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

Let ${\displaystyle (\rho ,V)}$ be a complex representation of G.[1] ${\displaystyle \rho }$ is said to be smooth if V is a union of ${\displaystyle V^{K}}$ where K runs over all open compact subgroups K. ${\displaystyle \rho }$ is said to be admissible if it is smooth and ${\displaystyle V^{K}}$ is finite-dimensional for any open compact subgroup K.

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

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

The contravariant functor

${\displaystyle (\rho ,V)\mapsto ({\widetilde {\rho }},{\widetilde {V}})}$

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

• ${\displaystyle \rho }$ is admissible.
• ${\displaystyle {\widetilde {\rho }}}$ is admissible.[2]
• The canonical G-module map ${\displaystyle \rho \to {\widetilde {\widetilde {\rho }}}}$ is an isomorphism.

When ${\displaystyle \rho }$ is admissible, ${\displaystyle \rho }$ is irreducible if and only if ${\displaystyle {\widetilde {\rho }}}$ is irreducible.

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

Hecke algebra of a locally profinite group

See also: Hecke algebra of a locally compact group

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

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

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

${\displaystyle \rho (f)=\int _{G}f(g)\rho (g)d\mu (g).}$

Thus, we have the functor ${\displaystyle \rho \mapsto \rho }$ from the category of smooth representations of ${\displaystyle G}$ to the category of non-degenerate ${\displaystyle {\mathfrak {H}}(G)}$-modules. Here, "non-degenerate" means ${\displaystyle \rho ({\mathfrak {H}}(G))V=V}$. 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