In mathematics, a locally profinite group is a hausdorff topological group in which every neighborhood of the identity element contains a compact open subgroup. Equivalently, a locally profinite group is a topological group that is hausdorff locally compact and totally disconnected. Moreover, a locally profinite group is compact if and only if it is profinite; this explains the terminology. Basic examples of locally profinite groups are discrete groups and p-adic Lie group. Non-examples are real Lie groups which have no small subgroup property.
In a locally profinite group, a closed subgroup is locally profinite, and every compact subgroup is contained in an open compact subgroup.
Important examples of locally profinite groups come from algebraic number theory. Let F be a non-archimedean local field. Then both F and
F
×
are locally profinite. More generally, the matrix ring
M
n
(
F
)
and the general linear group
GL
n
(
F
)
are locally profinite. Another example of a locally profinite group is the absolute Weil group of a non-archimedean local field: this is in contrast to the fact that the absolute Galois group of such is profinite (in particular compact).
Let G be a locally profinite group. Then a group homomorphism
ψ
:
G
→
C
×
is continuous if and only if it has open kernel.
Let
(
ρ
,
V
)
be a complex representation of G.
ρ
is said to be smooth if V is a union of
V
K
where K runs over all open compact subgroups K.
ρ
is said to be admissible if it is smooth and
V
K
is finite-dimensional for any open compact subgroup K.
We now make a blanket assumption that
G
/
K
is at most countable for all open compact subgroups K.
The dual space
V
∗
carries the action
ρ
∗
of G given by
⟨
ρ
∗
(
g
)
α
,
v
⟩
=
⟨
α
,
ρ
∗
(
g
−
1
)
v
⟩
. In general,
ρ
∗
is not smooth. Thus, we set
V
~
=
⋃
K
(
V
∗
)
K
where
K
is acting through
ρ
∗
and set
ρ
~
=
ρ
∗
. The smooth representation
(
ρ
~
,
V
~
)
is then called the contragredient or smooth dual of
(
ρ
,
V
)
.
The contravariant functor
(
ρ
,
V
)
↦
(
ρ
~
,
V
~
)
from the category of smooth representations of G to itself is exact. Moreover, the following are equivalent.
ρ
is admissible.
ρ
~
is admissible.
The canonical G-module map
ρ
→
ρ
~
~
is an isomorphism.
When
ρ
is admissible,
ρ
is irreducible if and only if
ρ
~
is irreducible.
The countability assumption at the beginning is really necessary, for there exists a locally profinite group that admits an irreducible smooth representation
ρ
such that
ρ
~
is not irreducible.
Let
G
be a unimodular locally profinite group such that
G
/
K
is at most countable for all open compact subgroups K, and
ρ
a left Haar measure on
G
. Let
C
c
∞
(
G
)
denote the space of locally constant functions on
G
with compact support. With the multiplicative structure given by
(
f
∗
h
)
(
x
)
=
∫
G
f
(
g
)
h
(
g
−
1
x
)
d
μ
(
g
)
C
c
∞
(
G
)
becomes not necessarily unital associative
C
-algebra. It is called the Hecke algebra of G and is denoted by
H
(
G
)
. The algebra plays an important role in the study of smooth representations of locally profinite groups. Indeed, one has the following: given a smooth representation
(
ρ
,
V
)
of G, we define a new action on V:
ρ
(
f
)
=
∫
G
f
(
g
)
ρ
(
g
)
d
μ
(
g
)
.
Thus, we have the functor
ρ
↦
ρ
from the category of smooth representations of
G
to the category of non-degenerate
H
(
G
)
-modules. Here, "non-degenerate" means
ρ
(
H
(
G
)
)
V
=
V
. Then the fact is that the functor is an equivalence.
