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 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
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
- See also: Hecke algebra of a locally compact group
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
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