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❇✐❛➟♦✇✐❡➺❛ ✷✵✶✻

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s ❝❛♥♦♥✐❝❛❧❧②

r❡❧❛t❡❞ t♦ ❛ W ∗✲❛❧❣❡❜r❛

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③ ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s ❯♥✐✈❡rs✐t② ✐♥ ❇✐❛➟②st♦❦

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❘❊❋❊❘❊◆❈❊❙✿

✶ ❆✳ ❖❞③✐❥❡✇✐❝③✱ ❚✳ ❘❛t✐✉✳ ❇❛♥❛❝❤ ▲✐❡✲P♦✐ss♦♥ s♣❛❝❡s ❛♥❞

r❡❞✉❝t✐♦♥✱ ❈♦♠♠✳ ▼❛t❤✳ P❤②s✳✱ ✷✹✸ ✭✷✵✵✸✮ ✶✲✺✹

✷ ❆✳ ❖❞③✐❥❡✇✐❝③✱ ❆✳ ❙❧✐➺❡✇s❦❛✳ ❇❛♥❛❝❤ ▲✐❡ ❣r♦✉♣♦✐❞s ❛ss♦❝✐❛t❡❞

t♦ W ∗✲❛❧❣❡❜r❛✱ t♦ ❛♣♣❡❛r ✐♥ ❏✳ ❙②♠♣❧✳ ●❡♦♠✳

✸ ❆✳ ❖❞③✐❥❡✇✐❝③✱ ●✳ ❏❛❦✐♠♦✇✐❝③✱ ❆✳ ❙❧✐➺❡✇s❦❛✳ ❇❛♥❛❝❤✲▲✐❡

❛❧❣❡❜r♦✐❞s ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❣r♦✉♣♦✐❞ ♦❢ ♣❛rt✐❛❧❧② ✐♥✈❡rt✐❜❧❡ ❡❧❡♠❡♥ts ♦❢ ❛ W ∗✲❛❧❣❡❜r❛✳ ❏✳●❡♦♠✳P❤②s✳✱ ✾✺ ✭✷✵✶✺✮ ✶✵✽✲✶✷✻

✹ ❆✳ ❖❞③✐❥❡✇✐❝③✱ ●✳ ❏❛❦✐♠♦✇✐❝③✱ ❆✳ ❙❧✐➺❡✇s❦❛✳ ❋✐❜r❡✲✇✐s❡ ❧✐♥❡❛r

P♦✐ss♦♥ str✉❝t✉r❡s r❡❧❛t❡❞ t♦ W ∗✲❛❧❣❡❜r❛ ✭✐♥ ♣r❡♣❛r❛t✐♦♥✮

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❇❛♥❛❝❤ P♦✐ss♦♥ ♠❛♥✐❢♦❧❞

P ✲ ❇❛♥❛❝❤ ♠❛♥✐❢♦❧❞ ✭♠♦❞❡❧❡❞ ♦♥ ♥♦♥✲r❡✢❡❦s✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡ ✐♥ ❣❡♥❡r❛❧✮

• π ∈ Γ∞(2 T ∗∗P)
• Γ∞(T ∗P) ∋ α → #α := π(α, ·) ∈ Γ∞(2 T ∗∗P)

T ∗P T ∗∗P T ♭P TP

✻ ✻ ✲ ✲

# # , ✭✶✮

• T ∗P ⊃ T ♭P ✲ q✉❛s✐ ❇❛♥❛❝❤ ✈❡❝t♦r s✉❜❜✉♥❞❧❡ ✭✇✐t❤♦✉t ❇❛♥❛❝❤

❝♦♠♣❧❡♠❡♥t ✐♥ ❣❡♥❡r❛❧✮

• #T ♭P = TP
• P∞(P) := {f ∈ C∞(P) : #d

f ∈ Γ∞TP} P∞(P) ✲ P♦✐ss♦♥ ❛❧❣❡❜r❛ ✇✐t❤ r❡s♣❡❝t t♦ {f, g} := π(d f, dg}

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❇❛♥❛❝❤✲▲✐❡ ❣r♦✉♣♦✐❞

G, B ✲ ❇❛♥❛❝❤ ♠❛♥✐❢♦❧❞ ✇✐t❤ ❍❛✉s❞♦r✛ ✉♥❞❡r❧②✐♥❣ t♦♣♦❧♦❣② ❇❛♥❛❝❤✲▲✐❡ ❣r♦✉♣♦✐❞ G ⇒ B✿

✶ s♦✉r❝❡ ♠❛♣ s : G → B ❛♥❞ t❛r❣❡t ♠❛♣ t : G → B ✲

s✉❜♠❡rs✐♦♥s

✷ ♣r♦❞✉❝t m : G(2) → G

m(g, h) =: gh, ❞❡✜♥❡❞ ♦♥ t❤❡ s❡t ♦❢ ❝♦♠♣♦s❛❜❧❡ ♣❛✐rs G(2) := {(g, h) ∈ G × G : s(g) = t(h)},

✸ ✐❞❡♥t✐t② s❡❝t✐♦♥ ε : B → G ✲ ✐♠♠❡rs✐♦♥ ✹ ✐♥✈❡rs❡ ♠❛♣ ι : G → G✱ ι ◦ ι = id✱ ❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❇❛♥❛❝❤✲▲✐❡ ❣r♦✉♣♦✐❞

✇❤✐❝❤ s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s✿ s(gh) = s(h), t(gh) = t(g), ✭✷✮ k(gh) = (kg)h, ✭✸✮ ε(t(g))g = g = gε(s(g)), ✭✹✮ ι(g)g = ε(s(g)), gι(g) = ε(t(g)), ✭✺✮ ✇❤❡r❡ g, k, h ∈ G. ❚❤❡ s❤♦rt❡r ❞❡✜♥✐t✐♦♥✿ ❆ ❣r♦✉♣♦✐❞ ✐s ❛ s♠❛❧❧ ❝❛t❡❣♦r② ✇✐t❤ ✐♥✈❡rt✐❜❧❡ ♠♦r♣❤✐s♠s✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❇❛♥❛❝❤✲▲✐❡ ❛❧❣❡❜r♦✐❞

❆ ❇❛♥❛❝❤✲▲✐❡ ❛❧❣❡❜r♦✐❞ ♦♥ ❇❛♥❛❝❤ ♠❛♥✐❢♦❧❞ M ✐s ❛ ❇❛♥❛❝❤ ✈❡❝t♦r ❜✉♥❞❧❡ q : A → M t♦❣❡t❤❡r ✇✐t❤✿

✶ a : A → TM ✭❛♥❝❤♦r ♠❛♣✮ ✷ [ , ] : ΓA × ΓA → ΓA ✭▲✐❡ ❜r❛❝❦❡t✮ s✉❝❤ t❤❛t

[X, fY ] = f[X, Y ] + a(X)(f)Y a([X, Y ]) = [a(X), a(Y )] ❢♦r ❛❧❧ X, Y ∈ ΓA, f ∈ C∞(M)✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❋✐❜r❡✲✇✐s❡ ❧✐♥❡❛r P♦✐ss♦♥ str✉❝t✉r❡

❋✐❜r❡✲✇✐s❡ ❧✐♥❡❛r P♦✐ss♦♥ str✉❝t✉r❡ ❖♥❡ t❛❦❡s ❛s ❛ ✭s✉❜✮ P♦✐ss♦♥ ♠❛♥✐❢♦❧❞ P = E✱ ✇❤❡r❡ E ✐s t❤❡ t♦t❛❧ s♣❛❝❡ ♦❢ ❛ ❇❛♥❛❝❤ ✈❡❝t♦r ❜✉♥❞❧❡ E

q

→ M s✉❝❤ t❤❛t

• P∞(E) ⊃ P ∞

lin(E) ✲ ✜❜r❡✲✇✐s❡ ❧✐♥❡❛r

• P∞(E) ⊃ P ∞

B (E) ✲ ❝♦♥st❛♥t ♦♥ t❤❡ ✜❜r❡s ♦❢ q

• {P ∞

B (E), P ∞ B (E)} = 0

• {P ∞

B (E), P ∞ lin(E)} ⊂ P ∞ B (E)

• {P ∞

lin(E), P ∞ lin(E)} ⊂ P ∞ lin(E)

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

W ∗✲❛❧❣❡❜r❛

❆ C∗✲❛❧❣❡❜r❛ M ✐s ❝❛❧❧❡❞ W ∗✲❛❧❣❡❜r❛ ✭✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛✮ ✐❢ t❤❡r❡ ❡①✐sts ❛ ❇❛♥❛❝❤ s♣❛❝❡ M∗ s✉❝❤ t❤❛t (M∗)∗ = M, ✭✻✮ M∗ ✲ ♣r❡❞✉❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ♦❢ M✱ M∗ ⊂ M∗ σ(M, M∗) ✲ t♦♣♦❧♦❣② ♦♥ M

• M∗ ∋ ρ ≥ 0 and ρ = 1

ρ ✲ st❛t❡ ✭♥♦r♠❛❧✮ ♦❢ t❤❡ q✉❛♥t✉♠ s②st❡♠

• L(M) ∋ p ⇔

p2 = p = p∗ ∈ M L(M) ✲ ❧❛tt✐❝❡ ❝♦♠♣❧❡t❡ ✐♥ σ(M, M∗)✲t♦♣♦❧♦❣② L(M) ✲ ✒q✉❛♥t✉♠ ❧♦❣✐❝✑ ✭♣r♦♣♦s✐t✐♦♥s ❝❛❧❝✉❧✉s✮

• ♠♦r♣❤✐s♠ ♦❢ ❧❛tt✐❝❡s Σ : B → L(M) ≡ q✉❛♥t✉♠ ♦❜s❡r✈❛❜❧❡s

❊①❛♠♣❧❡✿ M = L∞(M) M∗ = L1(M) ✲ st❛♥❞❛r❞ ♠♦❞❡❧ ♦❢ q✉❛♥t✉♠ ♠❡❝❤❛♥✐❝s

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❚❤❡ ❇❛♥❛❝❤✲▲✐❡ P♦✐ss♦♥ str✉❝t✉r❡ ♦♥ M∗

• Df(ρ), Dg(ρ) ∈ M ❢♦r f, g ∈ C∞(M∗)
• ad∗M∗ ⊂ M∗ ✱

ad∗ : M∗ → M∗ and adXY := [X, Y ] for X, Y ∈ M

• (M, [·, ·])✲ ❇❛♥❛❝❤✲▲✐❡ ❛❧❣❡❜r❛

❍❡♥❝❡ ♦♥❡ ❤❛s ▲✐❡✲P♦✐ss♦♥ ❜r❛❝❦❡t {f, g}LP (ρ) := ρ, [Df(ρ), Dg(ρ)] ❍❛♠✐❧t♦♥✐❛♥ ❡q✉❛t✐♦♥ d dtρ = ad∗

DH(ρ)ρ

❢♦r ❛ ❍❛♠✐❧t♦♥✐❛♥ H ∈ C∞(M∗) ❊①❛♠♣❧❡✿ M = L∞(M) M∗ = L1(M) ✭✶✮ d

dtρ = [DH(ρ), ρ] ✲ ♥♦♥✲❧✐♥❡❛r ✈♦♥ ◆❡✉♠❛♥♥✲▲✐♦✉✈✐❧❧❡ ❡q✉❛t✐♦♥

✭✷✮ ❚❤❡ ❝❛s❡s ♦❢ t❤❡ ✐♥✜♥✐t❡ ❚♦❞❛ ❧❛tt✐❝❡ ❛♥❞ t❤❡ ♥♦♥✲❧✐♥❡❛r ❙❝❤rö❞✐♥❣❡r ❝❛♥ ❜❡ ❛❧s♦ ✇r✐tt❡♥ ✐♥ t❤✐s ✇❛②✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

• r♦✉♣♦✐❞ ♦❢ ♣❛rt✐❛❧❧② ✐♥✈❡rt✐❜❧❡ ❡❧❡♠❡♥ts ♦❢ W ∗✲❛❧❣❡❜r❛ M

▲❡❢t s✉♣♣♦rt l(x) ∈ L(M) ✭r✐❣❤t s✉♣♣♦rt r(x) ∈ L(M)✮ ♦❢ x ∈ M ✐s t❤❡ ❧❡❛st ♣r♦❥❡❝t✐♦♥ ✐♥ M✱ s✉❝❤ t❤❛t l(x)x = x (resp. x r(x) = x). ✭✼✮ ■❢ x ∈ M ✐s s❡❧❢❛❞❥♦✐♥t✱ t❤❡♥ s✉♣♣♦rt s(x) s(x) := l(x) = r(x). P♦❧❛r ❞❡❝♦♠♣♦s✐t✐♦♥ ❢♦r x ∈ M x = u|x|, ✭✽✮ ✇❤❡r❡ u ∈ M ✐s ♣❛rt✐❛❧ ✐s♦♠❡tr② ❛♥❞ |x| := √ x∗x ∈ M+✱ s✉❝❤ t❤❛t l(x) = s(|x∗|) = uu∗, r(x) = s(|x|) = u∗u.

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

• r♦✉♣♦✐❞ ♦❢ ♣❛rt✐❛❧❧② ✐♥✈❡rt✐❜❧❡ ❡❧❡♠❡♥ts ♦❢ W ∗✲❛❧❣❡❜r❛ M

▲❡t G(pMp) ❜❡ t❤❡ ❣r♦✉♣ ♦❢ ❛❧❧ ✐♥✈❡rt✐❜❧❡ ❡❧❡♠❡♥ts ✐♥ W ∗✲s✉❜❛❧❣❡❜r❛ pMp ⊂ M✳ ❲❡ ❞❡✜♥❡ t❤❡ s❡t G(M) ♦❢ ♣❛rt✐❛❧❧② ✐♥✈❡rt✐❜❧❡ ❡❧❡♠❡♥ts ✐♥ M G(M) := {x ∈ M; |x| ∈ G(pMp), where p = s(|x|)} ❘❡♠❛r❦✿ G(M) M ✐♥ ❛ ❣❡♥❡r❛❧ ❝❛s❡✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

• r♦✉♣♦✐❞ ♦❢ ♣❛rt✐❛❧❧② ✐♥✈❡rt✐❜❧❡ ❡❧❡♠❡♥ts ♦❢ W ∗✲❛❧❣❡❜r❛ M

Pr♦♣♦s✐t✐♦♥ ❚❤❡ s❡t G(M) ✇✐t❤

✶ t❤❡ s♦✉r❝❡ ❛♥❞ t❛r❣❡t ♠❛♣s s, t : G(M) → L(M)

s(x) := r(x), t(x) := l(x),

✷ t❤❡ ♣r♦❞✉❝t ❞❡✜♥❡❞ ❛s t❤❡ ♣r♦❞✉❝t ✐♥ M ♦♥ t❤❡ s❡t

G(M)(2) := {(x, y) ∈ G(M) × G(M); s(x) = t(y)},

✸ t❤❡ ✐❞❡♥t✐t② s❡❝t✐♦♥ ε : L(M) ֒

→ G(M) ❛s t❤❡ ✐♥❝❧✉s✐♦♥ ♠❛♣✱

✹ t❤❡ ✐♥✈❡rs❡ ♠❛♣ ι : G(M) → G(M) ❞❡✜♥❡❞ ❜②

ι(x) := |x|−1u∗, ✐s ❛ ❣r♦✉♣♦✐❞ ♦✈❡r L(M)✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

• r♦✉♣♦✐❞ ♦❢ ♣❛rt✐❛❧❧② ✐s♦♠❡tr✐❡s ♦❢ W ∗✲❛❧❣❡❜r❛ M

Pr♦♣♦s✐t✐♦♥ ❚❤❡ s❡t ♦❢ ♣❛rt✐❛❧ ✐s♦♠❡tr✐❡s U(M) ⊂ M ✐s t❤❡ ✇✐❞❡ s✉❜❣r♦✉♣♦✐❞ U(M) ⇒ L(M) ♦❢ t❤❡ ❣r♦✉♣♦✐❞ G(M) ⇒ L(M)✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❖r❜✐ts

• ■♥♥❡r ❛❝t✐♦♥ I : U(M) ∗ L(M) → L(M)

Ixp := xpι(x), s(x) = p ♦♥ t❤❡ ❧❛tt✐❝❡ L(M) ❣✐✈❡s t❤❡ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥✿ p ∼ q ⇔ q ∈ Op.

• ❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss [p] ♦❢ p ✐♥ s❡♥s❡ ♦❢ ▼✉rr❛②✲✈♦♥ ◆❡✉♠❛♥♥ ✐s

t❤❡ ♦r❜✐t Op ♦❢ p ∈ L(M)✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❖r❜✐ts

❘❡♠❛r❦ ❚❤❡ ▼✉rr❛②✲✈♦♥ ◆❡✉♠❛♥♥ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ W ∗✲❛❧❣❡❜r❛s ❞✐r❡❝t❧② ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ♦r❜✐ts ♦❢ t❤❡ ✐♥♥❡r ❛❝t✐♦♥ ♦❢ U(M) ⇒ L(M) ♦r G(M) ⇒ L(M) ♦♥ t❤❡ ❧❛tt✐❝❡ ♦❢ ♣r♦❥❡❝t✐♦♥s L(M)✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

G(M) ⇒ L(M) ❛s ❛ ❝♦♠♣❧❡① ❇❛♥❛❝❤✲▲✐❡ ❣r♦✉♣♦✐❞

❋♦r p ∈ L(M) ❧❡t ✉s ❞❡✜♥❡ t❤❡ s✉❜s❡t ♦❢ L(M) Πp := {q ∈ L(M) : M = qM ⊕ (1 − p)M} t❤❡♥ p = xp − yp ∈ qMp ⊕ (1 − p)Mp✳ ❚❤❡ ❛❜♦✈❡ ❞❡✜♥❡s t❤❡ ❜✐❥❡❝t✐♦♥ ϕp : Πp ˜ → (1 − p)Mp ❛♥❞ s❡❝t✐♦♥ σp : Πp → t−1(Πp) ❜② σp(q) := xp, ϕp(q) := yp.

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

G(M) ⇒ L(M) ❛s ❛ ❝♦♠♣❧❡① ❇❛♥❛❝❤✲▲✐❡ ❣r♦✉♣♦✐❞

■♥ ♦r❞❡r t♦ ✜♥❞ t❤❡ tr❛♥s✐t✐♦♥ ♠❛♣s ϕp ◦ ϕ−1

p′ : ϕp′(Πp ∩ Πp′) → ϕp(Πp ∩ Πp′)

✐♥ t❤❡ ❝❛s❡ Πp ∩ Πp′ = ∅ ♦♥❡ ❤❛s ❢♦r q ∈ Πp ∩ Πp′ t❤❡ s♣❧✐tt✐♥❣s M = qM ⊕ (1 − p)M = pM ⊕ (1 − p)M M = qM ⊕ (1 − p′)M = p′M ⊕ (1 − p′)M. ✭✾✮ ❛♥❞ ✇❡ ♦❜t❛✐♥ yp′ = (ϕp′ ◦ ϕ−1

p )(yp) = (b + dyp)ι(a + cyp),

✇❤❡r❡ a = p′p✱ b = (1 − p′)p✱ c = p′(1 − p) ❛♥❞ d = (1 − p′)(1 − p)✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

G(M) ⇒ L(M) ❛s ❛ ❝♦♠♣❧❡① ❇❛♥❛❝❤✲▲✐❡ ❣r♦✉♣♦✐❞

❚❤❡♦r❡♠ ❚❤❡ ❢❛♠✐❧② ♦❢ ♠❛♣s (Πp, ϕp) p ∈ L(M) ❞❡✜♥❡s ❛ ❝♦♠♣❧❡① ❛♥❛❧②t✐❝ ❛t❧❛s ♦♥ ❛ L(M)✳ ❚❤✐s ❛t❧❛s ✐s ♠♦❞❡❧❡❞ ❜② t❤❡ ❢❛♠✐❧② ♦❢ ❇❛♥❛❝❤ s♣❛❝❡s (1 − p)Mp✱ ✇❤❡r❡ p ∈ L(M)✳ ❋❛❝t✿ ■❢ p′ ∈ Op t❤❡♥ (1 − p)Mp ∼ = (1 − p′)Mp′✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

G(M) ⇒ L(M) ❛s ❛ ❝♦♠♣❧❡① ❇❛♥❛❝❤✲▲✐❡ ❣r♦✉♣♦✐❞

❋♦r ♣r♦❥❡❝t✐♦♥s ˜ p, p ∈ L(M) ✇❡ ❞❡✜♥❡ t❤❡ s❡t Ω˜

pp := t−1(Π˜ p) ∩ s−1(Πp)

❛♥❞ t❤❡ ♠❛♣ ψ˜

pp : Ω˜ pp → (1 − ˜

p)M˜ p ⊕ ˜ pMp ⊕ (1 − p)Mp ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛② ψ˜

pp(x) := (ϕ˜ p(t(x)), ι(σ˜ p(t(x)))xσp(s(x)), ϕp(s(x))) = (y˜ p, z˜ pp, yp) .

❚r❛♥s✐t✐♦♥ ♠❛♣s yp′ = (b + dyp)ι(a + cyp), zp′ ˜

p′ = (a + cyp)zp˜ pι(˜

a + ˜ c˜ y˜

p)

˜ y˜

p′ = (˜

b + ˜ d˜ y˜

p)ι(˜

a + ˜ c˜ y˜

p),

✭✶✵✮ ✇❤❡r❡ (yp′, zp′ ˜

p′, ˜

y˜

p′) = (ψp′ ˜ p′ ◦ ψ−1 p˜ p )(yp, zp˜ p, ˜

y˜

p)

˜ a = ˜ p′˜ p✱ ˜ b = (1 − ˜ p′)˜ p✱ ˜ c = ˜ p′(1 − ˜ p) ❛♥❞ ˜ d = (1 − ˜ p′)(1 − ˜ p)✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

G(M) ⇒ L(M) ❛s ❛ ❝♦♠♣❧❡① ❇❛♥❛❝❤✲▲✐❡ ❣r♦✉♣♦✐❞

❚❤❡♦r❡♠ ❚❤❡ ❢❛♠✐❧② ♦❢ ♠❛♣s (Ω˜

pp, ψ˜ pp)

˜ p, p ∈ L(M) ❞❡✜♥❡s ❛ ❝♦♠♣❧❡① ❛♥❛❧②t✐❝ ❛t❧❛s ♦♥ t❤❡ ❣r♦✉♣♦✐❞ G(M)✳ ❚❤❡ ❝♦♠♣❧❡① ❇❛♥❛❝❤ ♠❛♥✐❢♦❧❞ str✉❝t✉r❡ ♦❢ G(M) ❤❛s t②♣❡ G✱ ✇❤❡r❡ G ✐s t❤❡ s❡t ♦❢ ❇❛♥❛❝❤ s♣❛❝❡s (1 − ˜ p)M˜ p ⊕ ˜ pMp ⊕ (1 − p)Mp ✐♥❞❡①❡❞ ❜② t❤❡ ♣❛✐r ♦❢ ❡q✉✐✈❛❧❡♥t ♣r♦❥❡❝t✐♦♥s ♦❢ L(M)✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

Pr♦♣♦s✐t✐♦♥ ❚❤❡ ❝♦♠♣❧❡① ❇❛♥❛❝❤ ❣r♦✉♣♦✐❞ G(M) ⇒ L(M) ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ ✉♥❞❡r❧②✐♥❣ t♦♣♦❧♦❣② ✐s ❛ s❡♣❛r❛❜❧❡ ✭❍❛✉s❞♦r✛✮ t♦♣♦❧♦❣✐❝❛❧ ❣r♦✉♣♦✐❞✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

■♥ ♦r❞❡r t♦ ✐♥✈❡st✐❣❛t❡ t❤❡ ❇❛♥❛❝❤✲▲✐❡ ❣r♦✉♣♦✐❞ str✉❝t✉r❡ ♦❢ G(M) ⇒ L(M) ✐t ✐s ❡♥♦✉❣❤ t♦ r❡str✐❝t ♦✉rs❡❧❢s t♦ Lp0(M) := {p ∈ L(M) : p ∼ p0} = Op0 Gp0(M) := t−1(Lp0(M)) ∩ s−1(Lp0(M)). Pr♦♣♦s✐t✐♦♥ ❚❤❡ ❇❛♥❛❝❤✲▲✐❡ ❣r♦✉♣♦✐❞ G(M) ⇒ L(M) ✐s ❛ ❞✐s❥♦✐♥t ✉♥✐♦♥ ♦❢ ❇❛♥❛❝❤✲▲✐❡ s✉❜❣r♦✉♣♦✐❞s Gp0(M) ⇒ Lp0(M)✱ p0 ∈ L(M)✱ ✇❤✐❝❤ ❛r❡ ✐ts ❝❧♦s❡❞✲♦♣❡♥ ❇❛♥❛❝❤ s✉❜❣r♦✉♣♦✐❞s✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

Pr✐♥❝✐♣❛❧ ❜✉♥❞❧❡ P0

❲❡ ❝♦♥s✐❞❡r P0 := s−1(p0) ❛s t❤❡ t♦t❛❧ s♣❛❝❡ ♦❢ t❤❡ G0✲♣r✐♥❝✐♣❛❧ ❜✉♥❞❧❡ π0 := t|P0 : P0 → Lp0(M)✱ ✇❤❡r❡ G0 ✐s t❤❡ ❇❛♥❛❝❤✲▲✐❡ ❣r♦✉♣ G(p0Mp0) ♦❢ t❤❡ ✐♥✈❡rt✐❜❧❡ ❡❧❡♠❡♥ts ♦❢ t❤❡ W ∗✲s✉❜❛❧❣❡❜r❛ p0Mp0✳ ❚❤❡ ❢r❡❡ r✐❣❤t ❛❝t✐♦♥s ♦❢ G0 ♦♥ P0 ❛♥❞ ♦♥ P0 × P0 ❛r❡ ❞❡✜♥❡❞ ❜② κ : P0 × G0 ∋ (η, g) → ηg ∈ P0 ✭✶✶✮ ❛♥❞ ❜② κ2 : P0 × P0 × G0 ∋ (η, ξ, g) → (ηg, ξg) ∈ P0 × P0, ✭✶✷✮ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ❛❜♦✈❡ ❛❧❧♦✇s ✉s t♦ ❞❡✜♥❡ t❤❡ q✉♦t✐❡♥t ❣r♦✉♣♦✐❞

P0×P0 G0

⇒ P0/G0 ♦❢ t❤❡ ♣❛✐r ❣r♦✉♣♦✐❞ P0 × P0 ⇒ P0✱ ✇❤✐❝❤ ❜② ❞❡✜♥✐t✐♦♥ ✐s t❤❡ ❣❛✉❣❡ ❣r♦✉♣♦✐❞ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ G0✲♣r✐♥❝✐♣❛❧ ❜✉♥❞❧❡ π0 : P0 → P0/G0 ∼ = Lp0(M)✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

• ❛✉❣❡ ❣r♦✉♣♦✐❞ ♦❢ P0

❚❤❡ ❝♦♠♣❧❡① ❛♥❛❧②t✐❝ ♠❛♣s φ : P0 × P0 G0 ∋ η, ξ → ηξ−1 ∈ Gp0(M) ✭✶✸✮ ϕ : P0/G0 ∋ η → ηη−1 ∈ Lp0(M) ✭✶✹✮ ❞❡✜♥❡ t❤❡ ✐s♦♠♦r♣❤✐s♠

P0×P0 G0

Gp0(M) P0/G0 Lp0(M)

❄ ❄ ❄ ❄ ✲ ✲

s t φ ϕ , ✭✶✺✮ ♦❢ ❇❛♥❛❝❤✲▲✐❡ ❣r♦✉♣♦✐❞s✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❆t✐②❛❤ s❡q✉❡♥❝❡ ♦❢ t❤❡ ❣r♦✉♣♦✐❞ G(M) ⇒ L(M)

❖♥❡ ❤❛s J (M) G(M) L(M) ×R L(M) L(M), L(M) L(M)

❄ ❄ ❄ ❄ ❄ ❄ ✲ ✲ ✲ ✲

֒ → (t, s) ∼ ∼ ✭✶✻✮ ✇❤❡r❡

• J (M) := ker (t, s) = {x ∈ G(M);

t(x) = s(x)} ✐s t❤❡ ✐♥♥❡r ✭t♦t❛❧② ✐♥tr❛♥s✐t✐✈❡✮ s✉❜❣r♦✉♣♦✐❞ ♦❢ G(M) ⇒ L(M)✱

• L(M) ×R L(M) ∋ (q, p) iff q ∼ p✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❆t✐②❛❤ s❡q✉❡♥❝❡ ♦❢ ❛❧❣❡❜r♦✐❞s

❯s✐♥❣ t❤❡ ❛❜♦✈❡ ❢✉♥❝t♦r✐❛❧ ❝♦rr❡s♣♦♥❞❡♥❝❡ ✇❡ ♦❜t❛✐♥ t❤❡ s❤♦rt ❡①❛❝t s❡q✉❡♥❝❡ ♦❢ ❛❧❣❡❜r♦✐❞s AJ (M) AG(M) TL(M) L(M). L(M) L(M)

❄ ❄ ❄ ✲ ✲ ✲ ✲

ι a ∼ ∼ Tt Tt ✭✶✼✮

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❆t✐②❛❤ s❡q✉❡♥❝❡ ♦❢ ❛❧❣❡❜r♦✐❞s

Pr♦♣♦s✐t✐♦♥ ❚❤❡ ❆t✐②❛❤ s❡q✉❡♥❝❡ ✭✶✼✮ ✐s ✐s♦♠♦r♣❤✐❝ t♦ A(M) ML(M) T (M) L(M), L(M) L(M)

❄ ❄ ❄ ✲ ✲ ✲ ✲

ι a ∼ ∼ ✭✶✽✮ ✇❤❡r❡ A(M) := {(x, q) ∈ M × L(M) : x ∈ qMq} ML(M) := {(x, q) ∈ M × L(M) : x ∈ Mq} T (M) := {(x, q) ∈ M × L(M) : x ∈ (1 − q)Mq}.

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❆t✐②❛❤ s❡q✉❡♥❝❡ ♦❢ ❛❧❣❡❜r♦✐❞s

Pr♦♣♦s✐t✐♦♥ ❚❤❡ s❤♦rt ❡①❛❝t s❡q✉❡♥❝❡ Ap0(M) ML

p0(M)

Tp0(M) Lp0(M), Lp0(M) Lp0(M)

❄ ❄ ❄ ✲ ✲ ✲ ✲

ι a ∼ ∼ ✭✶✾✮ ♦❢ ❇❛♥❛❝❤✲▲✐❡ ❛❧❣❡❜r♦✐❞s✱ ❞❡✜♥❡❞ ❛s t❤❡ r❡str✐❝t✐♦♥ ♦❢ ✭✶✽✮ t♦ Lp0(M)✱ ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ❆t✐②❛❤ s❡q✉❡♥❝❡ p0Mp0 ×AdG0 P0 TP0/G0 T(P0/G0) P0/G0 P0/G0 P0/G0

❄ ❄ ❄ ✲ ✲ ✲ ✲

ι ∼ ∼ a ✭✷✵✮

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

Pr♦♦❢✿ ❚❤❡ ♠❛♣s IA : p0Mp0 ×AdG0 P0 ∋ x, η → (ηxη−1, ηη−1) ∈ Ap0(M) ✭✷✶✮ IM : Mp0 × P0 G0 ∋ ϑ, η → (ϑη−1, ηη−1) ∈ ML

p0(M)

✭✷✷✮ IT : Mp0 × P0 TG0 ∋ ϑ, η → ((1 − ηη−1)ϑη−1, ηη−1) ∈ Tp0(M) ✭✷✸✮ ❞❡✜♥❡ ✐s♦♠♦r♣❤✐s♠s ❜❡t✇❡❡♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❡❝t♦r ❜✉♥❞❧❡s ❛♣♣❡❛r✐♥❣ ✐♥ t❤❡ ❞✐❛❣r❛♠s ✭✷✵✮ ❛♥❞ ✭✶✾✮ ❛♥❞ t❤❡② ❝♦♠♠✉t❡ ✇✐t❤ t❤❡ ❤♦r✐③♦♥t❛❧ ❛rr♦✇s ♦❢ t❤❡s❡ ❞✐❛❣r❛♠s✳ ❲❡ ❛❧s♦ ❤❛✈❡ Lp0(M) ∼ = P0/G0✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❈♦♦r❞✐♥❛t❡ ❞❡s❝r✐♣t✐♦♥

(v, η) ∈ Mp0 × P0 ∼ = TP0 v = d

dtη(t)|t=0

❖♥❡ ❤❛s✿ η = (p + yp)zpp0 v = [ap + (p + yp)bp]zpp0 ✇❤❡r❡ ap = d dtyp(t)|t=0, bp = d dtzpp0(t)|t=0z−1

pp0

ap = (v − η(pη)−1v)(pη)−1 ✭✷✹✮ bp = pv(pη)−1 ✭✷✺✮ yp = η(pη)−1 − p ✭✷✻✮ zpp0 = pη. ✭✷✼✮

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❆❧❣❡❜r♦✐❞ ❜r❛❝❦❡t ❛♥❞ ❛♥❝❤♦r ♠❛♣

Pr♦♣♦s✐t✐♦♥ ✭✐✮ ❚❤❡ ❛♥❝❤♦r ♠❛♣ a : AL(M) → TL(M) ❛❝ts ♦♥ X = ap ∂

∂yp + bp ∂ ∂zpp0 ∈ Γ∞ML(M) ❛s ❢♦❧❧♦✇s

a(X) = ap ∂ ∂yp ; ✭✷✽✮ ✭✐✐✮ ❚❤❡ ✈❡rt✐❝❛❧ ♣❛rt ♦❢ X ✐s ❣✐✈❡♥ ❜② bp

∂ ∂zpp0 ❀

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

Pr♦♣♦s✐t✐♦♥ ❝♦♥t✳ ✭✐✐✐✮ ❚❤❡ ▲✐❡ ❜r❛❝❦❡t ♦❢ X1, X2 ∈ Γ∞ML(M) ❛ss✉♠❡s t❤❡ ❢♦r♠ [X1, X2] = ap ∂ ∂yp + bp ∂ ∂zpp , ✭✷✾✮ ✇❤❡r❡ ap = ∂a2p ∂yp , a1p

• −

∂a1p ∂yp , a2p

• ❛♥❞

bp = ∂b2p ∂yp , a1p

• −

∂b1p ∂yp , a2p

• + [b2p, b1p].

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

Pr❡❞✉❛❧ ❆t✐②❛❤ s❡q✉❡♥❝❡ ♦❢ t❤❡ ❣r♦✉♣♦✐❞ G(M) ⇒ L(M)

M = (M∗)∗ A∗(M) ⊂ A∗(M) ← → (qMq)∗ ⊃ (qMq)∗ ∼ = qM∗q A∗G(M) ⊂ A∗G(M) ← → (Mq)∗ ⊃ (qM)∗ ∼ = M∗q T∗L(M) ⊂ TL∗(M) ← → ((1 − q)Mq)∗ ⊃ ((1 − q)Mq)∗ ∼ = qM∗(1 − q) ✇❤❡r❡ ❢♦r ❡✈❡r② x ∈ M R∗

aϕ, x := ϕ, ax

L∗

aϕ, x := ϕ, xa ,

✭✸✵✮ ❘❡♠❛r❦✿ R∗

aM∗ ⊂ M∗,

L∗

aM∗ ⊂ M∗✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

Pr❡❞✉❛❧ ❆t✐②❛❤ s❡q✉❡♥❝❡ ♦❢ t❤❡ ❣r♦✉♣♦✐❞ G(M) ⇒ L(M)

❙♦✱ ♦♥❡ ❤❛s T∗L(M) A∗G(M) A∗J (M) L(M) L(M) L(M)

❄ ❄ ❄ ✲ ✲ ✲ ✲

a∗ ι∗ ∼ ∼ ✭✸✶✮

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

Pr❡❞✉❛❧ ❆t✐②❛❤ s❡q✉❡♥❝❡ ♦❢ t❤❡ ❣r♦✉♣♦✐❞ G(M) ⇒ L(M)

❚❤❡ s❤♦rt ❡①❛❝t s❡q✉❡♥❝❡ T∗Lp0(M) A∗Gp0(M) A∗Jp0(M) Lp0(M) Lp0(M) Lp0(M)

❄ ❄ ❄ ✲ ✲ ✲ ✲

a∗ ι∗ ∼ ∼ ✭✸✷✮ ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ♣r❡❞✉❛❧ ❆t✐②❛❤ s❡q✉❡♥❝❡ T∗(P0/G0) T∗P0/G0 p0M∗p0 ×AdG∗

0 P0

P0/G0, P0/G0 P0/G0

❄ ❄ ❄ ✲ ✲ ✲ ✲

a∗ ι∗ ∼ ∼ ✭✸✸✮

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❋✐❜r❡✲✇✐s❡ ❧✐♥❡❛r s✉❜ P♦✐ss♦♥ str✉❝t✉r❡ ♦❢ t❤❡ ♣r❡❞✉❛❧ ❆t✐②❛❤ s❡q✉❡♥❝❡

• ❲❡❛❦ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡ ♦❢ T∗P0 ∼

= p0M∗ × P0 (ϕ, η) ∈ T∗P0, ξ(ϕ,η) = (ϕ, η, θ, ϑ) ω(ϕ,η)(ξ1

(ϕ,η), ξ2 (ϕ,η)) = θ1, ϑ2 − θ2, ϑ1.

✭✸✹✮ ❇② ω(ϕ,η)(ξ(ϕ,η), ·) : T(ϕ,η)(p0M∗ × P0) → T ∗

(ϕ,η)(p0M∗ × P0)

✭✸✺✮ ♦♥❡ ❞❡✜♥❡s t❤❡ ❜✉♥❞❧❡ ♠♦r♣❤✐s♠ ♭ : T(T∗P0) ֒ → T ∗(T∗P0) ✇❤❡r❡ T ♭(T∗P0) := ♭(T(T∗P0)) T ∗(T∗P0) ✐s ❛ ♣r♦♣❡r ❇❛♥❛❝❤ ✈❡❝t♦r s✉❜❜✉♥❞❧❡✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

▼♦♠❡♥t✉♠ ♠❛♣

• ▼♦♠❡♥t✉♠ ♠❛♣ J0 : T∗P0 → p0M∗p0 s✉❝❤ t❤❛t

ω(ξx, ·) = −dγ, ξx = −dJ0, x ✭✸✻✮ ✇❤❡r❡ ξx(f) = d dtf(exp(−tx)ϕ, η exp(tx))|t=0 ❢♦r x ∈ p0Mp0✳ ❖♥❡ ❤❛s J0(ϕ, η) = ϕη i.e. J0(ϕ, η), x = ϕ, ηx ❢♦r ❛♥② x ∈ p0Mp0✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

• ❋♦r f ∈ C∞(T∗P0) ♦♥❡ ❤❛s

∂f ∂η (ϕ, η) ∈ (Mp0)∗ ❛♥❞ ∂f ∂ϕ(ϕ, η) ∈ (p0M∗)∗ = Mp0 ✳

• ❚❤✉s ❢♦r f, g ∈ C∞(T∗P0)♦♥❡ ❞❡✜♥❡s t❤❡ ❜r❛❝❦❡t

{f, g} = ∂g ∂η, ∂f ∂ϕ − ∂f ∂η , ∂g ∂ϕ ✭✸✼✮ ✇❤✐❝❤ ✐s ❜✐❧✐♥❡❛r✱ ❛♥t✐✲s②♠♠❡tr✐❝ ❛♥❞ s❛t✐s✜❡s t❤❡ ▲❡✐❜♥✐③ ♣r♦♣❡rt② ❜✉t ♥♦t s❛t✐s✜❡s t❤❡ ❏❛❝♦❜✐ ✐❞❡♥t✐t② ❢♦r ❛r❜✐tr❛r② s♠♦♦t❤ ❢✉♥❝t✐♦♥s✳

• ❚❤❡r❡❢♦r❡ ✇❡ ❞❡✜♥❡

P∞(T∗P0) :=

• f ∈ C∞(T∗P0) : ∂f

∂η (ϕ, η) ∈ (Mp0)∗ ⊂ (Mp0)∗

• .

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

• ❙✐♥❝❡ T ♭(T∗P0) T ∗(T∗P0) t❤❡ ❜✉♥❞❧❡ ♠❛♣

# : T ♭(T∗P0) → T(T∗P0)✱ t❤❡ ✐♥✈❡rs❡ t♦ ♭ : T(T∗P0) ֒ → T ∗(T∗P0)✱ ✐s ♥♦t ❞❡✜♥❡❞ ♦♥ t❤❡ ✇❤♦❧❡ ♦❢ T ∗(T∗P0)✱ ✐t ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ❛ s✉❜ P♦✐ss♦♥ ♠♦r♣❤✐s♠✳

• P∞(T∗P0) ⊃ P∞

G0(T∗P0) ✲ t❤❡ P♦✐ss♦♥ s✉❜❛❧❣❡❜r❛ ♦❢

G0✲✐♥✈❛r✐❛♥t ❢✉♥❝t✐♦♥s P∞

G0(T∗P0) ∼

= P∞(T∗P0/G0) (P∞(T∗P0/G0), {·, ·}G0) ✲ ❛ P♦✐ss♦♥ ❛❧❣❡❜r❛ {F, G}G0 := {F ◦ π∗G0, G ◦ π∗G0} , ✭✸✽✮ ✇❤❡r❡ π∗G0 : T∗P0 → T∗P0/G0✳

• ▲✐❡✲P♦✐ss♦♥ ❜r❛❝❦❡t ♦❢ F, G ∈ C∞(p0M∗p0)✱ ∂F

∂β (β) ∈ p0Mp0

{F, G}LP (β) :=

• β,

∂F ∂β (β), ∂G ∂β (β)

• ✭✸✾✮

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

Pr♦♣♦s✐t✐♦♥ ✭✐✮ ❖♥❡ ❤❛s t❤❡ s✉r❥❡❝t✐✈❡ P♦✐ss♦♥ s✉❜♠❡rs✐♦♥s✿ T∗P0 T∗P0/G0 p0M∗p0

• ✠

❅ ❅ ❅ ❅ ❅ ❅ ❘

π∗G0 J0 ✭✹✵✮ ♦❢ t❤❡ ✇❡❛❦ s②♠♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞ (T∗P0, ω) ♦♥ t❤❡ s✉❜ P♦✐ss♦♥ ♠❛♥✐❢♦❧❞ (T∗P0/G0, {·, ·}G0) ❛♥❞ t❤❡ ❇❛♥❛❝❤ ▲✐❡✲P♦✐ss♦♥ s♣❛❝❡ (p0Mp0, {·, ·}LP )✳ ✭✐✐✮ ❚❤❡ P♦✐ss♦♥ s✉❜❛❧❣❡❜r❛s J∗

0(C∞(p0M∗p0) ❛♥❞

π∗

∗G0(P∞(T∗P0/G0)) = P∞ G0(T∗P0) ♦❢ P∞(T∗P0) ❛r❡ ♣♦❧❛r

♦♥❡ t♦ ❛♥♦t❤❡r ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✇❡❛❦ s②♠♣❧❡❝t✐❝ ❢♦r♠ ω✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❚❤❡♦r❡♠ ✭✐✮ ❚❤❡ ❇❛♥❛❝❤ ✈❡❝t♦r ❜✉♥❞❧❡s ♠❛♣ ι∗ : T∗P0/G0 → p0M∗p0 ×Ad∗

G0 P0 ✐s ❛ P♦✐ss♦♥ s✉❜♠❡rs✐♦♥✳

✭✐✐✮ ❖♥❡ ❤❛s ker ι∗ = J−1

0 (0)/G0✱ ✇❤❡r❡ J−1 0 (0)/G0 ✐s t❤❡ ✇❡❛❦

s②♠♣❧❡❝t✐❝ ❧❡❛❢ ✐♥ T∗P0/G0 ♦❜t❛✐♥❡❞ ❜② t❤❡ ▼❛rs❞❡♥✲❲❡✐♥st❡✐♥ s②♠♣❧❡❝t✐❝ r❡❞✉❝t✐♦♥ ♣r♦❝❡❞✉r❡✳ ❚❤❡ ♣r❡❞✉❛❧ ❛♥❝❤♦r ♠❛♣ a∗ : T∗(P0/G0) ֒ → T∗P0/G0 ✐s ❛♥ ✐♠♠❡rs✐♦♥ ✇❤✐❝❤ ❞❡✜♥❡s t❤❡ ❜✉♥❞❧❡ ✐s♦♠♦r♣❤✐s♠ T∗(P0/G0) ∼ = J−1

0 (0)/G0✱ ✇❤❡r❡ t❤❡ ♣r❡❝♦t❛♥❣❡♥t ❜✉♥❞❧❡

T∗(P0/G0) ✐s ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ ❝❛♥♦♥✐❝❛❧ ✇❡❛❦ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❚❤❡♦r❡♠ ❆❧❧ ❣r♦✉♣♦✐❞s ✐♥ t❤❡ ❢r♦♥t ♦❢ t❤❡ s❤♦rt ❡①❛❝t s❡q✉❡♥❝❡ T∗

• P0×

P0 G0

• T∗P0/G0

P0× P0 G0 P0 G0

❄ ❄ ❄ ❄ ✲ ✲ ❳❳❳❳❳❳ ❳ ③ ❳❳❳❳❳❳❳ ❳ ③ ❳❳❳❳❳❳❳ ❳ ❳❳❳❳❳❳❳ ❳ ❳❳❳❳❳❳❳ ❳ ❳❳❳❳❳❳❳ ❳

a∗

2

id

T∗P0×T∗P0 G0

T∗P0/G0

P0×P0 G0 P0 G0

❄ ❄ ❄ ❄ ✲ ✲ ❳❳❳❳❳❳ ❳ ③ ❳❳❳❳❳❳ ❳ ③ ❳❳❳❳❳❳❳❳ ❳ ❳❳❳❳❳❳❳❳ ❳ ❳❳❳❳❳❳❳ ❳ ❳❳❳❳❳❳❳ ❳

ι∗

2

[π∗]

P0×p0M∗p0×P0 G0

P0/G0

P0×P0 G0 P0 G0 ,

❄ ❄ ❄ ❄ ✲ ✲

❛r❡ t❤❡ s✉❜ P♦✐ss♦♥ VB✲❣r♦✉♣♦✐❞s ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❤♦r✐③♦♥t❛❧ ❛rr♦✇s ♦❢ ✐ts ❞❡✜♥❡ P♦✐ss♦♥ ♠♦r♣❤✐s♠s✳

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❚❍❆◆❑ ❨❖❯

❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③

• ❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳