Screening operators and Parabolic inductions for Affine W-algebras (with an appendix by Shigenori Nakatsuka)

Abstract: (Affine) $\mathcal{W}$-algebras are a family of vertex algebras defined by the generalized Drinfeld-Sokolov reductions associated with a finite-dimensional reductive Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, a nilpotent element $f$ in $[\mathfrak{g},\mathfrak{g}]$, a good grading $\Gamma$ and a symmetric invariant bilinear form $\kappa$ on $\mathfrak{g}$. We introduce free field realizations of $\mathcal{W}$-algebras by using Wakimoto representations of affine Lie algebras, where $\mathcal{W}$-algebras are described as the intersections of kernels of screening operators. We call these Wakimoto free fields realizations of $\mathcal{W}$-algebras. As applications, under certain conditions that are valid in all cases of type $A$, we construct parabolic inductions for $\mathcal{W}$-algebras, which we expect to induce the parabolic inductions of finite $\mathcal{W}$-algebras defined by Premet and Losev. In type $A$, we show that our parabolic inductions are a chiralization of the coproducts for finite $\mathcal{W}$-algebras defined by Brundan-Kleshchev. In type $BCD$, we are able to obtain some generalizations of the coproducts in some special cases. This paper also contains an appendix by Shigenori Nakatsuka on the compatibility of screening operators with Miura maps.