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

Heng Sun University of Toronto, Toronto, ON, Canada
Available Formats:
Electronic ISBN: 978-1-4704-0336-2
Product Code: MEMO/156/743.E
63 pp
List Price: $51.00 MAA Member Price:$45.90
AMS Member Price: $30.60 Click above image for expanded view Spectral Decomposition of a Covering of$GL(r)$: the Borel case Heng Sun University of Toronto, Toronto, ON, Canada Available Formats:  Electronic ISBN: 978-1-4704-0336-2 Product Code: MEMO/156/743.E 63 pp  List Price:$51.00 MAA Member Price: $45.90 AMS Member Price:$30.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 1562002
MSC: Primary 11; Secondary 22;

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 $\lambda$.

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

• Chapters
• Introduction
• 1. Preliminaries
• 2. Local intertwining operators
• 3. Spectrum associated with the diagonal subgroup
• 4. Contour integration (after MW)
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Volume: 1562002
MSC: Primary 11; Secondary 22;

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 $\lambda$.