On the Langlands-Kazhdan correspondence. (Q922591)

Let \(F\) be a non-archimedean local field, \(p\) the characteristic of its residue field. The subject of this paper is the correspondence between irreducible \(p\)-dimensional complex representations of the Galois group \(\Gamma_ F\) of \(F\) and supercuspidal representations of \(\text{GL}(p,F)\). Let \(\rho\) be an irreducible representation of \(\Gamma_ F\) induced by a character of \(\Gamma_ L\), where \(L\) is a ramified extension of \(F\) of degree \(p\) (the case where \(L/F\) is unramified is not treated here). In this case there is a corresponding supercuspidal representation \(\pi\) of \(\text{GL}(p,F)\), after Kazhdan [cf. \textit{G. Henniart}, Ann. Math. (2) 123, 145--203 (1986; Zbl 0588.12010)]. This representation \(\pi\) is identified, in the present paper, with one of the supercuspidal representations constructed by Carayol. The proof is given by showing the equality of \(\epsilon\)-factors, using tame base change. It is assumed that \(\rho\) is determined by the \(\epsilon\)-factors \(\epsilon\) (\(\rho\otimes \sigma)\), \(\sigma\) running through the irreducible representations of \(W_ F\) of degree \(<p\). The case of a primitive representation of \(\Gamma_ F\) is also treated.