Automorphic functions twisted by representations, unitary and non-unitary

Automorphic functions that are twisted by unitary or non-unitary representations occur naturally in many areas, most notably in spectral theory, number theory and mathematical physics. In addition, already Selberg promoted the idea to investigate twisted automorphic functions alongside their untwisted relatives and to exploit their interaction in order to understand their properties. While during the decades the focus has been on unitary twists, the investigations recently turned to non-unitary twists as well. I will report on the status of an ongoing project to investigate simultaneously unitarily and non-unitarily twisted automorphic functions with the help of ergodic methods and with a view towards period functions and a classical-quantum correspondence.