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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

148 Proofi With the

148 Proofi With the notation of Proposition3.1, we take the H-invariant open neighbourhood A = q(~e) of [HI in G/H and define the local section a : A --* G by = xH(g), g e H. o Set z "- g[H], g e H.~e. Then z e A and by Proposition 3.1, we have a(hz) = a(hg[H D = xH(hgh -1) = hxH(g)h -~ = ha(z)h -1. | Remark 3.3 The local section given by Lemma 3.3 is H-equivariant with respect to left translations by H on G/H and the action of H = Hc on G. Definition 3.1 We call a smooth local section a : A C G/H --* G satisfying the invari- ance properties of Lemma 3.3, an admissible section of G over G/ H. As an immediate corollary of Lemma 3.3 we have Lemma 3.5 Let a : A -* G be an admissible section of G over G/H. Let K be a closed subgroup of H, and suppose g e N(K) and g[H] E A. Then /.Prom Lemma 3.5 follows e C(K) Proposition 3.4 Let M be a G-manifold and ~ C M be a G orbit. Suppose S = {S=tx G a} is a family of slices for c~. Let a : A C G/G= ~ G be an admissible section of G over G/Gx. Let pa : S= x A --* a(A).Sx be the map defined by pe,(y, a) = a(a)y. For all y e Sx and a = g[H] e A, with g e N(G~), we have p (y,a) C(Gy).y 4 Structure of compact abelian subgroups of a compact Lie group In this section we start by recalling some facts about the maximal torus theory of compact connected Lie groups. We then show how these results may be usefully extended to compact non-connected Lie groups. Our basic references for compact Lie groups are BrScker and Dieck[3] and Adams[I]. For p > 0, let T p denold the p-fold product of S 1 and give T p the associated product group structure. We refer to T p, or any Lie group isomorphic to T p, as a p-dimensional torus group. A compact Lie group is a torus group if and only if it is a connected abelian Lie group. Recall that if G is a compact Lie group then the rank of G, denoted rk(G), is defined to be the dimension of any maximal torus subgroup of G. It follows from BrScker and Dieck[3, I,(4.14)], that if H is a compact abelian subgroup of G then H is isomorphic to T p x A, where dim(H) = p ~ rk(G) and A is a finite abelian group, isomorphic to a product of cyclic groups. For our purposes, we need to investigate the structure of compact abelian subgroups of G in case G is not connected. First we recall some elementary results on Lie groups with a topological generator.

149 Definition 4.1 Let 11 be a closed subgroup of G. We say 11 is monogenic if there exists g E G such that 11 is the closure of the subgroup of G generated by g. We call g a (topological) generator of 11. Lemma 4.1 A subgroup It of G is monogenie if and only if11 is isomorphic to the product of a torus group with a cyclic group. Proof: If 11 is monogenic, 11 is compact abelian and obviously 11/11 ° c~ ZI " for some p > 0. Hence 11 -~ T r × Z~, where r = dim(//). For the converse, see [1, Proposition 4.4]. l Since G o is a normal subgroup of G, G/GO has the structure of a finite group. Let H : G ~ G/G ° denote the quotient map and set P = G/G °. Let ,9 = S(G) denote the set of all cyclic subgroups of P. Given X E S, let IXI denote the order of X. Thus, if IXl = p, x z,,. Lemma 4,2 Let X E S and set IX[ = p. Then there ezists u E G such that 1. The group J generated by u is finite and isomorphic to Zp. 2. H(J) = X. Proof: Choose v E G such that v, v2,...,v p-I ~ G o but v p E G °. Let K be the group generated by v and apply Lemma 4.1. II Definition 4.2 Let X E S. We call an element u E G satisfying the conclusions of Lemma ,t.Y, a representative generator of X. Definition 4.3 Let X E S. An abelian subgroup K of G is of type X if K ~- T" x Zq, where q -- IXl and H(K) -- X. Remark 4.1 Let K be an abelian subgroup of G wlfich is of type X, X E S. Then K contains at least one representative generator of X. Definition 4.4 Let X E S. Let K be an abelian subgroup of G which is of type X. We say K is X-maximal if K is not a proper subgroup of any abelian subgroup of G of type X. The next result is presumably well-known to experts 1 but as we have been unable to locate a suitable reference, we have included a proof. Theorem 4.1 Let X E S(G) and K1, I(2 be X-maximal subgroups of G. Then I. dim(K1) = dim(K2). ~. I(1 and 1(2 are confl~gate subgroups of G. Remark 4.2 Theorem 4.1 is a generalisation of the fundamental theorem on conjugacy of maximal tori in compact connected Lie groups. 1Tudor Ratiu tells me that the result is known to Duistermv~t

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