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.

## Contents

* 1 Examples
* 2 Representations of a locally profinite group
* 3 Hecke algebra of a locally profinite group
* 4 Notes
* 5 References

## 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× are locally profinite. More generally, the matrix ring Mn(F) and the general linear group GLn(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× is continuous if and only if it has open kernel.

Let (ρ,V) be a complex representation of G.[1] ρ is said to be smooth if V is a union of VK where K runs over all open compact subgroups K. ρ is said to be admissible if it is smooth and VK 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.[2]
* 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.

## 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 μ a left Haar measure on G. Let Cc∞(G) denote the space of locally constant functions on G with compact support. With the multiplicative structure given by

(f*h)(x)=∫Gf(g)h(g−1x)dμ(g)

Cc∞(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)=∫Gf(g)ρ(g)dμ(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.[3]

## Notes

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.

## 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,

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Locally profinite group. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Locally_profinite_group&oldid=4157500"