4The Langlands correspondence
IV Topics in Number Theory
4.1 Representations of groups
The adjectives we need are fairly general. The group
GL
n
(
F
) contains a profinite
open subgroup
GL
n
(
O
F
). The general theory applies to any topological group
with a profinite open subgroup K, with G/K countable.
Definition
(Smooth representation)
.
A smooth representation of
G
is a contin-
uous representation of
G
over
C
, where
C
is given the discrete topology. That
is, it is a pair (
π, V
) where
V
is a complex vector space and
π
:
G → GL
C
(
V
) a
homomorphism such that for every v ∈ V , the stabilizer of v in G is open.
Note that we can replace
C
with any field, but we like
C
. Typically,
V
is an
infinite-dimensional vector space. To retain some sanity, we often desire the
following property:
Definition
(Admissible representation)
.
We say (
π, V
) is admissible if for every
open compact subgroup
K ⊆ G
the fixed set
V
K
=
{v ∈ V
:
π
(
g
)
v
=
v ∀g ∈ K}
is finite-dimensional.
Example.
Take
G
=
GL
2
(
F
). Then
P
1
(
F
) has a right action of
G
by linear
transformations. In fact, we can write P
1
(F ) as
P
1
(F ) =
∗ ∗
0 ∗
\G.
Let
V
be the space of all locally constant functions
f
:
P
1
(
F
)
→ C
. There are
lots of such functions, because
P
1
(
F
) is totally disconnected. However, since
P
1
(F ) is compact, each such function can only take finitely many values.
We let
π(g)f = (x 7→ f(xg)).
It is not difficult to see that this is an infinite-dimensional admissible representa-
tion.
Of course, any finite-dimensional representation is an example, but
GL
n
(
F
)
does not have very interesting finite-dimensional representations.
Proposition.
Let
G
=
GL
n
(
F
). If (
π, V
) is a smooth representation with
dim V < ∞, then
π = σ ◦ det
for some σ : F
×
→ GL
C
(V ).
So these are pretty boring.
Proof. If V =
L
d
i=1
Ce
i
, then
ker π =
d
\
i=1
(stabilizers of e
i
)
is open. It is also a normal subgroup, so
ker π ⊇ K
m
= {g ∈ GL
n
(O) : g ≡ I mod $
m
}
for some
m
, where
$
is a uniformizer of
F
. In particular,
ker π
contains
E
ij
(
x
)
for some
i 6
=
j
and
x
, which is the matrix that is the identity except at entry
(i, j), where it is x.
But since
ker π
is normal, conjugation by diagonal matrices shows that it
contains all
E
ij
(
x
) for all
x ∈ F
and
i 6
=
j
. For any field, these matrices generate
SL
n
(F ). So we are done.
So the interesting representations are all infinite-dimensional. Fortunately, a
lot of things true for finite-dimensional representations also hold for these. For
example,
Lemma
(Schur’s lemma)
.
Let (
π, V
) be an irreducible representation. Then
every endomorphism of V commuting with π is a scalar.
In particular, there exists ω
π
: Z(G) → C
×
such that
π(zg) = ω
π
(z)π(g)
for all z ∈ Z(G) and g ∈ G. This is called the central character.
At this point, we are already well-equipped to state a high-level description
of the local Langlands correspondence.
Theorem (Harris–Taylor, Henniart). There is a bijection
irreducible, admissible
representations of GL
n
(F )
←→
semi-simple n-dimensional
representations of L
F
.
In the next section, we will introduce the Hecke algebra, which allows us to
capture these scary infinite dimensional representations of
GL
n
(
F
) in terms of
something finite-dimensional.
Afterwards, we are going to state the Langlands classification of irreducible
admissible representations of
GL
n
(
F
). We can then state a detailed version of
the local Langlands correspondence in terms of this classification.