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.
Important examples of locally profinite groups come from algebraic number theory. Let F be a non-archimedean local field. Then both F and F × are locally profinite. More generally, the matrix ring M n ( F ) and the general linear group 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).
Let G be a locally profinite group. Then a group homomorphism ψ : G → C × is continuous if and only if it has open kernel.
Let ( ρ , V ) be a complex representation of G. ρ is said to be smooth if V is a union of V K where K runs over all open compact subgroups K. ρ is said to be admissible if it is smooth and V K is finite-dimensional for any open compact subgroup K.
We now make a blanket assumption that G / K is at most countable for all open compact subgroups K.
The dual space V ∗ carries the action ρ ∗ of G given by ⟨ ρ ∗ ( g ) α , v ⟩ = ⟨ α , ρ ∗ ( g − 1 ) v ⟩ . In general, ρ ∗ is not smooth. Thus, we set V ~ = ⋃ K ( V ∗ ) K where K is acting through ρ ∗ and set ρ ~ = ρ ∗ . The smooth representation ( ρ ~ , V ~ ) is then called the contragredient or smooth dual of ( ρ , V ) .
The contravariant functor
( ρ , V ) ↦ ( ρ ~ , V ~ ) from the category of smooth representations of G to itself is exact. Moreover, the following are equivalent.
ρ is admissible. ρ ~ is admissible.The canonical G-module map ρ → ρ ~ ~ is an isomorphism.When ρ is admissible, ρ is irreducible if and only if ρ ~ is irreducible.
The countability assumption at the beginning is really necessary, for there exists a locally profinite group that admits an irreducible smooth representation ρ such that ρ ~ is not irreducible.
Let G be a unimodular locally profinite group such that G / K is at most countable for all open compact subgroups K, and ρ a left Haar measure on G . Let C c ∞ ( G ) denote the space of locally constant functions on G with compact support. With the multiplicative structure given by
( f ∗ h ) ( x ) = ∫ G f ( g ) h ( g − 1 x ) d μ ( g ) C c ∞ ( G ) becomes not necessarily unital associative C -algebra. It is called the Hecke algebra of G and is denoted by H ( G ) . 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 ( ρ , V ) of G, we define a new action on V:
ρ ( f ) = ∫ G f ( g ) ρ ( g ) d μ ( g ) . Thus, we have the functor ρ ↦ ρ from the category of smooth representations of G to the category of non-degenerate H ( G ) -modules. Here, "non-degenerate" means ρ ( H ( G ) ) V = V . Then the fact is that the functor is an equivalence.
