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Spectral Decomposition of a Covering of $GL(r)$: the Borel case
 
Heng Sun University of Toronto, Toronto, ON, Canada
Front Cover for Spectral Decomposition of a Covering of $GL(r)$: the Borel case
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
Front Cover for Spectral Decomposition of a Covering of $GL(r)$: the Borel case
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  • Front Cover for Spectral Decomposition of a Covering of $GL(r)$: the Borel case
  • Back Cover for Spectral Decomposition of a Covering of $GL(r)$: the Borel case
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 \(\lambda\).

    Readership

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

  • Table of Contents
     
     
    • 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 \(\lambda\).

Readership

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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