The local Jacquet-Langlands correspondence via Fourier analysis
نویسندگان
چکیده
Let F be a locally compact non-Archimedean field, and let B/F be a division algebra of dimension 4. The JacquetLanglands correspondence provides a bijection between smooth irreducible representations π′ of B× of dimension > 1 and irreducible cuspidal representations of GL2(F ). We present a new construction of this bijection in which the preservation of epsilon factors is automatic. This is done by constructing a family of pairs (L, ρ), where L ⊂ M2(F ) × B is an order and ρ is a finitedimensional representation of a certain subgroup of GL2(F )×B× containing L×. Let π ⊗ π′ be an irreducible representation of GL2(F )×B×; we show that π⊗π′ contains such a ρ if and only if π is cuspidal and corresponds to π̌′ under Jacquet-Langlands, and also that every π and π′ arises this way. The agreement of epsilon factors is reduced to a Fourier-analytic calculation on a finite ring quotient of L.
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