Shalika newforms for GL(n)

Abstract

Let (pi,V) be a generic irreducible representation of a general linear group over a p-adic field. Jacquet, Piatetski-Shapiro, and Shalika gave an open compact subgroup K, so that the subspace V^K consisting of v in V fixed by K is one-dimensional. If pi has a Shalika model Lambda, then we call vectors in Lambda(V) the Shalika forms of pi, and those in Lambda(V^{K}) the Shalika newforms. In this article, we give a method to determine all values of the Shalika newforms on the mirabolic subgroup in the case where pi is supercuspidal. Using this result, we give another Shalika form with nice properties, which is not fixed by K in the case where the character defining the Shalika model is ramified.