Hecke algebra of a locally compact group

Summary

Let $K$ be a compact open subgroup of a locally compact group $G$. The Hecke algebra of $(G, K)$ is the algebra of compactly supported $K$-bi-invariant functions on $G$ under convolution. It has a basis given by the indicator functions of the double cosets $K g K$. The $K$-fixed vectors of a smooth representation of $G$ form a module over it. This makes it a main tool for representations of $p$-adic groups. For $G = \mathrm{GL}_n(\Q_p)$ and $K = \mathrm{GL}_n(\Z_p)$ it is commutative, and it is closely related to the Hecke operators on modular forms.