4 1. INTRODUCTION above, agrees with the action of CG by multiplication on the space of conjugacy classes in G. Thus, this identification and the choice of the set of simple roots for $ determine a homomorphism CG W(T, G) denoted by c *- wc. For any subgroup C C CG we denote by WQ the image in W(T, G) of C under this homomorphism. Let SWc be the subtorus of T whose Lie algebra is twc, the subspace of t pointwise fixed under the action of wc- When C is the cyclic group generated by c we denote e»c by e»c &ndSWc by SWc. Through the identification of C$ and CG, we have CG represented as a group of automorphisms of A coming from a group of diagram automorphisms of DV(G) preserving the coroot integers ga. Let C C CG be a subgroup. Denote by Ac, resp. A^, the quotient of A, resp. A v , under the action of C. For each orbit a G Ac we define g^ to be riaQa where n-a is the cardinality of the orbit a and ga is the coroot integer associated to any a in this orbit. 1.2. The case of commuting pairs in a simply connected group In [1], the first author proved that, if x and y are commuting elements in a compact, simply connected group G, then there is a maximal torus T C G containing both x and y. Furthermore, two pairs of elements in T are conjugate in G if and only if they are conjugate by an element of W. Thus, the moduli space of conjugacy classes of commuting pairs of elements in G is identified with (T x T)/W. The torus T is the quotient of t by the lattice 2mQy', where Q v is generated by A v , and W is the group of isometries of t generated by reflections in the hyperplanes through the origin defined by the elements of A v . The group W acts on t preserving the lattice 2iriQy and hence there is an induced W-action on T. This is the model result that we carry over into all the other cases we study. 1.3. c-pairs Next, let us consider a compact, connected, simple, but not necessarily simply connected group K with G as simply connected covering. The first invariant of a K-bundle £ over T2 is the characteristic class w(£) G H2(T2 7Ti(K)) = 7Ti(K). We identify TT\{K) with a subgroup of the center CG of G, so that w(£) G CG. If £ has a flat connection and if x, y G K are the holonomy images of the standard generators of the fundamental group of the two-torus, then for any lifts x G G, resp. y G G, of x, resp. y, we have [x,y] = w(£). Our classification results in this case are simplified by three assumptions: (1) We fix the topological type of the bundle £, or equivalently the class w(£) = ceCG. (2) We assume that £ does not lift to a bundle over any non-trivial covering group of K. (3) We classify flat connections on £ up to restricted gauge equivalence, i.e., up to G-gauge equivalence. In this case, it turns out that restricted gauge equivalence is the same as if-equivalence. Translating these conditions gives the following equivalent group theoretic prob- lem. Let G be simple, and let c G CG. A pair of elements (x, y) in G is said to be a c-pair if [x, y] c. We classify c-pairs up to simultaneous conjugation by elements ofG.
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