The essentially tame local Langlands correspondence, II: totally ramified representations

TY - JOUR

T1 - The essentially tame local Langlands correspondence, II: totally ramified representations

AU - Bushnell, C J

AU - Henniart, G

PY - 2005/7

Y1 - 2005/7

N2 - Let F be a non-Archimedean local field. Let G(n)(et)(F) be the set of equivalence classes of irreducible, n-dimensional representations of the Weil group WF of F which are essentially tame. Let A(n)(et)(F) be the set of equivalence classes of irreducible, essentially tame, super-cuspidal representations of GL(n)(F). The Langlands correspondence induces a canonical bijection L : G(n)(et)(F) -> A(n)(et)(F). We continue the programme of describing this map in terms of explicit descriptions of the sets G(m)(et)(F) and A(n)(et)(F). These descriptions are in terms of admissible pairs (E/F,xi), consisting of a tamely ramified field extension E/F of degree n and a quasicharacter xi of E-x subject to certain technical conditions. If P-n(F) is the set of isomorphism classes of admissible pairs of degree n, we have explicit bijections P-n(F) congruent to G(n)(et)(F) and P-n(F) congruent to A(n)(et)(F). In an earlier paper we showed that, if sigma is an element of G(n)(et)(F) corresponds to an admissible pair (E/F,xi), then L(sigma) corresponds to the admissible pair (E/F,mu xi), for a certain tamely ramified character mu of E-x. In this paper, we determine the character mu when E/F is totally ramified

AB - Let F be a non-Archimedean local field. Let G(n)(et)(F) be the set of equivalence classes of irreducible, n-dimensional representations of the Weil group WF of F which are essentially tame. Let A(n)(et)(F) be the set of equivalence classes of irreducible, essentially tame, super-cuspidal representations of GL(n)(F). The Langlands correspondence induces a canonical bijection L : G(n)(et)(F) -> A(n)(et)(F). We continue the programme of describing this map in terms of explicit descriptions of the sets G(m)(et)(F) and A(n)(et)(F). These descriptions are in terms of admissible pairs (E/F,xi), consisting of a tamely ramified field extension E/F of degree n and a quasicharacter xi of E-x subject to certain technical conditions. If P-n(F) is the set of isomorphism classes of admissible pairs of degree n, we have explicit bijections P-n(F) congruent to G(n)(et)(F) and P-n(F) congruent to A(n)(et)(F). In an earlier paper we showed that, if sigma is an element of G(n)(et)(F) corresponds to an admissible pair (E/F,xi), then L(sigma) corresponds to the admissible pair (E/F,mu xi), for a certain tamely ramified character mu of E-x. In this paper, we determine the character mu when E/F is totally ramified

U2 - 10.1112/S0010437X05001363

DO - 10.1112/S0010437X05001363

M3 - Article

VL - 141

SP - 979

EP - 1011

JO - COMPOSITIO MATHEMATICA

JF - COMPOSITIO MATHEMATICA

IS - 4

ER -