Feigin-Frenkel duality is the isomorphism between the principal \(\mathcal {W}\) -algebras of a simple Lie algebra \(\mathfrak {g}\) and its Langlands dual Lie algebra \( ^L\mathfrak {g}\) . A generalization of this duality to a larger family of \(\mathcal {W}\) -algebras called hook-type was recently conjectured by Gaiotto and Rapčák and proved by the first two authors. It says that the affine cosets of two different hook-type \(\mathcal {W}\) -(super)algebras are isomorphic. A natural question is whether the duality between the affine cosets can be enhanced to reconstruct one W-algebra from the other. There is a convolution operation that maps a hook-type \(\mathcal {W}\) -algebra \(\mathcal {W}\) to a certain relative semi-infinite cohomology of \(\mathcal {W}\) tensored with a suitable kernel VOA. The first two authors conjectured previously that this cohomology is isomorphic to the Feigin-Frenkel dual hook-type \(\mathcal {W}\) -algebra. Our main result is a proof of this conjecture.
