Conjugacy classes in loop groups and \(G\)-bundles on elliptic curves. (English) Zbl 0992.20034

Baranovsky, Vladimir; Ginzburg, Victor

Int. Math. Res. Not. 1996, No. 15, 733-751 (1996).

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

Keywords:

complex algebraic groups; orbits; twisted conjugation actions; twisted conjugacy classes; elliptic curves; connected semisimple algebraic groups; semistable holomorphic principal bundles