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 $$\operatorname{M}_n(F)$$ 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. $$\rho$$ 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 $$\left\langle \rho^*(g) \alpha, v \right\rangle = \left\langle \alpha, \rho^*(g^{-1}) v \right\rangle$$. 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. When $$\rho$$ is admissible, $$\rho$$ is irreducible if and only if $$\widetilde{\rho}$$ is irreducible.
 * $$\rho$$ is admissible.
 * $$\widetilde{\rho}$$ is admissible.
 * The canonical G-module map $$\rho \to \widetilde{\widetilde{\rho}}$$ is an isomorphism.

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

$$C^\infty_c(G)$$ 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.