Definition
Locally profinite group
A locally compact totally disconnected Hausdorff group, equivalently one with compact-open subgroups forming an identity basis.
Definition
A locally profinite group is a Hausdorff topological group that is locally compact and totally disconnected.
By van Dantzig's theorem, this is equivalent to requiring that the identity have a neighborhood basis consisting of compact open subgroups. A compact locally profinite group is a profinite group.
Examples
- The additive and multiplicative groups of a nonarchimedean local field are locally profinite.
- If is an algebraic group over such a field , then is locally profinite.
- Every discrete group is locally profinite: the trivial subgroup is compact and open.
Representation theory
A representation on a discrete vector space is smooth exactly when every vector is fixed by some compact open subgroup. Compact-open fixed spaces define admissibility, and compactly supported locally constant functions form the Hecke algebra.
References
- David van Dantzig, “Zur topologischen Algebra. III. Brouwersche und Cantorsche Gruppen,” Compositio Mathematica 3 (1936), 408–426. Numdam.
- George W. Mackey, “Induced representations of locally compact groups I,” Annals of Mathematics 55 (1952), 101–139.