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 F × {\displaystyle F^{\times }} are locally profinite. More generally, the matrix ring M n ( F ) {\displaystyle \operatorname {M} _{n}(F)} and the general linear group GL n ( F ) {\displaystyle \operatorname {GL} _{n}(F)} 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 ψ : G → C × {\displaystyle \psi :G\to \mathbb {C} ^{\times }} is continuous if and only if it has open kernel. Let ( ρ , V ) {\displaystyle (\rho ,V)} be a complex representation of G. ρ {\displaystyle \rho } is said to be smooth if V is a union of V K {\displaystyle V^{K}} where K runs over all open compact subgroups K. ρ {\displaystyle \rho } is said to be admissible if it is smooth and V K {\displaystyle V^{K}} is finite-dimensional for any open compact subgroup K. We now make a blanket assumption that G / K {\displaystyle G/K} is at most countable for all open compact subgroups K. The dual space V ∗ {\displaystyle V^{*}} carries the action ρ ∗ {\displaystyle \rho ^{*}} of G given by ⟨ ρ ∗ ( g ) α , v ⟩ = ⟨ α , ρ ∗ ( g − 1 ) v ⟩ {\displaystyle \left\langle \rho ^{*}(g)\alpha ,v\right\rangle =\left\langle \alpha ,\rho ^{*}(g^{-1})v\right\rangle } . In general, ρ ∗ {\displaystyle \rho ^{*}} is not smooth. Thus, we set V ~ = ⋃ K ( V ∗ ) K {\displaystyle {\widetilde {V}}=\bigcup _{K}(V^{*})^{K}} where K {\displaystyle K} is acting through ρ ∗ {\displaystyle \rho ^{*}} and set ρ ~ = ρ ∗ {\displaystyle {\widetilde {\rho }}=\rho ^{*}} . The smooth representation ( ρ ~ , V ~ ) {\displaystyle ({\widetilde {\rho }},{\widetilde {V}})} is then called the contragredient or smooth dual of ( ρ , V ) {\displaystyle (\rho ,V)} . The contravariant functor
( ρ , V ) ↦ ( ρ ~ , V ~ ) {\displaystyle (\rho ,V)\mapsto ({\widetilde {\rho }},{\widetilde {V}})}
from the category of smooth representations of G to itself is exact. Moreover, the following are equivalent.
ρ {\displaystyle \rho } is admissible.
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