Integral Representation of Whittaker Functions

Here W is in the Whittaker model W(π, ψ) of a unitary generic representation π of GL(n, F) and W ′ in W(π ′, ψ) where π ′ is a unitary generic representation of GL(n − 1, F) (see below for unexplained notations). One of the difficulties of the theory is that the representations π and π ′ need not be tempered. Thus one is led to consider holomorphic fiber bundles of representations (πu) and (π ′ u′), for instance, non-unitary principal series. Correspondingly, the functions W = Wu and W ′ = W ′ u′ depend also on u and u′. They are associated with sections of the fiberbundle of representations at hand. Rather than standard sections (with a constant restriction to the maximal compact subgroup), we consider convolutions of standard sections with smooth functions of compact support. It is difficult to prove that the integrals (s, Wu, W ′ u′) are meromorphic functions of (s, u, u ′). The elaborate technics of [JS] were designed to go around this difficulty. In particular, there, the analytic properties of the integral as functions of s, as well as their functional equations, were found to be equivalent to a family of identities (depending on (u, u′)) which were then established by analytic continuation with respect to the parameters (u, u′). In the present note, we first find integral representations for Wu and W ′ u′ which converge for all values of the parameters (u, u ′). In particular, it is easy to obtain estimates for Wu and W ′ u′ which are uniform in (u, u ′). Then, using these integral representations, we show that the integrals at hand are meromorphic functions of (s, u, u′). This being established, one can use the methods of [JS] to prove the functional equations. We do not repeat this step here because it is now much easier: one first proves that the integrals are meromorphic functions