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Local coefficients as Artin factors for real groups. (English) Zbl 0674.10027

This paper deals with the local constant that appears in the functional equation of the automorphic L-function of Langlands. Let G be the real points of a quasisplit reductive algebraic group over \({\mathbb{R}}\), \(P=MAN\) a standard parabolic of G, \(\sigma\) a generic representation of M; the local constant C(\(\sigma)\) (ignoring other data) is defined by the Schiffman-Knapp-Stein intertwining operator and Whittaker function. The author proves that \[ C(\sigma)=(i)^{2m+p}\prod^{n}_{i=1}\epsilon (a_ is,r_ i)\frac{L(1-a_ is,\tilde r_ i)}{L(a_ is,r_ i)} \] where L is the Artin L-function and \(\epsilon\) is the root number. As an application the author gives a new proof of the function equation of the Rankin-Selberg L-function attached to pairs of cusp forms.

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E30 Analysis on real and complex Lie groups
11R39 Langlands-Weil conjectures, nonabelian class field theory
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
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