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 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. 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.  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. 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. Notes

References
Corinne Blondel, Basic representation theory of reductive p-adic groups

Topological groups