Whittaker spaces for reducible unitary principal series representations of $\widetilde{SL_2(F)}$

TL;DR

This paper computes the dimensions of Whittaker functional spaces for reducible unitary principal series representations of the n-fold cover of SL_2(F), analyzing how these dimensions vary with different characters and covers.

Contribution

It provides explicit calculations of Whittaker space dimensions for reducible genuine principal series of the metaplectic cover of SL_2(F), including ramified and unramified cases.

Findings

Dimensions of Whittaker spaces are explicitly computed.

Dimensions vary with the Whittaker character modification.

Action of the twisted Kazhdan-Patterson cover on summands is determined.

Abstract

Let $F$ be a $p$-adic field containing the full group of $n^{th}$ roots of 1 and let $ \widetilde{SL_2(F)}$ be the $n$-fold cover of $SL_2(F)$ constructed by Kubota. 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 $\widetilde{SL_2(F)}$. 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 $GL_2(F)$ on the two summands. We emphasize that our main results addresses both ramified and unramified representations and do not rely on the assumption that the cover is tame.