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

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Important examples of locally profinite groups come from algebraic number theory. Let F be a non-archimedean local field. Then both F and   are locally profinite. More generally, the matrix ring   and the general linear group   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

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Let G be a locally profinite group. Then a group homomorphism   is continuous if and only if it has open kernel.

Let   be a complex representation of G.[1]   is said to be smooth if V is a union of   where K runs over all open compact subgroups K.   is said to be admissible if it is smooth and   is finite-dimensional for any open compact subgroup K.

We now make a blanket assumption that   is at most countable for all open compact subgroups K.

The dual space   carries the action   of G given by  . In general,   is not smooth. Thus, we set   where   is acting through   and set  . The smooth representation   is then called the contragredient or smooth dual of  .

The contravariant functor

 

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

  •   is admissible.
  •   is admissible.[2]
  • The canonical G-module map   is an isomorphism.

When   is admissible,   is irreducible if and only if   is irreducible.

The countability assumption at the beginning is really necessary, for there exists a locally profinite group that admits an irreducible smooth representation   such that   is not irreducible.

Hecke algebra of a locally profinite group

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Let   be a unimodular locally profinite group such that   is at most countable for all open compact subgroups K, and   a left Haar measure on  . Let   denote the space of locally constant functions on   with compact support. With the multiplicative structure given by

 

  becomes not necessarily unital associative  -algebra. It is called the Hecke algebra of G and is denoted by  . 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   of G, we define a new action on V:

 

Thus, we have the functor   from the category of smooth representations of   to the category of non-degenerate  -modules. Here, "non-degenerate" means  . Then the fact is that the functor is an equivalence.[3]

Notes

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  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

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  • 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], vol. 335, 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