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.
Categories: math.AG