I am a mathematician. I am studying and working on Hecke pairs which I am going to give the related definitions in the following. But first let me explain what I am looking for to learn by asking this question.

I have observed that the following algebraic notions naturally appears in many different areas of mathematics such as number theory, representation theory, geometric group theory, hyperbolic 3-manifolds, etc. Even Hecke pairs appear in the Bost-Connes quantum statistical approach to class field theory of $\mathbb{Q}$. The basic definitions involved in Hecke pairs are rather elementary, so I suspect these notion may have appeared in various areas of theoretical physics as well. The things is people (even in mathematics) use different terminology to describe these notions, so one cannot Google to find all instances of Hecke pairs in other sciences.

Therefore I give several elementary definitions about Hecke pairs, to see if they ring any bells and hopefully to get some clues how they appear in physics.

Two subgroups $H$ and $K$ of a (discrete) group $G$ are called commensurable if $H\cap K$ is a finite index subgroup of both $H$ and $K$. One easily checks that commensurability is an equivalence relation among subgroups.

A subgroup $H$ of a group $G$ is called a Hecke subgroup of $G$ if $H$ is commensurable with $gHg^{-1}$ for all $g\in G$. One observes that this condition is milder than normality condition for subgroups, so some author refer to $H$ as an almost normal subgroup of $G$. There are also at least two more name for the same notion!

Let me give one more definition. Given a subgroup $H$ of $G$, we define the commensurator of $H$ in $G$ by the following formula: $$ \operatorname{Comm}_G(H) := \{ g\in G\ |\ H\ \mathrm{and\ }gHg^{-1}\ \mathrm{are\ comensurable} \} $$ Obviously, $H$ is always a Hecke subgroup of $Comm_G(H)$, and so $H$ is a Hecke subgroup of $G$ if $G=\operatorname{Comm}_G(H)$.


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