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

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

(d2) d(gzg -1,gyg-z) =

(d2) d(gzg -1,gyg-z) = d(z, y). 146 If H is a closed subgroup of G, we have an action of T(H) on G defined by restric- tion of the action of T(G). Any T(G)-equivaxiant riemannian metric on G is, of course, T(H)-equlvariant. In the sequel, we always assume a T(G)-equlvariamt metric on G, what- ever the subgroup H. Most of our results hold, however, with the weaker assumption of T(H)-equlvariance. We adopt the convention that if ~ is a subset of G then H. ~ denotes the Hr-orbit of ~. In particular, H .g will be the coset gH. Lemma 3.1 Let H be a closed subgroup of G. Then I. H = T(H).e. 2. T(H). = Ho 3. /f~ = {Evly E H} is a family of slices for the T(H)-action on G, hEeh -1 = Ee, h E H. Proof: Statements 1 and 2 are trivial. Statement 3 follows by property ($2) of slices. • Lernma 3.2 Let ~ = {~v]Y e H} be a family of slices for the T(H)-action on G. Let y E H and set U = T(H). ~u" Then 1. U is independent of the choice of y E H 2. U=H.Ey. S. Hr acts yreely on V. 4. For all z E U, the Hr-orbit Hr .z = zH meets Eu at precisely one point. Furthermore, this intersection is transversal. Proof: All statements follow from basic properties of slices. • Proposition 3.1 Let H be a closed subgroup of G and ~ = {EulY E H} be a family of slices for the action ofT(H) on G. There e~sts a smooth map X H : H. ~e ~ Ee satisfying: 1. 9H = xn(g)H, all g E H. Ec 2. xH(hgh-lt) = hxH(g)h -1, all t,h E H, g E H. Ee Further, if we replace H by a subgroup K of H with K ° = H °, then X K = xHI K. Ee In particular, X K = X H on a neighbourhood of E, in H. E,. Proof.- Let g E H. ~e. Using Lemma 3.2(4), we define xg(g) to be the unique point of intersection of gH with Ee. Since the intersection is transverse, this construction defines X n as a smooth map from H • ~e to ~e. By construction, X H satisfies 1. We claim that Xg(hgh-lt) = hxg(g)h -1, all h,t e H. By definition of X H, hgh-ltH = xH(hgh-]t)H Now hgh-ltH = hgh-lH = hgHh -a = hxH(g)Hh -1 = hxY(g)h-XH. By Lemma 3.1(3), hxn(g)h -x E ~e and so, by uniqueness of xH(g), hxH(g)h -1 = xH(hgh-lt). The final assertion of the proposition is an immediate consequence of our constructions. •

147 ttemark 3.1 It follows by our methods (see also [7]) that for g near H, xH(g) is the unique point minimising distance from the coset gH to e, relative to the T(G)-equivaxiant metric on G. Lemma 3.3 Suppose that K C H are elosed subgroups of G. Then the map X H : H. ~e --~ ~e given by Proposition 3.1 satisfies for all g E N(K) n H. ~e. x'(g) e c(z), Proof." Let g E N(K) N H.~e, k E K. Since K C H, it follows by Proposition 3.1 that xH(kgk -1) = kxH(g)k -1 But kgk -1 = gk ~, for some k I E K, since g E N(K). Again, by Proposition 3.1, we have xH(gk ') = xH(g). Hence, xH(g) ---- xH(kgk -1) = kxH(g)k -1, all k e K. That is, xH(g) e c(g). • As an immediate corollary of Lemma 3.3, we have the following useful result describing the identity component of the normaliser of H in G. Proposition 3.2 Let H be a closed subgroup of G. Then I. N(n) ° = 0 (N(n)IH) ° (c(n)l(Z(n))) °. Remark 3.2 In general, it is false that N(H) = C(H).H. Let L(K) denote the Lie algebra of the closed subgroup K of G. As a simple conse- quence of Proposition 3.2 we have Lemma 3.4 Let H be a closed subgroup of G. Let E denote the orthogonal complement of L(H) in L(N(H)) relative to the T(G)-equivariant metric on G. Then E c L(C(U)) If H is a dosed subgroup of G, we have an action of H on G/H defined by left- translation: h(g[H]) = hg[H]. The action of He on G drops down to the action by left translations on G/H whilst Hr drops down to the trivial action on G/H. In particular, the quotient map q : G ~ G/H is equivariant with respect to these actions in the sense that if p = (a, b) E T(H) then for all g E G, a, b E H we have q((a,b)g) = aq(g) Proposition 3.3 There exists an open H-invariant neighbourhood A of [H] in G/TI and a smooth local section a : A C G/H --* G such that for all h E H and x E A we have o(hx) = ho(x)h -1

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