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Relative trace formula and simple algebras. (English) Zbl 0623.22016

Let E/F be a quadratic extension of global fields, G a reductive F-group. A cuspidal \(G({\mathbb{A}}_ E)\)-module is said to be distinguished if it has an integrable function whose integral over \(G(F)/G({\mathbb{A}}_ F)\) is nonzero. Let G be the multiplicative group of a central simple algebra divided by centre and G’ be GL(n) divided by centre. Suppose the \(G({\mathbb{A}}_ E)\)-module \(\pi\) corresponds by Deligne-Kazhdan correspondence to the \(G'({\mathbb{A}}_ E)\)-module \(\pi\) ’, by a good choice of the function f defining the kernel of the right regular representation the author obtains a simple relative trace formula which he uses to prove under certain ramification assumptions that \(\pi\) is distinguished if and only if \(\pi\) ’ is distinguished.
Reviewer: K.Lai

MSC:

22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
11R39 Langlands-Weil conjectures, nonabelian class field theory
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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