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 ${\mathrm{M}}_n(F)$ and the general linear group ${\mathrm{GL}}_n(F)$ 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. $\rho$ 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 $⟨{\rho }^{*}(g)\alpha ,v⟩=⟨\alpha ,{\rho }^{*}(g^{-1})v⟩$. In general, ${\rho }^{*}$ is not smooth. Thus, we set $\tilde{V}=\bigcup _K(V^{*}{)}^K$ where $K$ is acting through ${\rho }^{*}$ and set $\tilde{\rho }={\rho }^{*}$. The smooth representation $(\tilde{\rho },\tilde{V})$ is then called the contragredient or smooth dual of $(\rho ,V)$.

The contravariant functor

$(\rho ,V)\mapsto (\tilde{\rho },\tilde{V})$

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

• $\rho$ is admissible.
• $\tilde{\rho }$ is admissible.
• The canonical G-module map $\rho \to \widetilde{\stackrel{~}{\rho }}$ is an isomorphism.

When $\rho$ is admissible, $\rho$ is irreducible if and only if $\tilde{\rho }$ 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 $\tilde{\rho }$ 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 $\mu$ a left Haar measure on $G$. Let $C_c^{\mathrm{\infty }}(G)$ denote the space of locally constant functions on $G$ with compact support. With the multiplicative structure given by

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

$C_c^{\mathrm{\infty }}(G)$ 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:

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

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.

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,