ON NON VANISHING OF I-FUNCTIONS BY FREYDOON SHAHIDI*

The nonvanishing of Hecke Z-functions at the line Re (s) = 1 has proved to be useful in the theory of uniform distribution of primes. One of the general-izations of this fact is due to H. Jacquet and J. A. Shalika [4], who proved the nonvanishing of the L-functions considered in [2]. The following theorem generalizes this result to the /.-functions attached to the pairs of cusp forms on GLn x GLm (cf. [3]). It appears to have an application in the classification of automorphic forms on GLn (communications with H. Jacquet and J. A. Shalika). Let F be a number field and denote by A its ring of adeles. Fix two posi-tive integers m and n. Let n and n ' be two cuspidal representations of GLn(A) and GLm(A). Fix a complex number s. Write n = ® v irv and n ' = ® v ^ » where nv and 7r'v denote the irth components of IT and n ' at each place v of F, respectively. Let S be the finite set of all ramified places, including the infinite ones. For every finite place v, H. Jacquet, I. I. Piatetski-Shapiro, and J. A. Shalika have defined (cf. [3]) a local L-function L(s, nv x TT'V). Let Ls(s,n x TT') = YiL(s> «v x K)-Put i = (- 1)1 / 2. Then we have THEOREM. LS( \ + it,-n x TT') = £ 0 for St G R. OUTLINE OF THE PROOF. The proof follows the general principle of applying Eisenstein series to L-functions which is due to R. P. Langlands [5] (same as in [4]). Put G = GLn + m and M = GLn x GLm. Consider M as a Levi factor of a maximal standard parabolic subgroup of G. Choose $ in the space of °7T = ir 0 7r', where 9r denotes the contragredient of it. Extend <p to <p, a function on G(A), as in [7]. Put *,W = 8j-1/2(rt?u), where P = MN, g = kp, p G P(A), and k G K. Here K = n ^ Kv is a maximal compact subgroup of G(A) such that Kv = G(Ov) for every finite v. Now se