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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 $$\mathcal E$$.

##### MSC:
 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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