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

Zbl 0809.11032
Full Text:

References:

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