Double-coset Hecke algebra

For a group $G$ and subgroup $\Gamma$ such that every double coset has finitely many orbits under left multiplication by $\Gamma$, this algebra consists of complex $\Gamma$-bi-invariant functions on $G$ supported on finitely many double cosets. Its convolution isThe double coset indicator functions form a basis. On invariant modular functions it acts on the right by $f\ast h={\sum }_{\Gamma x}h(x)f{|}_{k}x$. For $G=G{L}_2(\mathbb{Q}{)}^{+}$ and $\Gamma =S{L}_2(\mathbb{Z})$, rational conjugation of finite-index modular subgroups verifies the finiteness condition. The determinant-$n$ Hecke operator uses the indicator of all integral determinant-$n$ matrices, not only a single double coset when $n$ is composite.