Universal deformation rings of a special class of modules over generalized Brauer tree algebras

Let $\Bbbk$ be an algebraically closed field and $\Lambda$ a generalized Brauer tree algebra over $\Bbbk$. We compute the universal deformation rings of the periodic string modules over $\Lambda$. Moreover, for a specific class of generalized Brauer tree algebras $\Lambda(n,\overline{m})$, we classify the universal deformation rings of the modules lying in $\Omega$-stable components $\mathfrak{C}$ of the stable Auslander-Reiten quiver provided that $\mathfrak{C}$ contains at least one simple module. Our approach uses several tools and techniques from the representation theory of Brauer graph algebras. Notably, we leverage Duffield's work on the Auslander-Reiten theory of these algebras and Opper-Zvonareva's results on derived equivalences between Brauer graph algebras.