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 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.

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 $\operatorname {M}_{n}(F)$ 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 $\psi :G\to {\mathbb {C}}^{\times }$ 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 $\langle \rho ^{*}(g)\alpha ,v\rangle =\langle \alpha ,\rho ^{*}(g^{{-1}})v\rangle$. In general, $\rho ^{*}$ is not smooth. Thus, we set $\widetilde {V}=\bigcup _{K}(V^{*})^{K}$ 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

Let $G$ be a unimodular locally profinite group such that $G/K$ is at most countable for all open compact subgroups K, and $\rho$ a left Haar measure on $G$. Let $C_{c}^{\infty }(G)$ denote the space of locally constant functions on $G$ with compact support. With the multiplicative structure given by

$(f*h)(x)=\int _{G}f(g)h(g^{{-1}}x)d\mu (g)$

$C_{c}^{\infty }(G)$ becomes not necessarily unital associative $\mathbb {C}$-algebra. It is called the Hecke algebra of G and is denoted by ${\mathfrak {H}}(G)$. 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 ${\mathfrak {H}}(G)$-modules. Here, "non-degenerate" means $\rho ({\mathfrak {H}}(G))V=V$. 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, MR 2234120
• Milne, J.S. (1988), Canonical models of (mixed) Shimura varieties and automorphic vector bundles, MR 1044823