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.[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 \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.

  • \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

See also: Hecke algebra of a locally compact 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.[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), Vol. 335, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Berlin, New York: Springer-Verlag, doi:10.1007/3-540-31511-X. ISBN 978-3-540-31486-8. MR 2234120
  • Milne, James S. (1988), "Canonical models of (mixed) Shimura varieties and automorphic vector bundles,", MR 1044823