= Double-coset Hecke algebra
= Hecke algebra of double cosets
{c}
{synonym}
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 is
$$
(h_1*h_2)(g)=\sum_{\Gamma x\in\Gamma\backslash G}h_1(gx^{-1})h_2(x).
$$
The <double coset> indicator functions form a basis. On invariant modular <functions> it acts on the right by $f*h=\sum_{\Gamma x}h(x)f|_kx$. For $G=GL_2(\mathbb Q)^+$ and $\Gamma=SL_2(\mathbb Z)$, <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.
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