On the Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras

We prove a part of the Cachazo-Douglas-Seiberg-Witten conjecture uniformly for any simple Lie algebra $\mathfrak{g}$. The main ingredients in the proof are: Garland's result on the Lie algebra cohomology of $\hat{\mathfrak{u}} := \mathfrak{g}\otimes t\mathbb{C}[t]$; Kostant's result on the `diagonal' cohomolgy of $\hat{\mathfrak{u}}$ and its connection with abelian ideals in a Borel subalgebra of $\mathfrak{g}$; and a certain deformation of the singular cohomology of the infinite Grassmannian introduced by Belkale-Kumar.