Representations of compact Lie groups and their orbit spaces · Representations of compact Lie...

Representations of compact Lie groupsand their orbit spaces

Claudio Gorodski

Encounters in GeometryHotel la Plage, Cabo Frio

June 3-7, 2013

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Basic setting

We consider an effective orthogonal representation

ρ : G → O(V )

of a compact (possibly disconnected) Lie group G on a finite-dimensionalreal Euclidean space V .

View X = V /G as a metric space:

d(Gv1,Gv2) = inf{d(v1, gv2) : g ∈ G}

(realized by length of minimizing geodesic orthogonal to orbits)Note that X = Cone(S(V )/G) and S(V )/G is the “unit sphere” in X .

Main question: What kind of algebraic invariants of ρ can be recoveredfrom X?

The cohomogeneityThe invariant subspaces (in particular, the irreducibility)But NOT the dimension

Main definition: ρi : Gi → O(Vi ) for i = 1, 2, are calledquotient-equivalent if V1/G1, V2/G2 are isometric.

Basic setting

ρ : G → O(V )

d(Gv1,Gv2) = inf{d(v1, gv2) : g ∈ G}

(realized by length of minimizing geodesic orthogonal to orbits)

Note that X = Cone(S(V )/G) and S(V )/G is the “unit sphere” in X .

Basic setting

ρ : G → O(V )

d(Gv1,Gv2) = inf{d(v1, gv2) : g ∈ G}

Basic setting

ρ : G → O(V )

d(Gv1,Gv2) = inf{d(v1, gv2) : g ∈ G}

Basic setting

ρ : G → O(V )

d(Gv1,Gv2) = inf{d(v1, gv2) : g ∈ G}

The cohomogeneity

The invariant subspaces (in particular, the irreducibility)But NOT the dimension

Basic setting

ρ : G → O(V )

d(Gv1,Gv2) = inf{d(v1, gv2) : g ∈ G}

The cohomogeneityThe invariant subspaces (in particular, the irreducibility)

But NOT the dimension

Basic setting

ρ : G → O(V )

d(Gv1,Gv2) = inf{d(v1, gv2) : g ∈ G}

Basic setting

ρ : G → O(V )

d(Gv1,Gv2) = inf{d(v1, gv2) : g ∈ G}

Polar representations

Exs 1 and 3: reductions to finite groups:

polar representations (base ofhierarchy, from our point of view)

X = V /G = W /H, H finite⇒ Xreg = Vreg/G is flat

O’Neill ⇒ horiz distr of Vreg → Vreg/G integrable

Leaves are t.g. ⇒ integral mfld extend to subspace Σ ⊂ V

Σ is the normal space to a principal orbit; meets all G -orbits orthogonally:section; Σ ∼= W and dim Σ = dimX

H ∼= NG (Σ)/ZG (Σ) is a finite group

Exs 1 and 3: reductions to finite groups: polar representations

(base ofhierarchy, from our point of view)

Exs 1 and 3: reductions to finite groups: polar representations (base ofhierarchy, from our point of view)

X = V /G = W /H, H finite

⇒ Xreg = Vreg/G is flat

Σ is the normal space to a principal orbit;

meets all G -orbits orthogonally:section; Σ ∼= W and dim Σ = dimX

Σ is the normal space to a principal orbit; meets all G -orbits orthogonally:

section; Σ ∼= W and dim Σ = dimX

Σ is the normal space to a principal orbit; meets all G -orbits orthogonally:section;

Σ ∼= W and dim Σ = dimX

Copolarity

Ex 4: Σ ⊂ V contains normal spaces to principal orbits it meets:

generalized section [GOT]; dim Σ ≥ dimX

W ∼= Σ, H ∼= NG (Σ)/ZG (Σ)

dim Σ− dimX = dimH; for a minimal generalized section Σ, this numberis called the copolarity of ρ : G → O(V ).

Question. Does a minimal reduction always come from a minimalgeneralized section?

Copolarity

Ex 4: Σ ⊂ V contains normal spaces to principal orbits it meets:generalized section [GOT];

dim Σ ≥ dimX

W ∼= Σ, H ∼= NG (Σ)/ZG (Σ)

Copolarity

Ex 4: Σ ⊂ V contains normal spaces to principal orbits it meets:generalized section [GOT]; dim Σ ≥ dimX

W ∼= Σ, H ∼= NG (Σ)/ZG (Σ)

Copolarity

W ∼= Σ, H ∼= NG (Σ)/ZG (Σ)

Copolarity

W ∼= Σ, H ∼= NG (Σ)/ZG (Σ)

dim Σ− dimX = dimH;

for a minimal generalized section Σ, this numberis called the copolarity of ρ : G → O(V ).

Copolarity

W ∼= Σ, H ∼= NG (Σ)/ZG (Σ)

Copolarity

W ∼= Σ, H ∼= NG (Σ)/ZG (Σ)

Properties of polar representations

They are classified by Dadok:

for connected group, all orbit-equivalent toisotropy representations of symmetric spaces

Orbits yield isoparametric foliationChevalley restriction theorem

If ρ : G → O(V ) is polar then ∂X 6= ∅

Proposition. If V1/G1 = V2/G2 and ∂(V1/G◦1 ) = ∅ then

dimV1 ≤ dimV2. So ρ1 can admit a non-trivial reduction only if∂(V1/G

◦1 ) 6= ∅ (only if ∂(V1/G1) 6= ∅)

The Weyl group of a symmetric space is a Coxeter group

Proposition. If V1/G1 = V2/G2 and G1 is connected then G2/G◦2 acts by

reflections in subspaces of codimension 1 on V2/G◦2 (in fact, its image in

Iso(V2/G◦2 ) is a Coxeter group)

If ρ : G → O(V ) is polar then X = V /G is a good orbifold; in particular,

S(V )/G is a good orbifold

Not true in general; x ∈ X is an orbifold point iff the slice repr atp ∈ π−1(x) is polar [LT]: Xorb ⊃ Xreg

We have a classification of when S(V )/G is a good Riemannian orbifold.

They are classified by Dadok: for connected group, all orbit-equivalent toisotropy representations of symmetric spaces

◦1 ) 6= ∅ (only if ∂(V1/G1) 6= ∅)

Orbits yield isoparametric foliation

Chevalley restriction theorem

◦1 ) 6= ∅ (only if ∂(V1/G1) 6= ∅)

dimV1 ≤ dimV2.

So ρ1 can admit a non-trivial reduction only if∂(V1/G

◦1 ) 6= ∅ (only if ∂(V1/G1) 6= ∅)

◦1 ) 6= ∅

(only if ∂(V1/G1) 6= ∅)

◦1 ) 6= ∅ (only if ∂(V1/G1) 6= ∅)

If ρ : G → O(V ) is polar then X = V /G is a good orbifold;

in particular,

◦1 ) 6= ∅ (only if ∂(V1/G1) 6= ∅)

Not true in general;

x ∈ X is an orbifold point iff the slice repr atp ∈ π−1(x) is polar [LT]: Xorb ⊃ Xreg

◦1 ) 6= ∅ (only if ∂(V1/G1) 6= ∅)

Not true in general; x ∈ X is an orbifold point iff the slice repr atp ∈ π−1(x) is polar [LT]:

Xorb ⊃ Xreg

◦1 ) 6= ∅ (only if ∂(V1/G1) 6= ∅)

Application: isometric actions on spheres with good orbifold quotients

Theorem. (Reduced case) Let ρ : G → O(V ) be non-polar, with Gconnected.

Let τ : H → O(W ) be a minimal reduction of ρ. Assume thatthe orbit space of the induced isometric action on the unit sphereX = S(V )/G (= S(W )/H) is a good Riemannian orbifold. Then alsoS(W )/H◦ is a good Riemannian orbifold and τ |H◦ is either a Hopf actionwith ` ≥ 2 summands

U1 C⊕ · · · ⊕ CSp1 H⊕ · · · ⊕H

or a pseudo-Hopf action which is a doubling representation

U2 C2 ⊕ C2

Sp2 H2 ⊕H2

Sp2U1 H2 ⊕H2

Theorem. (Reduced case) Let ρ : G → O(V ) be non-polar, with Gconnected. Let τ : H → O(W ) be a minimal reduction of ρ.

Assume thatthe orbit space of the induced isometric action on the unit sphereX = S(V )/G (= S(W )/H) is a good Riemannian orbifold. Then alsoS(W )/H◦ is a good Riemannian orbifold and τ |H◦ is either a Hopf actionwith ` ≥ 2 summands

U1 C⊕ · · · ⊕ CSp1 H⊕ · · · ⊕H

U2 C2 ⊕ C2

Sp2 H2 ⊕H2

Sp2U1 H2 ⊕H2

Theorem. (Reduced case) Let ρ : G → O(V ) be non-polar, with Gconnected. Let τ : H → O(W ) be a minimal reduction of ρ. Assume thatthe orbit space of the induced isometric action on the unit sphereX = S(V )/G (= S(W )/H) is a good Riemannian orbifold.

Then alsoS(W )/H◦ is a good Riemannian orbifold and τ |H◦ is either a Hopf actionwith ` ≥ 2 summands

U1 C⊕ · · · ⊕ CSp1 H⊕ · · · ⊕H

U2 C2 ⊕ C2

Sp2 H2 ⊕H2

Sp2U1 H2 ⊕H2

Theorem. (Reduced case) Let ρ : G → O(V ) be non-polar, with Gconnected. Let τ : H → O(W ) be a minimal reduction of ρ. Assume thatthe orbit space of the induced isometric action on the unit sphereX = S(V )/G (= S(W )/H) is a good Riemannian orbifold. Then alsoS(W )/H◦ is a good Riemannian orbifold

and τ |H◦ is either a Hopf actionwith ` ≥ 2 summands

U1 C⊕ · · · ⊕ CSp1 H⊕ · · · ⊕H

U2 C2 ⊕ C2

Sp2 H2 ⊕H2

Sp2U1 H2 ⊕H2

Theorem. (Reduced case) Let ρ : G → O(V ) be non-polar, with Gconnected. Let τ : H → O(W ) be a minimal reduction of ρ. Assume thatthe orbit space of the induced isometric action on the unit sphereX = S(V )/G (= S(W )/H) is a good Riemannian orbifold. Then alsoS(W )/H◦ is a good Riemannian orbifold and τ |H◦ is either a Hopf actionwith ` ≥ 2 summands

U1 C⊕ · · · ⊕ CSp1 H⊕ · · · ⊕H

U2 C2 ⊕ C2

Sp2 H2 ⊕H2

Sp2U1 H2 ⊕H2

Theorem. (Reduced case) Let ρ : G → O(V ) be non-polar, with Gconnected. Let τ : H → O(W ) be a minimal reduction of ρ. Assume thatthe orbit space of the induced isometric action on the unit sphereX = S(V )/G (= S(W )/H) is a good Riemannian orbifold. Then alsoS(W )/H◦ is a good Riemannian orbifold and τ |H◦ is either a Hopf actionwith ` ≥ 2 summands

U1 C⊕ · · · ⊕ CSp1 H⊕ · · · ⊕H

U2 C2 ⊕ C2

Sp2 H2 ⊕H2

Sp2U1 H2 ⊕H2

Non-reduced case

Theorem. Let ρ : G → O(V ) be non-reduced and non-polar, where G isconnected.

Assume that X = S(V )/G is a good Riemannian orbifold.If ρ is irreducible then its cohomogeneity is 3, its connected minimalreduction is (U1,C ⊕ C), and it is one of

SO2 × Spin9 R2 ⊗R R16 −U2 × Spn C2 ⊗C C2n n ≥ 2Sp1 × Spn S3(C2) ⊗H C2n n ≥ 2

If ρ is not irreducible then it has exacly two irreducible summands, and wehave the following possibilities:

Representations of cohomogeneity 3

SOn Rn ⊕ Rn n ≥ 3U1 × SUn × U1 Cn ⊕ Cn n ≥ 2Sp1 × Spn × Sp1 Hn ⊕ Hn n ≥ 2

or an orbit-equivalent subgroup action, with conn min reduction (U1, C ⊕ C).Representations of cohomogeneity 4

SUn Cn ⊕ Cn n ≥ 3Un Cn ⊕ Cn n ≥ 3

SpnSp1 Hn ⊕ Hn n ≥ 3U1 × Spn × U1 Hn ⊕ Hn n ≥ 2

Spin9 R16 ⊕ R16 −

with connected minimal reduction (U2, C2 ⊕ C2).

(Spn, Hn ⊕ Hn) (n ≥ 3) of cohom 6 with conn min reduction (Sp2, H2 ⊕ H2).

(SpnU1, Hn ⊕ Hn) (n ≥ 3) of cohom 5 with conn min reduct (Sp2U1, H2 ⊕ H2).

Non-reduced case

Theorem. Let ρ : G → O(V ) be non-reduced and non-polar, where G isconnected. Assume that X = S(V )/G is a good Riemannian orbifold.

If ρ is irreducible then its cohomogeneity is 3, its connected minimalreduction is (U1,C ⊕ C), and it is one of

Spin9 R16 ⊕ R16 −

Non-reduced case

Spin9 R16 ⊕ R16 −

Non-reduced case

Spin9 R16 ⊕ R16 −

Non-reduced case

or an orbit-equivalent subgroup action, with conn min reduction (U1, C ⊕ C).

Spin9 R16 ⊕ R16 −

Non-reduced case

Spin9 R16 ⊕ R16 −

Non-reduced case

Spin9 R16 ⊕ R16 −

Non-reduced case

Spin9 R16 ⊕ R16 −

Application: representations with toric connected reductions

Theorem. Let ρ : G → O(V ) be irreducible, non-polar, non-reduced withH connected. Let τ : H → O(W ) be a non-trivial minimal reduction of ρ.If H◦ acts reducibly on W then H◦ is a torus T k and its action on W canbe identified with that of the maximal torus of SUk+1 on Ck+1; inparticular, cohom(ρ) = k + 2. Moreover, such ρ can be classified:

ρ is one of the non-polar irreducible representations of cohomogeneity three:

SO2 × Spin9 R2 ⊗R R16 −U2 × Spn C2 ⊗C C2n n ≥ 2SU2 × Spn S3(C2) ⊗H C2n n ≥ 2

The group G is the semisimple factor of an irreducible polar representationof Hermitian type such that action of G is not orbit-equivalent to the polarrepresentation:

SUn S2Cn n ≥ 3SUn Λ2Cn n = 2p ≥ 6

SUn × SUn Cn ⊗C Cn n ≥ 3E6 C27 −

ρ is one of the two exceptions: SO3 ⊗ G2, SO4 ⊗ Spin7.

Note. We can prove that if dimH ≤ 6 then H◦ always acts reducibly onW .On the other hand, (U3 × Sp2,C

3 ⊗C C4) reduces to(SO3 × U2,R

3 ⊗R R4), and SO3 × U2 is 7-dimensional.

Discuss case k = 1.

Theorem. Let ρ : G → O(V ) be irreducible, non-polar, non-reduced withH connected.

Let τ : H → O(W ) be a non-trivial minimal reduction of ρ.If H◦ acts reducibly on W then H◦ is a torus T k and its action on W canbe identified with that of the maximal torus of SUk+1 on Ck+1; inparticular, cohom(ρ) = k + 2. Moreover, such ρ can be classified:

Discuss case k = 1.

Theorem. Let ρ : G → O(V ) be irreducible, non-polar, non-reduced withH connected. Let τ : H → O(W ) be a non-trivial minimal reduction of ρ.

If H◦ acts reducibly on W then H◦ is a torus T k and its action on W canbe identified with that of the maximal torus of SUk+1 on Ck+1; inparticular, cohom(ρ) = k + 2. Moreover, such ρ can be classified:

Discuss case k = 1.

Theorem. Let ρ : G → O(V ) be irreducible, non-polar, non-reduced withH connected. Let τ : H → O(W ) be a non-trivial minimal reduction of ρ.If H◦ acts reducibly on W

then H◦ is a torus T k and its action on W canbe identified with that of the maximal torus of SUk+1 on Ck+1; inparticular, cohom(ρ) = k + 2. Moreover, such ρ can be classified:

Discuss case k = 1.

Theorem. Let ρ : G → O(V ) be irreducible, non-polar, non-reduced withH connected. Let τ : H → O(W ) be a non-trivial minimal reduction of ρ.If H◦ acts reducibly on W then H◦ is a torus T k and its action on W canbe identified with that of the maximal torus of SUk+1 on Ck+1;

inparticular, cohom(ρ) = k + 2. Moreover, such ρ can be classified:

Discuss case k = 1.

Theorem. Let ρ : G → O(V ) be irreducible, non-polar, non-reduced withH connected. Let τ : H → O(W ) be a non-trivial minimal reduction of ρ.If H◦ acts reducibly on W then H◦ is a torus T k and its action on W canbe identified with that of the maximal torus of SUk+1 on Ck+1; inparticular, cohom(ρ) = k + 2.

Moreover, such ρ can be classified:ρ is one of the non-polar irreducible representations of cohomogeneity three:

Discuss case k = 1.

Note. We can prove that if dimH ≤ 6 then H◦ always acts reducibly onW .

On the other hand, (U3 × Sp2,C3 ⊗C C4) reduces to

(SO3 × U2,R3 ⊗R R4), and SO3 × U2 is 7-dimensional.

Discuss case k = 1.

References

[1] C. G., C. Olmos and R. Tojeiro, Copolarity of isometric actions. Trans.Amer. Math. Soc. 356 (2004), 1585–1608.

[2] A. Lytchak, Geometric resolution of singular Riemannian foliations.Geom. Dedicata 149, 379–395 (2010).

[3] C. G. and A. Lytchak, On orbit spaces of representations of compactLie groups. To appear in J. Reine Angew. Math.

[4] C. G. and A. Lytchak, Representations whose connected minimalreduction is toric. To appear in Proc. Amer. Math. Soc..

[5] C. G. and A. Lytchak, Isometric actions on spheres with a goodorbifold quotient. In preparation.

Representations of compact Lie groups and their orbit spaces · Representations of compact Lie...

Documents

Transcript of Representations of compact Lie groups and their orbit spaces · Representations of compact Lie...

Symmetry and Orbit Detection via Lie-Algebra Votingmisha/ReadingSeminar/Papers/Shi16.pdf · where one can easily sum or subtract inﬁnitesimal elements. A Lie group and its Lie algebra

Lie Algebra - Fermilab

Dynamics in the study of discrete subgroups of Lie groupsrpotrie/documentos/pdfs/VCLAM.pdf · Schottky groups (representations of the free group in k generators), Faithful, discrete

Transiting extra-solar planets: spin-orbit misalignment ...suto/myresearch/kasi06_rossiter.pdf · Transiting extra-solar planets: spin-orbit misalignment and rings Transiting extra-solar

SEMI-GROUPS AND REPRESENTATIONS OF LIE GROUPSsunsite.ubc.ca › DigitalMathArchive › Langlands › pdf › ... · 2001-05-13 · Semi-groups and representations of Lie groups 6

Chapter 1 Representations - Homepages of UvA/FNWI staff · 2015-02-05 · Chapter 1 Representations 1.1 Representations of Groups and Algebras A representation of a group Gon a ﬁnite-dimensional

Orbit and Constellation Design

Unitary Representations of Nilpotent Super Lie groups · Basic Deﬁnitions and Notation Let Gbe a Lie group and Hbe a Hilbert space.A unitary representation π of Gin His a map π

Lecture 9. Permutation Representations

ε-FACTORS OF WEIL-DELIGNE REPRESENTATIONS

Weyl Group Representations and Unitarity of …apantano/pdf_documents/windsor.pdfLet G be a Lie group and let g be its Lie algebra. For all x 2 G, there is an inner automorphism Int(x):

Semisimple Lie Algebras Math 649, 2013 Lie algebra cohomology

I. Automorphic representations and Galois representationsgarrett/harris/FirstThree... · 2005-02-03 · I. Automorphic representations and Galois representations I will be lecturing

Ellipsoidal Representations Regarding Correlations

Semisimple Lie Algebras Math 649, 2013

benjaminlharris.weebly.combenjaminlharris.weebly.com › uploads › 1 › 0 › 9 › ...sets_of_reductive_l… · WAVE FRONT SETS OF REDUCTIVE LIE GROUP REPRESENTATIONS BENJAMIN

Lecture 14: Fourier representations - MIT … · Fourier Representations October 27, 2011 Fourier Representations Fourier series represent signals in terms of sinusoids. ... piano

Restrictions of unitary representations: Examples and

Spin-orbit splitting of 13ΛC

TOPOLOGICAL INVARIANTS OF ANOSOV REPRESENTATIONS