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
$\lambda$.

Readership

Graduate students and research mathematicians interested in number
theory, and the Langlands program.