Smooth SL(m, C) SL(n, C)-actions on the (2m + 2n 1)-sphere

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1 Smooth SLm, C) SLn, C)-actions on the 2m + 2n 1)-sphere Kazuo Mukōyama Abstract We shall study smooth SLm, C) SLn, C)-actions on the natural 2m+2n-1)-sphere whose restricted action to the maximal compact subgroup is standard. We shall show that such an action is characterized by a smooth R 2 -action on the 3-sphere and consequently there exist uncountably many topologically distinct such actions. 1 Introduction. Let ψ be the orthogonal SUm) SUn)-action on the natural 2m+2n 1)- sphere. If m, n 2, then the action has codimension-one principal orbits with SUm 1) SUn 1) as the principal isotropy subgroup. The action restricted to the principal isotropy subgroup has the fixed point set F diffeomorphic to S 3. In this paper, we shall study smooth SLm, C) SLn, C)-actions on the sphere for m, n 4 whose restricted action to the maximal compact subgroup SUm) SUn) is ψ. The characterizaton of such actions of non-compact simple Lie groups on a standard sphere was first introduced by Asoh [1] and improved by [6], [3] and [7]. SLm, R) SLn, R)-actions and SLm, H) SLn, H)-actions are studied by [8] and [9], respectively. In 2, we shall give some subgroups of SLm, C) SLn, C) using in the later. In 3, we shall give examples of smooth SLm, C) SLn, C)-actions on the 2m + 2n 1)-sphere with one open orbit. The other sections, we shall construct such actions on the sphere with plural open orbits. 2Mathematics Subject Classification. Primary 57S2. Key words and phrases. Non-conpact Lie group, smooth action. 1

2 2 Certain subgroups of SLm, C) SLn, C). Let Ln), L n), Nn) and N n) denote the closed connected subgroups of SLn, C) consisting of matrices in the form ,, and respectively. Let σ be the automorphism of SLn, C) defined by σa) = A 1. Then the restriction σ SUn) is the identity on SUn) and we see σl n)) = Ln) and σn n)) = Nn). We have the following lemma. The proof is same as [9]. Hence we omit it. Lemma 2.1. Suppose m 4 and n 4. Let S be a closed subgroup of SLm, C) SLn, C) such that the natural SUm) SUn)-acton on the homogeneous space SLm, C) SLn, C))/S has at most three isotropy types SUm) SUn 1), SUm 1) SUn) and SUm 1) SUn 1). Then 1) If S contains SUm) SUn 1), then SLm, C) Ln) S SLm, C) Nn). 2) If S contains SUm 1) SUn), then Lm) SLn, C) S Nm) SLn, C). 3) If S contains SUm 1) SUn 1) but S does not contain SUm) SUn 1) nor SUm 1) SUn), then Lm) Ln) S Nm) Nn). Here the results are described up to an equivalence under automorphisms of SLm, C) SLn, C) which leave invariant the subgroup SUm) SUn). Now we also define the closed connected subgroups N n a) of Nn) and Na : b) of Nm) Nn) consisting of the matrices in the form exp aθ.., exp aθ.., exp bθ..,, respectively. Here θ R and a, b are complex numbers satisfying a, b), ). 3 Twisted linear actions. Let α, β be complex numbers satisfying Re α >, Re β >. Then there is the smooth action Φ α,β of SLm, C) SLn, C) on S 2m+2n 1 defined by Φ α,β A, B), u v) = expαθ)au expβθ)bv, 3.1) 2

3 where A, B) SLm, C) SLn, C), u v S 2m+2n 1 C m C n and θ is a certain real number. The restricted SUm) SUn)-action is the standard action ψ. The action Φ α,β is called the twisted linear action. We see the action Φ α,β has just three orbits; One is an open orbit and the others are compact orbits. Especially, if Re α = Re β, then the righthand side of 3.1) is presented as follows. exp 1 + ic 1 ) log Au Bv) )Au exp 1 + ic 2 ) log Au Bv) )Bv, where c 1, c 2 are certain real numbers. We denote the action by Φ c1,c 2 ). The isotropy types of this action are N1 + ic 1 : 1 + ic 2 ) at the open orbit and N m 1 + ic 1 ) SLn, C) and SLm, C) N n 1 + ic 2 ) at the compact orbits. We say two actions Φ c1,c 2 ), Φ c 1,c 2 ) of SLm, C) SLn, C) on S 2m+2n 1 are equivalent if there exists an SLm, C) SLn, C)-equivariant diffeomorphism. Considering the restriction of the equivariant diffeomorphism to the two compact orbits, we have the following. Lemma 3.1. If two actions Φ c1,c 2 ), Φ c 1,c 2 ) are equivalent, then c 1, c 2 ) = c 1, c 2). Corollary 3.2. There exist non-countably many non-equivalent smooth SLm, C) SLn, C)-actions on S 2m+2n 1 with one open orbit whose restricted SUm) SUn)-action is ψ. 4 Smooth SLm, C) SLn, C)-actions on S 2m+2n 1 We consider smooth SLm, C) SLn, C)-actions on S 2m+2n 1 whose restricted SUm) SUn)-action is ψ. The action ψ has codimension-one principal orbits with SUm 1) SUn 1) as the principal isotropy subgroup and has two singular orbits. We put F = {ue 1 + ve m+1 u 2 + v 2 = 1}, where e 1,, e m, e m+1,, e m+n is the orthonormal standard basis of C m C n. Then F is the fixed points set of the restricted SUm 1) SUn 1)-action of ψ on S 2m+2n 1. Hereafter we define G = SLm, C) SLn, C), K = SUm) SUn), H = SUm 1) SUn 1). Let Φ : G S 2m+2n 1 S 2m+2n 1 be a smooth G-action whose restricted K-action is ψ. We denote by F S) the fixed point set of the restricted S-action of the action Φ on S 2m+2n 1. Then we see that F = F H). Lemma 4.1. For any smooth G-action on S 2m+2n 1 whose restricted K-action is ψ, the set F coincides with F Lm) Ln)), F L m) Ln)), F Lm) L n)) or F L m) L n)). The proof is same as [8],[9]. By this lemma and the automorphism σ of G, we may assume F = F Lm) Ln)) for the action Φ. Moreover, we have the following. 3

4 Lemma 4.2. Let Φ : G S 2m+2n 1 S 2m+2n 1 be a smooth G-action whose restricted K-action is ψ and F = F Lm) Ln)). Suppose x F. Then 1) If G x contains SUm 1) SUn), then G x = N n 1 + ic 1 ) SLm, C) for some c 1 R 2) If G x contains SUm) SUn 1), then G x = SLm, C) N n 1 + ic 2 ) for some c 2 R Proof. We shall show the case 2). By Lemma 2.1, we have SLm, C) Ln) G x SLm, C) Nn). By the dimension of the orbit, we have dim G x dim SLm, C) Nn)). Suppose dim G x = dim SLm, C) Ln)). Then the orbit Gx) is 2n dimensional and must contains a principal orbit of the restricted K-action ψ. This is a contradiction. Hence we obtain the result. We denote the elements D 1,c1 θ), D 2,c2 θ), D 1 t) and D 2 t) by exp 1 + ic 1 )θ... D 1,c1 θ) =. xi m 1 C SLm,C)SUm 1)), exp 1 + ic 2 )θ... D 2,c2 θ) =. yi n 1 C SLn,C)SUn 1)), D 1 t) = diagt, t 1, 1,, 1) N SUm) SUm 1)), D 2 t) = diagt, t 1, 1,, 1) N SUn) SUn 1)), respectively, where θ, c 1, c 2 R, t U1), x m 1 exp1+ic 1 )θ = 1, y n 1 exp1+ ic 2 )θ = 1, C A B) is the centralizer of B in A and N A B) is the normalizer of B in A. We put D 1,c1 R) = {D 1,c1 θ) θ R}, D 2,c2 R) = {D 2,c2 θ) θ R}, Mc 1, c 2 ) = D 1,c1 R) D 2,c2 R) T = {D 1 t 1 ), D 2 t 2 )) t 1, t 2 U1)}. Now we have the following decompositions by the slight deformation of the Iwasawa decompositions of the Lie groups SLm, C) and SLn, C). SLm, C) = SUm)D 1,c1 R)Lm), SLn, C) = SUn)D 2,c2 R)Ln), 4.1) where c 1, c 2 are any real numbers. 4

5 Let Φ : G S 2m+2n 1 S 2m+2n 1 be a smooth G-action whose restricted K-action is ψ and F = F Lm) Ln)). Then by Lemma 4.2 we have G e1 N m 1 + ic 1 ), G em+1 N n 1 + ic 2 ) for some real numbers c 1, c 2. The restricted Mc 1, c 2 )-action on F implies the R 2 -action Φ Mc1,c 2) : R 2 F F defined by Φ Mc1,c 2 )θ 1, θ 2 ), u, v)) = ΦD 1,c1 θ 1 ), D 2,c2 θ 2 )), u, v)), where u, v) = ue 1 ve m+1 F. We can describe Φ Mc1,c 2 )θ 1, θ 2 ), u, v)) = aθ 1, θ 2, u, v), bθ 1, θ 2, u, v)) for certain complex valued functions a, b. Since D s,cs θ s )D s t s ) = D s t s )D s,cs θ s ) s = 1, 2), we have aθ 1, θ 2, t 1 u, t 2 v) = t 1 aθ 1, θ 2, u, v), bθ 1, θ 2, t 1 u, t 2 v) = t 2 bθ 1, θ 2, u, v), for any t 1, t 2 ) T. Hence we see that there exist smooth complex functions α, β on R 2 F satisfying aθ 1, θ 2, u, v) = uαθ 1, θ 2, u, v), bθ 1, θ 2, u, v) = vβθ 1, θ 2, u, v). the functions α, β satisfy the following relations: for any t 1, t 2 ) T. We also have αθ 1, θ 2, t 1 u, t 2 v) = αθ 1, θ 2, u, v), βθ 1, θ 2, t 1 u, t 2 v) = βθ 1, θ 2, u, v), α,, u, v) = β,, u, v) = 1. Theorem 4.3. Let Φ, Φ : G S 2m+2n 1 S 2m+2n 1 be smooth G-actions whose restricted K-action is ψ and F = F Lm) Ln)). Suppose Φ Mc1,c 2 ) = Φ Mc 1,c 2 ) and c 1, c 2 ) = c 1, c 2). Then Φ = Φ Proof. For the action Φ we have real numbers c 1, c 2 satisfying G e1 N m 1 + ic 1 ), G em+1 N n 1 + ic 2 ). Using the numbers c 1, c 2, we decompose SLm, C), SLn, C) as 4.1). Let g = g 1, g 2 ) G and decompose g s as Then we have g s = k s D s,cs θ s )l s s = 1, 2). Φg 1, g 2 ), ue 1 ve m+1 ) = αuk 1 e 1 βvk 2 e m+1. 5

6 We also have for uv. Hence we obtain Φg 1, g 2 ), ue 1 ve m+1 ) θ 1 = log g 1ue 1 ue 1, θ 2 = log g 2ve m+1 ve m+1, = α exp 1 + ic 1 ) log g 1ue 1 ue 1 )g 1ue 1 β exp 1 + ic 2 ) log g 2ve m+1 ve m+1 )g 2ve m+1 for u >, v >. Here α = αlog g 1ue 1 ue 1, log g 2ve m+1 ve m+1, ue 1, ve m+1 ), β = βlog g 1ue 1 ue 1, log g 2ve m+1 ve m+1, ue 1, ve m+1 ). For any u v S 2m+2n 1 satisfying u = and v =, there exists k 1, k 2) K and u >, v > such that u = k 1ue 1 and v = k 2ve m+1. Since Φg 1, g 2 ), u v) = Φg 1 k 1, g 2 k 2), ue 1 ve m+1 ), we obtain Φg 1, g 2 ), u v) = α exp 1 + ic 1 ) log g ) 1u g 1 u β exp 1 + ic 2 ) log g ) 2v g 2 v, u v where α = αlog g 1u u, log g 2v, u, v ), v β = βlog g 1u u, log g 2v, u, v ). v For u v =, we see Φg 1, g 2 ), u ) = exp 1 + ic 1 ) log g 1 u )g 1 u, Φg 1, g 2 ), v) = exp 1 + ic 2 ) log g 2 v )g 2 v. Hence we see that Φ = Φ if Φ Mc1,c 2 ) = Φ Mc 1,c 2 ) and c 1, c 2 ) = c 1, c 2). 5 Induced smooth R 2 -action on F Let Φ : G S 2m+2n 1 S 2m+2n 1 be a smooth G-action whose restricted K-action is ψ and F = F Lm) Ln)) and let Φ Mc1,c 2 ) be the induced R 2 - action on F. We denote X = u, v) F and Θ = θ 1, θ 2 ) R 2. Then the action Φ c1,c 2 ) is described by Φ Mc1,c 2)Θ, X) = uαθ, X), vβθ, X)) 6

7 for certain complex valued smooth functions α, β on R 2 F satisfying the equations i) αθ, ψt 1, t 2 ), X)) = αθ, X), βθ, ψt 1, t 2 ), X)) = βθ, X), ii) αo, X) = βo, X) = 1, where t 1, t 2 ) T see 4). Since Φ Mc1,c 2 ) is the R 2 -action, we also obtain the equations iii) αθ + Θ, X) = αθ, αθ, X)u, βθ, X)v))αΘ, X), iv) βθ + Θ, X) = βθ, αθ, X)u, βθ, X)v))βΘ, X). By these equations, we have αθ, X), βθ, X) for any Θ, X) R 2 F. Let α, β be complex valued smooth functions on R 2 F satisfying the four conditions above and c 1, c 2 be any real numbers. We define a mapping Φ : G S 2m+2n 1 S 2m+2n 1 by where Φg 1, g 2 ), u v) = α exp 1 + ic 1 ) log g 1u u )g 1u β exp 1 + ic 2 ) log g 2v v )g 2v α = αlog g 1u u, log g 2v, u, v ), v β = βlog g 1u u, log g 2v, u, v ) v for any u v S 2m+2n 1 satisfying u and v = and Φg 1, g 2 ), u ) = exp 1 + ic 1 ) log g 1 u )g 1 u, Φg 1, g 2 ), v) = exp 1 + ic 2 ) log g 2 v )g 2 v. By a routine work, we have Proposition 5.1. Φ is an abstract G-action on S 2m+2n 1. However this action is not smooth in general. Proposition 5.2. If α = β = 1, then Φ is not smooth. The proof is same as [8]. In the rest of this paper, we construct some smooth G-actions on S 2m+2n 1. 6 Construction of smooth G-actions on S 2m+2n Certain smooth actions of SLm, C) on C m For any real number c, we define the map ϕ m,c : SLm, C) C m C m by ϕ m,c A, u) = s expic log s)au, 7

8 where 2 s =. 1 u Au u 2 ) 2 ) 1 2 Then we see that ϕ m,c is a smooth SLm, C)-action on C m and the three sets {u C m u < 1}, {u C m u = 1}, {u C m u > 1} are invariant under the action ϕ m,c. Moreover, if u = 1, then we have and if u = 1, then we have exp ic log s) 1 ϕ m,c A, u) 2 ϕ 1 m,ca, u) = 1 u 2 Au ϕ m,c A, u) = exp 1 + ic) log Au )Au. We put D 2m = {u C m u < 1} and define the map h : D 2m D 2m by hu) = exp ic log1 u 2 )) 1 u 2 u. Then the map h is an equivariant diffeomorphism of D 2m with the action ϕ m,c onto C m with the linear action. Now we define a smooth R-action φ m,c : R C C by φ m,c θ, z)e 1 = ϕ m,c D 1,c θ), ze 1 ). We see that the R-action φ m,c on C corresponds to the following vector field η 1 on C : η 1 = 1 z z ic)z d dz + 1 ic) z d ), d z where we identify C with R 2 by z = x + iy and put d dz = 1 2 x i ) d, y d z = 1 2 x + i y 6.2 Construction of smooth G-actions on S 2m+2n 1 First, we shall construct a smooth one-parameter group on F. Let k, l be diffeomorphisms of C S 1 onto open sets of F defined by ) z kz, t) =, 4t, 16 + z z 2 ) 4t lz, t) =, z z z 2 ). 8

9 Let ρx) be a smooth real valued even function such that ρx) = 1 for x 1, ρx) = for x 2, < ρx) < 1 for 1 < x < 2 and monotone decreasing at 1 < x < 2 and let σz) be the smooth function on C defined by σz) = ρ z ). Let η 2 be the tangent vector field induced from the R 2 -action Φ M,) on F. Namely, we put η 2 = u 2 v v + v ) v 2 u v u + ū ). ū Define a tangent vector field ξ on F by ξ = k σz 2 )η 1 + O) + l σz 2 )η 1 + O) + σ 12 5 u2 ) σ6u 2 ))η 2, where O is the zero vetor field on S 1. Let φ be the one-parameter group on F corresponding to the vector field ξ. Since the vector field ξ is invariant under the standard T - action on F, we see that ψd 1 t 1 ), D 2 t 2 )), φθ, u, v))) = φθ, ψd 1 t 1 ), D 2 t 2 )), u, v))) 6.1) for t 1, t 2 ) T. Now we define a smooth R 2 -action ϕ on F by φθ 1, u, v)) for u 2 < 1 9, ϕθ 1, θ 2 ), u, v)) = φθ 2, u, v)) for u 2 > 8 9 φaθ 1 + bθ 2, u, v)) for 1 9 < u 2 < 8 9, where a, b are real numbers. Since ξ = for 1 9 < u 2 < 1 6 and 5 6 < u 2 < 8 9, we see that ϕ is smooth on F. By the use of 6.1), we can put ϕθ 1, θ 2 ), u, v)) = uαθ 1, θ 2, u, v), vβθ 1, θ 2, u, v)) for some complex valued functons α, β on R 2 F, where the functons satisfy four conditions in 4. Thus we can construct an abstract G-action Φ on S 2m+2n 1 defined in 4 Now we show that Φ is smooth on neighborhoods of u = and v =. Let k : C m S 2n 1 S 2m+2n 1 be the diffeomorphism onto an open set of S 2m+2n 1 defined by Then we see ku, v) = ΦA, B), u v) = k 1 u 4 v u u 2 ϕ m,c A, ) 4 u), 1 1 u 2 Bv Bv 9

10 for u 2 < Hence Φ is smooth on a neighborhood of u =. Similarly, we may show that the smoothness of Φ on a neighborhood of v =. Therefore we see that the G-action Φ on S 2m+2n 1 is smooth. By this G-action, we obtain the following: Theorem 6.1. There exists a smooth SLm, C) SLn, C)-action on S 2m+2n 1 such that it has five open orbits and the isotropy types of them are the following : Lm) L n 1) two orbits), N1 + ic)b : a), N m 1) Ln) two orbits) for any real numbers a, b, c such that a, b), ). Corollary 6.2. There are uncountably many topologically distinct smooth SLm, C) SLn, C)-actons on S 2m+2n 1 with five open orbits whose restrict SUm) SUn)-action is ψ. Remark 6.3. By an exchange of the third term of the vector field ξ, we may obtain another smooth SLm, C) SLn, C)-acton on S 2m+2n 1 with 4k + 1) open orbits. Acknowledgement. The author wishes to express his gratitude to Professor Fuichi Uchida who offered this problem and many helpful suggestions and advice. References [1] T. Asoh, On smooth SL2, C) actions on 3-manifolds, Osaka J. Math ), [2] G.E. Bredon, Introduction to compact transformation groups, Pure and Applied Math. 46, Academic Press, [3] K. Mukōyama, Smooth Sp2, R)-actions on the 4-sphere, Tôhoku Math. J ), [4] R.S. Palais, A global formulation of the Lie theory of transformation groups, Memoirs of Amer. Math. Soc ). [5] F. Uchida, Real analytic SLn, R) actions on spheres, Tôhoku Math. J ), [6] F. Uchida, On smooth SO p, q)-actions on S p+q 1, Osaka J. of Math ), [7] F.Uchida: On smooth SO p, q)-actions on S p+q 1, II, Tôhoku Math. J ), [8] F. Uchida, On smooth SLm, R) SLn, R) actions on S m+n 1, Interdiscip. Inf. Sci. 22), [9] S. Kuroki, On the construction of smooth SLm, H) SLn, H) actions on S 4m+n) 1, Bull. of Yamagata Univ. Nat. Sci ),

11 [1] K. Mukōyama, Smooth SLm, C) SLn, C)-actions on the 2m+2n 1)- sphere and on the complex projective m + n 1)-space, in preparation Kazuo Mukōyama Tokyo Metropolitan College of Aeronautical Engineering 17-1, Minami-senju 8-chome Arakawa-ku, Tokyo Japan kmukoyam@kouku-k.ac.jp) 11