Spectral Decomposition of a Covering of GL(r): The Borel Case

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 $\lambda$.