Locally profinite group explained

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 ^{x} are locally profinite. More generally, the matrix ring \operatorname{M} _{n(F)} and the general linear group \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

\psi: G \to C ^{x} is continuous if and only if it has open kernel.

Let

(\rho, V) be a complex representation of G.^[1] \rho is said to be smooth if V is a union of V ^{K} where K runs over all open compact subgroups K. \rho 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 \rho ^{*} of G given by \left\langle \rho ^{*(g)} \alpha, v \right\rangle = \left\langle \alpha, \rho ^{*(g} ^{-1} ) v \right\rangle . In general, \rho ^{*} is not smooth. Thus, we set \widetilde{V} = c up _{K} (V ^{*)} ^{K} where K is acting through \rho ^{*} and set \widetilde{\rho} = \rho ^{*} . The smooth representation (\widetilde{\rho}, \widetilde{V}) is then called the contragredient or smooth dual of (\rho, V) .

The contravariant functor

(\rho, V) \mapsto (\widetilde{\rho}, \widetilde{V}) from the category of smooth representations of G to itself is exact. Moreover, the following are equivalent.\rho is admissible.\widetilde{\rho} is admissible.^[2]

• The canonical G-module map

\rho \to \widetilde{\widetilde{\rho}} is an isomorphism.When \rho is admissible, \rho is irreducible if and only if \widetilde{\rho} is irreducible.

The countability assumption at the beginning is really necessary, for there exists a locally profinite group that admits an irreducible smooth representation

\rho such that \widetilde{\rho} is not irreducible.

Hecke algebra of a locally profinite group

See also: Hecke algebra of a locally compact group.

Let

G be a unimodular locally profinite group such that G/K is at most countable for all open compact subgroups K, and \mu a left Haar measure on G . Let C ^{inf ty} _{c(G)} denote the space of locally constant functions on G with compact support. With the multiplicative structure given by(f * h)(x) = \int _{G} f(g) h(g ^{-1} x) d \mu(g) C ^{inf ty} _{c(G)} becomes not necessarily unital associative C -algebra. It is called the Hecke algebra of G and is denoted by ak{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 (\rho, V) of G, we define a new action on V:\rho(f) = \int _{G} f(g) \rho(g) d\mu(g). Thus, we have the functor \rho \mapsto \rho from the category of smooth representations of G to the category of non-degenerate ak{H}(G) -modules. Here, "non-degenerate" means \rho( ak{H}(G))V=V . Then the fact is that the functor is an equivalence.^[3]

References

• Corinne Blondel, Basic representation theory of reductive p-adic groups

Notes and References

1. We do not put a topology on V; so there is no topological condition on the representation.
2. Blondel, Corollary 2.8.
3. Blondel, Proposition 2.16.