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Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

Geometric Quantization and Models of Semisimple Lie Groups

Meng-Kiat Chuah

Department of Mathematics National Tsing Hua University Hsinchu, Taiwan

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 1. Introduction

G connected real semisimple Lie group. Construct its unitary representations. Geometric Quantization (Kostant 1970): G-invariant symplectic manifold unitary G-representation on Hilbert space. Symplectic structure provides inner product.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 1. Introduction

G connected real semisimple Lie group. Construct its unitary representations. Geometric Quantization (Kostant 1970): G-invariant symplectic manifold unitary G-representation on Hilbert space. Symplectic structure provides inner product.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 1. Introduction

G connected real semisimple Lie group. Construct its unitary representations. Geometric Quantization (Kostant 1970): G-invariant symplectic manifold unitary G-representation on Hilbert space. Symplectic structure provides inner product.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 2. Symplectic Manifolds and Group Actions

Symplectic Manifold: (X, ω), ω closed nondegenerate 2-form. Examples: coadjoint orbit X ⊂ g∗, cotangent bundle X = T ∗M. Suppose G acts on X. Given ξ ∈ g, define infinitesimal vector field ξ♯ on X by (ξ♯f)(x) = d

dt|t=0f(etξx)

for all f ∈ C∞(X) and x ∈ X.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 2. Symplectic Manifolds and Group Actions

Symplectic Manifold: (X, ω), ω closed nondegenerate 2-form. Examples: coadjoint orbit X ⊂ g∗, cotangent bundle X = T ∗M. Suppose G acts on X. Given ξ ∈ g, define infinitesimal vector field ξ♯ on X by (ξ♯f)(x) = d

dt|t=0f(etξx)

for all f ∈ C∞(X) and x ∈ X.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 2. Symplectic Manifolds and Group Actions

Symplectic Manifold: (X, ω), ω closed nondegenerate 2-form. Examples: coadjoint orbit X ⊂ g∗, cotangent bundle X = T ∗M. Suppose G acts on X. Given ξ ∈ g, define infinitesimal vector field ξ♯ on X by (ξ♯f)(x) = d

dt|t=0f(etξx)

for all f ∈ C∞(X) and x ∈ X.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 3. Moment Map

Suppose G acts on symplectic manifold (X, ω). Let Φ : X − → g∗ be a G-equivariant map. For each ξ ∈ g, define (Φ, ξ) ∈ C∞(X) by x → (Φ(x), ξ). Call Φ a moment map if for all ξ ∈ g, d(Φ, ξ) = ω(ξ♯). If G semisimple, moment map exists and is unique. If moment map exists, call the action Hamiltonian.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 3. Moment Map

Suppose G acts on symplectic manifold (X, ω). Let Φ : X − → g∗ be a G-equivariant map. For each ξ ∈ g, define (Φ, ξ) ∈ C∞(X) by x → (Φ(x), ξ). Call Φ a moment map if for all ξ ∈ g, d(Φ, ξ) = ω(ξ♯). If G semisimple, moment map exists and is unique. If moment map exists, call the action Hamiltonian.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 3. Moment Map

Suppose G acts on symplectic manifold (X, ω). Let Φ : X − → g∗ be a G-equivariant map. For each ξ ∈ g, define (Φ, ξ) ∈ C∞(X) by x → (Φ(x), ξ). Call Φ a moment map if for all ξ ∈ g, d(Φ, ξ) = ω(ξ♯). If G semisimple, moment map exists and is unique. If moment map exists, call the action Hamiltonian.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

Examples

If X is a coadjoint orbit of G, its moment map X − → g∗ is the inclusion. If G = X = T 2 is 2-torus and ω is Haar measure, then action is not Hamiltonian.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 4. Symplectic Reduction

Hamiltonian G-action on (X, ω), moment map Φ : X − → g∗. Gξ stabilizer of ξ ∈ g∗, let Xξ = Φ−1(ξ)/Gξ. Suppose Xξ manifold (e.g. free action and G compact). Xξ

π

← − Φ−1(ξ)

i

− → X Marsden, Weinstein 1974: symplectic form ωξ on Xξ , π∗ωξ = i∗ω Symplectic Reduction (X, ω) (Xξ, ωξ)

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 4. Symplectic Reduction

Hamiltonian G-action on (X, ω), moment map Φ : X − → g∗. Gξ stabilizer of ξ ∈ g∗, let Xξ = Φ−1(ξ)/Gξ. Suppose Xξ manifold (e.g. free action and G compact). Xξ

π

← − Φ−1(ξ)

i

− → X Marsden, Weinstein 1974: symplectic form ωξ on Xξ , π∗ωξ = i∗ω Symplectic Reduction (X, ω) (Xξ, ωξ)

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 4. Symplectic Reduction

Hamiltonian G-action on (X, ω), moment map Φ : X − → g∗. Gξ stabilizer of ξ ∈ g∗, let Xξ = Φ−1(ξ)/Gξ. Suppose Xξ manifold (e.g. free action and G compact). Xξ

π

← − Φ−1(ξ)

i

− → X Marsden, Weinstein 1974: symplectic form ωξ on Xξ , π∗ωξ = i∗ω Symplectic Reduction (X, ω) (Xξ, ωξ)

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 4. Symplectic Reduction

Hamiltonian G-action on (X, ω), moment map Φ : X − → g∗. Gξ stabilizer of ξ ∈ g∗, let Xξ = Φ−1(ξ)/Gξ. Suppose Xξ manifold (e.g. free action and G compact). Xξ

π

← − Φ−1(ξ)

i

− → X Marsden, Weinstein 1974: symplectic form ωξ on Xξ , π∗ωξ = i∗ω Symplectic Reduction (X, ω) (Xξ, ωξ)

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 5. Geometric Quantization

G-invariant K¨ ahler (symplectic) manifold (X, ω), suppose [ω] ∈ H2(X, Z). Exists line bundle L over X with connection ∇ and Hermitian structure , . Chern class of L is [ω], curvature of ∇ is ω. Hermitian structure is G-invariant.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 5. Geometric Quantization

G-invariant K¨ ahler (symplectic) manifold (X, ω), suppose [ω] ∈ H2(X, Z). Exists line bundle L over X with connection ∇ and Hermitian structure , . Chern class of L is [ω], curvature of ∇ is ω. Hermitian structure is G-invariant.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

Geometric Quantization

Section s of L is called holomorphic if ∇vs = 0 for all anti-holomorphic vector fields v. Find suitable G-invariant measure µ on X. Section s of L is called square-integrable if

• Xs, sµ < ∞.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

Geometric Quantization

Section s of L is called holomorphic if ∇vs = 0 for all anti-holomorphic vector fields v. Find suitable G-invariant measure µ on X. Section s of L is called square-integrable if

• Xs, sµ < ∞.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

Geometric Quantization

H = H(X) Hilbert space of square-integrable holomorphic sections (or Dolbeault cohomology) on L. Note H depends on ∇, and ∇ depends on ω. G-action on X leads to unitary G-representation on H. Study H.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

Geometric Quantization

H = H(X) Hilbert space of square-integrable holomorphic sections (or Dolbeault cohomology) on L. Note H depends on ∇, and ∇ depends on ω. G-action on X leads to unitary G-representation on H. Study H.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

Geometric Quantization

H = H(X) Hilbert space of square-integrable holomorphic sections (or Dolbeault cohomology) on L. Note H depends on ∇, and ∇ depends on ω. G-action on X leads to unitary G-representation on H. Study H.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 6. Philosophy of Quantization

Coadjoint orbits of G irreducible unitary G-representations. Works well with integral semisimple orbits. elliptic orbits discrete series. hyperbolic orbits principal series.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 6. Philosophy of Quantization

Coadjoint orbits of G irreducible unitary G-representations. Works well with integral semisimple orbits. elliptic orbits discrete series. hyperbolic orbits principal series.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 6. Philosophy of Quantization

Coadjoint orbits of G irreducible unitary G-representations. Works well with integral semisimple orbits. elliptic orbits discrete series. hyperbolic orbits principal series.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 7. Quantization commutes with Reduction

Guillemin, Sternberg 1982 Recall symplectic reduction (X, ω) (Xξ, ωξ). Use ξ ∈ g∗ to construct H(X)ξ such that H(X)ξ = H(Xξ). Usually, H(X)ξ ⊂ H(X) subrepresentation. More generally, Riemann-Roch theory.

• Ref. R. Sjamaar, Bull AMS 33 (1996).

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 7. Quantization commutes with Reduction

Guillemin, Sternberg 1982 Recall symplectic reduction (X, ω) (Xξ, ωξ). Use ξ ∈ g∗ to construct H(X)ξ such that H(X)ξ = H(Xξ). Usually, H(X)ξ ⊂ H(X) subrepresentation. More generally, Riemann-Roch theory.

• Ref. R. Sjamaar, Bull AMS 33 (1996).

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 7. Quantization commutes with Reduction

Guillemin, Sternberg 1982 Recall symplectic reduction (X, ω) (Xξ, ωξ). Use ξ ∈ g∗ to construct H(X)ξ such that H(X)ξ = H(Xξ). Usually, H(X)ξ ⊂ H(X) subrepresentation. More generally, Riemann-Roch theory.

• Ref. R. Sjamaar, Bull AMS 33 (1996).

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 7. Quantization commutes with Reduction

Guillemin, Sternberg 1982 Recall symplectic reduction (X, ω) (Xξ, ωξ). Use ξ ∈ g∗ to construct H(X)ξ such that H(X)ξ = H(Xξ). Usually, H(X)ξ ⊂ H(X) subrepresentation. More generally, Riemann-Roch theory.

• Ref. R. Sjamaar, Bull AMS 33 (1996).

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 8. Bigger Manifold Bigger Representation

Symplectic Induction Symplectic fibration over symplectic manifold sum of several representations. G-invariant symplectic fibration X over coadjoint orbit, H(X) provides several unitary G-irreducibles.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 8. Bigger Manifold Bigger Representation

Symplectic Induction Symplectic fibration over symplectic manifold sum of several representations. G-invariant symplectic fibration X over coadjoint orbit, H(X) provides several unitary G-irreducibles.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 9. Fibration over Semisimple Coadjoint Orbit

Integral semisimple λ ∈ g∗, coadjoint orbit G/Gλ. Let Xλ = G/(Gλ, Gλ) × V , dim V = dim Gλ/(Gλ, Gλ). Fiber of Xλ − → G/Gλ is complexification of subgroup of Cartan. dim Xλ even. If G has compact Cartan, Xλ ֒ → GC/(P, P) complex. Equip Xλ with G-invariant symplectic form, expect H(Xλ) to provide several unitary G-irreducibles.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 9. Fibration over Semisimple Coadjoint Orbit

Integral semisimple λ ∈ g∗, coadjoint orbit G/Gλ. Let Xλ = G/(Gλ, Gλ) × V , dim V = dim Gλ/(Gλ, Gλ). Fiber of Xλ − → G/Gλ is complexification of subgroup of Cartan. dim Xλ even. If G has compact Cartan, Xλ ֒ → GC/(P, P) complex. Equip Xλ with G-invariant symplectic form, expect H(Xλ) to provide several unitary G-irreducibles.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 9. Fibration over Semisimple Coadjoint Orbit

Integral semisimple λ ∈ g∗, coadjoint orbit G/Gλ. Let Xλ = G/(Gλ, Gλ) × V , dim V = dim Gλ/(Gλ, Gλ). Fiber of Xλ − → G/Gλ is complexification of subgroup of Cartan. dim Xλ even. If G has compact Cartan, Xλ ֒ → GC/(P, P) complex. Equip Xλ with G-invariant symplectic form, expect H(Xλ) to provide several unitary G-irreducibles.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 9. Fibration over Semisimple Coadjoint Orbit

Integral semisimple λ ∈ g∗, coadjoint orbit G/Gλ. Let Xλ = G/(Gλ, Gλ) × V , dim V = dim Gλ/(Gλ, Gλ). Fiber of Xλ − → G/Gλ is complexification of subgroup of Cartan. dim Xλ even. If G has compact Cartan, Xλ ֒ → GC/(P, P) complex. Equip Xλ with G-invariant symplectic form, expect H(Xλ) to provide several unitary G-irreducibles.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 9. Fibration over Semisimple Coadjoint Orbit

Integral semisimple λ ∈ g∗, coadjoint orbit G/Gλ. Let Xλ = G/(Gλ, Gλ) × V , dim V = dim Gλ/(Gλ, Gλ). Fiber of Xλ − → G/Gλ is complexification of subgroup of Cartan. dim Xλ even. If G has compact Cartan, Xλ ֒ → GC/(P, P) complex. Equip Xλ with G-invariant symplectic form, expect H(Xλ) to provide several unitary G-irreducibles.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 10. Compact Lie Groups

Suppose G compact. Unitary irreducibles parametrized by highest weights ξ in dominant Weyl chamber. Model of G (Gelfand, Zelevinsky) A unitary G-representation H, every irreducible occurs once. Use H(Xλ) to construct a model of G.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 10. Compact Lie Groups

Suppose G compact. Unitary irreducibles parametrized by highest weights ξ in dominant Weyl chamber. Model of G (Gelfand, Zelevinsky) A unitary G-representation H, every irreducible occurs once. Use H(Xλ) to construct a model of G.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 10. Compact Lie Groups

Suppose G compact. Unitary irreducibles parametrized by highest weights ξ in dominant Weyl chamber. Model of G (Gelfand, Zelevinsky) A unitary G-representation H, every irreducible occurs once. Use H(Xλ) to construct a model of G.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 11. Cells

h ⊂ g Cartan, ∆s ⊂ h∗ simple roots. A subset S of ∆s defines a cell, σ = {λ ∈ h∗ ; (λ, S) > 0 , (λ, ∆s\S) = 0}. Dominant Weyl chamber is disjoint union of cells. cells {σ} ← → parabolic subgroups {P} of GC. May write Gλ = Gσ and Xλ = Xσ, where λ ∈ σ.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 11. Cells

h ⊂ g Cartan, ∆s ⊂ h∗ simple roots. A subset S of ∆s defines a cell, σ = {λ ∈ h∗ ; (λ, S) > 0 , (λ, ∆s\S) = 0}. Dominant Weyl chamber is disjoint union of cells. cells {σ} ← → parabolic subgroups {P} of GC. May write Gλ = Gσ and Xλ = Xσ, where λ ∈ σ.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 11. Cells

h ⊂ g Cartan, ∆s ⊂ h∗ simple roots. A subset S of ∆s defines a cell, σ = {λ ∈ h∗ ; (λ, S) > 0 , (λ, ∆s\S) = 0}. Dominant Weyl chamber is disjoint union of cells. cells {σ} ← → parabolic subgroups {P} of GC. May write Gλ = Gσ and Xλ = Xσ, where λ ∈ σ.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 11. Cells

h ⊂ g Cartan, ∆s ⊂ h∗ simple roots. A subset S of ∆s defines a cell, σ = {λ ∈ h∗ ; (λ, S) > 0 , (λ, ∆s\S) = 0}. Dominant Weyl chamber is disjoint union of cells. cells {σ} ← → parabolic subgroups {P} of GC. May write Gλ = Gσ and Xλ = Xσ, where λ ∈ σ.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 12. Models of Compact Lie Groups

Theorem Recall moment map Φ : Xσ − → g∗.

• 1. Im(Φ) ∩ h∗ contained in σ.
• 2. G-rep with highest wgt ξ occurs in H iff ξ ∈ Im(Φ).

Can choose ωσ on Xσ such that Im(Φ) ∩ h∗ = σ. Hence

σ H(Xσ) is a model.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 12. Models of Compact Lie Groups

Theorem Recall moment map Φ : Xσ − → g∗.

• 1. Im(Φ) ∩ h∗ contained in σ.
• 2. G-rep with highest wgt ξ occurs in H iff ξ ∈ Im(Φ).

Can choose ωσ on Xσ such that Im(Φ) ∩ h∗ = σ. Hence

σ H(Xσ) is a model.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 13. [Q, R] = 0

There are left G and right Tσ actions on X. e.g. X = GC/N has left G and right T actions. Perform symplectic reduction wrt right Tσ action, ξ ∈ t∗

σ.

Get G-invariant symplectic manifold (Xξ, ωξ). Quantize it, get G-rep H(Xξ). Theorem Quantization commutes with reduction As G-representation, H(Xξ) = H(X)ξ. RHS is G-subrepresentation of H(X) wrt ξ.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 13. [Q, R] = 0

There are left G and right Tσ actions on X. e.g. X = GC/N has left G and right T actions. Perform symplectic reduction wrt right Tσ action, ξ ∈ t∗

σ.

Get G-invariant symplectic manifold (Xξ, ωξ). Quantize it, get G-rep H(Xξ). Theorem Quantization commutes with reduction As G-representation, H(Xξ) = H(X)ξ. RHS is G-subrepresentation of H(X) wrt ξ.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 13. [Q, R] = 0

There are left G and right Tσ actions on X. e.g. X = GC/N has left G and right T actions. Perform symplectic reduction wrt right Tσ action, ξ ∈ t∗

σ.

Get G-invariant symplectic manifold (Xξ, ωξ). Quantize it, get G-rep H(Xξ). Theorem Quantization commutes with reduction As G-representation, H(Xξ) = H(X)ξ. RHS is G-subrepresentation of H(X) wrt ξ.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 14. Non-compact Groups

Suppose G non-compact. Do not know all its irreducible unitary representations. Modify notion of model: Certain family of irreducibles. Holomorphic discrete model: Every holomorphic discrete series occurs once. Regular principal model: Every regular principal series occurs once.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 14. Non-compact Groups

Suppose G non-compact. Do not know all its irreducible unitary representations. Modify notion of model: Certain family of irreducibles. Holomorphic discrete model: Every holomorphic discrete series occurs once. Regular principal model: Every regular principal series occurs once.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 14. Non-compact Groups

Suppose G non-compact. Do not know all its irreducible unitary representations. Modify notion of model: Certain family of irreducibles. Holomorphic discrete model: Every holomorphic discrete series occurs once. Regular principal model: Every regular principal series occurs once.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 14. Non-compact Groups

Suppose G non-compact. Do not know all its irreducible unitary representations. Modify notion of model: Certain family of irreducibles. Holomorphic discrete model: Every holomorphic discrete series occurs once. Regular principal model: Every regular principal series occurs once.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 15. Elliptic Orbit

Suppose G has compact Cartan subgroup. λ integral element of compact Cartan. Elliptic orbit G/Gλ. G/Gλ ֒ → GC/P complex. Xλ ֒ → GC/(P, P) complex. Holomorphic fibration Xλ − → G/Gλ, fiber is complexification of subgroup of Cartan.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 15. Elliptic Orbit

Suppose G has compact Cartan subgroup. λ integral element of compact Cartan. Elliptic orbit G/Gλ. G/Gλ ֒ → GC/P complex. Xλ ֒ → GC/(P, P) complex. Holomorphic fibration Xλ − → G/Gλ, fiber is complexification of subgroup of Cartan.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 15. Elliptic Orbit

Suppose G has compact Cartan subgroup. λ integral element of compact Cartan. Elliptic orbit G/Gλ. G/Gλ ֒ → GC/P complex. Xλ ֒ → GC/(P, P) complex. Holomorphic fibration Xλ − → G/Gλ, fiber is complexification of subgroup of Cartan.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 15. Elliptic Orbit

Suppose G has compact Cartan subgroup. λ integral element of compact Cartan. Elliptic orbit G/Gλ. G/Gλ ֒ → GC/P complex. Xλ ֒ → GC/(P, P) complex. Holomorphic fibration Xλ − → G/Gλ, fiber is complexification of subgroup of Cartan.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 16. Discrete Series

Harish-Chandra: Since G has compact Cartan, it has non-empty discrete series. Subrepresentations of L2(G). Harish-Chandra parameter Θξ+ρ. Schmid: Construct Θξ+ρ via harmonic Dolbeault (0, q)-forms Hq(G/Gλ). q = q(ξ)

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 16. Discrete Series

Harish-Chandra: Since G has compact Cartan, it has non-empty discrete series. Subrepresentations of L2(G). Harish-Chandra parameter Θξ+ρ. Schmid: Construct Θξ+ρ via harmonic Dolbeault (0, q)-forms Hq(G/Gλ). q = q(ξ)

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 16. Discrete Series

Harish-Chandra: Since G has compact Cartan, it has non-empty discrete series. Subrepresentations of L2(G). Harish-Chandra parameter Θξ+ρ. Schmid: Construct Θξ+ρ via harmonic Dolbeault (0, q)-forms Hq(G/Gλ). q = q(ξ)

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 17. Holomorphic Discrete Model

Quantize (Xλ, ωλ): Harmonic Dolbeault (0, q)-forms in L, Hq(Xλ, ωλ). Theorem As G-representation, Hq((Xλ, ωλ)ξ) = (Hq(Xλ, ωλ))ξ = Θξ+ρ. For G Hermitian and suitable ωλ,

• σ H0(Xσ, ωσ) is a holomorphic discrete model of G.

(recall σ are the cells)

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 17. Holomorphic Discrete Model

Quantize (Xλ, ωλ): Harmonic Dolbeault (0, q)-forms in L, Hq(Xλ, ωλ). Theorem As G-representation, Hq((Xλ, ωλ)ξ) = (Hq(Xλ, ωλ))ξ = Θξ+ρ. For G Hermitian and suitable ωλ,

• σ H0(Xσ, ωσ) is a holomorphic discrete model of G.

(recall σ are the cells)

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 18. Hyperbolic Orbit and Principal Series

Suppose G split. Cartan subgroup MA. M finite abelian, A Euclidean. Let λ ∈ a∗ ⊂ g∗ generic. Hyperbolic orbit G/Gλ = G/MA, M finite. Principal series I(σ ⊗ ν) = IndG

MAN(σ ⊗ ν ⊗ 1).

f : G − → C f(gman) = σ(m)e(ν+ρ)(a)f(g) ,

• K |f(k)|2 dk < ∞.

Call I(σ ⊗ ν) regular if ν ∈ a∗ regular.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 18. Hyperbolic Orbit and Principal Series

Suppose G split. Cartan subgroup MA. M finite abelian, A Euclidean. Let λ ∈ a∗ ⊂ g∗ generic. Hyperbolic orbit G/Gλ = G/MA, M finite. Principal series I(σ ⊗ ν) = IndG

MAN(σ ⊗ ν ⊗ 1).

f : G − → C f(gman) = σ(m)e(ν+ρ)(a)f(g) ,

• K |f(k)|2 dk < ∞.

Call I(σ ⊗ ν) regular if ν ∈ a∗ regular.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 18. Hyperbolic Orbit and Principal Series

Suppose G split. Cartan subgroup MA. M finite abelian, A Euclidean. Let λ ∈ a∗ ⊂ g∗ generic. Hyperbolic orbit G/Gλ = G/MA, M finite. Principal series I(σ ⊗ ν) = IndG

MAN(σ ⊗ ν ⊗ 1).

f : G − → C f(gman) = σ(m)e(ν+ρ)(a)f(g) ,

• K |f(k)|2 dk < ∞.

Call I(σ ⊗ ν) regular if ν ∈ a∗ regular.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 19. Regular Principal Model

X = G × a fibers over hyperbolic orbit G/MA. Want H(X) to be a regular principal model: Every regular principal series representation I(σ ⊗ ν)

• ccurs once in H(X).

Problems: X no complex structure. ν ∈ a∗ continuous parameter. Method: Real Polarization, Direct Integral.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 19. Regular Principal Model

X = G × a fibers over hyperbolic orbit G/MA. Want H(X) to be a regular principal model: Every regular principal series representation I(σ ⊗ ν)

• ccurs once in H(X).

Problems: X no complex structure. ν ∈ a∗ continuous parameter. Method: Real Polarization, Direct Integral.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 19. Regular Principal Model

X = G × a fibers over hyperbolic orbit G/MA. Want H(X) to be a regular principal model: Every regular principal series representation I(σ ⊗ ν)

• ccurs once in H(X).

Problems: X no complex structure. ν ∈ a∗ continuous parameter. Method: Real Polarization, Direct Integral.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

• 20. Regular Principal Model

Theorem Again [Q, R] = 0. For suitable ω, H(X, ω) =

ˆ M

⊕

a∗

+ I(σ ⊗ ν) dν

is a regular principal model.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan

The End. Thank You.

Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups