Unramified Whittaker functions on the metaplectic group

Abstract:

Kazhdan (unpublished), Shintani [Sh] and Casselman and Shalika [CS] computed explicitly the unramified Whittaker function of a quasisplit $p$-adic group. This is the main local ingredient used in the Rankin-Selberg-Shimura method, which yielded interesting results in the study of Euler products such as $L(s,\pi \otimes \pi ’)$ by Jacquet and Shalika [JS] (here $\pi ,\pi ’$ are cuspidal $GL(n,{A_F})$-modules), and $L(s,\pi ,r)$ by [F] (here $\pi$ is a cuspidal $GL(n,{A_E})$-module, $E$ is a quadratic extension of the global field $F$, and $r$ is the twisted tensor representation of the dual group of $\operatorname {Res}_{E/F}GL(n))$. Our purpose here is to generalize Shintani’s computation [Sh] from the context of $GL(n)$ to that of the metaplectic $r$-fold covering group $\tilde G$ of $GL(n)$ (see $[{\mathbf {F’}},{\mathbf {FK}}]$).