Resumen de W -algebras and Whittaker categories
This article is concerned with Whittaker models in geometric representation theory, and gives applications to the study of affine W-algebras. The main new innovation connects Whittaker models to invariants for compact-open subgroups of the loop group. This method, which has a counterpart for p-adic groups, settles a conjecture of Gaitsgory in the categorical setting. This method shows that Whittaker sheaves in geometric representation theory admit t-structures, as had previously been observed in some special cases. We then apply this method to the setting of affine W-algebras. We study a new family of modules for affine W-algebras, which can be thought of as analogues of certain tautological (“generalized vaccuum”) modules over the Kac-Moody algebra. Using the above t-structure, we obtain an affine analogue of Skryabin’s theorem that connects affine W-algebras and Whittaker models. This theorem allows various geometric methods to be used to study affine W-algebras. As one such application, we offer a new proof of one of Arakawa’s foundational results in the theory.
