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A lemma on highly ramified \(\epsilon\)-factors. (English) Zbl 0541.12010
A standard property of the \(\epsilon\)-factor attached to a representation of the Weil group W of a local field F is the following: given \(\sigma_ 1\) and \(\sigma_ 2\) with the same determinant, let \(\sigma_ 1\otimes \chi\) and \(\sigma_ 2\otimes \chi\) be the tensor products with a character \(\chi\) of \(F^{\times}\) (identified with a character of W); then, if the conductor of \(\chi\) is sufficiently big, the \(\epsilon\)- factors for \(\sigma_ 1\otimes \chi\) and \(\sigma_ 2\otimes \chi\) coincide. Similarly, to every pair of irreducible representations of general linear groups, one can attach an \(\epsilon\)-factor: it is established that it has the corresponding property.

MSC:
11S37 Langlands-Weil conjectures, nonabelian class field theory
22E50 Representations of Lie and linear algebraic groups over local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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References:
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