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).
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$\begingroup$ 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? $\endgroup$ Commented Oct 10, 2013 at 18:24
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$\begingroup$ Yes, the dimension of the moduli space is $(g−1) \dim_{\mathbb{C}} G$. $\endgroup$– abxCommented Oct 11, 2013 at 18:08
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$\begingroup$ @abx As far as understand, Ramanathan's papers are about holomorphic principal $G$-bundles. Is there a reference on the moduli space of ALL principal $G$-bundles over a surface (not necessarily equipped with a complex structure)? This might be a very big space but still would like to know whether such space has been studied in the literature, for say $G=\text{U}(1)$. $\endgroup$– QGravityCommented May 21 at 3:47
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1$\begingroup$ @QGravity: I am not sure I understand the question. The topological classification of $G$-bundles is well-known. For instance, $\operatorname{U}(1) $-bundles are determined by their degree (given by the first Chern class $c_1\in H^2(X,\mathbb{Z})=\mathbb{Z}$). $\endgroup$– abxCommented May 21 at 7:05