---
tag: 61Z8MZ2M28
kind: definition
name: Hecke algebra of a locally compact group
msc: [22D20, 20C08, "22E50"]
defines: 6H3WCCHZB5
uses: [AHV05ZBCYW, VY0HXD3GC4, 2Z29JN3VES, QYCFH68RD1, 0R6EWFKRCF]
sources:
  - url: https://en.wikipedia.org/wiki/Hecke_algebra_of_a_pair
---

## Definition

Let $G$ be a [[locally compact group|AHV05ZBCYW]] with a left [[Haar measure|VY0HXD3GC4]] $\mu$, and let $K$ be a [[compact|0R6EWFKRCF]] [[open subgroup|QYCFH68RD1]] of $G$.
The *Hecke algebra* $\mathcal{H}(G, K)$ is the set of functions $f \colon G \to \C$ such that:

1. $f(k_1 g k_2) = f(g)$ for all $g \in G$ and $k_1, k_2 \in K$;
2. $f$ has compact support.

Its product is [[convolution|2Z29JN3VES]]:
$$(f_1 * f_2)(x) = \int_G f_1(y) f_2(y^{-1} x) \, d\mu(y).$$

## Remarks

Each double coset $KgK$ is compact and open, and a compact set meets only finitely many of them.
So the indicator functions $\mathbf{1}_{KgK}$ form a basis of $\mathcal{H}(G, K)$ as a vector space.
The function $\mu(K)^{-1} \mathbf{1}_K$ is the unit element.

Example: $G$ discrete and $K = \set{e}$, with $\mu$ the counting measure.
Both clauses say only that $f$ is finitely supported, and $\mathbf{1}_{\set{a}} * \mathbf{1}_{\set{b}} = \mathbf{1}_{\set{ab}}$.
So $\mathcal{H}(G, \set{e})$ is the group algebra $\C[G]$.

Example: $G = \mathrm{GL}_2(\Q_p)$ and $K = \mathrm{GL}_2(\Z_p)$.
Then $\mathcal{H}(G, K)$ is commutative.
It is generated by the indicators of $K \operatorname{diag}(p, 1) K$ and $K \operatorname{diag}(p, p) K$ and the inverse of the latter.

Non-example: for $G = \Q_p$ and $K = \Z_p$, the indicator of $p \Z_p$ fails clause 1, since it is not constant on the coset $\Z_p$.

When $G$ is totally disconnected, the union of the $\mathcal{H}(G, K)$ over all compact open $K$ is the full Hecke algebra $\mathcal{H}(G)$ of locally constant compactly supported functions.