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Conjugacy classes in loop groups and \(G\)-bundles on elliptic curves. (English) Zbl 0992.20034
Authors’ introduction: Let \(\mathbb C[[z]]\) be the ring of formal power series and \(\mathbb C((z))\) the field of formal Laurent power series, the field of fractions of \(\mathbb C[[z]]\). Given a complex algebraic group \(G\), we will write \(G((z))\) for the group of \(\mathbb C((z))\)-rational points of \(G\), thought of as a formal “loop group”, and \(a(z)\) for an element of \(G((z))\). Let \(q\) be a fixed nonzero complex number. Define a “twisted” conjugation action of \(G((z))\) on itself by the formula \[ g(z): a(z)\mapsto{^ga}=g(q\cdot z)\cdot a(z)\cdot g(z)^{-1}. \] We are concerned with the problem of classifying the orbits of the twisted conjugation action. If \(q=1\), twisted conjugation becomes the ordinary conjugation, and the problem reduces to the classification of conjugacy classes in \(G((z))\).
In this paper we are interested in the case \(|q|<1\). Let \(G[[z]]\subset G((z))\) be the subgroup of \(\mathbb C[[z]]\)-points of \(G\). A twisted conjugacy class in \(G( (z))\) is called integral if it contains an element of \(G[[z]]\). Introduce the elliptic curve \({\mathcal E}=\mathbb C^*/q^\mathbb Z\).
Our main result is the following. Theorem 1.2. Let \(G\) be a complex connected semisimple algebraic group. Then there is a natural bijection between the set of integral twisted conjugacy classes in \(G((z))\) and the set of isomorphism classes of semistable holomorphic principal \(G\)-bundles on \(\mathcal E\).

20G20 Linear algebraic groups over the reals, the complexes, the quaternions
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14H52 Elliptic curves
20E45 Conjugacy classes for groups
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