WHITTAKER SPACES FOR REDUCIBLE UNITARY PRINCIPAL SERIES REPRESENTATIONS OF SL₂(F)

Let F be a p-adic field containing the full group of n^th roots of 1 and let G͂ be the n-fold cover of SL₂(F) constructed by Kubota [On automorphic functions and the reciprocity law in a number field, Kyoto University, Tokyo, 1969]. In this paper we compute the dimension of the space of Whittaker functionals of the two irreducible summands inside a reducible unitary genuine principal series representation of G͂. We also show how these dimensions change when the Whittaker character is modified. As an application we determine the action of the twisted Kazhdan-Patterson n-fold cover of GL₂(F) on the two summands. We emphasize that our main results address both ramified and unramified representations and do not rely on the assumption that the cover is tame.