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. 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). 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.