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Let $X$ be a compact Riemann surface with genus $2$. Can you give me examples of stable principal $SL(2)$-bundles on $X$?

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up vote 2 down vote accepted

Take any irreducible flat $SU(2)$-bundle over your surface.

Edit 1: Explicitly, take two noncommuting rotations $a, b$ of order $>2$ in $SO(3)$ and lift them (arbitrarily) to elements $A, B$ in $SU(2)$, which is a 2-fold cover of $SO(3)$. Now, let $\pi$ be the fundamental group of the genus 2 surface $S$, $$ \pi=<a_1, b_1, a_2, b_2| [a_1,b_1][a_2,b_2]=1>. $$ Let $F_2$ be the free group on two generators $x, y$. Define epimorphism $\pi\to F_2$ by sending $a_1$ to $x$, $b_1$ to $1$, $a_2$ ot $y$ and $b_2$ to $1$. Lastly, consider the homomorphism $F_2\to SU(2)$ sending $x$ to $A$ and $y$ to $B$. Let $h: \pi\to SU(2)$ be the composition of the above homomorphisms. Using the associated bundle construction, see my answer in

Compatibility of a connection and metric

we obtain a flat principal $SU(2)$-bundle over the surface $S$ with holonomy $h$. Embedding $SU(2)$ in $G=SL(2,C)$, we obtain a holomorphic principal $G$-bundle over $S$. In order to see that this bundle is stable note that the flat $SU(2)$-bundle is irreducible (due to our choice of $A, B$); we can now apply the theorem of Narasimhan-Seshadri that the corresponding rank 2 holomorphic bundle is stable. Therefore, the corresponding principal $G$-bundle is stable as well.

Edit 2: You can find the statement and references to proofs of Narasimhan-Seshadri theorem in (google is your friend!). I am using above the "easy" direction of the theorem, namely that vector bundles associated with unitary representations of $\pi_1$ are semistable; they are stable provided that the unitary representation is irreducible, i.e., admits no proper invariant subspace. In other words, the associated flat bundle is irreducible as a flat bundle.

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Could you describe explicitly it? – ArthurStuart Oct 8 '13 at 13:07
I'm sorry, I have some questions: are $a,b$ in $SU(2)$? What is surface group? Is $F_2 \simeq \mathbb{Z}_2$? What type of associate bundle have I to consider? Thank's. – ArthurStuart Oct 8 '13 at 13:48
I added details with answers to your questions. – studiosus Oct 8 '13 at 17:52
I'm sorry, but why we have to take the order of $a,b$ $>2$? – ArthurStuart Oct 8 '13 at 20:14
I'm sorry, but why we have to take the order of $a,b$ $>2$? And in you construiction of associated bundle you obtain a flat vector bundle not a principal $G$-bundle... – ArthurStuart Oct 8 '13 at 22:46

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