I know there is a rich theory of automorphic forms of weight $k \in \frac{1}{2} \mathbb{Z}$ (e.g. Dedekind Eta).
Is there any reason, apart from being very technical, that automorphic forms of other non-integral weight have not been studied in much detail e.g. of weight $1/5$?
As far as I am aware applications of automorphic forms are found across a wide variety of maths - in string theory for example. Is there a call from these areas for non-integer automorphic forms to be studied or not?
This follows from the MO question here https://mathoverflow.net/questions/52996/modular-forms-of-fractional-weight but since this was asked 5 years ago I was hoping for some updates.
There have been recent developments detailing mock modular forms and quantum modular forms, but these seem to me to be a less obvious route than studying different weights.