Localization for affine <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi mathvariant="script">W</mml:mi></mml:math>-algebras

We prove a localization theorem for affine W-algebras in the spirit of Beilinson–Bernstein and Kashiwara–Tanisaki. More precisely, any non-critical regular weight λ, we identify λ-monodromic Whittaker D-modules on enhanced flag variety with full subcategory Category O W-algebra. To essential image our functor, provide new realization using Iwahori–Whittaker modules corresponding Kac–Moody algebra. Using these methods, also obtain proof Arakawa's character formulae simple positive energy representations