Vinberg's $\theta$-groups and rigid connections

Abstract

Let $G$ be a simple complex group of adjoint type. In his unpublished work, Z. Yun associated to each $\theta$-group $(G_0,{\mathfrak{g}}_1)$ and a vector $X\in {\mathfrak{g}}_1$ a flat $G$-connection ${\nabla }^X$ on ${\mathbb{P}}^1-\{0,\infty \}$, generalizing the construction of Frenkel and Gross in [FG]. In this paper we study the local monodromy of those flat $G$-connections and compute the de Rham cohomology of ${\nabla }^X$ with values in the adjoint representations of $G$. In particular, we show that in many cases the connection ${\nabla }^X$ is cohomologically rigid.

arXiv commentRewrite introduction and typos corrected. To appear in IMRN