$C_2$-cofiniteness of logarithmic $W$-algebras
Let $\mathfrak{g}$ be a finite-dimensional simple complex Lie algebra with root lattice $Q$ and lacing number $t$, equipped with an invariant bilinear form normalized so that long roots have squared length two. We prove $C_2$-cofiniteness of the Feigin--Tipunin logarithmic $W$-algebra $\W_p(Q)$ for $p=tm\geq2$ and $m\geq h^\vee_{{}^L\g}-1$, where $h^\vee_{{}^L\g}$ is the dual Coxeter number of the Langlands dual. Drinfeld--Sokolov reduction of induced affine Weyl modules gives graded generators for the multiplicity spaces of the conformal group action. Their degree bounds show that the…
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