The triplet VOA ($A_1$ type Feigin-Tipunin VOA) is one of the most famous examples of $C_2$-cofinite and irrational VOA. However, there are not much known about the Feigin-Tipunin VOA $W(p)_Q$, the $ADE$ type generalization of the triplet VOA.
In this talk, we will give the geometric construction and character formulas of $W(p)_Q$ that conjectured in the paper of Feigin-Tipunin.
Moreover, we will give a $W$-algebraic conditioning of $C_2$-cofiniteness of $W(p)_Q$ under the expectable assumptions of simpleness. In the case of $A_2$ type with a fixed $p$, this conditioning enables us to prove the $C_2$-cofiniteness of $W(p)_Q$ much easier than direct calculation.