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 the p-adic Lie groups.
Non-examples are real Lie groups, which have the 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.
More generally, the matrix ring
and the general linear group
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.
is continuous if and only if it has open kernel.
where K runs over all open compact subgroups K.
is finite-dimensional for any open compact subgroup K. We now make a blanket assumption that
is at most countable for all open compact subgroups K. The dual space
is then called the contragredient or smooth dual of
The contravariant functor from the category of smooth representations of G to itself is exact.
The countability assumption at the beginning is really necessary, for there exists a locally profinite group that admits an irreducible smooth representation
be a unimodular locally profinite group such that
is at most countable for all open compact subgroups K, and
a left Haar measure on
denote the space of locally constant functions on
becomes not necessarily unital associative
It is called the Hecke algebra of G and is denoted by
The algebra plays an important role in the study of smooth representations of locally profinite groups.
of G, we define a new action on V: Thus, we have the functor
from the category of smooth representations of
Here, "non-degenerate" means
Then the fact is that the functor is an equivalence.