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

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Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage, Cabo Frio June 3-7, 2013 Claudio Gorodski Representations of compact Lie groups and their orbit spaces

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Page 1: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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

Page 2: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 3: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 4: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 5: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 6: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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 cohomogeneity

The 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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 7: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 8: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 9: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 10: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 11: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 12: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 13: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 14: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 15: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 16: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 17: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 18: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 19: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 20: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 21: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 22: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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?

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 23: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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?

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 24: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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?

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 25: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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?

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 26: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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?

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 27: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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?

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 28: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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?

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 29: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 30: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 31: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

Properties of polar representations

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

Orbits yield isoparametric foliation

Chevalley 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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 32: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 33: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 34: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 35: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 36: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 37: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 38: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 39: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 40: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 41: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 42: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 43: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 44: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 45: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 46: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 47: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 48: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 49: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 50: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 51: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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).

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 52: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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).

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 53: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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).

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 54: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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).

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 55: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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).

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 56: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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).

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 57: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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).

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 58: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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).

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 59: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 60: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 61: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 62: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 63: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 64: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 65: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 66: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 67: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 68: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 69: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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,C3 ⊗C C4) reduces to

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

Discuss case k = 1.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 70: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 71: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Page 72: Representations of compact Lie groups and their orbit spaces · Representations of compact Lie groups and their orbit spaces Claudio Gorodski Encounters in Geometry Hotel la Plage,

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.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces