# Moduli space of stable principal $G$-bundles

We have this Mumford's theorem: Let $X$ be a Riemann surface of genus $g$, $G$ a simple Lie group. We can consider a principal stable $G$-bundles over $X$ (say $\xi$), where $rk(\xi)=r$ and $deg(\xi)=d$. For fixed $X$, $r$ , $d$, there exists a connected moduli space $M^g$ of $S$-equivalence classes of rank $r$ , degree $d$ semistable bundles over $X$, which is a complex projective variety, having dimension $r^2(g - 1) + 1$ when $g \ge 2$. (For reference look at this: http://www.math.columbia.edu/~thaddeus/papers/odense.pdf, page $4$). I'm interested in stable principal $G$-bundles over $X$. So does exist a version of this theorem that says us what is the dimension of moduli space of stable $G$-bundles over $X$?

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Yes. The standard reference is : A. Ramanathan: Moduli for principal bundles over algebraic curves, I and II, Proc. Indian Acad. Sci. Math. Sci. 106 (1996) 301–328 and 421–449. The dimension of the moduli space is $(g-1)\dim G$ (note that the moduli space you seem to consider is that of $GL(r,\mathbb{C})$-bundles, and $GL(r,\mathbb{C})$ is not simple).
But if I want that $G$ is a simple Lie group, is it true that the dimention of the moduli space is $(g-1)dim(G)$?... I think $dim_{\mathbb{C}}(G)$, right? –  Oscar1778 Oct 10 '13 at 18:24
Yes, the dimension of the moduli space is $(g−1) \dim_{\mathbb{C}} G$. –  abx Oct 11 '13 at 18:08