Maybe I can answer this question by myself now. I did some literature research and find some papers and notes illustrating $P^{1}$ to establish various theory Lectures on Hall algebras The author talks about Hall algebra of coherent sheaves on $P^{1}$, relation with representation theory of $U_{q}(\hat{sl_{2}})$ and also The Hall algebra of the category of coherent sheaves on the projective line talks about similar facts.
Twisted rings of differential operators on the projective line and the Beilinson-Bernstein theorem. It is a master thesis by Koushik Panda. He established $P^{1}$(flag variety of $sl_{2}$) version of Beilinson-Bernstein localization. His treatment is very detailed.
t-stabilities and t-structures on triangulated categoriesillustrates classifications of t-structures on $D^{b}(Coh{P^{1}})$ Introduction to coherent sheaves on weighted projective lines by Chen-Xiaowu and Henning Krause. Very expository notes for coherent sheaves, Tilting theory, derived category of $P^{1}$