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 $F^\times$ are locally profinite. More generally, the matrix ring $\operatorname_n(F)$ and the general linear group $\operatorname_n(F)$ 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 $\psi: G \to \mathbb^\times$ is continuous if and only if it has open kernel.

Let $(\rho, V)$ be a complex representation of ''G''. $\rho$ 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 $\left\langle \rho^*(g) \alpha, v \right\rangle = \left\langle \alpha, \rho^*(g^) v \right\rangle$. In general, $\rho^*$ is not smooth. Thus, we set $\widetilde = \bigcup_K (V^*)^K$ where $K$ is acting through $\rho^*$ and set $\widetilde = \rho^*$. The smooth representation $(\widetilde, \widetilde)$ is then called the contragredient or smooth dual of $(\rho, V)$.

The contravariant functor
:$(\rho, V) \mapsto (\widetilde, \widetilde)$
from the category of smooth representations of ''G'' to itself is exact. Moreover, the following are equivalent.
* $\rho$ is admissible.
* $\widetilde$ is admissible.
* The canonical ''G''-module map $\rho \to \widetilde$ is an isomorphism.
When $\rho$ is admissible, $\rho$ is irreducible if and only if $\widetilde$ 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 $\widetilde$ is not irreducible.

Hecke algebra of a locally profinite 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 $C^\infty_c(G)$ denote the space of locally constant functions on $G$ with compact support. With the multiplicative structure given by
:$(f * h)(x) = \int_G f(g) h(g^ x) d \mu(g)$
$C^\infty_c(G)$ becomes not necessarily unital associative $\mathbb$-algebra. It is called the Hecke algebra of ''G'' and is denoted by $\mathfrak(G)$. 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 $\mathfrak(G)$-modules. Here, "non-degenerate" means $\rho(\mathfrak(G))V=V$. Then the fact is that the functor is an equivalence.Blondel, Proposition 2.16.

Notes

References

*Corinne Blondel
Basic representation theory of reductive ''p''-adic groups
*
*{{citation , last1=Milne , first1=James S. , authorlink1=James Milne (mathematician) , title=Canonical models of (mixed) Shimura varieties and automorphic vector bundles, , year=1988 , MR=1044823
Topological groups