4The Langlands correspondence
IV Topics in Number Theory
4.2 Hecke algebras
Let G, K be as before.
Notation.
We write
C
∞
c
(
G
) for the vector space of locally constant functions
f : G → C of compact support.
Definition (Hecke algebra). The Hecke algebra is defined to be
H(G, K) = {ϕ ∈ C
∞
c
(G) : ϕ(kgk
0
) = ϕ(g) for all k, k
0
∈ K}.
This is spanned by the characteristic functions of double cosets KgK.
This algebra comes with a product called the convolution product. To define
this, we need the Haar measure on
G
. This is a functional
C
∞
c
(
G
)
→ C
, written
f 7→
Z
G
f(g) dµ(g),
that is invariant under left translation, i.e. for all h ∈ G,we have
Z
f(hg) dµ(g) =
Z
f(g) dµ(g).
To construct the Haar measure, we take
µ
(1
K
) = 1. Then if
K
0
⊆ K
is an open
subgroup, then it is of finite index, and since we want
µ
(1
xK
0
) =
µ
(1
K
0
), we
must have
µ(1
K
0
) =
1
(K : K
0
)
.
We then set
µ
(1
xK
0
) =
µ
(1
K
0
) for any
x ∈ G
, and since these form a basis of the
topology, this defines µ.
Definition (Convolution product). The convolution product on H(G, K) is
(ϕ ∗ ϕ
0
)(g) =
Z
G
ϕ(x)ϕ
0
(x
−1
g) dµ(x).
Observe that this integral is actually a finite sum.
It is an exercise to check that this is a C-algebra with unit
e
K
=
1
µ(K)
1
K
,
Now if (
π, V
) is a smooth representation, then for all
v ∈ V
and
ϕ ∈ H
(
G, K
),
consider the expression
π(ϕ)v =
Z
G
ϕ(g)π(g)v dµ(g).
Note that since the stabilizer of
v
is open, the integral is actually a finite sum,
so we can make sense of it. One then sees that
π(ϕ)π(ϕ
0
) = π(ϕ ∗ ϕ
0
).
This would imply
V
is a
H
(
G, K
)-module, if the unit acted appropriately. It
doesn’t, however, since in fact
π
(
ϕ
) always maps into
V
K
. Indeed, if
k ∈ K
,
then
π(k)π(ϕ)v =
Z
G
ϕ(g)π(kg)v dµ(g) =
Z
ϕ(g)π(g) dµ(g) = π(ϕ)v,
using that ϕ(g) = ϕ(k
−1
g) and dµ(g) = dµ(k
−1
g).
So our best hope is that
V
K
is an
H
(
G, K
)-module, and one easily checks
that
π
(
e
K
) indeed acts as the identity. We also have a canonical projection
π(e
K
) : V V
K
.
In good situations, this Hecke module determines V .
Proposition.
There is a bijection between isomorphism classes of irreducible ad-
missible (
π, V
) with
V
K
6
= 0 and isomorphism classes of simple finite-dimensional
H(G, K)-modules, which sends (π, V ) to V
K
with the action we described.
If we replace K by a smaller subgroup K
0
⊆ K, then we have an inclusion
H(G, K) → H(G, K
0
),
which does not take
e
K
to
e
K
0
. We can then form the union of all of these, and
let
H(G) = lim
−→
K
H(G, K)
which is an algebra without unit. Heuristically, the unit should be the delta
function concentrated at the identity, but that is not a function.
This
H
(
G
) acts on any smooth representation, and we get an equivalence of
categories
smooth
G-representations
←→
non-degenerate
H(G)-modules
.
The non-degeneracy condition is V = H(G)V .
Note that if
ϕ ∈ H
(
G
) and (
π, V
) is admissible, then the
rank π
(
ϕ
)
< ∞
,
using that
V
K
is finite-dimensional. So the trace is well-defined. The character
of (π, V ) is then the map
ϕ 7→ tr π(ϕ)
In this sense, admissible representations have traces.