2 x n') where now n' must run over appropriate cuspidalrepresentations of GL TO (A) with m = 1,2,3,4. The control of these triple productsis an application of the Langlands-Shahidi method of analysing F-functions and