1.4. COMMUTING TRIPLES 5 THEOREM 1.3.1. Let G be simple, and let c G CG. Then the moduli space of c-pairs of elements in G, modulo simultaneous conjugation, is homeomorphic to (SWc xSWc)/W(Sw^G). In a very closely related form, this theorem was first proved by Schweigert [16]. In Theorem 1.6.2 below, we shall describe S c and W(SWc, G) in terms of the extended coroot diagram of G and the action of c on this diagram. 1.4. Commuting triples Next, we let G be simple and we turn to flat G-bundles over the 3-torus. The holonomy of such a connection around the standard basis of the fundamental group of the torus is a commuting triple (x, y, z) in G well-defined up to simultaneous conjugation. Let TQ denote the moduli space of conjugacy classes of commuting triples in G. In general, TQ has several components even though there is only one topological type for a G-bundle over T3. THEOREM 1.4.1. Let G be simple. For any k 1 dividing at least one of the coroot integers ga we set I(k) = {a G A : k \ ga}, and we let S(k) C T be the subtorus with Ue{S(k)) = t(k)= p | Kera. ael(k) Note that dim S(k) is one less than the number of a such that k\ga. Then: (1) For each commuting triple (x,y,z) in G, there is a unique integer k 1 dividing at least one of the coroot integers ga such that S(k) is conjugate to a maximal torus for Z(x,y,z). The integer k is called the order of (2) The order is a conjugacy class invariant and defines a locally constant function on TQ. We define the order of a component X of TQ to be the value of this function on X. (3) If k 1 divides at least one of the ga, there are exactly cp(k) components ofTc of order k, where cp is the Euler (p-function. Given a component X ofTG, let dx = \ dimX + 1. Then x (4) Each component of TQ of order k is homeomorphic to (S{k) x S(k) x S(k)) IW(S{k),G). (5) Let TTf e be orthogonal projection from it to it(k). For a G A, 7Tk(av) is non-zero if and only if a £ I(k). Thus 7Tk determines an embedding of A v Iv{k) into it(k). Let Qw(k) C it(k) be the lattice generated by this image. The torus S(k) is the quotient oft(k) by the lattice 2mQy (k). The group W(S(k), G) is the group of isometries oft(k) generated by reflections in the hyperplanes 7r^(av)-L for a G A I(k). Results along these lines have been obtained independently by Kac-Smilga [11]. In Section 1.7 we state a result which shows that the image of A v I(k)w in it(k) is the extended set of simple coroots of a root system &(t(k)) on ii(k) and explain how to derive its extended coroot diagram from the extended coroot diagram DV(G) and the coroot integers ga.
Previous Page Next Page