CHAPTER 1 Introduction Let K be a compact, connected and semisimple Lie group. This paper describes the moduli space of isomorphism classes of flat connections on principal if-bundles over the two-torus and the three-torus. There are two motivations for this study. The first is the relation between flat if-bundles over the two-torus and holomorphic principal bundles over an elliptic curve with structure group the complexification of K, The second is to give a proof of a conjecture of Witten concerning the moduli space of flat if-bundles over the three-torus, used in [20]. Of course, a flat bundle is completely determined by its holonomy representa- tion, so that the problem of classifying flat bundles over two- and three-tori becomes the question of classifying ordered pairs and triples of commuting elements in if, up to simultaneous conjugation. If two elements commute in if, then any lifts of them to the universal cover of K commute up to an element of the center. Thus we shall work in the simply connected cover G of K and study pairs and triples of elements in G which commute up to the center, hence the name "almost commuting." This is the form in which we attack the question. Our point of view is that the extended Dynkin diagram of G, the action of TT\ (if) on this diagram, and the coroot integers associated to this diagram completely determine the answer in a manner which we shall describe below. Notation used throughout this paper © G is a compact, connected and simply connected Lie group, and in particular G is semi-simple, T is a maximal torus of G, and t = Lie(T). o CG denotes the center of G. If G is simple, then CG is cyclic except for G = Spin(4n), n 2. An element of CSpin(4n) whose image in 50(4n) is nontrivial will be called exotic. © For any subset I C G , ZQ{X) denotes the centralizer of X in G. We will denote ZQ(X) by Z(X) when G is clear from the context. o If S is a torus in G, not necessarily maximal, we let W(S, G) be the finite group NG(S)/ZG(S) and we call it the Weyl group of S in G. If s = Lie(S), then we set W{s,G) = W(S,G). Given x,y G G, we define the commutator [x,y] = xyx~1y~~1 and denote conju- gation by x as ix(y) = x y = xyx~l. One convention concerning subtori used throughout the paper is the following. Let i be a vector space with a positive definite inner product (•, •), let A C t be a lattice, such that (v,w) G Z for all v,w G A, and let T = t/A be the associated torus. For any subtorus S C T let s C t be its tangent space. Let s1- be the perpendicular subspace to s. Then s1- is the tangent space of another subtorus Sf C T and Fs = S D S' is a finite group. We denote by S the quotient torus l
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