# Non-Generic Unramified Representations in Metaplectic Covering Groups

## TL;DR

This paper investigates conditions under which unramified representations of metaplectic covering groups lack nonzero Whittaker functions, proposing a conjecture and proving it for specific groups.

## Contribution

It formulates a general conjecture on unramified characters leading to no Whittaker functions and proves it for certain groups like $GL_n^{(r)}$ and $G_2^{(r)}$.

## Key findings

- Conjecture holds for $GL_n^{(r)}$ with $r extgreater n-1$
- Conjecture holds for $G_2^{(r)}$ when $r
e 2$
- Provides conditions for non-generic unramified representations

## Abstract

Let $G^{(r)}$ denote the metaplectic covering group of the linear algebraic group $G$. In this paper we study conditions on unramified representations of the group $G^{(r)}$ not to have a nonzero Whittaker function. We state a general Conjecture about the possible unramified characters $\chi$ such that the unramified sub-representation of $Ind_{B^{(r)}}^{G^{(r)}}\chi\delta_B^{1/2}$ will have no nonzero Whittaker function. We prove this Conjecture for the groups $GL_n^{(r)}$ with $r\ge n-1$, and for the exceptional groups $G_2^{(r)}$ when $r\ne 2$.