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 formula_1 are locally profinite. More generally, the matrix ring formula_2 and the general linear group formula_3 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 formula_4 is continuous if and only if it has open kernel. 
Let formula_5 be a complex representation of "G". formula_6 is said to be smooth if "V" is a union of formula_7 where "K" runs over all open compact subgroups "K". formula_6 is said to be admissible if it is smooth and formula_7 is finite-dimensional for any open compact subgroup "K".
We now make a blanket assumption that formula_10 is at most countable for all open compact subgroups "K".
The dual space formula_11 carries the action formula_12 of "G" given by formula_13. In general, formula_12 is not smooth. Thus, we set formula_15 where formula_16 is acting through formula_12 and set formula_18. The smooth representation formula_19 is then called the contragredient or smooth dual of formula_5.
The contravariant functor
from the category of smooth representations of "G" to itself is exact. Moreover, the following are equivalent.
When formula_6 is admissible, formula_6 is irreducible if and only if formula_23 is irreducible. 
The countability assumption at the beginning is really necessary, for there exists a locally profinite group that admits an irreducible smooth representation formula_6 such that formula_23 is not irreducible.
Hecke algebra of a locally profinite group.
Let formula_30 be a unimodular locally profinite group such that formula_10 is at most countable for all open compact subgroups "K", and formula_32 a left Haar measure on formula_30. Let formula_34 denote the space of locally constant functions on formula_30 with compact support. With the multiplicative structure given by
formula_34 becomes not necessarily unital associative formula_38-algebra. It is called the Hecke algebra of "G" and is denoted by formula_39. 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 formula_5 of "G", we define a new action on "V":
Thus, we have the functor formula_42 from the category of smooth representations of formula_30 to the category of non-degenerate formula_39-modules. Here, "non-degenerate" means formula_45. Then the fact is that the functor is an equivalence.