Shift system and its applications

We introduce a new concept named shift system. This is a purely Lie algebraic setting to develop the geometric representation theory of Feigin-Tipunin construction. After reformulating the discussion in past works of the second author under this new setting, as an application, we extend almost all the main results of these works to the (multiplet) principal W-algebra at positive integer level associated with a simple Lie algebra $\mathfrak{g}$ and Lie superalgebra $\mathfrak{osp}(1|2n)$, respectively. This paper also contains an appendix by Myungbo Shim on the relationship between Feigin-Tipunin construction and recent quantum field theories.