aa r X i v : . [ m a t h . M G ] M a r LIFTING ISOMETRIES OF ORBIT SPACES
RICARDO A. E. MENDES
Abstract.
Given an orthogonal representation of a compact group,we show that any element of the connected component of the isom-etry group of the orbit space lifts to an equivariant isometry of theoriginal Euclidean space. Introduction
Given a metric space X , an isometry of X is defined as a bijectionthat preserves the distance function. The set Isom( X ) of all isometriesof X forms a group under composition, which is called the isometrygroup of X .An early result [DW28] (see also [KN96, pages 46–50]) states that if X is connected and locally compact, then Isom( X ), endowed with thecompact-open topology, is also locally compact, and the isotropy groupIsom( X ) x is compact for each x ∈ X . Moreover, if X is itself compact,then so is Isom( X ). Shortly thereafter, Myers and Steenrod provedthat Isom( X ) is a Lie transformation group when X is a Riemannianmanifold, see [MS39], [Kob95, page 41]. Generalizing this, Fukaya andYamaguchi have shown [FY94] that Isom( X ) is a Lie group whenever X is an Alexandrov space .In the present article, we consider a special class of Alexandrovspaces. Namely, given a compact group G acting linearly by isome-tries on a Euclidean vector space V , the orbit space X = V /G is anAlexandrov space with curvature bounded from below by 0. A natu-ral subgroup of Isom(
V /G ) consists of the isometries induced by the G -equivariant isometries of V , that is, isometries that commute withthe G -action (see also Remark 2 below). Our main result is that, up totaking connected components, all isometries of V /G arise in this way:
Theorem A.
Let V be an orthogonal representation space of the com-pact group G . Then, any element in the connected component Isom(
V /G ) of the Lie group Isom(
V /G ) lifts to a G -equivariant isometry of V . Mathematics Subject Classification.
Key words and phrases.
Orbit space, isometry group.
The proof consists of showing that every isometry of
V /G that lifts toa diffeomorphism actually lifts to an isometry; and that every isometryin Isom(
V /G ) lifts to a diffeomorphism. The second statement followsfrom two results about the smooth structure of V /G (in the sense of G.Schwarz), namely [AR17], and [Sch80, Corollary (2.4)].We note that elements of Isom(
V /G ) \ Isom(
V /G ) do not necessarilylift, and that Theorem A does not generalize from Euclidean spaces togeneral Riemannian manifolds. See Section 2 for details.If the G -representation V has trivial factors, translations in such di-rections induce a closed, non-compact subgroup of Isom( V /G ). On theother had, in the absence of trivial factors, every isometry of
V /G mustfix (the image of) the origin [GL14, Section 5.1], thus Isom(
V /G ) =Isom(
SV /G ) is compact, where SV ⊂ V denotes the unit sphere. Ingeneral, Isom( SV /G ), identified with the isometries fixing the origin,forms a maximal compact subgroup of Isom(
V /G ).Recall that every compact, connected Lie group is, up to a finitecover, a product of a torus and compact, simply-connected, simple Liegroups [BtD95, Theorem 8.1]. The latter are classified, and fall intotwo types: classical (SU( n ), Spin( n ), and Sp( n )), and exceptional ( G , F , E , E , and E ). As a corollary of Theorem A, we obtain thefollowing structure result: Theorem B.
Let V be a representation of a compact Lie group G .Then, up to a finite cover, Isom(
SV /G ) is isomorphic to a product ofa torus and compact simple Lie groups of classical type. As another corollary of Theorem A, we obtain:
Theorem C.
Let V be an irreducible orthogonal representation of acompact Lie group G . Then Isom(
V /G ) has rank at most one. For a more detailed analysis of the case where V is irreducible, seethe end of Section 3.This article is organized as follows. Section 2 contains the proof ofTheorem A and a few remarks. In Section 3 we extract some conse-quences of Theorem A with the help of Schur’s Lemma, and in partic-ular prove Theorems B and C. Acknowledgements.
We would like to thank C. Gorodski and A.Lytchak for discussions regarding the lifting of isometries that led tothis project. 2.
Lifting isometries
The proof of Theorem A uses the smooth structure of orbit spaces,in the sense of G. Schwarz [Sch80]. Given a G -manifold M , recall that IFTING ISOMETRIES OF ORBIT SPACES 3 a function
M/G → R is defined to be smooth if its composition withthe projection M → M/G is a smooth function on M in the usualsense; and that a map M/G → M ′ /G ′ is smooth if the pull back ofsmooth functions are smooth. Proof of Theorem A.
Let ϕ ∈ Isom(
V /G ) . First we note that ϕ (aswell as the isometries in a path connecting ϕ to the identity) may beassumed to fix the origin, see [GL14, Section 5.1].By [AR17], the action of Isom( V /G ) on V /G is smooth in thesense of Schwarz, and so, by [Sch80, Corollary (2.4)], ϕ lifts to a G -equivariant diffeomorphism ˜ ϕ of V that fixes the origin. To finish theproof it suffices to show that the differential d ˜ ϕ at the origin is or-thogonal and lifts ϕ .Let λ >
0, and denote by ˜ r λ : V → V multiplication by λ , and by r λ : V /G → V /G the induced map on
V /G . Since ϕ fixes the origin,it commutes with r λ , so that ˜ r − λ ˜ ϕ ˜ r λ is again a G -equivariant diffeo-morphism of V lifting ϕ = r − λ ϕr λ . Taking λ →
0, this converges tothe differential d ˜ ϕ , which is then a linear G -equivariant lift of ϕ . Thisimplies that d ˜ ϕ takes the unit sphere SV to itself, and in particularit is an orthogonal transformation. (cid:3) Next we mention ways in which Theorem A can and cannot be ex-tended.
Remark . The condition that ϕ be in the connected component ofIsom( V /G ) in the statement of Theorem A is necessary. For an exampleof a non-liftable isometry, consider G = SO(2) × SO(3) and its naturalproduct action on R × R . The quotient is a sector of angle π/ , ∞ ) × [0 , ∞ ), and its only non-trivial isometry is thereflection across the bisecting ray. This does not lift to an isometry of R × R , because any such lift would have to swap two singular orbitsisometric to a 2-sphere and a circle, which is clearly impossible. A moresubtle example is the outer tensor product action of G on R ⊗ R . Herethe quotient ( R ⊗ R ) /G is isometric to a sector of angle π/
4, and theisotropy groups along the two boundary rays are isomorphic to SO(2)and SO(2) × Z (see third line of Table E in [GWZ08]). This meansthat the corresponding singular orbits are not diffeomorphic, so thatthe non-trivial isometry of ( R ⊗ R ) /G cannot lift. Remark . There is a natural class of isometries of V , larger thanthe group of G -equivariant isometries considered in Theorem A, whichdescend to isometries of V /G . It is given by the normalizer N ( G ) ofthe image of G in O( V ). Remark 1 above shows that in general notevery isometry of the quotient lifts to N ( G ). R. A. E. MENDES
However, when G is finite, N ( G ) does induce all isometries ϕ of V /G that fix the origin, that is, all isometries of
SV /G . Indeed, theprojection π : SV → SV /G and the map ϕ ◦ π : SV → SV /G areuniversal orbifold coverings, hence ϕ lifts to an isometry f : SV → SV (see [Thu80, Chapter 13] and [Lan20]). Since the image of G in O( V )and its conjugate by f are orbit-equivalent finite subgroups of O( V ),they must coincide, that is, f normalizes G . Remark . The conclusion of Theorem A may fail if the Euclidean space V (or the round sphere SV ) is replaced with a general Riemannianmanifold. Indeed, consider the torus M = S × S with coordinates x, y , endowed with a warped product metric dx + ρ ( x ) dy . Then G = S acts by isometries, with orbit space M/G = S . If the warpingfactor ρ is not constant, that is, if the G -orbits do not all have thesame length, then not every isometry in Isom( M/G ) = S lifts to anisometry of M .3. Computing the connected component of the isometrygroup
Let V be an orthogonal representation of the compact group G , andassume for simplicity that V has no trivial factors, which is equivalentto Isom( V /G ) being compact. Denote by Isom G ( V ) the group of all G -equivariant isometries of V . Any element in this group descendsto an isometry of the quotient, so we have a group homomorphismIsom G ( V ) → Isom(
V /G ). Theorem A implies that the restriction to theidentity component Isom G ( V ) gives a surjective group homomorphism(1) p : Isom G ( V ) → Isom(
V /G ) . This points to a strategy for computing the isomorphism type of Isom(
V /G ) ,namely:(a) Determine the group Isom G ( V ) , and(b) Determine the normal subgroup ker( p ).Task (a) is easily achieved using Schur’s Lemma (see, for example,[BtD95, page 69]). More precisely, one first decomposes V ∼ = k M i =1 V ⊕ n i i where V i are pair-wise non-isomorphic irreducible G -representations.Each V i is of real, complex, or quaternionic type, according to whetherthe skew field Hom G ( V i , V i ) is isomorphic to R , C , or H (see [BtD95, IFTING ISOMETRIES OF ORBIT SPACES 5
II.6]). One then has the corresponding decompositionIsom G ( V ) = k Y i =1 H i with each H i isomorphic to SO( n i ), U( n i ), or Sp( n i ), according to thetype of V i . Proof of Theorem B.
First assume V has no trivial factors. By thediscussion above, the Lie algebra of Isom( SV /G ) = Isom(
V /G ) is thequotient of a Lie algebra h by an ideal I , where h is the direct sum of aAbelian factor a with a finite number of simple Lie algebras isomorphicto so ( n ), su ( n ), or sp ( n ). Then I must be equal to the direct sum ofsome subspace of a with a finite collection of the simple factors of h ,by uniqueness of simple ideals. Thus the quotient h /I has the desiredform.For general V , one uses a similar argument as above, the only differ-ence being that one only considers those G -equivariant isometries of V that fix the origin. (cid:3) Task (b) can be approached using the characterization of ker( p ) asthe largest subgroup L of Isom G ( V ) with the property that the naturalaction of L × G on V is orbit-equivalent to the original G -action. Thisleads to the following relationship with the presence of boundary (thatis, codimension-one strata) in the orbit space: Proposition 4.
Let G be a compact group, V an orthogonal G -representationwithout trivial factors, and p : Isom G ( V ) → Isom(
V /G ) the surjec-tive homomorphism considered in (1) above. Assume G and G × ker( p ) have distinct images in O( V ) . Then V /G has boundary.Proof.
Let x ∈ SV be a principal point for the actions of both G andˆ G := G × ker( p ). Since these actions are orbit-equivalent, we have inparticular G · x = ˆ G · x . These are naturally identified with G/G x and ˆ G/ ˆ G x , which, together with the assumption that G and ˆ G havedistinct images in O( V ), implies that the principal isotropy group ˆ G x acts non-trivially on V . In other words, the fixed-point set V ˆ G x is aproper subspace of V , so that we have a non-trivial Luna–Richardsontype reduction [LR79] VG = V ˆ G = V ˆ G x N ˆ G ( ˆ G x ) . By [GL14, Proposition 1.1], the orbit space
V /G has boundary. (cid:3)
R. A. E. MENDES
Example . Consider the Hopf action of G = U(1) on V = R = C .As a representation, V decomposes as the direct sum of two copies ofthe irreducible representation, of complex type, of G on R = C . BySchur’s Lemma, Isom G ( V ) is isomorphic to U(2). The quotient SV /G is isometric to the 2-sphere of radius 1 /
2, and hence Isom(
V /G ) =Isom( SV /G ) is isomorphic to SO(3). Thus the map p in Equation (1)corresponds to the quotient of U(2) by its center: p : U(2) → PSU(2) ∼ = SO(3) . Even though ker( p ) is non-trivial and SV /G has no boundary, this doesnot contradict Proposition 4, because G and G × ker( p ) have the sameimage in O(4).We conclude by specializing the discussion in this section to the casewhere V is irreducible, and in particular prove Theorem C.If V is of real type, then Isom( V /G ) is a quotient of SO(1), hencetrivial. That is, Isom( V /G ) is a finite group.If V is of complex type, then Isom( V /G ) is a quotient of U(1), henceeither trivial, or isomorphic to U(1). Both cases occur. For example, if G is finite, then Isom( V /G ) ∼ = U(1), because G and U(1) × G cannotbe orbit-equivalent. On the other hand, the representation of U(1) × G on V is still irreducible of complex type, and Isom( V / (U(1) × G )) istrivial.If V is of quaternionic type, then Isom( V /G ) is a quotient of Sp(1),hence either trivial, or isomorphic to either Sp(1) or SO(3). It canbe shown that Isom( V /G ) is trivial only when V /G is isometric to[0 , ∞ ). The proof uses Proposition 4 and the classification of irreduciblerepresentations of simple Lie groups whose orbit spaces have boundary,see [GKW] for details. References [AR17] Marcos M. Alexandrino and Marco Radeschi. Smoothness of isometricflows on orbit spaces and applications.
Transform. Groups , 22(1):1–27,2017.[BtD95] Theodor Br¨ocker and Tammo tom Dieck.
Representations of compact Liegroups , volume 98 of
Graduate Texts in Mathematics . Springer-Verlag,New York, 1995. Translated from the German manuscript, Correctedreprint of the 1985 translation.[DW28] D. van Dantzig and B. L. van der Waerden. ¨Uber metrisch homogener¨aume.
Abh. Math. Sem. Univ. Hamburg , 6(1):367–376, 1928.[FY94] Kenji Fukaya and Takao Yamaguchi. Isometry groups of singular spaces.
Math. Z. , 216(1):31–44, 1994.[GKW] Claudio Gorodski, Andreas Kollross, and Burkhard Wilking. Actions onpositively curved manifolds and boundary in the orbit space.
Preprint . IFTING ISOMETRIES OF ORBIT SPACES 7 [GL14] Claudio Gorodski and Alexander Lytchak. On orbit spaces of representa-tions of compact Lie groups.
J. Reine Angew. Math. , 691:61–100, 2014.[GWZ08] Karsten Grove, Burkhard Wilking, and Wolfgang Ziller. Positively curvedcohomogeneity one manifolds and 3-Sasakian geometry.
J. DifferentialGeom. , 78(1):33–111, 2008.[KN96] Shoshichi Kobayashi and Katsumi Nomizu.
Foundations of differentialgeometry. Vol. I . Wiley Classics Library. John Wiley & Sons, Inc., NewYork, 1996. Reprint of the 1963 original, A Wiley-Interscience Publica-tion.[Kob95] Shoshichi Kobayashi.
Transformation groups in differential geometry .Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reprint of the1972 edition.[Lan20] Christian Lange. Orbifolds from a metric viewpoint.
Geometriae Dedi-cata , 2020.[LR79] D. Luna and R. W. Richardson. A generalization of the Chevalley re-striction theorem.
Duke Math. J. , 46(3):487–496, 1979.[MS39] S. B. Myers and N. E. Steenrod. The group of isometries of a Riemannianmanifold.
Ann. of Math. (2) , 40(2):400–416, 1939.[Sch80] Gerald W. Schwarz. Lifting smooth homotopies of orbit spaces.
Inst.Hautes ´Etudes Sci. Publ. Math. , (51):37–135, 1980.[Thu80] W. Thurston. The geometry and topology of threemanifolds. lecture notesfrom princeton university. 1979–1980.
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