5 years ago

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

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

lemmas. 150 Our proof is

lemmas. 150 Our proof is very similar to that of Theorem 4.21 in [1]. We break the proof into two Lemma 4.3 Let X E S and suppose Z is a cyclic subgroup of G such that (1) [ZI = IX]; and (2) H(Z) = X. Iet T1, 7"2 be tonal subgroups of G o contained in C(Z). Suppose that T1 × Z and T2 x Z are X-mazimal. Then 7"1 x Z, T2 × Z are conjugate subgroups of G. Proof: It follows from our assumptions that T1, T2 C C(Z) °. Since T1 × Z, 7"2 x Z axe X-maximal, 7'1, 7"2 axe maximal tort in C(Z) °. Hence, T1 and T2 axe conjugate subgroups of C(Z) °. That is, there exists t 6 C(Z) ° C G o such that tTlt -1 = T2. Since t 6 C(Z), it follows that t(T1 x Z)t -I = T2 x Z. • Lemma 4.4 Let X 6 S and K be an X-maximal subgroup of G. Let u be a topological generator of K. Suppose that u lies in the connected component G* of G. Then for every z E G*, there exists t E G O such that x E tKt -I. Proof: Consider the adjoint representation of K on L(G). Since K is abelian, K acts trivially on L(K) C L(G). Moreover, since K is X-maximal, the trivial factor of the K representation on L(G) is precisely L(K). We may therefore decompose the K represen- tation L( G) as m i=l where each ~ is a non-trivial irreducible real representation of K. Since K is abelian, dim(R) = 1,2, all i. If dim(R) = 1, the action of K on Y~ is given as multiplication by exp(2riSi(k)), where/~i : K ~ R./Z is a non-trivial homomorphism taking the values 0, 1/2. If dim(V~) = 2, the representation of K on Vi is complex and, identifying ~ with C, the action of K on Y~ is given as multiplication by exp(2riOi(k)), where 81 : K ~ tL/Z is a non-trivial homomorphism. Note that since u is a topological generator of K, we have (D) 81(u) ~ 0, i = 1,...,m. We now complete the proof of the Lemma by using the Lefschetz fixed point argument used by Adams[l] in his proof of Theorem 4.21 (op. cit). Specifically, let G* be the Lie subgroup of G generated by u and G °. Given x E G °, consider the map f : G*/K --* G°/K defined by f(u[K]) = xu[K]. To prove the lemma, it suffices to show that f has a fixed point. By the Lefschetz fixed point theorem it it is enough to prove that the Lefschtez index A(f) ~ 0. By standard homotopy arguments (see [1]), one can reduce to showing that A(]0) ~ 0, where ]0 : G)/K --' G'/H is induced from the map f0 : G ° --~ G ° given by g ~ uzu -1. The proof that A(]0) ~ 0 uses (D) and follows the corresponding computation in [1, Theorem 4.21]. We omit details. • Proof of Theorem 4.1: Let IHI = p and choose Ul,UZ E G such that Ki = K~ x Zi, where Zi is generated by ul, i = 1,2. If Z1 ~ Z2, it follows by Lemma 4.4 that there exists t E G o such that u2 E tKlt -1. Suppose u2 = tfilt -1, fil E IQ. Then fil has order p and K1 is generated by K ° and ill. Hence K1 is conjugate to tK°t -1 x Z2. Now apply Lemma 4.3. • Definition 4.5 Let H e S(G). We define r~G,H) to be the dimension of an H-maximal subgroup of G.

151 Remark 4.3 It follows from Theorem 4.1 that rk(G, H) is well defined for all H 6 S(G). Proposition 4.1 Let H 6 S(G) be of order p. Let K be a monogenic compact abelian subgroup of G such that II(K) = H. Then 1. K ~- T r x Zr~, where q > 1 is the number of connected components of K n G °. ~. There ezists an H-mazimal subgroup A" of G with ~" D_ I(. Proof: Since K is monogenic it follows from Lemma 4.1 that K c~ T r x Zs, for some s > 0. But H(K) = H, so p divides s and (1) is proved. To prove (2), choose any//-maximal subgroup J of G. Choose a topological generator u of K. By Lemma 4.4, there exists t E G o such that u 6. tJt -1. Hence, K is contained in the//-maximal subgroup tJt -1 of G. • Example 4.1 Let G be the n-fold direct product of 0(2). Then G/G ° ~ Z~. For f. 0 < r < n, let fir(r) be the subgroup of G generated by (e, e .... ,e,s,... ,x), where ~; is any element of 0(2) \ S0(2). Then fir(r) 6. S(G), IH(r)l = 2 and rk(G, fir(r)) = n - r. Example 4.2 Let n > 2. Let us denote the non-trivial cyclic subgroup of O(n)/SO(n) by fir. Obviously fir c~ Z2. Then rk(SO(n)) if n is even rk(O(n),II) = rk(SO(n)) - 1 if n is odd 5 Equivariant dynamics on a group orbit In this section we review and extend work of Field[6, 9] and Krupa[14] on the structure of equivariant diffeomorphisms and flows on a G orbit. We start by considering the case of diffeomorphisms. Lemma 5.1 Let H be a closed subgroup of G and G act by left translations on G/H. Then 1. Every G-equivariant map f : G/tt --, G/H is smooth. 2. Diff~°G ( G/ H ) is naturally isomorphic to Diff~N(H)( N ( H) / H ). 3. Diff~G (G/H ) and Diff~N(H)(N(H)/H ) are naturally anti-isomorphic to N ( tt ) ] It. Proof: Statement 1 follows easily using the method of proof of Lemma 2.1. State- ment 2 is trivial. Suppose f E Diff~G(G/H ). Then f([H]) = 9[H], for some g E N(H). Clearly, g[H] E N(H)/H depends only on f and not on the particular choice of g 6 N(H). Conversely, if g 6 N(H) and we define f([H]) = g[H], f extends uniquely to a G-equlvariant diffeomorphism of G/H. These constructions define a natural bijection between Diff~°G(G/H) and N(H)/H. It is easy to verify that this bijection is an anti- isomorphism of groups. • Given f 6 Di/]~G (G/H ) and x 6 G/H, define Ol(x ) = closure{f"(z)ln > 0}

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