Representations of compact Lie groups and their orbit spaces(Joint work with Alexander Lytchak)

Claudio Gorodski

Geometry and Lie theory. Applications to classical and quantum mechanicsDedicated to Eldar Straume on his 70th birthday.

Norwegian University of Science and Technology, TrondheimNovember 3-4, 2016

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

s-representations

Consider:

a symmetric pair (L,G);the ±1-eigenspace decomposition l = g + V ;the isotropy representation of the symmetric space L/G or, equivalently, theadjoint representation of G on V (a so called s-representation);a maximal Abelian subalgebra Σ of V (Cartan subspace).

Then:Σ meets all G -orbits in V ;Σ is orthogonal to every orbit it meets;the largest subquotient of G acting on Σ is a finite groupW = NG (Σ)/ZG (Σ) generated by reflections;the inclusion Σ→ V induces an isometry between the orbit spacesΣ/W = V /G ;it follows that V /G and S(V )/G are good Riemannian orbifolds of constantcurvature 0 and 1, resp.

Concrete example: (SL(n,R),SO(n)) and SO(n)-conjugation of n× n realsymmetric matrices.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

s-representations

Consider:a symmetric pair (L,G);

the ±1-eigenspace decomposition l = g + V ;the isotropy representation of the symmetric space L/G or, equivalently, theadjoint representation of G on V (a so called s-representation);a maximal Abelian subalgebra Σ of V (Cartan subspace).

Then:Σ meets all G -orbits in V ;Σ is orthogonal to every orbit it meets;the largest subquotient of G acting on Σ is a finite groupW = NG (Σ)/ZG (Σ) generated by reflections;the inclusion Σ→ V induces an isometry between the orbit spacesΣ/W = V /G ;it follows that V /G and S(V )/G are good Riemannian orbifolds of constantcurvature 0 and 1, resp.

Concrete example: (SL(n,R),SO(n)) and SO(n)-conjugation of n× n realsymmetric matrices.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

s-representations

Consider:a symmetric pair (L,G);the ±1-eigenspace decomposition l = g + V ;

the isotropy representation of the symmetric space L/G or, equivalently, theadjoint representation of G on V (a so called s-representation);a maximal Abelian subalgebra Σ of V (Cartan subspace).

Then:Σ meets all G -orbits in V ;Σ is orthogonal to every orbit it meets;the largest subquotient of G acting on Σ is a finite groupW = NG (Σ)/ZG (Σ) generated by reflections;the inclusion Σ→ V induces an isometry between the orbit spacesΣ/W = V /G ;it follows that V /G and S(V )/G are good Riemannian orbifolds of constantcurvature 0 and 1, resp.

Concrete example: (SL(n,R),SO(n)) and SO(n)-conjugation of n× n realsymmetric matrices.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

s-representations

Consider:a symmetric pair (L,G);the ±1-eigenspace decomposition l = g + V ;the isotropy representation of the symmetric space L/G or, equivalently, theadjoint representation of G on V (a so called s-representation);

a maximal Abelian subalgebra Σ of V (Cartan subspace).

Then:Σ meets all G -orbits in V ;Σ is orthogonal to every orbit it meets;the largest subquotient of G acting on Σ is a finite groupW = NG (Σ)/ZG (Σ) generated by reflections;the inclusion Σ→ V induces an isometry between the orbit spacesΣ/W = V /G ;it follows that V /G and S(V )/G are good Riemannian orbifolds of constantcurvature 0 and 1, resp.

Concrete example: (SL(n,R),SO(n)) and SO(n)-conjugation of n× n realsymmetric matrices.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

s-representations

Consider:a symmetric pair (L,G);the ±1-eigenspace decomposition l = g + V ;the isotropy representation of the symmetric space L/G or, equivalently, theadjoint representation of G on V (a so called s-representation);a maximal Abelian subalgebra Σ of V (Cartan subspace).

Then:Σ meets all G -orbits in V ;Σ is orthogonal to every orbit it meets;the largest subquotient of G acting on Σ is a finite groupW = NG (Σ)/ZG (Σ) generated by reflections;the inclusion Σ→ V induces an isometry between the orbit spacesΣ/W = V /G ;it follows that V /G and S(V )/G are good Riemannian orbifolds of constantcurvature 0 and 1, resp.

Concrete example: (SL(n,R),SO(n)) and SO(n)-conjugation of n× n realsymmetric matrices.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

s-representations

Consider:a symmetric pair (L,G);the ±1-eigenspace decomposition l = g + V ;the isotropy representation of the symmetric space L/G or, equivalently, theadjoint representation of G on V (a so called s-representation);a maximal Abelian subalgebra Σ of V (Cartan subspace).

Then:Σ meets all G -orbits in V ;

Σ is orthogonal to every orbit it meets;the largest subquotient of G acting on Σ is a finite groupW = NG (Σ)/ZG (Σ) generated by reflections;the inclusion Σ→ V induces an isometry between the orbit spacesΣ/W = V /G ;it follows that V /G and S(V )/G are good Riemannian orbifolds of constantcurvature 0 and 1, resp.

Concrete example: (SL(n,R),SO(n)) and SO(n)-conjugation of n× n realsymmetric matrices.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

s-representations

Consider:a symmetric pair (L,G);the ±1-eigenspace decomposition l = g + V ;the isotropy representation of the symmetric space L/G or, equivalently, theadjoint representation of G on V (a so called s-representation);a maximal Abelian subalgebra Σ of V (Cartan subspace).

Then:Σ meets all G -orbits in V ;Σ is orthogonal to every orbit it meets;

the largest subquotient of G acting on Σ is a finite groupW = NG (Σ)/ZG (Σ) generated by reflections;the inclusion Σ→ V induces an isometry between the orbit spacesΣ/W = V /G ;it follows that V /G and S(V )/G are good Riemannian orbifolds of constantcurvature 0 and 1, resp.

Concrete example: (SL(n,R),SO(n)) and SO(n)-conjugation of n× n realsymmetric matrices.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

s-representations

Consider:a symmetric pair (L,G);the ±1-eigenspace decomposition l = g + V ;the isotropy representation of the symmetric space L/G or, equivalently, theadjoint representation of G on V (a so called s-representation);a maximal Abelian subalgebra Σ of V (Cartan subspace).

Then:Σ meets all G -orbits in V ;Σ is orthogonal to every orbit it meets;the largest subquotient of G acting on Σ is a finite groupW = NG (Σ)/ZG (Σ) generated by reflections;

the inclusion Σ→ V induces an isometry between the orbit spacesΣ/W = V /G ;it follows that V /G and S(V )/G are good Riemannian orbifolds of constantcurvature 0 and 1, resp.

Concrete example: (SL(n,R),SO(n)) and SO(n)-conjugation of n× n realsymmetric matrices.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

s-representations

Consider:a symmetric pair (L,G);the ±1-eigenspace decomposition l = g + V ;the isotropy representation of the symmetric space L/G or, equivalently, theadjoint representation of G on V (a so called s-representation);a maximal Abelian subalgebra Σ of V (Cartan subspace).

Then:Σ meets all G -orbits in V ;Σ is orthogonal to every orbit it meets;the largest subquotient of G acting on Σ is a finite groupW = NG (Σ)/ZG (Σ) generated by reflections;the inclusion Σ→ V induces an isometry between the orbit spacesΣ/W = V /G ;

it follows that V /G and S(V )/G are good Riemannian orbifolds of constantcurvature 0 and 1, resp.

Concrete example: (SL(n,R),SO(n)) and SO(n)-conjugation of n× n realsymmetric matrices.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

s-representations

Consider:a symmetric pair (L,G);the ±1-eigenspace decomposition l = g + V ;the isotropy representation of the symmetric space L/G or, equivalently, theadjoint representation of G on V (a so called s-representation);a maximal Abelian subalgebra Σ of V (Cartan subspace).

Then:Σ meets all G -orbits in V ;Σ is orthogonal to every orbit it meets;the largest subquotient of G acting on Σ is a finite groupW = NG (Σ)/ZG (Σ) generated by reflections;the inclusion Σ→ V induces an isometry between the orbit spacesΣ/W = V /G ;it follows that V /G and S(V )/G are good Riemannian orbifolds of constantcurvature 0 and 1, resp.

Concrete example: (SL(n,R),SO(n)) and SO(n)-conjugation of n× n realsymmetric matrices.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

s-representations

Consider:a symmetric pair (L,G);the ±1-eigenspace decomposition l = g + V ;the isotropy representation of the symmetric space L/G or, equivalently, theadjoint representation of G on V (a so called s-representation);a maximal Abelian subalgebra Σ of V (Cartan subspace).

Then:Σ meets all G -orbits in V ;Σ is orthogonal to every orbit it meets;the largest subquotient of G acting on Σ is a finite groupW = NG (Σ)/ZG (Σ) generated by reflections;the inclusion Σ→ V induces an isometry between the orbit spacesΣ/W = V /G ;it follows that V /G and S(V )/G are good Riemannian orbifolds of constantcurvature 0 and 1, resp.

Concrete example: (SL(n,R),SO(n)) and SO(n)-conjugation of n× n realsymmetric matrices.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Polar representations

Let G be a compact Lie group.

Definition 1

A representation ρ : G → O(V ) is called polar if there exists a subspace Σ,called a section, that meets all G-orbits and meets them always orthogonally.

Definition 2

A representation ρ : G → O(V ) is called polar if there exists a representationτ : H → O(W ) of a finite group H such that V /G = W /H.

Theorem (Dadok 1985)

Eevery polar representation (Def 1) of a connected compact Lie group isorbit-equivalent to an s-representation.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Polar representations

Let G be a compact Lie group.

Definition 1

A representation ρ : G → O(V ) is called polar if there exists a subspace Σ,called a section, that meets all G-orbits and meets them always orthogonally.

Definition 2

A representation ρ : G → O(V ) is called polar if there exists a representationτ : H → O(W ) of a finite group H such that V /G = W /H.

Theorem (Dadok 1985)

Eevery polar representation (Def 1) of a connected compact Lie group isorbit-equivalent to an s-representation.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Polar representations

Let G be a compact Lie group.

Definition 1

A representation ρ : G → O(V ) is called polar if there exists a subspace Σ,called a section, that meets all G-orbits and meets them always orthogonally.

Definition 2

A representation ρ : G → O(V ) is called polar if there exists a representationτ : H → O(W ) of a finite group H such that V /G = W /H.

Theorem (Dadok 1985)

Eevery polar representation (Def 1) of a connected compact Lie group isorbit-equivalent to an s-representation.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Polar representations

Let G be a compact Lie group.

Definition 1

A representation ρ : G → O(V ) is called polar if there exists a subspace Σ,called a section, that meets all G-orbits and meets them always orthogonally.

Definition 2

A representation ρ : G → O(V ) is called polar if there exists a representationτ : H → O(W ) of a finite group H such that V /G = W /H.

Theorem (Dadok 1985)

Eevery polar representation (Def 1) of a connected compact Lie group isorbit-equivalent to an s-representation.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Equivalence between definitions

Assume (G ,V ) reduces to a finite group action (H,W ).

Consider the projections

V W

V /G

πG∨

= W /H

πH∨

Since H is a finite group, πH is a local isometry on the regular set Wreg soVreg/G = Wreg/H is flat.

By O’Neill’s formula for Riemannian submersion πG : Vreg → Vreg/G :

K(X ,Y ) = K(X , Y ) + 3||∇vX Y ||2

Since both terms with sectional curvatures vanish, the horizontaldistribution H on Vreg , consisting of normal spaces to the principal orbits,is integrable with totally geodesic leaves.

The converse essentially follows from the fact that a minimizing geodesicbetween G -orbits can be taken contained in Σ.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Equivalence between definitions

Assume (G ,V ) reduces to a finite group action (H,W ).

Consider the projections

V W

V /G

πG∨

= W /H

πH∨

Since H is a finite group, πH is a local isometry on the regular set Wreg soVreg/G = Wreg/H is flat.

By O’Neill’s formula for Riemannian submersion πG : Vreg → Vreg/G :

K(X ,Y ) = K(X , Y ) + 3||∇vX Y ||2

Since both terms with sectional curvatures vanish, the horizontaldistribution H on Vreg , consisting of normal spaces to the principal orbits,is integrable with totally geodesic leaves.

The converse essentially follows from the fact that a minimizing geodesicbetween G -orbits can be taken contained in Σ.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Equivalence between definitions

Assume (G ,V ) reduces to a finite group action (H,W ).

Consider the projections

V W

V /G

πG∨

= W /H

πH∨

Since H is a finite group, πH is a local isometry on the regular set Wreg soVreg/G = Wreg/H is flat.

By O’Neill’s formula for Riemannian submersion πG : Vreg → Vreg/G :

K(X ,Y ) = K(X , Y ) + 3||∇vX Y ||2

Since both terms with sectional curvatures vanish, the horizontaldistribution H on Vreg , consisting of normal spaces to the principal orbits,is integrable with totally geodesic leaves.

The converse essentially follows from the fact that a minimizing geodesicbetween G -orbits can be taken contained in Σ.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Equivalence between definitions

Assume (G ,V ) reduces to a finite group action (H,W ).

Consider the projections

V W

V /G

πG∨

= W /H

πH∨

Since H is a finite group, πH is a local isometry on the regular set Wreg soVreg/G = Wreg/H is flat.

By O’Neill’s formula for Riemannian submersion πG : Vreg → Vreg/G :

K(X ,Y ) = K(X , Y ) + 3||∇vX Y ||2

Since both terms with sectional curvatures vanish, the horizontaldistribution H on Vreg , consisting of normal spaces to the principal orbits,is integrable with totally geodesic leaves.

The converse essentially follows from the fact that a minimizing geodesicbetween G -orbits can be taken contained in Σ.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Equivalence between definitions

Assume (G ,V ) reduces to a finite group action (H,W ).

Consider the projections

V W

V /G

πG∨

= W /H

πH∨

Since H is a finite group, πH is a local isometry on the regular set Wreg soVreg/G = Wreg/H is flat.

By O’Neill’s formula for Riemannian submersion πG : Vreg → Vreg/G :

K(X ,Y ) = K(X , Y ) + 3||∇vX Y ||2

Since both terms with sectional curvatures vanish, the horizontaldistribution H on Vreg , consisting of normal spaces to the principal orbits,is integrable with totally geodesic leaves.

The converse essentially follows from the fact that a minimizing geodesicbetween G -orbits can be taken contained in Σ.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Equivalence between definitions

Assume (G ,V ) reduces to a finite group action (H,W ).

Consider the projections

V W

V /G

πG∨

= W /H

πH∨

Since H is a finite group, πH is a local isometry on the regular set Wreg soVreg/G = Wreg/H is flat.

By O’Neill’s formula for Riemannian submersion πG : Vreg → Vreg/G :

K(X ,Y ) = K(X , Y ) + 3||∇vX Y ||2

Since both terms with sectional curvatures vanish, the horizontaldistribution H on Vreg , consisting of normal spaces to the principal orbits,is integrable with totally geodesic leaves.

The converse essentially follows from the fact that a minimizing geodesicbetween G -orbits can be taken contained in Σ.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Equivalence between definitions

Assume (G ,V ) reduces to a finite group action (H,W ).

Consider the projections

V W

V /G

πG∨

= W /H

πH∨

Since H is a finite group, πH is a local isometry on the regular set Wreg soVreg/G = Wreg/H is flat.

By O’Neill’s formula for Riemannian submersion πG : Vreg → Vreg/G :

K(X ,Y ) = K(X , Y ) + 3||∇vX Y ||2

Since both terms with sectional curvatures vanish, the horizontaldistribution H on Vreg , consisting of normal spaces to the principal orbits,is integrable with totally geodesic leaves.

The converse essentially follows from the fact that a minimizing geodesicbetween G -orbits can be taken contained in Σ.

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Basic setting

For ρ : G → O(V ), the orbit space X = V /G (or S(V )/G) is anAlexandrov space of curvature bounded below, stratified by orbit-type.

Main question: How much of ρ can be recovered from X ?

Main definition: ρi : Gi → O(Vi ) for i = 1, 2, are calledquotient-equivalent if V1/G1, V2/G2 are isometric. If dim G2 < dim G1,then ρ2 is called a reduction of ρ1; the minimal dimension of such a G2 iscalled the abstract copolarity of ρ1.

Existence and uniqueness; hierarchy.

Luna-Richardson reduction: V H/(NG (H)/H) = V /G where H is aprincipal isotropy group. (Representations of connected groups withnon-trivial principal isotropy representations were classified byW.-C. Hsiang and W.-Y. Hsiang in 1970.)

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Basic setting

For ρ : G → O(V ), the orbit space X = V /G (or S(V )/G) is anAlexandrov space of curvature bounded below, stratified by orbit-type.

Main question: How much of ρ can be recovered from X ?

Main definition: ρi : Gi → O(Vi ) for i = 1, 2, are calledquotient-equivalent if V1/G1, V2/G2 are isometric. If dim G2 < dim G1,then ρ2 is called a reduction of ρ1; the minimal dimension of such a G2 iscalled the abstract copolarity of ρ1.

Existence and uniqueness; hierarchy.

Luna-Richardson reduction: V H/(NG (H)/H) = V /G where H is aprincipal isotropy group. (Representations of connected groups withnon-trivial principal isotropy representations were classified byW.-C. Hsiang and W.-Y. Hsiang in 1970.)

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Basic setting

For ρ : G → O(V ), the orbit space X = V /G (or S(V )/G) is anAlexandrov space of curvature bounded below, stratified by orbit-type.

Main question: How much of ρ can be recovered from X ?

Main definition: ρi : Gi → O(Vi ) for i = 1, 2, are calledquotient-equivalent if V1/G1, V2/G2 are isometric. If dim G2 < dim G1,then ρ2 is called a reduction of ρ1; the minimal dimension of such a G2 iscalled the abstract copolarity of ρ1.

Existence and uniqueness; hierarchy.

Luna-Richardson reduction: V H/(NG (H)/H) = V /G where H is aprincipal isotropy group. (Representations of connected groups withnon-trivial principal isotropy representations were classified byW.-C. Hsiang and W.-Y. Hsiang in 1970.)

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Basic setting

For ρ : G → O(V ), the orbit space X = V /G (or S(V )/G) is anAlexandrov space of curvature bounded below, stratified by orbit-type.

Main question: How much of ρ can be recovered from X ?

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

If dim G2 < dim G1,then ρ2 is called a reduction of ρ1; the minimal dimension of such a G2 iscalled the abstract copolarity of ρ1.

Existence and uniqueness; hierarchy.

Luna-Richardson reduction: V H/(NG (H)/H) = V /G where H is aprincipal isotropy group. (Representations of connected groups withnon-trivial principal isotropy representations were classified byW.-C. Hsiang and W.-Y. Hsiang in 1970.)

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Basic setting

For ρ : G → O(V ), the orbit space X = V /G (or S(V )/G) is anAlexandrov space of curvature bounded below, stratified by orbit-type.

Main question: How much of ρ can be recovered from X ?

Main definition: ρi : Gi → O(Vi ) for i = 1, 2, are calledquotient-equivalent if V1/G1, V2/G2 are isometric. If dim G2 < dim G1,then ρ2 is called a reduction of ρ1; the minimal dimension of such a G2 iscalled the abstract copolarity of ρ1.

Existence and uniqueness; hierarchy.

Luna-Richardson reduction: V H/(NG (H)/H) = V /G where H is aprincipal isotropy group. (Representations of connected groups withnon-trivial principal isotropy representations were classified byW.-C. Hsiang and W.-Y. Hsiang in 1970.)

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Basic setting

For ρ : G → O(V ), the orbit space X = V /G (or S(V )/G) is anAlexandrov space of curvature bounded below, stratified by orbit-type.

Main question: How much of ρ can be recovered from X ?

Main definition: ρi : Gi → O(Vi ) for i = 1, 2, are calledquotient-equivalent if V1/G1, V2/G2 are isometric. If dim G2 < dim G1,then ρ2 is called a reduction of ρ1; the minimal dimension of such a G2 iscalled the abstract copolarity of ρ1.

Existence and uniqueness; hierarchy.

Luna-Richardson reduction: V H/(NG (H)/H) = V /G where H is aprincipal isotropy group. (Representations of connected groups withnon-trivial principal isotropy representations were classified byW.-C. Hsiang and W.-Y. Hsiang in 1970.)

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Basic setting

For ρ : G → O(V ), the orbit space X = V /G (or S(V )/G) is anAlexandrov space of curvature bounded below, stratified by orbit-type.

Main question: How much of ρ can be recovered from X ?

Main definition: ρi : Gi → O(Vi ) for i = 1, 2, are calledquotient-equivalent if V1/G1, V2/G2 are isometric. If dim G2 < dim G1,then ρ2 is called a reduction of ρ1; the minimal dimension of such a G2 iscalled the abstract copolarity of ρ1.

Existence and uniqueness; hierarchy.

Luna-Richardson reduction: V H/(NG (H)/H) = V /G where H is aprincipal isotropy group.

(Representations of connected groups withnon-trivial principal isotropy representations were classified byW.-C. Hsiang and W.-Y. Hsiang in 1970.)

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Basic setting

For ρ : G → O(V ), the orbit space X = V /G (or S(V )/G) is anAlexandrov space of curvature bounded below, stratified by orbit-type.

Main question: How much of ρ can be recovered from X ?

Main definition: ρi : Gi → O(Vi ) for i = 1, 2, are calledquotient-equivalent if V1/G1, V2/G2 are isometric. If dim G2 < dim G1,then ρ2 is called a reduction of ρ1; the minimal dimension of such a G2 iscalled the abstract copolarity of ρ1.

Existence and uniqueness; hierarchy.

Luna-Richardson reduction: V H/(NG (H)/H) = V /G where H is aprincipal isotropy group. (Representations of connected groups withnon-trivial principal isotropy representations were classified byW.-C. Hsiang and W.-Y. Hsiang in 1970.)

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Hsiang, Wu-Yi. Lie transformation groups and differential geometry.Differential geometry and differential equations (Shanghai, 1985), 34-52,Lecture Notes in Math., 1255, Springer, Berlin, 1987.39

Let M be a given compact, I - connected, homogeneous space,

G = ISO(M), K = ISO(M, Xo). Let V be the variety of focal

points of X o which can be decomposed into the union of

K-orbits. For each point X 1 6 V, it is natural to define

the variational cocompleteness of a shortest geodesic segment

, to be the codimension of the space of Killing vector

fields in the space of Jacobi vector fields, vanishing at

both X o and X I. Suggested by the remarkable accomplishment of

Bott and Samelson [ 4,5 ] in the case of compact symmetric spaces,

a more systematic study of the topological-geometric structure

of the focal variety together with the variational co-completeness

will certainly lead to valuable insights in correlating the

Lie structure of the pair (G,K) and the geometric-topological

structure of M = G/K such as the rational homotopy type of

G/K as well as that of its loop space. I believe that the

above approach will eventually provide a satisfactory

resolution of the following problem.

Problem [ii, Problem29]o Is it true that two homeomorphic compact

homogeneous spaces MI,M 2 are necessarily diffeomorphic? Or does

the above statement hold with only a few exceptions?

Remark: It is known to be affirmative for many special cases,

e.g., no exotic differentiable structures on topological

spheres can be homogeneous. However, it is actually

a near miss because the cQhomo~eneities of those exotic spheres

of the Kervaire type are, in fact, equal to I.

Example 2: Geometry Of triangles

Roughly speaking, geodesic interval is the basic geometric

object associated to a generic pair of points, and triangle

is the basic geometric object attached to a generic triple

of points. In the study of Euclidean, spherical and hyperbolic

geometries, the geometry of triangles is of central importance

and, in fact, they are commonly characterized by the same kind

of congruence axioms for intervals and triangles. Analytically,

a triangle has six basic "elements", namely, the size of its

three angles and the lengths of its three sides; and the fun-

damental functional relationships among the above six basic

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Straume, Eldar. On the invariant theory and geometry of compact lineargroups of cohomogeneity 3. Differential Geom. Appl. 4 (1994), no. 1, 1-23.

On the inuariant theory 3

Let G c O(n) b e a full linear group. A minimal reduction of G is a linear group

K c O(h), k minimal, such that a) Ii and G have isomorphic invariant rings (as graded algebras) and

b) ,!Y1/G and S”-r/K are isometric. We state our main results as follows.

Theorem A. Let (G, V), d im G > 0, be a full linear group of cohomogeneity c(G) = 3.

Then its minimal reduction is a linear group of dimension < 1, that is, k = 3 or 4. Moreover, if dim G > 1 then its homological dimension hd(G) is at most one.

Theorem B. Let (G, V), d im G > 0, be a full linear group with c(G) = 2 or 3. (i) If c(G) = 2 th en the minimal reduction of (G, V) is a crystallographic dihedral

reflection group. In particular, hd(G) = 0. (ii) If c(G) = 3 and G is connected, then hd(G) < 1 and moreover, hd(G) = 0 if G

is not a circle group. (iii) Assume c(G) = 3, d imG > 1 and G connected. Then SV/G is a disk with at

most 3 vertices. If the number of vertices is # 1 then SV/G is a geodesic “region” (i.e.

with geodesic boundary arcs) on a sphere of radius 1 or l/2. If there is only one vertex, then SV/G is “half” of an ovaloid of revolution whose top is a conical singularity of

total angle TIT.

Here is a brief summary of the various sections. In Section 1 we explain some basic concepts and general facts, including the reduction principle and the orbital distance

metric. Finite groups of O(3) are surveyed in Section 2 for convenience. We remark that most groups G with c(G) < 3 reduce to a finite group, and this includes all cases with

c(G) < 2. The proof of Theorem 1.3, given in Section 3, shows that these G are the

same as polar groups, using the terminology from [2] or [9]. In Sections 5-6 we show the

remaining groups have l-dimensional reductions, which in turn are analyzed in detail

in Section 4. The final proof of Theorem A and B is given at the end of Section 6. For

more detailed information, see Tables I-111.

We refer to Straume [14] for tables of compact connected linear groups with c(G) 6 3.

This classification will be our starting point.

1. Basic reductions

In this section we give basic facts about general compact linear groups and explain some reduction technique which may apply under favorable circumstances, essentially

when the principal isotropy type is nontrivial.

Let G and G’ be closed subgroups of O(n). W e say the groups are C-equivalent if, after replacing one of the groups by a conjugate group in O(n), they have precisely the

same orbits in IRn. This gives an equivalence relation on the set of conjugacy classes (K) of linear groups of a fixed degree n, and each C-equivalence class has a unique maximal element called C-maximal or simply a maximal linear group.

A slightly different type of relation is c-equivalence. G and G’ are c-equivalent if, modulo conjugation in O(n), G and G’ both lie in some group G” satisfying c(G) =

Question. (Alexandrino-Lytchak) Does the metric of a quotient space Xdetermine its smooth structure? In other words, given two manifolds withisometric actions (G ,M) and (H,N) and an isometry I : M/G → N/H, isI always a diffeomorphism?

For representations, this is equivalent to I inducing an isomorphismbetween the rings of invariants (Schwarz’s theorem).

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Straume, Eldar. On the invariant theory and geometry of compact lineargroups of cohomogeneity 3. Differential Geom. Appl. 4 (1994), no. 1, 1-23.

On the inuariant theory 3

Let G c O(n) b e a full linear group. A minimal reduction of G is a linear group

K c O(h), k minimal, such that a) Ii and G have isomorphic invariant rings (as graded algebras) and

b) ,!Y1/G and S”-r/K are isometric. We state our main results as follows.

Theorem A. Let (G, V), d im G > 0, be a full linear group of cohomogeneity c(G) = 3.

Then its minimal reduction is a linear group of dimension < 1, that is, k = 3 or 4. Moreover, if dim G > 1 then its homological dimension hd(G) is at most one.

Theorem B. Let (G, V), d im G > 0, be a full linear group with c(G) = 2 or 3. (i) If c(G) = 2 th en the minimal reduction of (G, V) is a crystallographic dihedral

reflection group. In particular, hd(G) = 0. (ii) If c(G) = 3 and G is connected, then hd(G) < 1 and moreover, hd(G) = 0 if G

is not a circle group. (iii) Assume c(G) = 3, d imG > 1 and G connected. Then SV/G is a disk with at

most 3 vertices. If the number of vertices is # 1 then SV/G is a geodesic “region” (i.e.

with geodesic boundary arcs) on a sphere of radius 1 or l/2. If there is only one vertex, then SV/G is “half” of an ovaloid of revolution whose top is a conical singularity of

total angle TIT.

Here is a brief summary of the various sections. In Section 1 we explain some basic concepts and general facts, including the reduction principle and the orbital distance

metric. Finite groups of O(3) are surveyed in Section 2 for convenience. We remark that most groups G with c(G) < 3 reduce to a finite group, and this includes all cases with

c(G) < 2. The proof of Theorem 1.3, given in Section 3, shows that these G are the

same as polar groups, using the terminology from [2] or [9]. In Sections 5-6 we show the

remaining groups have l-dimensional reductions, which in turn are analyzed in detail

in Section 4. The final proof of Theorem A and B is given at the end of Section 6. For

more detailed information, see Tables I-111.

We refer to Straume [14] for tables of compact connected linear groups with c(G) 6 3.

This classification will be our starting point.

1. Basic reductions

In this section we give basic facts about general compact linear groups and explain some reduction technique which may apply under favorable circumstances, essentially

when the principal isotropy type is nontrivial.

Let G and G’ be closed subgroups of O(n). W e say the groups are C-equivalent if, after replacing one of the groups by a conjugate group in O(n), they have precisely the

same orbits in IRn. This gives an equivalence relation on the set of conjugacy classes (K) of linear groups of a fixed degree n, and each C-equivalence class has a unique maximal element called C-maximal or simply a maximal linear group.

A slightly different type of relation is c-equivalence. G and G’ are c-equivalent if, modulo conjugation in O(n), G and G’ both lie in some group G” satisfying c(G) =

Question. (Alexandrino-Lytchak) Does the metric of a quotient space Xdetermine its smooth structure? In other words, given two manifolds withisometric actions (G ,M) and (H,N) and an isometry I : M/G → N/H, isI always a diffeomorphism?

For representations, this is equivalent to I inducing an isomorphismbetween the rings of invariants (Schwarz’s theorem).

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Straume, Eldar. On the invariant theory and geometry of compact lineargroups of cohomogeneity 3. Differential Geom. Appl. 4 (1994), no. 1, 1-23.

On the inuariant theory 3

Let G c O(n) b e a full linear group. A minimal reduction of G is a linear group

K c O(h), k minimal, such that a) Ii and G have isomorphic invariant rings (as graded algebras) and

b) ,!Y1/G and S”-r/K are isometric. We state our main results as follows.

Theorem A. Let (G, V), d im G > 0, be a full linear group of cohomogeneity c(G) = 3.

Then its minimal reduction is a linear group of dimension < 1, that is, k = 3 or 4. Moreover, if dim G > 1 then its homological dimension hd(G) is at most one.

Theorem B. Let (G, V), d im G > 0, be a full linear group with c(G) = 2 or 3. (i) If c(G) = 2 th en the minimal reduction of (G, V) is a crystallographic dihedral

reflection group. In particular, hd(G) = 0. (ii) If c(G) = 3 and G is connected, then hd(G) < 1 and moreover, hd(G) = 0 if G

is not a circle group. (iii) Assume c(G) = 3, d imG > 1 and G connected. Then SV/G is a disk with at

most 3 vertices. If the number of vertices is # 1 then SV/G is a geodesic “region” (i.e.

with geodesic boundary arcs) on a sphere of radius 1 or l/2. If there is only one vertex, then SV/G is “half” of an ovaloid of revolution whose top is a conical singularity of

total angle TIT.

Here is a brief summary of the various sections. In Section 1 we explain some basic concepts and general facts, including the reduction principle and the orbital distance

metric. Finite groups of O(3) are surveyed in Section 2 for convenience. We remark that most groups G with c(G) < 3 reduce to a finite group, and this includes all cases with

c(G) < 2. The proof of Theorem 1.3, given in Section 3, shows that these G are the

same as polar groups, using the terminology from [2] or [9]. In Sections 5-6 we show the

remaining groups have l-dimensional reductions, which in turn are analyzed in detail

in Section 4. The final proof of Theorem A and B is given at the end of Section 6. For

more detailed information, see Tables I-111.

We refer to Straume [14] for tables of compact connected linear groups with c(G) 6 3.

This classification will be our starting point.

1. Basic reductions

In this section we give basic facts about general compact linear groups and explain some reduction technique which may apply under favorable circumstances, essentially

when the principal isotropy type is nontrivial.

Let G and G’ be closed subgroups of O(n). W e say the groups are C-equivalent if, after replacing one of the groups by a conjugate group in O(n), they have precisely the

same orbits in IRn. This gives an equivalence relation on the set of conjugacy classes (K) of linear groups of a fixed degree n, and each C-equivalence class has a unique maximal element called C-maximal or simply a maximal linear group.

A slightly different type of relation is c-equivalence. G and G’ are c-equivalent if, modulo conjugation in O(n), G and G’ both lie in some group G” satisfying c(G) =

Question. (Alexandrino-Lytchak) Does the metric of a quotient space Xdetermine its smooth structure? In other words, given two manifolds withisometric actions (G ,M) and (H,N) and an isometry I : M/G → N/H, isI always a diffeomorphism?

For representations, this is equivalent to I inducing an isomorphismbetween the rings of invariants (Schwarz’s theorem).

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Riemannian orbifolds

Recall: A Riemannian orbifold is a metric space X such that every x ∈ Xhas a neighborhood U isometric to M/Γ, where M is a Riemannianmanifold and Γ is a finite group of isometries of M.

Theorem (G.-Lytchak 2016)

If X = S(V )/G a Riemannian orbifold, then X has constant curvature 1 or 4,or X is a complex or quaternionic weighted projective space. Moreover, suchactions can be listed.

Main tool in proof is the following characterization of orbifold points byLytchak and Thorbergsson: x ∈ X is an orbifold point iff X has locallybounded curvature near x iff the slice representation at a point projectingto x is polar.

If there is an isometry I : X → B, where B is a Riemannian orbifold, thenI is a diffeomorphism between the quotient smooth structure of X and theunderlying smooth orbifold structure of B. Hence an isometry betweenorbit spaces is smooth if these are Riemannian orbifolds[Alexandrino-Lytchak].

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Riemannian orbifolds

Recall: A Riemannian orbifold is a metric space X such that every x ∈ Xhas a neighborhood U isometric to M/Γ, where M is a Riemannianmanifold and Γ is a finite group of isometries of M.

Theorem (G.-Lytchak 2016)

If X = S(V )/G a Riemannian orbifold, then X has constant curvature 1 or 4,or X is a complex or quaternionic weighted projective space. Moreover, suchactions can be listed.

Main tool in proof is the following characterization of orbifold points byLytchak and Thorbergsson: x ∈ X is an orbifold point iff X has locallybounded curvature near x iff the slice representation at a point projectingto x is polar.

If there is an isometry I : X → B, where B is a Riemannian orbifold, thenI is a diffeomorphism between the quotient smooth structure of X and theunderlying smooth orbifold structure of B. Hence an isometry betweenorbit spaces is smooth if these are Riemannian orbifolds[Alexandrino-Lytchak].

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Riemannian orbifolds

Recall: A Riemannian orbifold is a metric space X such that every x ∈ Xhas a neighborhood U isometric to M/Γ, where M is a Riemannianmanifold and Γ is a finite group of isometries of M.

Theorem (G.-Lytchak 2016)

If X = S(V )/G a Riemannian orbifold, then X has constant curvature 1 or 4,or X is a complex or quaternionic weighted projective space. Moreover, suchactions can be listed.

Main tool in proof is the following characterization of orbifold points byLytchak and Thorbergsson: x ∈ X is an orbifold point iff X has locallybounded curvature near x iff the slice representation at a point projectingto x is polar.

If there is an isometry I : X → B, where B is a Riemannian orbifold, thenI is a diffeomorphism between the quotient smooth structure of X and theunderlying smooth orbifold structure of B. Hence an isometry betweenorbit spaces is smooth if these are Riemannian orbifolds[Alexandrino-Lytchak].

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Riemannian orbifolds

Recall: A Riemannian orbifold is a metric space X such that every x ∈ Xhas a neighborhood U isometric to M/Γ, where M is a Riemannianmanifold and Γ is a finite group of isometries of M.

Theorem (G.-Lytchak 2016)

If X = S(V )/G a Riemannian orbifold, then X has constant curvature 1 or 4,or X is a complex or quaternionic weighted projective space. Moreover, suchactions can be listed.

Main tool in proof is the following characterization of orbifold points byLytchak and Thorbergsson: x ∈ X is an orbifold point iff X has locallybounded curvature near x iff the slice representation at a point projectingto x is polar.

If there is an isometry I : X → B, where B is a Riemannian orbifold, thenI is a diffeomorphism between the quotient smooth structure of X and theunderlying smooth orbifold structure of B. Hence an isometry betweenorbit spaces is smooth if these are Riemannian orbifolds[Alexandrino-Lytchak].

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Riemannian orbifolds

Recall: A Riemannian orbifold is a metric space X such that every x ∈ Xhas a neighborhood U isometric to M/Γ, where M is a Riemannianmanifold and Γ is a finite group of isometries of M.

Theorem (G.-Lytchak 2016)

If X = S(V )/G a Riemannian orbifold, then X has constant curvature 1 or 4,or X is a complex or quaternionic weighted projective space. Moreover, suchactions can be listed.

Main tool in proof is the following characterization of orbifold points byLytchak and Thorbergsson: x ∈ X is an orbifold point iff X has locallybounded curvature near x iff the slice representation at a point projectingto x is polar.

If there is an isometry I : X → B, where B is a Riemannian orbifold, thenI is a diffeomorphism between the quotient smooth structure of X and theunderlying smooth orbifold structure of B.

Hence an isometry betweenorbit spaces is smooth if these are Riemannian orbifolds[Alexandrino-Lytchak].

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

Riemannian orbifolds

Recall: A Riemannian orbifold is a metric space X such that every x ∈ Xhas a neighborhood U isometric to M/Γ, where M is a Riemannianmanifold and Γ is a finite group of isometries of M.

Theorem (G.-Lytchak 2016)

If X = S(V )/G a Riemannian orbifold, then X has constant curvature 1 or 4,or X is a complex or quaternionic weighted projective space. Moreover, suchactions can be listed.

Main tool in proof is the following characterization of orbifold points byLytchak and Thorbergsson: x ∈ X is an orbifold point iff X has locallybounded curvature near x iff the slice representation at a point projectingto x is polar.

If there is an isometry I : X → B, where B is a Riemannian orbifold, thenI is a diffeomorphism between the quotient smooth structure of X and theunderlying smooth orbifold structure of B. Hence an isometry betweenorbit spaces is smooth if these are Riemannian orbifolds[Alexandrino-Lytchak].

Claudio Gorodski Representations of compact Lie groups and their orbit spaces

HAPPY BIRTHDAY, ELDAR!

Claudio Gorodski Representations of compact Lie groups and their orbit spaces