# Locally profinite group

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.

## Examples

Important examples of locally profinite groups come from algebraic number theory. Let *F* be a non-archimedean local field. Then both *F* and are locally profinite. More generally, the matrix ring and the general linear group 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).

## Representations of a locally profinite group

Let *G* be a locally profinite group. Then a group homomorphism is continuous if and only if it has open kernel.

Let be a complex representation of *G*.[1] is said to be smooth if *V* is a union of where *K* runs over all open compact subgroups *K*. is said to be admissible if it is smooth and 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 carries the action of *G* given by . In general, is not smooth. Thus, we set where is acting through and set . The smooth representation is then called the contragredient or smooth dual of .

The contravariant functor

from the category of smooth representations of *G* to itself is exact. Moreover, the following are equivalent.

- is admissible.
- is admissible.[2]
- The canonical
*G*-module map is an isomorphism.

When is admissible, is irreducible if and only if is irreducible.

The countability assumption at the beginning is really necessary, for there exists a locally profinite group that admits an irreducible smooth representation such that is not irreducible.

## Hecke algebra of a locally profinite group

Let be a unimodular locally profinite group such that is at most countable for all open compact subgroups *K*, and a left Haar measure on . Let denote the space of locally constant functions on with compact support. With the multiplicative structure given by

becomes not necessarily unital associative -algebra. 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. Indeed, one has the following: given a smooth representation of *G*, we define a new action on *V*:

Thus, we have the functor from the category of smooth representations of to the category of non-degenerate -modules. Here, "non-degenerate" means . Then the fact is that the functor is an equivalence.[3]

## Notes

- We do not put a topology on
*V*; so there is no topological condition on the representation. - Blondel, Corollary 2.8.
- Blondel, Proposition 2.16.

## References

- Corinne Blondel, Basic representation theory of reductive p-adic groups
- Bushnell, Colin J.; Henniart, Guy (2006),
*The local Langlands conjecture for GL(2)*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],**335**, Berlin, New York: Springer-Verlag, doi:10.1007/3-540-31511-X, ISBN 978-3-540-31486-8, MR 2234120 - Milne, J.S. (1988),
*Canonical models of (mixed) Shimura varieties and automorphic vector bundles*, MR 1044823