Spectral Decomposition of a Covering of GL(r)

Let $F$ be a number field and ${/bf A}$ the ring of adeles over $F$. Suppose $/overline{G({/bf A})}$ is a metaplectic cover of $G({/bf A})=GL(r,{/bf A})$ which is given by the $n$-th Hilbert symbol on ${/bf A}$. According to Langlands' theory of Eisenstein series, the decomposition of the right regular representation on $L^2/left(G(F)/backslash/overline{G({/bf A})}/right)$ can be understood in terms of the residual spectrum of Eisenstein series associated with cuspidal data on standard Levi subgroups $/overline{M}$. Under an assumption on the base field $F$, this paper calculates the spectrum associated with the diagonal subgroup $/overline{T}$. Specifically, the diagonal residual spectrum is at the point $/lambda=((r-1)/2n,(r-3)/2n,/cdots,(1-r)/2n)$.Each irreducible summand of the corresponding representation is the Langlands quotient of the space induced from an irreducible automorphic representation of $/overline{T}$, which is invariant under symmetric group $/mathfrak{S}_r$, twisted by an unramified character of $/overline{T}$ whose exponent is given by $/lambda$.