Vanishing of cohomology associated to quantized Drinfeld-Sokolov reduction

We prove a vanishing theorem of the cohomology arising from the two Quantized Drinfeld-Sokolov reductions (+'' and -‘’ reduction) introduced by Feigin-Frenkel and Frenkel-Kac-Wakimoto. As a consequence, the vanishing conjecture of Frenkel-Kac-Wakimoto is proved for the -'' reduction and partially for the +‘’ reduction.