Twisted Whittaker category on affine flags and category of representations of mixed quantum group

Abstract

We prove the twisted Whittaker category on the affine flag variety and the category of representations of the mixed quantum group are equivalent.(see (0.14) in [GL1]). A naive attempt is to use the Jacquet functor F KD from F l G to Gr T , namely, we !-pullback from F l G to F l B − and then take * -direct image to Gr T = F l T . However, the problem is that the image does not have a Ω W hit,Lus κ factorization module structure (except when κ is generic). For a coweight λ, the λ-component of