Quasi-split real groups and the Hitchin map International Meeting - - PowerPoint PPT Presentation

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Quasi-split real groups and the Hitchin map International Meeting - - PowerPoint PPT Presentation

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Higgs bundles Abelianization The HKR section Whats next? Quasi-split real groups and the Hitchin map International Meeting AMS/EMS/SPM Special session Higgs bundles and character varieties Porto, June 2015 Ana Pen-Nieto Mathematisches

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Higgs bundles Abelianization The HKR section What’s next?

Quasi-split real groups and the Hitchin map

International Meeting AMS/EMS/SPM Special session Higgs bundles and character varieties Porto, June 2015 Ana Peón-Nieto

Mathematisches Institut Ruprecht–Karls Universität Heidelberg

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

Index

1

G-Higgs bundles and the Hitchin map

2

Abelianization

3

The Hitchin–Kostant–Rallis section

4

What’s next?

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

G-Higgs bundles

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

G-Higgs bundles X connected smooth projective curve/C, g(X) ≥ 2.

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

G-Higgs bundles X connected smooth projective curve/C, g(X) ≥ 2. G real reductive Lie group

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

G-Higgs bundles X connected smooth projective curve/C, g(X) ≥ 2. G real reductive Lie group (GL(n, R), U(n)).

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

G-Higgs bundles X connected smooth projective curve/C, g(X) ≥ 2. G real reductive Lie group (GL(n, R), U(n)). H ≤ G maximal compact subgroup

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 8

Higgs bundles Abelianization The HKR section What’s next?

G-Higgs bundles X connected smooth projective curve/C, g(X) ≥ 2. G real reductive Lie group (GL(n, R), U(n)). H ≤ G maximal compact subgroup (O(n), U(n)).

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 9

Higgs bundles Abelianization The HKR section What’s next?

G-Higgs bundles X connected smooth projective curve/C, g(X) ≥ 2. G real reductive Lie group (GL(n, R), U(n)). H ≤ G maximal compact subgroup (O(n), U(n)). g = h ⊕ m Cartan decomposition,

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

G-Higgs bundles X connected smooth projective curve/C, g(X) ≥ 2. G real reductive Lie group (GL(n, R), U(n)). H ≤ G maximal compact subgroup (O(n), U(n)). g = h ⊕ m Cartan decomposition, θ Cartan involution.

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

G-Higgs bundles X connected smooth projective curve/C, g(X) ≥ 2. G real reductive Lie group (GL(n, R), U(n)). H ≤ G maximal compact subgroup (O(n), U(n)). g = h ⊕ m Cartan decomposition, θ Cartan involution. HC mC isotropy representation

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 12

Higgs bundles Abelianization The HKR section What’s next?

G-Higgs bundles X connected smooth projective curve/C, g(X) ≥ 2. G real reductive Lie group (GL(n, R), U(n)). H ≤ G maximal compact subgroup (O(n), U(n)). g = h ⊕ m Cartan decomposition, θ Cartan involution. HC mC isotropy representation (O(n) sym(n, R)).

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

G-Higgs bundles X connected smooth projective curve/C, g(X) ≥ 2. G real reductive Lie group (GL(n, R), U(n)). H ≤ G maximal compact subgroup (O(n), U(n)). g = h ⊕ m Cartan decomposition, θ Cartan involution. HC mC isotropy representation (O(n) sym(n, R)). Definition A G-Higgs bundle on X is a pair (E, φ)

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

G-Higgs bundles X connected smooth projective curve/C, g(X) ≥ 2. G real reductive Lie group (GL(n, R), U(n)). H ≤ G maximal compact subgroup (O(n), U(n)). g = h ⊕ m Cartan decomposition, θ Cartan involution. HC mC isotropy representation (O(n) sym(n, R)). Definition A G-Higgs bundle on X is a pair (E, φ) with E → X a holomorphic principal HC-bundle

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

G-Higgs bundles X connected smooth projective curve/C, g(X) ≥ 2. G real reductive Lie group (GL(n, R), U(n)). H ≤ G maximal compact subgroup (O(n), U(n)). g = h ⊕ m Cartan decomposition, θ Cartan involution. HC mC isotropy representation (O(n) sym(n, R)). Definition A G-Higgs bundle on X is a pair (E, φ) with E → X a holomorphic principal HC-bundle and φ ∈ H0(X, E(mC) ⊗ K) the Higgs field.

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

G-Higgs bundles X connected smooth projective curve/C, g(X) ≥ 2. G real reductive Lie group (GL(n, R), U(n)). H ≤ G maximal compact subgroup (O(n), U(n)). g = h ⊕ m Cartan decomposition, θ Cartan involution. HC mC isotropy representation (O(n) sym(n, R)). Definition A G-Higgs bundle on X is a pair (E, φ) with E → X a holomorphic principal HC-bundle and φ ∈ H0(X, E(mC) ⊗ K) the Higgs field. Examples

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 17

Higgs bundles Abelianization The HKR section What’s next?

G-Higgs bundles X connected smooth projective curve/C, g(X) ≥ 2. G real reductive Lie group (GL(n, R), U(n)). H ≤ G maximal compact subgroup (O(n), U(n)). g = h ⊕ m Cartan decomposition, θ Cartan involution. HC mC isotropy representation (O(n) sym(n, R)). Definition A G-Higgs bundle on X is a pair (E, φ) with E → X a holomorphic principal HC-bundle and φ ∈ H0(X, E(mC) ⊗ K) the Higgs field. Examples

  • 1. GL(n, C)R : E rk n vector bundle, φ : E → E ⊗ K.

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

G-Higgs bundles X connected smooth projective curve/C, g(X) ≥ 2. G real reductive Lie group (GL(n, R), U(n)). H ≤ G maximal compact subgroup (O(n), U(n)). g = h ⊕ m Cartan decomposition, θ Cartan involution. HC mC isotropy representation (O(n) sym(n, R)). Definition A G-Higgs bundle on X is a pair (E, φ) with E → X a holomorphic principal HC-bundle and φ ∈ H0(X, E(mC) ⊗ K) the Higgs field. Examples

  • 1. GL(n, C)R : E rk n vector bundle, φ : E → E ⊗ K.
  • 2. GL(n, R) : E ∼

= E ∗ rk n v.b.+symmetric form, φ symmetric.

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

G-Higgs bundles X connected smooth projective curve/C, g(X) ≥ 2. G real reductive Lie group (GL(n, R), U(n)). H ≤ G maximal compact subgroup (O(n), U(n)). g = h ⊕ m Cartan decomposition, θ Cartan involution. HC mC isotropy representation (O(n) sym(n, R)). Definition A G-Higgs bundle on X is a pair (E, φ) with E → X a holomorphic principal HC-bundle and φ ∈ H0(X, E(mC) ⊗ K) the Higgs field. Examples

  • 1. GL(n, C)R : E rk n vector bundle, φ : E → E ⊗ K.
  • 2. GL(n, R) : E ∼

= E ∗ rk n v.b.+symmetric form, φ symmetric.

  • 3. U(p, q) : E = V ⊕ W , φ = (β, γ) : V

γ

→ W ⊗ K, W

β

→ V ⊗ K.

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

The Hitchin map

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

The Hitchin map Definition The Hitchin map is a morphism hG : M(G) → BG := H0(X, ⊕iK di)

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

The Hitchin map Definition The Hitchin map is a morphism hG : M(G) → BG := H0(X, ⊕iK di) (E, φ) → (p1(φ), . . . , pr(φ))

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

The Hitchin map Definition The Hitchin map is a morphism hG : M(G) → BG := H0(X, ⊕iK di) (E, φ) → (p1(φ), . . . , pr(φ)) M(G) moduli space of ps. Higgs bundles

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

The Hitchin map Definition The Hitchin map is a morphism hG : M(G) → BG := H0(X, ⊕iK di) (E, φ) → (p1(φ), . . . , pr(φ)) M(G) moduli space of ps. Higgs bundles(∼ = Hom(π1, G)//G)

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

The Hitchin map Definition The Hitchin map is a morphism hG : M(G) → BG := H0(X, ⊕iK di) (E, φ) → (p1(φ), . . . , pr(φ)) M(G) moduli space of ps. Higgs bundles(∼ = Hom(π1, G)//G) r = rkRG, p1, . . . , pr ∈ C[mC]HC generators,

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

The Hitchin map Definition The Hitchin map is a morphism hG : M(G) → BG := H0(X, ⊕iK di) (E, φ) → (p1(φ), . . . , pr(φ)) M(G) moduli space of ps. Higgs bundles(∼ = Hom(π1, G)//G) r = rkRG, p1, . . . , pr ∈ C[mC]HC generators, di = deg pi.

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

The Hitchin map Definition The Hitchin map is a morphism hG : M(G) → BG := H0(X, ⊕iK di) (E, φ) → (p1(φ), . . . , pr(φ)) M(G) moduli space of ps. Higgs bundles(∼ = Hom(π1, G)//G) r = rkRG, p1, . . . , pr ∈ C[mC]HC generators, di = deg pi. Examples

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

The Hitchin map Definition The Hitchin map is a morphism hG : M(G) → BG := H0(X, ⊕iK di) (E, φ) → (p1(φ), . . . , pr(φ)) M(G) moduli space of ps. Higgs bundles(∼ = Hom(π1, G)//G) r = rkRG, p1, . . . , pr ∈ C[mC]HC generators, di = deg pi. Examples

  • 1. GL(n, C)R pi(x) = tr(xi) i = 1, . . . , n.

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

The Hitchin map Definition The Hitchin map is a morphism hG : M(G) → BG := H0(X, ⊕iK di) (E, φ) → (p1(φ), . . . , pr(φ)) M(G) moduli space of ps. Higgs bundles(∼ = Hom(π1, G)//G) r = rkRG, p1, . . . , pr ∈ C[mC]HC generators, di = deg pi. Examples

  • 1. GL(n, C)R pi(x) = tr(xi) i = 1, . . . , n.
  • 2. GL(n, R) : pi(x) = tr(xi), i = 1, . . . , n (split).

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

The Hitchin map Definition The Hitchin map is a morphism hG : M(G) → BG := H0(X, ⊕iK di) (E, φ) → (p1(φ), . . . , pr(φ)) M(G) moduli space of ps. Higgs bundles(∼ = Hom(π1, G)//G) r = rkRG, p1, . . . , pr ∈ C[mC]HC generators, di = deg pi. Examples

  • 1. GL(n, C)R pi(x) = tr(xi) i = 1, . . . , n.
  • 2. GL(n, R) : pi(x) = tr(xi), i = 1, . . . , n (split).
  • 3. U(p, q) : pi(x) = tr(x2i), i = 1, . . . , q (p > q).

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

Understanding the Hitchin map

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

Understanding the Hitchin map Two major steps:

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

Understanding the Hitchin map Two major steps:

1

Study of the fibers

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

Understanding the Hitchin map Two major steps:

1

Study of the fibers abelianization (complex groups).

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

Understanding the Hitchin map Two major steps:

1

Study of the fibers abelianization (complex groups).

2

Existence of a section

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 36

Higgs bundles Abelianization The HKR section What’s next?

Understanding the Hitchin map Two major steps:

1

Study of the fibers abelianization (complex groups).

2

Existence of a section Hitchin section (C/split groups)

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

Understanding the Hitchin map Two major steps:

1

Study of the fibers abelianization (complex groups).

2

Existence of a section Hitchin section (C/split groups), Hitchin–Kostant–Rallis section (real groups).

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

Abelianization

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

Abelianization Idea: Higgs bundles ↔ eigen-line bundles

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

Abelianization Idea: Higgs bundles ↔ eigen-line bundles Eigenvalues parametrized by

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 41

Higgs bundles Abelianization The HKR section What’s next?

Abelianization Idea: Higgs bundles ↔ eigen-line bundles Eigenvalues parametrized by (a) characteristic polynomials (unordered eigenvalues)

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 42

Higgs bundles Abelianization The HKR section What’s next?

Abelianization Idea: Higgs bundles ↔ eigen-line bundles Eigenvalues parametrized by (a) characteristic polynomials (unordered eigenvalues) spectral curve,

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

Abelianization Idea: Higgs bundles ↔ eigen-line bundles Eigenvalues parametrized by (a) characteristic polynomials (unordered eigenvalues) spectral curve, (b) diagonal matrices (ordered eigenvalues)

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

Abelianization Idea: Higgs bundles ↔ eigen-line bundles Eigenvalues parametrized by (a) characteristic polynomials (unordered eigenvalues) spectral curve, (b) diagonal matrices (ordered eigenvalues) cameral curve.

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

Abelianization Idea: Higgs bundles ↔ eigen-line bundles Eigenvalues parametrized by (a) characteristic polynomials (unordered eigenvalues) spectral curve, (b) diagonal matrices (ordered eigenvalues) cameral curve. (b) generalises to all reductive groups:

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

Abelianization Idea: Higgs bundles ↔ eigen-line bundles Eigenvalues parametrized by (a) characteristic polynomials (unordered eigenvalues) spectral curve, (b) diagonal matrices (ordered eigenvalues) cameral curve. (b) generalises to all reductive groups: diagonal matrices ↔ Cartan subalgebras dC

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 47

Higgs bundles Abelianization The HKR section What’s next?

Abelianization Idea: Higgs bundles ↔ eigen-line bundles Eigenvalues parametrized by (a) characteristic polynomials (unordered eigenvalues) spectral curve, (b) diagonal matrices (ordered eigenvalues) cameral curve. (b) generalises to all reductive groups: diagonal matrices ↔ Cartan subalgebras dC eigenbundles ↔ principal bundles of tori DC.

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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Higgs bundles Abelianization The HKR section What’s next?

Spectral vs. cameral covers

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 49

Higgs bundles Abelianization The HKR section What’s next?

Spectral vs. cameral covers (a) Spectral techniques (GL(2, C)):

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 50

Higgs bundles Abelianization The HKR section What’s next?

Spectral vs. cameral covers (a) Spectral techniques (GL(2, C)): (E, φ) char(φ) = λ2 + π∗a1λ + π∗a2 =: sa ∈ H0(|K|, π∗K)

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 51

Higgs bundles Abelianization The HKR section What’s next?

Spectral vs. cameral covers (a) Spectral techniques (GL(2, C)): (E, φ) char(φ) = λ2 + π∗a1λ + π∗a2 =: sa ∈ H0(|K|, π∗K)

  • Xa := {sa = 0} spectral curve

A.Peón-Nieto Quasi-split real groups and the Hitchin map

slide-52
SLIDE 52

Higgs bundles Abelianization The HKR section What’s next?

Spectral vs. cameral covers (a) Spectral techniques (GL(2, C)): (E, φ) char(φ) = λ2 + π∗a1λ + π∗a2 =: sa ∈ H0(|K|, π∗K)

  • Xa := {sa = 0} spectral curve ker(π∗φ − λId) = Lλ.

A.Peón-Nieto Quasi-split real groups and the Hitchin map

slide-53
SLIDE 53

Higgs bundles Abelianization The HKR section What’s next?

Spectral vs. cameral covers (a) Spectral techniques (GL(2, C)): (E, φ) char(φ) = λ2 + π∗a1λ + π∗a2 =: sa ∈ H0(|K|, π∗K)

  • Xa := {sa = 0} spectral curve ker(π∗φ − λId) = Lλ.

Theorem (Hitchin, 87) (E, φ) ↔ Lλ yields an isomorphism h−1(a) ∼ = Pic( Xa)

A.Peón-Nieto Quasi-split real groups and the Hitchin map

slide-54
SLIDE 54

Higgs bundles Abelianization The HKR section What’s next?

Spectral vs. cameral covers (a) Spectral techniques (GL(2, C)): (E, φ) char(φ) = λ2 + π∗a1λ + π∗a2 =: sa ∈ H0(|K|, π∗K)

  • Xa := {sa = 0} spectral curve ker(π∗φ − λId) = Lλ.

Theorem (Hitchin, 87) (E, φ) ↔ Lλ yields an isomorphism h−1(a) ∼ = Pic( Xa) (b) Cameral techniques(GL(2, C))

A.Peón-Nieto Quasi-split real groups and the Hitchin map

slide-55
SLIDE 55

Higgs bundles Abelianization The HKR section What’s next?

Spectral vs. cameral covers (a) Spectral techniques (GL(2, C)): (E, φ) char(φ) = λ2 + π∗a1λ + π∗a2 =: sa ∈ H0(|K|, π∗K)

  • Xa := {sa = 0} spectral curve ker(π∗φ − λId) = Lλ.

Theorem (Hitchin, 87) (E, φ) ↔ Lλ yields an isomorphism h−1(a) ∼ = Pic( Xa) (b) Cameral techniques(GL(2, C)) (E, φ) φss semisimple part

A.Peón-Nieto Quasi-split real groups and the Hitchin map

slide-56
SLIDE 56

Higgs bundles Abelianization The HKR section What’s next?

Spectral vs. cameral covers (a) Spectral techniques (GL(2, C)): (E, φ) char(φ) = λ2 + π∗a1λ + π∗a2 =: sa ∈ H0(|K|, π∗K)

  • Xa := {sa = 0} spectral curve ker(π∗φ − λId) = Lλ.

Theorem (Hitchin, 87) (E, φ) ↔ Lλ yields an isomorphism h−1(a) ∼ = Pic( Xa) (b) Cameral techniques(GL(2, C)) (E, φ) φss semisimple part up to conjugation φss ∈ K ⊕ K.

A.Peón-Nieto Quasi-split real groups and the Hitchin map

slide-57
SLIDE 57

Higgs bundles Abelianization The HKR section What’s next?

Spectral vs. cameral covers (a) Spectral techniques (GL(2, C)): (E, φ) char(φ) = λ2 + π∗a1λ + π∗a2 =: sa ∈ H0(|K|, π∗K)

  • Xa := {sa = 0} spectral curve ker(π∗φ − λId) = Lλ.

Theorem (Hitchin, 87) (E, φ) ↔ Lλ yields an isomorphism h−1(a) ∼ = Pic( Xa) (b) Cameral techniques(GL(2, C)) (E, φ) φss semisimple part up to conjugation φss ∈ K ⊕ K. a := h(E, φ) = (trφ, det φ) : X → K ⊕ K 2

A.Peón-Nieto Quasi-split real groups and the Hitchin map

slide-58
SLIDE 58

Higgs bundles Abelianization The HKR section What’s next?

Spectral vs. cameral covers (a) Spectral techniques (GL(2, C)): (E, φ) char(φ) = λ2 + π∗a1λ + π∗a2 =: sa ∈ H0(|K|, π∗K)

  • Xa := {sa = 0} spectral curve ker(π∗φ − λId) = Lλ.

Theorem (Hitchin, 87) (E, φ) ↔ Lλ yields an isomorphism h−1(a) ∼ = Pic( Xa) (b) Cameral techniques(GL(2, C)) (E, φ) φss semisimple part up to conjugation φss ∈ K ⊕ K. a := h(E, φ) = (trφ, det φ) : X → K ⊕ K 2 K ⊕ K → K ⊕ K 2

A.Peón-Nieto Quasi-split real groups and the Hitchin map

slide-59
SLIDE 59

Higgs bundles Abelianization The HKR section What’s next?

Spectral vs. cameral covers (a) Spectral techniques (GL(2, C)): (E, φ) char(φ) = λ2 + π∗a1λ + π∗a2 =: sa ∈ H0(|K|, π∗K)

  • Xa := {sa = 0} spectral curve ker(π∗φ − λId) = Lλ.

Theorem (Hitchin, 87) (E, φ) ↔ Lλ yields an isomorphism h−1(a) ∼ = Pic( Xa) (b) Cameral techniques(GL(2, C)) (E, φ) φss semisimple part up to conjugation φss ∈ K ⊕ K. a := h(E, φ) = (trφ, det φ) : X → K ⊕ K 2 K ⊕ K → K ⊕ K 2 (l, l′) → (l + l′, l · l′)

A.Peón-Nieto Quasi-split real groups and the Hitchin map

slide-60
SLIDE 60

Higgs bundles Abelianization The HKR section What’s next?

Spectral vs. cameral covers (a) Spectral techniques (GL(2, C)): (E, φ) char(φ) = λ2 + π∗a1λ + π∗a2 =: sa ∈ H0(|K|, π∗K)

  • Xa := {sa = 0} spectral curve ker(π∗φ − λId) = Lλ.

Theorem (Hitchin, 87) (E, φ) ↔ Lλ yields an isomorphism h−1(a) ∼ = Pic( Xa) (b) Cameral techniques(GL(2, C)) (E, φ) φss semisimple part up to conjugation φss ∈ K ⊕ K. a := h(E, φ) = (trφ, det φ) : X → K ⊕ K 2 K ⊕ K → K ⊕ K 2 (l, l′) → (l + l′, l · l′) Theorem (Donagi, 93, D-Gaitsgory 02) h−1(a) ∼ = H1( Xa, (C×)2)S2 where Xa := a∗K ⊕2 cameral cover

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 61

Higgs bundles Abelianization The HKR section What’s next?

Quasi-split real groups

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 62

Higgs bundles Abelianization The HKR section What’s next?

Quasi-split real groups Examples: complex groups,

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 63

Higgs bundles Abelianization The HKR section What’s next?

Quasi-split real groups Examples: complex groups, split (GL(n, R), Sp(2n, R), SO(n, n + 1), SO(n, n))

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 64

Higgs bundles Abelianization The HKR section What’s next?

Quasi-split real groups Examples: complex groups, split (GL(n, R), Sp(2n, R), SO(n, n + 1), SO(n, n)) SU(p, p), SU(p + 1, p), SO(p, p + 2)

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 65

Higgs bundles Abelianization The HKR section What’s next?

Quasi-split real groups Examples: complex groups, split (GL(n, R), Sp(2n, R), SO(n, n + 1), SO(n, n)) SU(p, p), SU(p + 1, p), SO(p, p + 2) What is special about them?

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 66

Higgs bundles Abelianization The HKR section What’s next?

Quasi-split real groups Examples: complex groups, split (GL(n, R), Sp(2n, R), SO(n, n + 1), SO(n, n)) SU(p, p), SU(p + 1, p), SO(p, p + 2) What is special about them? Most elements in mC have one dimensional eigenspaces (matrix groups).

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 67

Higgs bundles Abelianization The HKR section What’s next?

Quasi-split real groups Examples: complex groups, split (GL(n, R), Sp(2n, R), SO(n, n + 1), SO(n, n)) SU(p, p), SU(p + 1, p), SO(p, p + 2) What is special about them? Most elements in mC have one dimensional eigenspaces (matrix groups). In other words mreg ⊂ greg (any group).

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 68

Higgs bundles Abelianization The HKR section What’s next?

The Hitchin fibers for quasi-split real groups

  • G quasi-split real reductive group; θ Cartan involution.

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 69

Higgs bundles Abelianization The HKR section What’s next?

The Hitchin fibers for quasi-split real groups

  • G quasi-split real reductive group; θ Cartan involution.
  • dC ⊂ gC

Cartan subalgebra

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 70

Higgs bundles Abelianization The HKR section What’s next?

The Hitchin fibers for quasi-split real groups

  • G quasi-split real reductive group; θ Cartan involution.
  • dC ⊂ gC θ-inv. Cartan subalgebra

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 71

Higgs bundles Abelianization The HKR section What’s next?

The Hitchin fibers for quasi-split real groups

  • G quasi-split real reductive group; θ Cartan involution.
  • dC ⊂ gC θ-inv. Cartan subalgebra dC = tC ⊕ aC; aC ⊂ mC.

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 72

Higgs bundles Abelianization The HKR section What’s next?

The Hitchin fibers for quasi-split real groups

  • G quasi-split real reductive group; θ Cartan involution.
  • dC ⊂ gC θ-inv. Cartan subalgebra dC = tC ⊕ aC; aC ⊂ mC.
  • W Weyl group

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 73

Higgs bundles Abelianization The HKR section What’s next?

The Hitchin fibers for quasi-split real groups

  • G quasi-split real reductive group; θ Cartan involution.
  • dC ⊂ gC θ-inv. Cartan subalgebra dC = tC ⊕ aC; aC ⊂ mC.
  • W Weyl group
  • We note ⊕iK di = aC ⊗ K/W (a) ⊂ dC ⊗ K/W

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 74

Higgs bundles Abelianization The HKR section What’s next?

The Hitchin fibers for quasi-split real groups

  • G quasi-split real reductive group; θ Cartan involution.
  • dC ⊂ gC θ-inv. Cartan subalgebra dC = tC ⊕ aC; aC ⊂ mC.
  • W Weyl group
  • We note ⊕iK di = aC ⊗ K/W (a) ⊂ dC ⊗ K/W

Theorem (García-Prada, P-N.) Let a ∈ BG, define its associated cameral cover by

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 75

Higgs bundles Abelianization The HKR section What’s next?

The Hitchin fibers for quasi-split real groups

  • G quasi-split real reductive group; θ Cartan involution.
  • dC ⊂ gC θ-inv. Cartan subalgebra dC = tC ⊕ aC; aC ⊂ mC.
  • W Weyl group
  • We note ⊕iK di = aC ⊗ K/W (a) ⊂ dC ⊗ K/W

Theorem (García-Prada, P-N.) Let a ∈ BG, define its associated cameral cover by dC ⊗ K

  • dC ⊗ K/W

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 76

Higgs bundles Abelianization The HKR section What’s next?

The Hitchin fibers for quasi-split real groups

  • G quasi-split real reductive group; θ Cartan involution.
  • dC ⊂ gC θ-inv. Cartan subalgebra dC = tC ⊕ aC; aC ⊂ mC.
  • W Weyl group
  • We note ⊕iK di = aC ⊗ K/W (a) ⊂ dC ⊗ K/W

Theorem (García-Prada, P-N.) Let a ∈ BG, define its associated cameral cover by dC ⊗ K

  • X

a

dC ⊗ K/W

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 77

Higgs bundles Abelianization The HKR section What’s next?

The Hitchin fibers for quasi-split real groups

  • G quasi-split real reductive group; θ Cartan involution.
  • dC ⊂ gC θ-inv. Cartan subalgebra dC = tC ⊕ aC; aC ⊂ mC.
  • W Weyl group
  • We note ⊕iK di = aC ⊗ K/W (a) ⊂ dC ⊗ K/W

Theorem (García-Prada, P-N.) Let a ∈ BG, define its associated cameral cover by

  • Xa
  • dC ⊗ K
  • X

a

dC ⊗ K/W

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 78

Higgs bundles Abelianization The HKR section What’s next?

The Hitchin fibers for quasi-split real groups

  • G quasi-split real reductive group; θ Cartan involution.
  • dC ⊂ gC θ-inv. Cartan subalgebra dC = tC ⊕ aC; aC ⊂ mC.
  • W Weyl group
  • We note ⊕iK di = aC ⊗ K/W (a) ⊂ dC ⊗ K/W

Theorem (García-Prada, P-N.) Let a ∈ BG, define its associated cameral cover by

  • Xa
  • dC ⊗ K
  • X

a

dC ⊗ K/W

Then h−1(a) ∼ = H1( Xa, DC)W ,θ

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 79

Higgs bundles Abelianization The HKR section What’s next?

The Hitchin fibers for quasi-split real groups

  • G quasi-split real reductive group; θ Cartan involution.
  • dC ⊂ gC θ-inv. Cartan subalgebra dC = tC ⊕ aC; aC ⊂ mC.
  • W Weyl group
  • We note ⊕iK di = aC ⊗ K/W (a) ⊂ dC ⊗ K/W

Theorem (García-Prada, P-N.) Let a ∈ BG, define its associated cameral cover by

  • Xa
  • dC ⊗ K
  • X

a

dC ⊗ K/W

Then h−1(a) ∼ = H1( Xa, DC)W ,θ(regular)

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 80

Higgs bundles Abelianization The HKR section What’s next?

Two applications

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 81

Higgs bundles Abelianization The HKR section What’s next?

Two applications Corollary If G < G C is split then h−1

G (a) = h−1 G C(a)[2].

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 82

Higgs bundles Abelianization The HKR section What’s next?

Two applications Corollary If G < G C is split then h−1

G (a) = h−1 G C(a)[2].

Theorem (P-N.) If G = SU(p + 1, p), a ∈ BSU(p+1,p) generic

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 83

Higgs bundles Abelianization The HKR section What’s next?

Two applications Corollary If G < G C is split then h−1

G (a) = h−1 G C(a)[2].

Theorem (P-N.) If G = SU(p + 1, p), a ∈ BSU(p+1,p) generic h−1

G (a) is a bundle of

projective spaces over Pic(Y )

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 84

Higgs bundles Abelianization The HKR section What’s next?

Two applications Corollary If G < G C is split then h−1

G (a) = h−1 G C(a)[2].

Theorem (P-N.) If G = SU(p + 1, p), a ∈ BSU(p+1,p) generic h−1

G (a) is a bundle of

projective spaces over Pic(Y ) Y = {λ2p + a2λ2p−2 + · · · + a2p = 0}.

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 85

Higgs bundles Abelianization The HKR section What’s next?

The Hitchin–Kostant–Rallis section

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 86

Higgs bundles Abelianization The HKR section What’s next?

The Hitchin–Kostant–Rallis section Theorem (García-Prada, P-N., Ramanan) Let G be quasi-split. Then, there exists a section s : BG → M(G)smooth such that s(a) = (Ea, φa) is everywhere regular (φa(x) ∈ mreg).

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 87

Higgs bundles Abelianization The HKR section What’s next?

What’s next?

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 88

Higgs bundles Abelianization The HKR section What’s next?

What’s next?

1

“Non-abelianizable” case.

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 89

Higgs bundles Abelianization The HKR section What’s next?

What’s next?

1

“Non-abelianizable” case. Sheaves of non abelian groups involved e.g. for U(p, q), GL(p − q, C) × (C×)q

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 90

Higgs bundles Abelianization The HKR section What’s next?

What’s next?

1

“Non-abelianizable” case. Sheaves of non abelian groups involved e.g. for U(p, q), GL(p − q, C) × (C×)q Decompose M(G) into abelian M(Gqs) and non abelian M

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 91

Higgs bundles Abelianization The HKR section What’s next?

What’s next?

1

“Non-abelianizable” case. Sheaves of non abelian groups involved e.g. for U(p, q), GL(p − q, C) × (C×)q Decompose M(G) into abelian M(Gqs) and non abelian M e.g. h−1

U(p,q)(a) in terms of BGL(p − q, C) and h−1 U(p,p)(a).

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 92

Higgs bundles Abelianization The HKR section What’s next?

What’s next?

1

“Non-abelianizable” case. Sheaves of non abelian groups involved e.g. for U(p, q), GL(p − q, C) × (C×)q Decompose M(G) into abelian M(Gqs) and non abelian M e.g. h−1

U(p,q)(a) in terms of BGL(p − q, C) and h−1 U(p,p)(a).

2

Geometry of the HKR section.

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 93

Higgs bundles Abelianization The HKR section What’s next?

What’s next?

1

“Non-abelianizable” case. Sheaves of non abelian groups involved e.g. for U(p, q), GL(p − q, C) × (C×)q Decompose M(G) into abelian M(Gqs) and non abelian M e.g. h−1

U(p,q)(a) in terms of BGL(p − q, C) and h−1 U(p,p)(a).

2

Geometry of the HKR section. Image consists of Anosov representations

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 94

Higgs bundles Abelianization The HKR section What’s next?

What’s next?

1

“Non-abelianizable” case. Sheaves of non abelian groups involved e.g. for U(p, q), GL(p − q, C) × (C×)q Decompose M(G) into abelian M(Gqs) and non abelian M e.g. h−1

U(p,q)(a) in terms of BGL(p − q, C) and h−1 U(p,p)(a).

2

Geometry of the HKR section. Image consists of Anosov representations for minimal parabolic subgroups.

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 95

Higgs bundles Abelianization The HKR section What’s next?

What’s next?

1

“Non-abelianizable” case. Sheaves of non abelian groups involved e.g. for U(p, q), GL(p − q, C) × (C×)q Decompose M(G) into abelian M(Gqs) and non abelian M e.g. h−1

U(p,q)(a) in terms of BGL(p − q, C) and h−1 U(p,p)(a).

2

Geometry of the HKR section. Image consists of Anosov representations for minimal parabolic subgroups.

3

Langlands duality

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 96

Higgs bundles Abelianization The HKR section What’s next?

What’s next?

1

“Non-abelianizable” case. Sheaves of non abelian groups involved e.g. for U(p, q), GL(p − q, C) × (C×)q Decompose M(G) into abelian M(Gqs) and non abelian M e.g. h−1

U(p,q)(a) in terms of BGL(p − q, C) and h−1 U(p,p)(a).

2

Geometry of the HKR section. Image consists of Anosov representations for minimal parabolic subgroups.

3

Langlands duality Toy example: Hitchin (U(p, p) vs. Sp(2p, C)-Higgs bundles)

A.Peón-Nieto Quasi-split real groups and the Hitchin map

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SLIDE 97

Higgs bundles Abelianization The HKR section What’s next?

What’s next?

1

“Non-abelianizable” case. Sheaves of non abelian groups involved e.g. for U(p, q), GL(p − q, C) × (C×)q Decompose M(G) into abelian M(Gqs) and non abelian M e.g. h−1

U(p,q)(a) in terms of BGL(p − q, C) and h−1 U(p,p)(a).

2

Geometry of the HKR section. Image consists of Anosov representations for minimal parabolic subgroups.

3

Langlands duality Toy example: Hitchin (U(p, p) vs. Sp(2p, C)-Higgs bundles)

A.Peón-Nieto Quasi-split real groups and the Hitchin map