The semi-infinite cohomology of Weyl modules with two singular points

In their study of spherical representations of an affine Lie algebra at the critical level and of unramified opers, Frenkel and Gaitsgory introduced what they called the Weyl module $\mathbb{V}^\lambda$ corresponding to a dominant weight $\lambda$. This object plays an important role in the theory. In $\href{ https://doi.org/10.1007/s00220-022-04430-w}{[4]}$, we introduced a possible analogue $\mathbb{V}^{\lambda,\mu}_{2}$ of the Weyl module in the setting of opers with two singular points, and in the case of $\mathfrak{sl}(2)$ we proved that it has the ‘correct’ endomorphism ring. In this paper, we compute the semi-infinite cohomology of $\mathbb{V}^{\lambda,\mu}_{2}$ and we show that it does not share some of the properties of the semi-infinite cohomology of the Weyl module of Frenkel and Gaitsgory. For this reason, we introduce a new module $\tilde{\mathbb{V}}^{\lambda,\mu}_{2}$ which, in the case of $\mathfrak{sl}(2)$, enjoys all the expected properties of a Weyl module.