  We introduce Hecke algebras associated to discrete quantum groups with
commensurated quantum subgroups. We study their modular properties and the
associated Hecke operators. In order to investigate their analytic properties
we adapt the construction of the Schlichting completion to the quantum setting,
thus obtaining locally compact quantum groups with compact open quantum
subgroups. We study in detail a class of examples arising from quantum HNN
extensions.
