# Spectral Decomposition of a Covering of \(GL(r)\): the Borel case

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*Heng Sun*

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 \(\lambda\).