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Characterization of the local Langlands conjecture by \(\varepsilon\)-factors for pairs. (Caractérisation de la correspondance de Langlands locale par les facteurs \(\varepsilon\) de paires.) (French) Zbl 0810.11069

Let \(F\) be a non-archimedean local field with finite residue field. In the equal characteristic case G. Laumon, M. Rapoport and U. Stuhler [\(D\)-elliptic sheaves and the Langlands correspondence, Invent. Math. 113, 217-338 (1993; Zbl 0809.11032), Theorem 15.7] construct bijections \({\mathcal A}_ F^ 0(n) \leftrightarrow {\mathcal G}_ F^ 0 (n)\), \(\pi\mapsto \sigma_ \pi\) between irreducible supercuspidal representations of \(\text{GL}_ n(F)\) with finite central character and \(n\)-dimensional irreducible representations of the Galois group \(G_ F= \text{Gal} (\overline{F}/ F)\) resp. for all \(n\geq 1\). The main properties of these bijections are \[ L(\pi\times \pi',s)= L(\sigma_ \pi\otimes \sigma_{\pi'},s), \qquad \varepsilon(\pi\times \pi',s,\psi)= \varepsilon(\sigma_ \pi\otimes \sigma_{\pi'}, s,\psi), \] means that \(L\)- and \(\varepsilon\)-factors of pairs are preserved. In the paper under review the author proves the following uniqueness statement, which is stated in terms of the map \(\sigma\mapsto \pi_ \sigma\) in opposite direction:
Theorem 1.2: Let \(n\geq 2\) be an integer and let \(\sigma\in {\mathcal G}_ F^ 0 (n)\), \(\pi\in {\mathcal A}_ F^ 0 (n)\) have the property \[ \varepsilon(\pi\times \pi_ \tau, s,\psi)= \varepsilon(\sigma\otimes \tau,s,\psi) \] for all \(\tau\in {\mathcal G}_ F^ 0(r)\), \(r=1,\dots, n-1\). Then \(\pi\cong \pi_ \sigma\).
The proof is by reduction to
Theorem 1.1: Let \(n\geq 2\) be an integer and let \(\pi\), \(\pi'\) be irreducible generic representations of \(\text{GL}_ n(F)\). Then the equality of “Gamma factors” \(\gamma(\pi\times \tau,s,\psi)= \gamma(\pi'\times \tau, s,\psi)\) for all irreducible generic representations \(\tau\) of \(\text{GL}_{n-1}(F)\) implies \(\pi\cong \pi'\).
The last theorem does not depend on \(\text{char}(F)\) but uses the fact that irreducible generic representations \(\pi\) have a Whittaker model \({\mathcal W} (\pi,\psi)\) and that certain complex valued smooth functions \(H\) on \(\text{GL}_ n(F)\) vanish if the integration of \(H\) against Whittaker functions \(W\in {\mathcal W}(\pi, \psi)\) is always trivial (due to Jacquet, Piatetski-Shapiro, Shalika).
We note a misprint on p. 346, line 4, which sould read: \((\rho(g) W)^ \sim (h)= (\rho(g) W)(w_ n^ t h^{-1})\).
Reviewer: E.-W.Zink (Berlin)

MSC:

11S37 Langlands-Weil conjectures, nonabelian class field theory
22E50 Representations of Lie and linear algebraic groups over local fields

Citations:

Zbl 0809.11032
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References:

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