kac moody symmetric spacesmath.uni.lu/~rahm/ggn/slides_koehl.pdf · 2017-06-10 · examples:...

Kac–Moody symmetric spaces

Ralf Köhl

8 June 2017

Overview

1 Topologies on Lie groups

2 Topological Kac–Moody groups

3 Symmetric spaces

4 Kac–Moody symmetric spaces

Overview

3 Symmetric spaces

An open mapping theorem

Proposition 1

A surjective, continuous homomorphism

f : G → H

between Hausdorff topological groups where G is σ-compact and H is aBaire space, is open; moreover, H is locally compact.

Proof.

By hypothesis, G =⋃

n∈N Kn for certain compact sets Kn ⊆ G and thusH =

⋃n∈N f (Kn) with f (Kn) compact.

Since H is a Baire space, f (Kn) has non-empty interior for some n ∈ N,and thus H is locally compact. Moreover, f |Kn : Kn → f (Kn) is aquotient map.

Proposition 1

f : G → H

Proof.

Proposition 1

f : G → H

Proof.

Proposition 1

f : G → H

Proof.

[...]Let q : G → G/ ker(f ) be the quotient homomorphism andφ : G/ ker(f )→ H be the bijective continuous homomorphism inducedby f .

Then φ−1 ◦ f |Kn = q|Kn is continuous, whence φ−1|f (Kn) is continuous,φ−1 is a continuous homomorphism, and φ is a topologicalisomorphism.

Proposition 1

f : G → H

Proof.

[...]Let q : G → G/ ker(f ) be the quotient homomorphism andφ : G/ ker(f )→ H be the bijective continuous homomorphism inducedby f .Then φ−1 ◦ f |Kn = q|Kn is continuous, whence φ−1|f (Kn) is continuous,φ−1 is a continuous homomorphism, and φ is a topologicalisomorphism.

Colimits

A diagram in a category A is a covariant functor δ : I→ A from a smallcategory I to A.

A cone over a diagram δ : I→ A is a natural transformation φ : δ·→ γ

to a constant diagram γ.(That is, to a diagram γ : I→ A such that G := γ(i) = γ(j) for alli , j ∈ ob(I) and γ(α) = idG for each morphism α in I.)One can think of a cone as the object G ∈ ob(A), together with thefamily (φ(i))i∈ob(I) of morphisms φ(i) : δ(i)→ G , such that

φ(j) ◦ δ(α) = φ(i) for all i , j ∈ ob(I) and α ∈ Mor(i , j) .

A cone (G , (φi )i∈ob(I)) is called a colimit of δ if, for each cone(H, (ψi )i∈ob(I)), there is a unique morphism ψ : G → H such thatψ ◦ φi = ψi for all i ∈ ob(I).If it exists, a colimit is unique up to natural isomorphism.

Examples: direct limits, free products, amalgamated products A ∗C B

Colimits

A diagram in a category A is a covariant functor δ : I→ A from a smallcategory I to A.A cone over a diagram δ : I→ A is a natural transformation φ : δ

·→ γ

to a constant diagram γ.

(That is, to a diagram γ : I→ A such that G := γ(i) = γ(j) for alli , j ∈ ob(I) and γ(α) = idG for each morphism α in I.)One can think of a cone as the object G ∈ ob(A), together with thefamily (φ(i))i∈ob(I) of morphisms φ(i) : δ(i)→ G , such that

Colimits

·→ γ

to a constant diagram γ.(That is, to a diagram γ : I→ A such that G := γ(i) = γ(j) for alli , j ∈ ob(I) and γ(α) = idG for each morphism α in I.)

One can think of a cone as the object G ∈ ob(A), together with thefamily (φ(i))i∈ob(I) of morphisms φ(i) : δ(i)→ G , such that

Colimits

·→ γ

Colimits

·→ γ

Colimits

·→ γ

Amalgams

A diagram δ : I→ G of groups is called an amalgam of groups if

I is the category associated with some partially ordered set (J,≤)and

δ(α) is a monomorphism of groups for each morphism α in I.

The first condition means that ob(I) = J, and furthermore for a, b ∈ Jthere exists one (and only one) morphism a→ b if and only if a ≤ b.

A cone (G , (φi )i∈ob(I)) over δ in the category of abstract groups G iscalled an enveloping group of the amalgam and its colimit a universalenveloping group.

Example: A Coxeter group is by definition the universal envelopinggroup of the amalgam of its standard subgroups of ranks one and two.

Amalgams

Topologies on colimits

Let δ : I→ LCG be a diagram ofσ-compact locally compact groups Gi := δ(i) for i ∈ I := ob(I) andcontinuous homomorphisms φα := δ(α) : Gi → Gj for i , j ∈ I andα ∈ Mor(i , j)

such that I is countable.

Furthermore, let (G , (λi )i∈I ) be a colimit of the diagram δ in thecategory of abstract groups, with homomorphisms λi : Gi → G .

If there exists a locally compact Hausdorff group topology O on Gmaking λi : Gi → (G ,O) continuous for each i ∈ I , then

((G ,O), (λi )i∈I )

is a colimit of δ in the category oftopological groups,Hausdorff topological groups,locally compact groups,Lie groups (if Gi , (G ,O) are σ-compact Lie groups).

((G ,O), (λi )i∈I )

If there exists a locally compact Hausdorff group topology O on Gmaking λi : Gi → (G ,O) continuous for each i ∈ I ,

((G ,O), (λi )i∈I )

is a colimit of δ in the category oftopological groups,Hausdorff topological groups,locally compact groups,

Lie groups (if Gi , (G ,O) are σ-compact Lie groups).

((G ,O), (λi )i∈I )

Lie groups as colimits

Theorem 2 (Glöckner, Hartnick, K. 2010)

Let G be a simply connected compact/split real semisimple Lie groupwith Lie group topology O, let T be a maximal torus of G, letΣ = Σ(GC,TC) be its root system, and let Π be a system offundamental roots of Σ.

Let I be a small category with objects(

)and morphisms

{α} → {α, β}, for all α, β ∈ Π, and let δ : I→ LCG be a diagram with

δ({α}) = Gα := 〈Uα,U−α〉 ∩ G,

δ({α, β}) = Gαβ := 〈Uα,U−α,Uβ,U−β〉 ∩ G, and

δ({α} → {α, β}) = (Gα ↪→ Gαβ).

Then ((G ,O), (ιi )i∈(Π1)∪(Π

2)) is a colimit of δ in the category LIE of Lie

groups.

The fact that (G , (ιi )i∈(Π1)∪(Π

2)) is a colimit of δ in the category of

abstract groups goes back to Tits (1974).

Let G be a simply connected compact/split real semisimple Lie groupwith Lie group topology O, let T be a maximal torus of G, letΣ = Σ(GC,TC) be its root system, and let Π be a system offundamental roots of Σ.Let I be a small category with objects

)and morphisms

δ({α}) = Gα := 〈Uα,U−α〉 ∩ G,

δ({α} → {α, β}) = (Gα ↪→ Gαβ).

Then ((G ,O), (ιi )i∈(Π1)∪(Π

groups.

)and morphisms

δ({α}) = Gα := 〈Uα,U−α〉 ∩ G,

δ({α} → {α, β}) = (Gα ↪→ Gαβ).

Then ((G ,O), (ιi )i∈(Π1)∪(Π

groups.

)and morphisms

δ({α}) = Gα := 〈Uα,U−α〉 ∩ G,

δ({α} → {α, β}) = (Gα ↪→ Gαβ).

Then ((G ,O), (ιi )i∈(Π1)∪(Π

groups.

Overview

3 Symmetric spaces

Definition via colimits

Definition 3

∆ be an arbitrary Dynkin diagram without label ∞,

(Gα)α∈∆ be a family of copies of SL2(R),

(Gαβ){α,β}∈(∆2) be a family of appropriate simply connected split

real Lie groups, and

Gα → Gαβ the natural embeddings.

The (simply connected split real) topological Kac–Moody group G oftype ∆ is defined as the colimit of the above amalgam in the categoryof topological groups.

Abramenko–Mühlherr (1997) established the above approach in thecategory of abstract groups.The topological version is established by Hartnick–K.–Mars (2013)based on ideas by Kac–Peterson (1980s).

Definition 3

Abramenko–Mühlherr (1997) established the above approach in thecategory of abstract groups.

The topological version is established by Hartnick–K.–Mars (2013)based on ideas by Kac–Peterson (1980s).

Definition 3

Properties

A topological Kac–Moody group is Hausdorff and kω.(Kac–Peterson 1980s and Hartnick–K.–Mars 2013)

Subgroups of finite type carry the Lie group topology.(Hartnick–K.–Mars 2013)

The Iwasawa decomposition G = KAN is a homeomorphism.(Freyn–Hartnick–Horn–K. 2017)

A topological Kac–Moody group is Kazhdan.(Hartnick–K. 2015)

G/B+, G/B− form a topological twin building(in the sense of the theory by Kramer 2002).

Goal: Construct a symmetric space for a topological Kac–Moody group.

Properties

Overview

3 Symmetric spaces

A theorem by Loos

Theorem 4 (Loos 1967)

Let X be an affine symmetric space, and given x , y ∈ X denote by x · ythe point reflection of y at x. Then

µ : X × X → X : (x , y) 7→ x · y

is a C 1-map satisfying the following axioms:

1 for any x ∈ X one has x · x = x,

2 for any pair of points x , y ∈ X one has x · (x · y) = y ,

3 for any triple of points x , y , z ∈ X one has

x · (y · z) = (x · y) · (x · z),

4 every x ∈ X has a neighbourhood U such that x · y = y impliesy = x for all y ∈ U.

A theorem by Loos

Theorem 4 (continued)

Conversely, if X is a smooth manifold and µ : X × X → X is a C 1-mapsubject to the axioms above, then X is an affine symmetric space, andµ(x , y) is the point reflection of y at x.

If X is a Riemannian symmetric space, then the isometries of X areexactly the C 1-maps α : X → X satisfying α(x · y) = α(x) · α(y).

If X is moreover of the non-compact type, then instead of the localcondition it satisfies the global condition

4 x · y = y implies y = x for all y ∈ X.

Example: For any topological group G the assignment

G × G → G : (x , y) 7→ xy−1x

satisfies axioms 1, 2, 3.

A theorem by Loos

G × G → G : (x , y) 7→ xy−1x

A theorem by Loos

G × G → G : (x , y) 7→ xy−1x

A theorem by Loos

G × G → G : (x , y) 7→ xy−1x

One-parameter groups without C 1 hypothesis

Theorem 5 (Freyn, Hartnick, Horn, K. 2017)

Let (X , µ) be a topological space with continuous µ satisfying axioms1, 2, 3, 4.

Given x ∈ X let sx (y) := µ(x , y) and given a geodesic γ ⊂ Xlet

Tγ := {sp ◦ sq | p, q ∈ γ} ⊂ Aut(X , µ).

Then the following hold:

Tγ ∼= (R,+) is a one-parameter subgroup of Aut(X , µ).

Tγ acts sharply transitively on γ by Euclidean translations.

A geodesic γ ⊂ X is defined to be the image of a bijection

φ : R→ γ

such that

φ(2x − y) = µ(φ(x), φ(y)) for all x , y ∈ R.

Let (X , µ) be a topological space with continuous µ satisfying axioms1, 2, 3, 4. Given x ∈ X let sx (y) := µ(x , y) and given a geodesic γ ⊂ Xlet

Tγ := {sp ◦ sq | p, q ∈ γ} ⊂ Aut(X , µ).

φ : R→ γ

such that

Tγ := {sp ◦ sq | p, q ∈ γ} ⊂ Aut(X , µ).

φ : R→ γ

such that

Tγ := {sp ◦ sq | p, q ∈ γ} ⊂ Aut(X , µ).

φ : R→ γ

such that

Overview

3 Symmetric spaces

Properties of Kac–Moody symmetric spacesFreyn, Hartnick, Horn, K. 2017

G/K is a topological space with continuous multiplication

µ(gK , hK ) = gθ(g)−1θ(h)K

satisfying axioms 1, 2, 3, 4

(where G is a top. Kac–Moody group,K its subgroup generated by the maximal compact subgroups ofthe Gα ∼= SL2(R), and θ the involution of G fixing K pointwise.)

The maximal flats of G/K are in 1-1 correspondence to themaximal tori of G .

Aut(G/K , µ) = Aut(G ).

G/K admits a causal structure with the two halves of thetopological twin building as the future and past boundaries.

G/K admits a partial order, if Kostant convexity holds for G .(E.g., for type E10.)

satisfying axioms 1, 2, 3, 4

(where G is a top. Kac–Moody group,K its subgroup generated by the maximal compact subgroups ofthe Gα ∼= SL2(R), and θ the involution of G fixing K pointwise.)

satisfying axioms 1, 2, 3, 4 (where G is a top. Kac–Moody group,K its subgroup generated by the maximal compact subgroups ofthe Gα ∼= SL2(R), and θ the involution of G fixing K pointwise.)

kac moody symmetric spacesmath.uni.lu/~rahm/ggn/slides_koehl.pdf · 2017-06-10 · examples:...

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