Let X be a compact Riemann surface of genus g≥2\documentclass[12pt]{minimal}
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\begin{document}$$g\ge 2$$\end{document} and let n≥3\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 3$$\end{document} be an integer number. The group of automorphisms of the moduli space of vector bundles over X with rank n and trivial determinant is isomorphic to H1(X,Z/(n))⋊(Out(SL(n,C))×Aut(X))\documentclass[12pt]{minimal}
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\begin{document}$$H^1(X,{\mathbb {Z}}/(n))\rtimes ({{\,\textrm{Out}\,}}({{\,\textrm{SL}\,}}(n,{\mathbb {C}}))\times {{\,\textrm{Aut}\,}}(X))$$\end{document}. Several papers have studied the subvarieties of fixed points for the action of the unique outer involution of SL(n,C)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\textrm{SL}\,}}(n,{\mathbb {C}})$$\end{document} on this moduli space. In this paper, explicit descriptions of the fixed points for the actions of the elements of H1(X,Z/(n))\documentclass[12pt]{minimal}
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\begin{document}$$H^1(X,{\mathbb {Z}}/(n))$$\end{document}, H1(X,Z/(n))⋊Out(SL(n,C))\documentclass[12pt]{minimal}
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\begin{document}$$H^1(X,{\mathbb {Z}}/(n))\rtimes {{\,\textrm{Out}\,}}({{\,\textrm{SL}\,}}(n,{\mathbb {C}}))$$\end{document}, and Out(SL(n,C))×Aut(X)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\textrm{Out}\,}}({{\,\textrm{SL}\,}}(n,{\mathbb {C}}))\times {{\,\textrm{Aut}\,}}(X)$$\end{document} on the moduli space of rank n and trivial determinant vector bundles over X are provided. For the description of the fixed points for the action of the elements of Out(SL(n,C))×Aut(X)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\textrm{Out}\,}}({{\,\textrm{SL}\,}}(n,{\mathbb {C}}))\times {{\,\textrm{Aut}\,}}(X)$$\end{document}, the notion of Galois bundle is introduced. Specifically, Galois bundles over X admitting a nontrivial automorphism which commutes with the Galois structure are constructed associated with an involution σX\documentclass[12pt]{minimal}
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\begin{document}$$\sigma _X$$\end{document} of X. Finally, it is discussed how the description of fixed points for the action of the elements of [inline-graphic not available: see fulltext] is covered by the descriptions above.
