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 [math]\displaystyle{ F^\times }[/math] are locally profinite. More generally, the matrix ring [math]\displaystyle{ \operatorname{M}_n(F) }[/math] and the general linear group [math]\displaystyle{ \operatorname{GL}_n(F) }[/math] 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 [math]\displaystyle{ \psi: G \to \mathbb{C}^\times }[/math] is continuous if and only if it has open kernel.

Let [math]\displaystyle{ (\rho, V) }[/math] be a complex representation of G. [math]\displaystyle{ \rho }[/math] is said to be smooth if V is a union of [math]\displaystyle{ V^K }[/math] where K runs over all open compact subgroups K. [math]\displaystyle{ \rho }[/math] is said to be admissible if it is smooth and [math]\displaystyle{ V^K }[/math] is finite-dimensional for any open compact subgroup K.

We now make a blanket assumption that [math]\displaystyle{ G/K }[/math] is at most countable for all open compact subgroups K.

The dual space [math]\displaystyle{ V^* }[/math] carries the action [math]\displaystyle{ \rho^* }[/math] of G given by [math]\displaystyle{ \left\langle \rho^*(g) \alpha, v \right\rangle = \left\langle \alpha, \rho^*(g^{-1}) v \right\rangle }[/math]. In general, [math]\displaystyle{ \rho^* }[/math] is not smooth. Thus, we set [math]\displaystyle{ \widetilde{V} = \bigcup_K (V^*)^K }[/math] where [math]\displaystyle{ K }[/math] is acting through [math]\displaystyle{ \rho^* }[/math] and set [math]\displaystyle{ \widetilde{\rho} = \rho^* }[/math]. The smooth representation [math]\displaystyle{ (\widetilde{\rho}, \widetilde{V}) }[/math] is then called the contragredient or smooth dual of [math]\displaystyle{ (\rho, V) }[/math].

The contravariant functor

[math]\displaystyle{ (\rho, V) \mapsto (\widetilde{\rho}, \widetilde{V}) }[/math]

from the category of smooth representations of G to itself is exact. Moreover, the following are equivalent.

• [math]\displaystyle{ \rho }[/math] is admissible.
• [math]\displaystyle{ \widetilde{\rho} }[/math] is admissible.
• The canonical G-module map [math]\displaystyle{ \rho \to \widetilde{\widetilde{\rho}} }[/math] is an isomorphism.

When [math]\displaystyle{ \rho }[/math] is admissible, [math]\displaystyle{ \rho }[/math] is irreducible if and only if [math]\displaystyle{ \widetilde{\rho} }[/math] is irreducible.

The countability assumption at the beginning is really necessary, for there exists a locally profinite group that admits an irreducible smooth representation [math]\displaystyle{ \rho }[/math] such that [math]\displaystyle{ \widetilde{\rho} }[/math] is not irreducible.

Hecke algebra of a locally profinite group

Let [math]\displaystyle{ G }[/math] be a unimodular locally profinite group such that [math]\displaystyle{ G/K }[/math] is at most countable for all open compact subgroups K, and [math]\displaystyle{ \mu }[/math] a left Haar measure on [math]\displaystyle{ G }[/math]. Let [math]\displaystyle{ C^\infty_c(G) }[/math] denote the space of locally constant functions on [math]\displaystyle{ G }[/math] with compact support. With the multiplicative structure given by

[math]\displaystyle{ (f * h)(x) = \int_G f(g) h(g^{-1} x) d \mu(g) }[/math]

[math]\displaystyle{ C^\infty_c(G) }[/math] becomes not necessarily unital associative [math]\displaystyle{ \mathbb{C} }[/math]-algebra. It is called the Hecke algebra of G and is denoted by [math]\displaystyle{ \mathfrak{H}(G) }[/math]. 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 [math]\displaystyle{ (\rho, V) }[/math] of G, we define a new action on V:

[math]\displaystyle{ \rho(f) = \int_G f(g) \rho(g) d\mu(g). }[/math]

Thus, we have the functor [math]\displaystyle{ \rho \mapsto \rho }[/math] from the category of smooth representations of [math]\displaystyle{ G }[/math] to the category of non-degenerate [math]\displaystyle{ \mathfrak{H}(G) }[/math]-modules. Here, "non-degenerate" means [math]\displaystyle{ \rho(\mathfrak{H}(G))V=V }[/math]. Then the fact is that the functor is an equivalence.

References

• Corinne Blondel, Basic representation theory of reductive p-adic groups
• Bushnell, Colin J.; Henniart, Guy (2006), The local Langlands conjecture for GL(2), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 335, Berlin, New York: Springer-Verlag, doi:10.1007/3-540-31511-X, ISBN 978-3-540-31486-8 
• Milne, James S. (1988), Canonical models of (mixed) Shimura varieties and automorphic vector bundles,