Let
F
F
be a non-Archimedean local field (of characteristic
0
0
or
p
p
) with finite residue field of characteristic
p
p
. An irreducible smooth representation of the Weil group of
F
F
is called essentially tame if its restriction to wild inertia is a sum of characters. The set of isomorphism classes of irreducible, essentially tame representations of dimension
n
n
is denoted
G
n
e
t
(
F
)
\mathcal {G}^\mathrm {et}_n(F)
. The Langlands correspondence induces a bijection of
G
n
e
t
(
F
)
\mathcal {G}^\mathrm {et}_n(F)
with a certain set
A
n
e
t
(
F
)
\mathcal {A}^\mathrm {et}_n(F)
of irreducible supercuspidal representations of
G
L
n
(
F
)
\mathrm {GL}_n(F)
. We consider the set
P
n
(
F
)
P_n(F)
of isomorphism classes of certain pairs
(
E
/
F
,
ξ
)
(E/F,\xi )
, called “admissible”, consisting of a tamely ramified field extension
E
/
F
E/F
of degree
n
n
and a quasicharacter
ξ
\xi
of
E
×
E^\times
. There is an obvious bijection of
P
n
(
F
)
P_n(F)
with
G
n
e
t
(
F
)
\mathcal {G}^\mathrm {et}_n(F)
. Using the classification of supercuspidal representations and tame lifting, we construct directly a canonical bijection of
P
n
(
F
)
P_n(F)
with
A
n
e
t
(
F
)
\mathcal {A}^\mathrm {et}_n(F)
, generalizing and simplifying a construction of Howe (1977). Together, these maps give a canonical bijection of
G
n
e
t
(
F
)
\mathcal {G}^\mathrm {et}_n(F)
with
A
n
e
t
(
F
)
\mathcal {A}^\mathrm {et}_n(F)
. We show that one obtains the Langlands correspondence by composing the map
P
n
(
F
)
→
A
n
e
t
(
F
)
P_n(F) \to \mathcal {A}^\mathrm {et}_n(F)
with a permutation of
P
n
(
F
)
P_n(F)
of the form
(
E
/
F
,
ξ
)
↦
(
E
/
F
,
μ
ξ
ξ
)
(E/F,\xi )\mapsto (E/F,\mu _\xi \xi )
, where
μ
ξ
\mu _\xi
is a tamely ramified character of
E
×
E^\times
depending on
ξ
\xi
. This answers a question of Moy (1986). We calculate the character
μ
ξ
\mu _\xi
in the case where
E
/
F
E/F
is totally ramified of odd degree.