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Spectral Decomposition of a Covering of $$GL(r)$$: the Borel case
Heng Sun, University of Toronto, ON, Canada
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Memoirs of the American Mathematical Society
2002; 63 pp; softcover
Volume: 156
ISBN-10: 0-8218-2775-8
ISBN-13: 978-0-8218-2775-8
List Price: US$51 Individual Members: US$30.60
Institutional Members: US\$40.80
Order Code: MEMO/156/743

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 $$\lambda$$.