The twisted Drinfeld–Sokolov reduction conjecture for Weyl modules

Let $\mathfrak{g}$ be a Lie algebra, let $\widehat{\mathfrak g}$ be its affine algebra, let $d_{st}$ be the standard semi-infinite cohomology differential, let $\sigma^\lambda$ be spectral flow associated to a weight $\lambda$, and let $\chi$ be the usual character. Define

$$d_\lambda=d_{st}+\chi\circ\sigma^\lambda.$$

Let $M_{\Psi,\nu}$ be a Weyl module and let $\rho^\vee$ denote the dual Weyl vector. Twisted Drinfeld–Sokolov reduction conjecture. The $\lambda$-twisted reduction satisfies

$$H_{d_\lambda}(M_{\Psi,\nu})\cong L^{\Psi}\left(\gamma_{\nu-\Psi(\lambda+\rho^\vee)}\right)\cong L^{\Psi^{-1}}\left(\gamma_{\lambda-\Psi(\nu+\rho^\vee)}\right).$$

The claim is verified for $\lambda=0$ and $\mathfrak g=\mathfrak{sl}(2)$, while Euler–Poincaré character calculations support it for arbitrary $\lambda$; the general proof requires reduction beyond category $\mathcal O$ and remains open.