Differential operators on Hitchin variety

We introduce the notion of Hitchin variety over $\C$. Let $L$ be a holomorphic line bundle over a Hitchin variety $X$. We investigate the space of all global sections of sheaf of differential operators $\cat{D}^k (L)$ and symmetric powers of sheaf of first order differential operators $\cat{S}^k(\cat{D}^1 (L))$ over $X$ and show that for a projective Hithcin variety both the spaces are one dimensional. As an application, we show that the space $\cat{C}(L)$ of holomorphic connections on $L$ does not admit any non-constant regular function.