Vinberg’s θ-groups and rigid connections

Let G be a simple complex group of adjoint type. In his unpublished work, Z. Yun associated to each θ-group (G₀, g₁) and a vector X ∈ g₁ a flat G-connection ∇^X on the trivial G-bundle on P¹ −{0, ∞}, generalizing the construction of Frenkel and Gross in [5]. In this paper, we study the local monodromies of the flat G-connection ∇^X and compute the de Rham cohomology of ∇^X with value in the adjoint representation of G. In particular, we show that in many cases the connection ∇^X is cohomologically rigid.