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On local \(L\)-functions and normalized intertwining operators. (English) Zbl 1096.11019

In a tour de force at the beginning of the 21st century Henry Kim makes explicit the \(L\)-functions which appear in the constant term of the Eisenstein series in the cases \(B_n-1\), \(C_n-1\), \(D_n-1\), \(D_n-2\), \(D_n-3\), \(F_4-1\), \(F_4-2\), (xviii),(xxii) from Zbl 0231.20016, \(E_6-1\), \(E_6-2\), (x), (xxiv), \(E_7-1\), \(E_7-2\), \(E_7-3\), \(E_7-4\), (xi), (xxvi), (xxx), \(E_8-1\), \(E_8-2\), \(E_8-3\), \(E_8-4\), \(E_8-5\), (xiii), (xxviii), (xxxii), proves Shahidi’s holomorphy conjecture (7.1 in Zbl 0780.22005) for \(E\)-type groups except for the cases \(E_7-3\), \(E_8-3\), \(E_8-4\), (xxviii) and that the normalized local intertwining operators are holomorphic and non-vanishing for Re(s)\(\geq \frac{1}{2}\) except for the cases \(B_n-1\), \(D_n-1\), (xxx), \(E_8-4\), (xxxii), (xviii), (xxii), (xxiv), \(E_7-3\), (xxvi), \(E_8-3\), (xxviii).

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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