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Jan 28, 2024 188 likes •945 views

t ss rsss rt t rsss rtts r Prt ss

SLIDE 1

◗✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss❡s ❛♥❞ ❜✐rt❤ ❛♥❞ ❞❡❛t❤ ♣r♦❝❡ss❡s ♦♥ ♣❛rt✐t✐♦♥s

▼❛r❝✐♥ ➅✇✐❡❝❛

Pr♦❜❛❜✐❧✐t② ❛♥❞ ❆♥❛❧②s✐s ✶✺✳✵✺✳✷✵✶✼ ✲ ✶✾✳✵✺✳✷✵✶✼ ❇➛❞❧❡✇♦

✶ ✴ ✷✺

SLIDE 2

❇❛s❡❞ ♦♥ ✿

✶ ❲✳▼❛t②s✐❛❦✱ ▼✳ ➅✱ ❩♦♥❛❧ ♣♦❧②♥♦♠✐❛❧s ❛♥❞ ❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧

q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss✱ ❙t♦❝❤❛st✐❝ Pr♦❝❡ss✳ ❆♣♣❧✳✱ ✷✵✶✺✳

✷ ❲✳▼❛t②s✐❛❦✱ ▼✳ ➅✱ ❏♦r❞❛♥ ❛❧❣❡❜r❛s ❛♥❞ q✉❛♥t✉♠ ❇❡ss❡❧

♣r♦❝❡ss❡s✱ ■♥t✳ ▼❛t❤✳ ❘❡s✳ ◆♦t✳ ■▼❘◆✱ ✷✵✶✻✳

✷ ✴ ✷✺

SLIDE 3

❇✐❛♥❡✬s q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss

P❤✐❧✐♣♣❡ ❇✐❛♥❡ ✭✶✾✾✻✮ ✲ ❛ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛♥ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❇❡ss❡❧ ♣r♦❝❡ss ✭q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss✮✿ ❛ ▼❛r❦♦✈ ♣r♦❝❡ss ✭✐♥ ❛ ❝❧❛ss✐❝❛❧✱ ♥♦t ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ s❡♥s❡✮ ❧✐✈✐♥❣ ♦♥ ❛ s✉❜s❡t ♦❢ Rd✳ ■♥❣r❡❞✐❡♥ts✿

t❤❡ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣✿ H = Cn × R ✇✐t❤ t❤❡ ❣r♦✉♣ ❧❛✇

(z, w)(z′, w′) = (z+z′, w+w′+■♠

z′|z
), z, z′ ∈ Cn, w, w′ ∈ R,
t❤❡ ❢✉♥❝t✐♦♥ ψ(z, w) = iw − ✶

✷||z||✷ ♦♥ H ✇✐t❤ t❤❡ ♣r♦♣❡rt②✿

∀ t ≥ ✵✱ H ∋ g → ❡①♣[tψ(g)] ✐s ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✳ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t Qt : L✶(H) → L✶(H) Qtf (g) = ❡①♣[tψ(g)]f (g) ❡①t❡♥❞s t♦ ❛ s❡♠✐❣r♦✉♣ ♦❢ ✭❝♦♠♣❧❡t❡❧②✮ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥s ♦♥ ❈∗(H)✳

✸ ✴ ✷✺

SLIDE 4

❇✐❛♥❡✬s q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss

P❤✐❧✐♣♣❡ ❇✐❛♥❡ ✭✶✾✾✻✮ ✲ ❛ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛♥ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❇❡ss❡❧ ♣r♦❝❡ss ✭q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss✮✿ ❛ ▼❛r❦♦✈ ♣r♦❝❡ss ✭✐♥ ❛ ❝❧❛ss✐❝❛❧✱ ♥♦t ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ s❡♥s❡✮ ❧✐✈✐♥❣ ♦♥ ❛ s✉❜s❡t ♦❢ Rd✳ ■♥❣r❡❞✐❡♥ts✿

t❤❡ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣✿ H = Cn × R ✇✐t❤ t❤❡ ❣r♦✉♣ ❧❛✇

(z, w)(z′, w′) = (z+z′, w+w′+■♠

z′|z
), z, z′ ∈ Cn, w, w′ ∈ R,
t❤❡ ❢✉♥❝t✐♦♥ ψ(z, w) = iw − ✶

✷||z||✷ ♦♥ H ✇✐t❤ t❤❡ ♣r♦♣❡rt②✿

∀ t ≥ ✵✱ H ∋ g → ❡①♣[tψ(g)] ✐s ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✳ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t Qt : L✶(H) → L✶(H) Qtf (g) = ❡①♣[tψ(g)]f (g) ❡①t❡♥❞s t♦ ❛ s❡♠✐❣r♦✉♣ ♦❢ ✭❝♦♠♣❧❡t❡❧②✮ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥s ♦♥ ❈∗(H)✳

✸ ✴ ✷✺

SLIDE 5

❇✐❛♥❡✬s q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss

P❤✐❧✐♣♣❡ ❇✐❛♥❡ ✭✶✾✾✻✮ ✲ ❛ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛♥ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❇❡ss❡❧ ♣r♦❝❡ss ✭q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss✮✿ ❛ ▼❛r❦♦✈ ♣r♦❝❡ss ✭✐♥ ❛ ❝❧❛ss✐❝❛❧✱ ♥♦t ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ s❡♥s❡✮ ❧✐✈✐♥❣ ♦♥ ❛ s✉❜s❡t ♦❢ Rd✳ ■♥❣r❡❞✐❡♥ts✿

t❤❡ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣✿ H = Cn × R ✇✐t❤ t❤❡ ❣r♦✉♣ ❧❛✇

(z, w)(z′, w′) = (z+z′, w+w′+■♠

z′|z
), z, z′ ∈ Cn, w, w′ ∈ R,
t❤❡ ❢✉♥❝t✐♦♥ ψ(z, w) = iw − ✶

✷||z||✷ ♦♥ H ✇✐t❤ t❤❡ ♣r♦♣❡rt②✿

∀ t ≥ ✵✱ H ∋ g → ❡①♣[tψ(g)] ✐s ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✳ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t Qt : L✶(H) → L✶(H) Qtf (g) = ❡①♣[tψ(g)]f (g) ❡①t❡♥❞s t♦ ❛ s❡♠✐❣r♦✉♣ ♦❢ ✭❝♦♠♣❧❡t❡❧②✮ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥s ♦♥ ❈∗(H)✳

✸ ✴ ✷✺

SLIDE 6

❇✐❛♥❡✬s q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss

P❤✐❧✐♣♣❡ ❇✐❛♥❡ ✭✶✾✾✻✮ ✲ ❛ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛♥ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❇❡ss❡❧ ♣r♦❝❡ss ✭q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss✮✿ ❛ ▼❛r❦♦✈ ♣r♦❝❡ss ✭✐♥ ❛ ❝❧❛ss✐❝❛❧✱ ♥♦t ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ s❡♥s❡✮ ❧✐✈✐♥❣ ♦♥ ❛ s✉❜s❡t ♦❢ Rd✳ ■♥❣r❡❞✐❡♥ts✿

t❤❡ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣✿ H = Cn × R ✇✐t❤ t❤❡ ❣r♦✉♣ ❧❛✇

(z, w)(z′, w′) = (z+z′, w+w′+■♠

z′|z
), z, z′ ∈ Cn, w, w′ ∈ R,
t❤❡ ❢✉♥❝t✐♦♥ ψ(z, w) = iw − ✶

✷||z||✷ ♦♥ H ✇✐t❤ t❤❡ ♣r♦♣❡rt②✿

∀ t ≥ ✵✱ H ∋ g → ❡①♣[tψ(g)] ✐s ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✳ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t Qt : L✶(H) → L✶(H) Qtf (g) = ❡①♣[tψ(g)]f (g) ❡①t❡♥❞s t♦ ❛ s❡♠✐❣r♦✉♣ ♦❢ ✭❝♦♠♣❧❡t❡❧②✮ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥s ♦♥ ❈∗(H)✳

✸ ✴ ✷✺

SLIDE 7

❇✐❛♥❡✬s q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss

P❤✐❧✐♣♣❡ ❇✐❛♥❡ ✭✶✾✾✻✮ ✲ ❛ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛♥ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❇❡ss❡❧ ♣r♦❝❡ss ✭q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss✮✿ ❛ ▼❛r❦♦✈ ♣r♦❝❡ss ✭✐♥ ❛ ❝❧❛ss✐❝❛❧✱ ♥♦t ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ s❡♥s❡✮ ❧✐✈✐♥❣ ♦♥ ❛ s✉❜s❡t ♦❢ Rd✳ ■♥❣r❡❞✐❡♥ts✿

t❤❡ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣✿ H = Cn × R ✇✐t❤ t❤❡ ❣r♦✉♣ ❧❛✇

(z, w)(z′, w′) = (z+z′, w+w′+■♠

z′|z
), z, z′ ∈ Cn, w, w′ ∈ R,
t❤❡ ❢✉♥❝t✐♦♥ ψ(z, w) = iw − ✶

✷||z||✷ ♦♥ H ✇✐t❤ t❤❡ ♣r♦♣❡rt②✿

∀ t ≥ ✵✱ H ∋ g → ❡①♣[tψ(g)] ✐s ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✳ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t Qt : L✶(H) → L✶(H) Qtf (g) = ❡①♣[tψ(g)]f (g) ❡①t❡♥❞s t♦ ❛ s❡♠✐❣r♦✉♣ ♦❢ ✭❝♦♠♣❧❡t❡❧②✮ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥s ♦♥ ❈∗(H)✳

✸ ✴ ✷✺

SLIDE 8

❇✐❛♥❡✬s q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss

P❤✐❧✐♣♣❡ ❇✐❛♥❡ ✭✶✾✾✻✮ ✲ ❛ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛♥ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❇❡ss❡❧ ♣r♦❝❡ss ✭q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss✮✿ ❛ ▼❛r❦♦✈ ♣r♦❝❡ss ✭✐♥ ❛ ❝❧❛ss✐❝❛❧✱ ♥♦t ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ s❡♥s❡✮ ❧✐✈✐♥❣ ♦♥ ❛ s✉❜s❡t ♦❢ Rd✳ ■♥❣r❡❞✐❡♥ts✿

t❤❡ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣✿ H = Cn × R ✇✐t❤ t❤❡ ❣r♦✉♣ ❧❛✇

(z, w)(z′, w′) = (z+z′, w+w′+■♠

z′|z
), z, z′ ∈ Cn, w, w′ ∈ R,
t❤❡ ❢✉♥❝t✐♦♥ ψ(z, w) = iw − ✶

✷||z||✷ ♦♥ H ✇✐t❤ t❤❡ ♣r♦♣❡rt②✿

∀ t ≥ ✵✱ H ∋ g → ❡①♣[tψ(g)] ✐s ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✳ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t Qt : L✶(H) → L✶(H) Qtf (g) = ❡①♣[tψ(g)]f (g) ❡①t❡♥❞s t♦ ❛ s❡♠✐❣r♦✉♣ ♦❢ ✭❝♦♠♣❧❡t❡❧②✮ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥s ♦♥ ❈∗(H)✳

✸ ✴ ✷✺

SLIDE 9

❈❧❛ss✐❝❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss❡s ❢r♦♠ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ♦♥❡s

H ✲ ♥♦♥✲❛❜❡❧✐❛♥✱ s♦ ❈∗(H) ✲ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❛♥❞ (Qt)t ♦♥ ❈∗(H) ✐s t❤❡ s❡♠✐❣r♦✉♣ ♦❢ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ❍♦✇ ❝❛♥ ♦♥❡ ❝♦♥str✉❝t s♦♠❡ ❝❧❛ss✐❝❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss❡s ✐♥ s✉❝❤ s❡tt✐♥❣❄ ■❢

A ✲ ❛ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❈∗✲❛❧❣❡❜r❛✱
(Φt)t ✕ ❛ s❡♠✐❣r♦✉♣ ♦❢ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥s ♦♥ A✱
B ✕ ❛ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ A✳

❚❤❡♥✿

●❡❧❢❛♥❞ t❤❡♦r❡♠ ⇒ B ∼

= C✵ (σ(B))✱ ✇❤❡r❡ σ(B) ✕ t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ ♦❢ B❀

✐❢ B ✐s (Φt)t✲✐♥✈❛r✐❛♥t✱ t❤❡♥ t❤❡ r❡str✐❝t✐♦♥ ♦❢ (Φt)t t♦ B

❞❡✜♥❡s ❛ ▼❛r❦♦✈ s❡♠✐❣r♦✉♣ ♦♥ σ(B) ✭✈✐❛ t❤❡ ❘✐❡s③ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r❡♠ ❢♦r C✵✮

✹ ✴ ✷✺

SLIDE 10

❈❧❛ss✐❝❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss❡s ❢r♦♠ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ♦♥❡s

H ✲ ♥♦♥✲❛❜❡❧✐❛♥✱ s♦ ❈∗(H) ✲ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❛♥❞ (Qt)t ♦♥ ❈∗(H) ✐s t❤❡ s❡♠✐❣r♦✉♣ ♦❢ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ❍♦✇ ❝❛♥ ♦♥❡ ❝♦♥str✉❝t s♦♠❡ ❝❧❛ss✐❝❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss❡s ✐♥ s✉❝❤ s❡tt✐♥❣❄ ■❢

A ✲ ❛ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❈∗✲❛❧❣❡❜r❛✱
(Φt)t ✕ ❛ s❡♠✐❣r♦✉♣ ♦❢ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥s ♦♥ A✱
B ✕ ❛ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ A✳

❚❤❡♥✿

●❡❧❢❛♥❞ t❤❡♦r❡♠ ⇒ B ∼

= C✵ (σ(B))✱ ✇❤❡r❡ σ(B) ✕ t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ ♦❢ B❀

✐❢ B ✐s (Φt)t✲✐♥✈❛r✐❛♥t✱ t❤❡♥ t❤❡ r❡str✐❝t✐♦♥ ♦❢ (Φt)t t♦ B

❞❡✜♥❡s ❛ ▼❛r❦♦✈ s❡♠✐❣r♦✉♣ ♦♥ σ(B) ✭✈✐❛ t❤❡ ❘✐❡s③ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r❡♠ ❢♦r C✵✮

✹ ✴ ✷✺

SLIDE 11

❈❧❛ss✐❝❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss❡s ❢r♦♠ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ♦♥❡s

H ✲ ♥♦♥✲❛❜❡❧✐❛♥✱ s♦ ❈∗(H) ✲ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❛♥❞ (Qt)t ♦♥ ❈∗(H) ✐s t❤❡ s❡♠✐❣r♦✉♣ ♦❢ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ❍♦✇ ❝❛♥ ♦♥❡ ❝♦♥str✉❝t s♦♠❡ ❝❧❛ss✐❝❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss❡s ✐♥ s✉❝❤ s❡tt✐♥❣❄ ■❢

A ✲ ❛ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❈∗✲❛❧❣❡❜r❛✱
(Φt)t ✕ ❛ s❡♠✐❣r♦✉♣ ♦❢ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥s ♦♥ A✱
B ✕ ❛ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ A✳

❚❤❡♥✿

●❡❧❢❛♥❞ t❤❡♦r❡♠ ⇒ B ∼

= C✵ (σ(B))✱ ✇❤❡r❡ σ(B) ✕ t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ ♦❢ B❀

✐❢ B ✐s (Φt)t✲✐♥✈❛r✐❛♥t✱ t❤❡♥ t❤❡ r❡str✐❝t✐♦♥ ♦❢ (Φt)t t♦ B

❞❡✜♥❡s ❛ ▼❛r❦♦✈ s❡♠✐❣r♦✉♣ ♦♥ σ(B) ✭✈✐❛ t❤❡ ❘✐❡s③ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r❡♠ ❢♦r C✵✮

✹ ✴ ✷✺

SLIDE 12

❈❧❛ss✐❝❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss❡s ❢r♦♠ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ♦♥❡s

H ✲ ♥♦♥✲❛❜❡❧✐❛♥✱ s♦ ❈∗(H) ✲ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❛♥❞ (Qt)t ♦♥ ❈∗(H) ✐s t❤❡ s❡♠✐❣r♦✉♣ ♦❢ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ❍♦✇ ❝❛♥ ♦♥❡ ❝♦♥str✉❝t s♦♠❡ ❝❧❛ss✐❝❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss❡s ✐♥ s✉❝❤ s❡tt✐♥❣❄ ■❢

A ✲ ❛ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❈∗✲❛❧❣❡❜r❛✱
(Φt)t ✕ ❛ s❡♠✐❣r♦✉♣ ♦❢ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥s ♦♥ A✱
B ✕ ❛ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ A✳

❚❤❡♥✿

●❡❧❢❛♥❞ t❤❡♦r❡♠ ⇒ B ∼

= C✵ (σ(B))✱ ✇❤❡r❡ σ(B) ✕ t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ ♦❢ B❀

✐❢ B ✐s (Φt)t✲✐♥✈❛r✐❛♥t✱ t❤❡♥ t❤❡ r❡str✐❝t✐♦♥ ♦❢ (Φt)t t♦ B

❞❡✜♥❡s ❛ ▼❛r❦♦✈ s❡♠✐❣r♦✉♣ ♦♥ σ(B) ✭✈✐❛ t❤❡ ❘✐❡s③ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r❡♠ ❢♦r C✵✮

✹ ✴ ✷✺

SLIDE 13

❈❧❛ss✐❝❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss❡s ❢r♦♠ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ♦♥❡s

H ✲ ♥♦♥✲❛❜❡❧✐❛♥✱ s♦ ❈∗(H) ✲ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❛♥❞ (Qt)t ♦♥ ❈∗(H) ✐s t❤❡ s❡♠✐❣r♦✉♣ ♦❢ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ❍♦✇ ❝❛♥ ♦♥❡ ❝♦♥str✉❝t s♦♠❡ ❝❧❛ss✐❝❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss❡s ✐♥ s✉❝❤ s❡tt✐♥❣❄ ■❢

A ✲ ❛ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❈∗✲❛❧❣❡❜r❛✱
(Φt)t ✕ ❛ s❡♠✐❣r♦✉♣ ♦❢ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥s ♦♥ A✱
B ✕ ❛ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ A✳

❚❤❡♥✿

●❡❧❢❛♥❞ t❤❡♦r❡♠ ⇒ B ∼

= C✵ (σ(B))✱ ✇❤❡r❡ σ(B) ✕ t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ ♦❢ B❀

✐❢ B ✐s (Φt)t✲✐♥✈❛r✐❛♥t✱ t❤❡♥ t❤❡ r❡str✐❝t✐♦♥ ♦❢ (Φt)t t♦ B

❞❡✜♥❡s ❛ ▼❛r❦♦✈ s❡♠✐❣r♦✉♣ ♦♥ σ(B) ✭✈✐❛ t❤❡ ❘✐❡s③ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r❡♠ ❢♦r C✵✮

✹ ✴ ✷✺

SLIDE 14

❈❧❛ss✐❝❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss❡s ❢r♦♠ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ♦♥❡s

H ✲ ♥♦♥✲❛❜❡❧✐❛♥✱ s♦ ❈∗(H) ✲ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❛♥❞ (Qt)t ♦♥ ❈∗(H) ✐s t❤❡ s❡♠✐❣r♦✉♣ ♦❢ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ❍♦✇ ❝❛♥ ♦♥❡ ❝♦♥str✉❝t s♦♠❡ ❝❧❛ss✐❝❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss❡s ✐♥ s✉❝❤ s❡tt✐♥❣❄ ■❢

A ✲ ❛ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❈∗✲❛❧❣❡❜r❛✱
(Φt)t ✕ ❛ s❡♠✐❣r♦✉♣ ♦❢ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥s ♦♥ A✱
B ✕ ❛ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ A✳

❚❤❡♥✿

●❡❧❢❛♥❞ t❤❡♦r❡♠ ⇒ B ∼

= C✵ (σ(B))✱ ✇❤❡r❡ σ(B) ✕ t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ ♦❢ B❀

✐❢ B ✐s (Φt)t✲✐♥✈❛r✐❛♥t✱ t❤❡♥ t❤❡ r❡str✐❝t✐♦♥ ♦❢ (Φt)t t♦ B

❞❡✜♥❡s ❛ ▼❛r❦♦✈ s❡♠✐❣r♦✉♣ ♦♥ σ(B) ✭✈✐❛ t❤❡ ❘✐❡s③ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r❡♠ ❢♦r C✵✮

✹ ✴ ✷✺

SLIDE 15

❈❧❛ss✐❝❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss❡s ❢r♦♠ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ♦♥❡s

H ✲ ♥♦♥✲❛❜❡❧✐❛♥✱ s♦ ❈∗(H) ✲ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❛♥❞ (Qt)t ♦♥ ❈∗(H) ✐s t❤❡ s❡♠✐❣r♦✉♣ ♦❢ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ❍♦✇ ❝❛♥ ♦♥❡ ❝♦♥str✉❝t s♦♠❡ ❝❧❛ss✐❝❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss❡s ✐♥ s✉❝❤ s❡tt✐♥❣❄ ■❢

A ✲ ❛ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❈∗✲❛❧❣❡❜r❛✱
(Φt)t ✕ ❛ s❡♠✐❣r♦✉♣ ♦❢ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥s ♦♥ A✱
B ✕ ❛ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ A✳

❚❤❡♥✿

●❡❧❢❛♥❞ t❤❡♦r❡♠ ⇒ B ∼

= C✵ (σ(B))✱ ✇❤❡r❡ σ(B) ✕ t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ ♦❢ B❀

✐❢ B ✐s (Φt)t✲✐♥✈❛r✐❛♥t✱ t❤❡♥ t❤❡ r❡str✐❝t✐♦♥ ♦❢ (Φt)t t♦ B

❞❡✜♥❡s ❛ ▼❛r❦♦✈ s❡♠✐❣r♦✉♣ ♦♥ σ(B) ✭✈✐❛ t❤❡ ❘✐❡s③ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r❡♠ ❢♦r C✵✮

✹ ✴ ✷✺

SLIDE 16

❆ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ ❈∗(H)

❡t ❜❛❝❦ t♦ t❤❡ ❇✐❛♥❡✬s ❝♦♥str✉❝t✐♦♥✿
t❤❡ ❝♦♥✈♦❧✉t✐♦♥ s✉❜❛❧❣❡❜r❛ L✶(H/U(n)) ♦❢ r❛❞✐❛❧ ❢✉♥❝t✐♦♥s

✭f (z, w) = F(|z|, w)✮ ✐s ❝♦♠♠✉t❛t✐✈❡ ✲ ❡q✉✐✈❛❧❡♥t❧②✱ (U(n) ⋉ H, U(n)) ✐s ❛ ●❡❧❢❛♥❞ ♣❛✐r✱

t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ σ(❈∗(H/U(n))) ✐s ❡①♣❧✐❝✐t❧② ❦♥♦✇♥✿ ❛❧❧

✭♥♦♥✲tr✐✈✐❛❧✮ ❝♦♠♣❧❡① ❤♦♠♦♠♦r♣❤✐s♠s ❛r❡ ♦❢ t❤❡ ❢♦r♠ f →

f (g)φ(g−✶)❞g, ✇❤❡r❡ ❢✉♥❝t✐♦♥s φ ✭t❤❡ s♣❤❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ●❡❧❢❛♥❞ ♣❛✐r (U(n) ⋉ H, U(n))✮ ❛r❡ φτ,m(z, w) = m + n − ✶ m −✶ ❡①♣

iτw − |τ||z|✷

✷

L(n−✶)

m

|τ||z|✷

, τ = ✵✱ m ∈ N✱ ♦r φµ(z, w) = Γ(n)Jn−✶(µ|z|) µ|z| ✷ ✶−n , µ ≥ ✵,

✺ ✴ ✷✺

SLIDE 17

❆ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ ❈∗(H)

❡t ❜❛❝❦ t♦ t❤❡ ❇✐❛♥❡✬s ❝♦♥str✉❝t✐♦♥✿
t❤❡ ❝♦♥✈♦❧✉t✐♦♥ s✉❜❛❧❣❡❜r❛ L✶(H/U(n)) ♦❢ r❛❞✐❛❧ ❢✉♥❝t✐♦♥s

✭f (z, w) = F(|z|, w)✮ ✐s ❝♦♠♠✉t❛t✐✈❡ ✲ ❡q✉✐✈❛❧❡♥t❧②✱ (U(n) ⋉ H, U(n)) ✐s ❛ ●❡❧❢❛♥❞ ♣❛✐r✱

t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ σ(❈∗(H/U(n))) ✐s ❡①♣❧✐❝✐t❧② ❦♥♦✇♥✿ ❛❧❧

✭♥♦♥✲tr✐✈✐❛❧✮ ❝♦♠♣❧❡① ❤♦♠♦♠♦r♣❤✐s♠s ❛r❡ ♦❢ t❤❡ ❢♦r♠ f →

f (g)φ(g−✶)❞g, ✇❤❡r❡ ❢✉♥❝t✐♦♥s φ ✭t❤❡ s♣❤❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ●❡❧❢❛♥❞ ♣❛✐r (U(n) ⋉ H, U(n))✮ ❛r❡ φτ,m(z, w) = m + n − ✶ m −✶ ❡①♣

iτw − |τ||z|✷

✷

L(n−✶)

m

|τ||z|✷

, τ = ✵✱ m ∈ N✱ ♦r φµ(z, w) = Γ(n)Jn−✶(µ|z|) µ|z| ✷ ✶−n , µ ≥ ✵,

✺ ✴ ✷✺

SLIDE 18

❆ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ ❈∗(H)

❡t ❜❛❝❦ t♦ t❤❡ ❇✐❛♥❡✬s ❝♦♥str✉❝t✐♦♥✿
t❤❡ ❝♦♥✈♦❧✉t✐♦♥ s✉❜❛❧❣❡❜r❛ L✶(H/U(n)) ♦❢ r❛❞✐❛❧ ❢✉♥❝t✐♦♥s

✭f (z, w) = F(|z|, w)✮ ✐s ❝♦♠♠✉t❛t✐✈❡ ✲ ❡q✉✐✈❛❧❡♥t❧②✱ (U(n) ⋉ H, U(n)) ✐s ❛ ●❡❧❢❛♥❞ ♣❛✐r✱

t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ σ(❈∗(H/U(n))) ✐s ❡①♣❧✐❝✐t❧② ❦♥♦✇♥✿ ❛❧❧

✭♥♦♥✲tr✐✈✐❛❧✮ ❝♦♠♣❧❡① ❤♦♠♦♠♦r♣❤✐s♠s ❛r❡ ♦❢ t❤❡ ❢♦r♠ f →

f (g)φ(g−✶)❞g, ✇❤❡r❡ ❢✉♥❝t✐♦♥s φ ✭t❤❡ s♣❤❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ●❡❧❢❛♥❞ ♣❛✐r (U(n) ⋉ H, U(n))✮ ❛r❡ φτ,m(z, w) = m + n − ✶ m −✶ ❡①♣

iτw − |τ||z|✷

✷

L(n−✶)

m

|τ||z|✷

, τ = ✵✱ m ∈ N✱ ♦r φµ(z, w) = Γ(n)Jn−✶(µ|z|) µ|z| ✷ ✶−n , µ ≥ ✵,

✺ ✴ ✷✺

SLIDE 19

❆ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ ❈∗(H)

❡t ❜❛❝❦ t♦ t❤❡ ❇✐❛♥❡✬s ❝♦♥str✉❝t✐♦♥✿
t❤❡ ❝♦♥✈♦❧✉t✐♦♥ s✉❜❛❧❣❡❜r❛ L✶(H/U(n)) ♦❢ r❛❞✐❛❧ ❢✉♥❝t✐♦♥s

✭f (z, w) = F(|z|, w)✮ ✐s ❝♦♠♠✉t❛t✐✈❡ ✲ ❡q✉✐✈❛❧❡♥t❧②✱ (U(n) ⋉ H, U(n)) ✐s ❛ ●❡❧❢❛♥❞ ♣❛✐r✱

t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ σ(❈∗(H/U(n))) ✐s ❡①♣❧✐❝✐t❧② ❦♥♦✇♥✿ ❛❧❧

✭♥♦♥✲tr✐✈✐❛❧✮ ❝♦♠♣❧❡① ❤♦♠♦♠♦r♣❤✐s♠s ❛r❡ ♦❢ t❤❡ ❢♦r♠ f →

f (g)φ(g−✶)❞g, ✇❤❡r❡ ❢✉♥❝t✐♦♥s φ ✭t❤❡ s♣❤❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ●❡❧❢❛♥❞ ♣❛✐r (U(n) ⋉ H, U(n))✮ ❛r❡ φτ,m(z, w) = m + n − ✶ m −✶ ❡①♣

iτw − |τ||z|✷

✷

L(n−✶)

m

|τ||z|✷

, τ = ✵✱ m ∈ N✱ ♦r φµ(z, w) = Γ(n)Jn−✶(µ|z|) µ|z| ✷ ✶−n , µ ≥ ✵,

✺ ✴ ✷✺

SLIDE 20

❆ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ ❈∗(H)

❡t ❜❛❝❦ t♦ t❤❡ ❇✐❛♥❡✬s ❝♦♥str✉❝t✐♦♥✿
t❤❡ ❝♦♥✈♦❧✉t✐♦♥ s✉❜❛❧❣❡❜r❛ L✶(H/U(n)) ♦❢ r❛❞✐❛❧ ❢✉♥❝t✐♦♥s

✭f (z, w) = F(|z|, w)✮ ✐s ❝♦♠♠✉t❛t✐✈❡ ✲ ❡q✉✐✈❛❧❡♥t❧②✱ (U(n) ⋉ H, U(n)) ✐s ❛ ●❡❧❢❛♥❞ ♣❛✐r✱

t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ σ(❈∗(H/U(n))) ✐s ❡①♣❧✐❝✐t❧② ❦♥♦✇♥✿ ❛❧❧

✭♥♦♥✲tr✐✈✐❛❧✮ ❝♦♠♣❧❡① ❤♦♠♦♠♦r♣❤✐s♠s ❛r❡ ♦❢ t❤❡ ❢♦r♠ f →

f (g)φ(g−✶)❞g, ✇❤❡r❡ ❢✉♥❝t✐♦♥s φ ✭t❤❡ s♣❤❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ●❡❧❢❛♥❞ ♣❛✐r (U(n) ⋉ H, U(n))✮ ❛r❡ φτ,m(z, w) = m + n − ✶ m −✶ ❡①♣

iτw − |τ||z|✷

✷

L(n−✶)

m

|τ||z|✷

, τ = ✵✱ m ∈ N✱ ♦r φµ(z, w) = Γ(n)Jn−✶(µ|z|) µ|z| ✷ ✶−n , µ ≥ ✵,

✺ ✴ ✷✺

SLIDE 21

❆ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ ❈∗(H)

❡t ❜❛❝❦ t♦ t❤❡ ❇✐❛♥❡✬s ❝♦♥str✉❝t✐♦♥✿
t❤❡ ❝♦♥✈♦❧✉t✐♦♥ s✉❜❛❧❣❡❜r❛ L✶(H/U(n)) ♦❢ r❛❞✐❛❧ ❢✉♥❝t✐♦♥s

✭f (z, w) = F(|z|, w)✮ ✐s ❝♦♠♠✉t❛t✐✈❡ ✲ ❡q✉✐✈❛❧❡♥t❧②✱ (U(n) ⋉ H, U(n)) ✐s ❛ ●❡❧❢❛♥❞ ♣❛✐r✱

t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ σ(❈∗(H/U(n))) ✐s ❡①♣❧✐❝✐t❧② ❦♥♦✇♥✿ ❛❧❧

✭♥♦♥✲tr✐✈✐❛❧✮ ❝♦♠♣❧❡① ❤♦♠♦♠♦r♣❤✐s♠s ❛r❡ ♦❢ t❤❡ ❢♦r♠ f →

f (g)φ(g−✶)❞g, ✇❤❡r❡ ❢✉♥❝t✐♦♥s φ ✭t❤❡ s♣❤❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ●❡❧❢❛♥❞ ♣❛✐r (U(n) ⋉ H, U(n))✮ ❛r❡ φτ,m(z, w) = m + n − ✶ m −✶ ❡①♣

iτw − |τ||z|✷

✷

L(n−✶)

m

|τ||z|✷

, τ = ✵✱ m ∈ N✱ ♦r φµ(z, w) = Γ(n)Jn−✶(µ|z|) µ|z| ✷ ✶−n , µ ≥ ✵,

✺ ✴ ✷✺

SLIDE 22

❆ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ ❈∗(H)

❡t ❜❛❝❦ t♦ t❤❡ ❇✐❛♥❡✬s ❝♦♥str✉❝t✐♦♥✿
t❤❡ ❝♦♥✈♦❧✉t✐♦♥ s✉❜❛❧❣❡❜r❛ L✶(H/U(n)) ♦❢ r❛❞✐❛❧ ❢✉♥❝t✐♦♥s

✭f (z, w) = F(|z|, w)✮ ✐s ❝♦♠♠✉t❛t✐✈❡ ✲ ❡q✉✐✈❛❧❡♥t❧②✱ (U(n) ⋉ H, U(n)) ✐s ❛ ●❡❧❢❛♥❞ ♣❛✐r✱

t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ σ(❈∗(H/U(n))) ✐s ❡①♣❧✐❝✐t❧② ❦♥♦✇♥✿ ❛❧❧

✭♥♦♥✲tr✐✈✐❛❧✮ ❝♦♠♣❧❡① ❤♦♠♦♠♦r♣❤✐s♠s ❛r❡ ♦❢ t❤❡ ❢♦r♠ f →

f (g)φ(g−✶)❞g, ✇❤❡r❡ ❢✉♥❝t✐♦♥s φ ✭t❤❡ s♣❤❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ●❡❧❢❛♥❞ ♣❛✐r (U(n) ⋉ H, U(n))✮ ❛r❡ φτ,m(z, w) = m + n − ✶ m −✶ ❡①♣

iτw − |τ||z|✷

✷

L(n−✶)

m

|τ||z|✷

, τ = ✵✱ m ∈ N✱ ♦r φµ(z, w) = Γ(n)Jn−✶(µ|z|) µ|z| ✷ ✶−n , µ ≥ ✵,

✺ ✴ ✷✺

SLIDE 23

❆ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ ❈∗(H)

❡t ❜❛❝❦ t♦ t❤❡ ❇✐❛♥❡✬s ❝♦♥str✉❝t✐♦♥✿
t❤❡ ❝♦♥✈♦❧✉t✐♦♥ s✉❜❛❧❣❡❜r❛ L✶(H/U(n)) ♦❢ r❛❞✐❛❧ ❢✉♥❝t✐♦♥s

✭f (z, w) = F(|z|, w)✮ ✐s ❝♦♠♠✉t❛t✐✈❡ ✲ ❡q✉✐✈❛❧❡♥t❧②✱ (U(n) ⋉ H, U(n)) ✐s ❛ ●❡❧❢❛♥❞ ♣❛✐r✱

t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ σ(❈∗(H/U(n))) ✐s ❡①♣❧✐❝✐t❧② ❦♥♦✇♥✿ ❛❧❧

✭♥♦♥✲tr✐✈✐❛❧✮ ❝♦♠♣❧❡① ❤♦♠♦♠♦r♣❤✐s♠s ❛r❡ ♦❢ t❤❡ ❢♦r♠ f →

f (g)φ(g−✶)❞g, ✇❤❡r❡ ❢✉♥❝t✐♦♥s φ ✭t❤❡ s♣❤❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ●❡❧❢❛♥❞ ♣❛✐r (U(n) ⋉ H, U(n))✮ ❛r❡ φτ,m(z, w) = m + n − ✶ m −✶ ❡①♣

iτw − |τ||z|✷

✷

L(n−✶)

m

|τ||z|✷

, τ = ✵✱ m ∈ N✱ ♦r φµ(z, w) = Γ(n)Jn−✶(µ|z|) µ|z| ✷ ✶−n , µ ≥ ✵,

✺ ✴ ✷✺

SLIDE 24

❍❡✐s❡♥❜❡r❣ ❢❛♥

■s♦♠♦r♣❤✐s♠s✿ σ(❈∗(H/U)) ∼ = t❤❡ s❡t ♦❢ ❜♦✉♥❞❡❞ s♣❤❡r✐❝❛❧ ❢✉♥❝t✐♦♥s φ ∼ = {(τ, m|τ|) : τ ∈ R \ {✵}, m ∈ N} ∪ {(✵, µ) : µ ≥ ✵}

❋✐❣✉r❡ ✿ ❆ ❤♦♠❡♦♠♦r♣❤✐❝ ✐♠❛❣❡ ♦❢ t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ σ(❈∗(H/U)) ✕ t❤❡ st❛t❡ s♣❛❝❡ ♦❢ t❤❡ q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss

✻ ✴ ✷✺

SLIDE 25

❚r❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ❇✐❛♥❡✬s q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss

■❢ ① = (s, k|s|)✱ s < ✵✱ ❛♥❞ u = s + t < ✵✱ t❤❡♥

qt(①, ❞②) =

∞

Γ(n + l) Γ(n + k)(l − k)! u s n+k ✶ − u s l−k δ(u,−lu)(❞②).

■❢ ① = (s, k|s|) ✇✐t❤ s < ✵ ❛♥❞ t = −s✱ t❤❡♥

qt(①, ❞②) = ✶ Γ(n + k) ❡①♣

−y✶

t y✶ t n+k−✶ ✶ t (δ✵ ⊗ ▲❡❜) (❞②) ❢♦r ② = (y✵, y✶) ❢r♦♠ t❤❡ ❍❡✐s❡♥❜❡r❣ ❢❛♥ ✭▲❡❜ ❞❡♥♦t❡s t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✮✳

■❢ ① = (s, k|s|)✱ s < ✵✱ ❛♥❞ u = s + t > ✵✱ t❤❡♥

qt(①, ❞②) =

∞

l=✵

Γ(n + k + l) Γ(n + k)l! u t n+k −s t l δ(u,lu)(❞②).

✼ ✴ ✷✺

SLIDE 26

❚r❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ❇✐❛♥❡✬s q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss ✲ ❝♦♥t✳

■❢ ① = (✵, y✶) ✇✐t❤ y✶ ≥ ✵✱ t❤❡♥

qt(①, ❞②) =

∞

l=✵

✶ l! y✶ t l ❡①♣

−y✶

t

δ(t,lt)(❞②).
■❢ ① = (s, k|s|) ✇✐t❤ s > ✵✱ ❛♥❞ u = s + t✱ t❤❡♥

qt(①, ❞②) =

k

l=✵

k l s u l ✶ − s u k−l δ(u,lu)(❞②).

✽ ✴ ✷✺

SLIDE 27

▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❛♥❛❧♦❣✉❡ ♦❢ ❇✐❛♥❡✬s ❝♦♥str✉❝t✐♦♥

✳

❚❛❦❡ V ✲ ❛ s✐♠♣❧❡ ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛✳
❚❛❦❡ ✐ts ❝♦♠♣❧❡①✐✜❝❛t✐♦♥ W = V C✳
❇✉✐❧❞ ❛ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣ H = W × R✳
❚❛❦❡ U = (❙tr(W ) ∩ U(W ))✵✱ t❤❡♥ (U ⋉ H, U) ✐s ❛ ●❡❧❢❛♥❞

♣❛✐r ✐✳❡✳ L✶(H/U) ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛✳ ❚❤✐s ✇✐❧❧ r❡s✉❧t ✐♥ ❛ ▼❛r❦♦✈ s❡♠✐❣r♦✉♣ ♦❢ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ♦♥ t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ ♦❢ ❈∗(H/U)✱ ✇❤✐❝❤ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ ❈∗(H)✳ ❋✐♥❛❧❧②✱ ✇❡ ✇✐❧❧ ✜♥❞ ❛♥ ❡♠❜❡❞❞✐♥❣ ♦❢ t❤❡ s♣❡❝tr✉♠ ✐♥t♦ ❛ s✉❜s❡t ♦❢ ❛ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ✭❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❍❡✐s❡♥❜❡r❣ ❢❛♥✮✳ ❚❤✉s ✇❡ ✇✐❧❧ ♦❜t❛✐♥ ❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss ✭t❤❡ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss✮✳

✾ ✴ ✷✺

SLIDE 28

▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❛♥❛❧♦❣✉❡ ♦❢ ❇✐❛♥❡✬s ❝♦♥str✉❝t✐♦♥

✳

❚❛❦❡ V ✲ ❛ s✐♠♣❧❡ ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛✳
❚❛❦❡ ✐ts ❝♦♠♣❧❡①✐✜❝❛t✐♦♥ W = V C✳
❇✉✐❧❞ ❛ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣ H = W × R✳
❚❛❦❡ U = (❙tr(W ) ∩ U(W ))✵✱ t❤❡♥ (U ⋉ H, U) ✐s ❛ ●❡❧❢❛♥❞

♣❛✐r ✐✳❡✳ L✶(H/U) ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛✳ ❚❤✐s ✇✐❧❧ r❡s✉❧t ✐♥ ❛ ▼❛r❦♦✈ s❡♠✐❣r♦✉♣ ♦❢ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ♦♥ t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ ♦❢ ❈∗(H/U)✱ ✇❤✐❝❤ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ ❈∗(H)✳ ❋✐♥❛❧❧②✱ ✇❡ ✇✐❧❧ ✜♥❞ ❛♥ ❡♠❜❡❞❞✐♥❣ ♦❢ t❤❡ s♣❡❝tr✉♠ ✐♥t♦ ❛ s✉❜s❡t ♦❢ ❛ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ✭❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❍❡✐s❡♥❜❡r❣ ❢❛♥✮✳ ❚❤✉s ✇❡ ✇✐❧❧ ♦❜t❛✐♥ ❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss ✭t❤❡ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss✮✳

✾ ✴ ✷✺

SLIDE 29

▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❛♥❛❧♦❣✉❡ ♦❢ ❇✐❛♥❡✬s ❝♦♥str✉❝t✐♦♥

✳

❚❛❦❡ V ✲ ❛ s✐♠♣❧❡ ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛✳
❚❛❦❡ ✐ts ❝♦♠♣❧❡①✐✜❝❛t✐♦♥ W = V C✳
❇✉✐❧❞ ❛ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣ H = W × R✳
❚❛❦❡ U = (❙tr(W ) ∩ U(W ))✵✱ t❤❡♥ (U ⋉ H, U) ✐s ❛ ●❡❧❢❛♥❞

♣❛✐r ✐✳❡✳ L✶(H/U) ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛✳ ❚❤✐s ✇✐❧❧ r❡s✉❧t ✐♥ ❛ ▼❛r❦♦✈ s❡♠✐❣r♦✉♣ ♦❢ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ♦♥ t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ ♦❢ ❈∗(H/U)✱ ✇❤✐❝❤ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ ❈∗(H)✳ ❋✐♥❛❧❧②✱ ✇❡ ✇✐❧❧ ✜♥❞ ❛♥ ❡♠❜❡❞❞✐♥❣ ♦❢ t❤❡ s♣❡❝tr✉♠ ✐♥t♦ ❛ s✉❜s❡t ♦❢ ❛ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ✭❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❍❡✐s❡♥❜❡r❣ ❢❛♥✮✳ ❚❤✉s ✇❡ ✇✐❧❧ ♦❜t❛✐♥ ❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss ✭t❤❡ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss✮✳

✾ ✴ ✷✺

SLIDE 30

▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❛♥❛❧♦❣✉❡ ♦❢ ❇✐❛♥❡✬s ❝♦♥str✉❝t✐♦♥

✳

❚❛❦❡ V ✲ ❛ s✐♠♣❧❡ ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛✳
❚❛❦❡ ✐ts ❝♦♠♣❧❡①✐✜❝❛t✐♦♥ W = V C✳
❇✉✐❧❞ ❛ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣ H = W × R✳
❚❛❦❡ U = (❙tr(W ) ∩ U(W ))✵✱ t❤❡♥ (U ⋉ H, U) ✐s ❛ ●❡❧❢❛♥❞

♣❛✐r ✐✳❡✳ L✶(H/U) ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛✳ ❚❤✐s ✇✐❧❧ r❡s✉❧t ✐♥ ❛ ▼❛r❦♦✈ s❡♠✐❣r♦✉♣ ♦❢ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ♦♥ t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ ♦❢ ❈∗(H/U)✱ ✇❤✐❝❤ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ ❈∗(H)✳ ❋✐♥❛❧❧②✱ ✇❡ ✇✐❧❧ ✜♥❞ ❛♥ ❡♠❜❡❞❞✐♥❣ ♦❢ t❤❡ s♣❡❝tr✉♠ ✐♥t♦ ❛ s✉❜s❡t ♦❢ ❛ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ✭❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❍❡✐s❡♥❜❡r❣ ❢❛♥✮✳ ❚❤✉s ✇❡ ✇✐❧❧ ♦❜t❛✐♥ ❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss ✭t❤❡ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss✮✳

✾ ✴ ✷✺

SLIDE 31

▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❛♥❛❧♦❣✉❡ ♦❢ ❇✐❛♥❡✬s ❝♦♥str✉❝t✐♦♥

✳

❚❛❦❡ V ✲ ❛ s✐♠♣❧❡ ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛✳
❚❛❦❡ ✐ts ❝♦♠♣❧❡①✐✜❝❛t✐♦♥ W = V C✳
❇✉✐❧❞ ❛ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣ H = W × R✳
❚❛❦❡ U = (❙tr(W ) ∩ U(W ))✵✱ t❤❡♥ (U ⋉ H, U) ✐s ❛ ●❡❧❢❛♥❞

♣❛✐r ✐✳❡✳ L✶(H/U) ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛✳ ❚❤✐s ✇✐❧❧ r❡s✉❧t ✐♥ ❛ ▼❛r❦♦✈ s❡♠✐❣r♦✉♣ ♦❢ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ♦♥ t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ ♦❢ ❈∗(H/U)✱ ✇❤✐❝❤ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ ❈∗(H)✳ ❋✐♥❛❧❧②✱ ✇❡ ✇✐❧❧ ✜♥❞ ❛♥ ❡♠❜❡❞❞✐♥❣ ♦❢ t❤❡ s♣❡❝tr✉♠ ✐♥t♦ ❛ s✉❜s❡t ♦❢ ❛ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ✭❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❍❡✐s❡♥❜❡r❣ ❢❛♥✮✳ ❚❤✉s ✇❡ ✇✐❧❧ ♦❜t❛✐♥ ❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss ✭t❤❡ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss✮✳

✾ ✴ ✷✺

SLIDE 32

▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❛♥❛❧♦❣✉❡ ♦❢ ❇✐❛♥❡✬s ❝♦♥str✉❝t✐♦♥

✳

❚❛❦❡ V ✲ ❛ s✐♠♣❧❡ ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛✳
❚❛❦❡ ✐ts ❝♦♠♣❧❡①✐✜❝❛t✐♦♥ W = V C✳
❇✉✐❧❞ ❛ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣ H = W × R✳
❚❛❦❡ U = (❙tr(W ) ∩ U(W ))✵✱ t❤❡♥ (U ⋉ H, U) ✐s ❛ ●❡❧❢❛♥❞

♣❛✐r ✐✳❡✳ L✶(H/U) ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛✳ ❚❤✐s ✇✐❧❧ r❡s✉❧t ✐♥ ❛ ▼❛r❦♦✈ s❡♠✐❣r♦✉♣ ♦❢ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ♦♥ t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ ♦❢ ❈∗(H/U)✱ ✇❤✐❝❤ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ ❈∗(H)✳ ❋✐♥❛❧❧②✱ ✇❡ ✇✐❧❧ ✜♥❞ ❛♥ ❡♠❜❡❞❞✐♥❣ ♦❢ t❤❡ s♣❡❝tr✉♠ ✐♥t♦ ❛ s✉❜s❡t ♦❢ ❛ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ✭❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❍❡✐s❡♥❜❡r❣ ❢❛♥✮✳ ❚❤✉s ✇❡ ✇✐❧❧ ♦❜t❛✐♥ ❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss ✭t❤❡ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss✮✳

✾ ✴ ✷✺

SLIDE 33

▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❛♥❛❧♦❣✉❡ ♦❢ ❇✐❛♥❡✬s ❝♦♥str✉❝t✐♦♥

✳

❚❛❦❡ V ✲ ❛ s✐♠♣❧❡ ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛✳
❚❛❦❡ ✐ts ❝♦♠♣❧❡①✐✜❝❛t✐♦♥ W = V C✳
❇✉✐❧❞ ❛ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣ H = W × R✳
❚❛❦❡ U = (❙tr(W ) ∩ U(W ))✵✱ t❤❡♥ (U ⋉ H, U) ✐s ❛ ●❡❧❢❛♥❞

♣❛✐r ✐✳❡✳ L✶(H/U) ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛✳ ❚❤✐s ✇✐❧❧ r❡s✉❧t ✐♥ ❛ ▼❛r❦♦✈ s❡♠✐❣r♦✉♣ ♦❢ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ♦♥ t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ ♦❢ ❈∗(H/U)✱ ✇❤✐❝❤ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ ❈∗(H)✳ ❋✐♥❛❧❧②✱ ✇❡ ✇✐❧❧ ✜♥❞ ❛♥ ❡♠❜❡❞❞✐♥❣ ♦❢ t❤❡ s♣❡❝tr✉♠ ✐♥t♦ ❛ s✉❜s❡t ♦❢ ❛ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ✭❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❍❡✐s❡♥❜❡r❣ ❢❛♥✮✳ ❚❤✉s ✇❡ ✇✐❧❧ ♦❜t❛✐♥ ❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss ✭t❤❡ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss✮✳

✾ ✴ ✷✺

SLIDE 34

▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❛♥❛❧♦❣✉❡ ♦❢ ❇✐❛♥❡✬s ❝♦♥str✉❝t✐♦♥

✳

❚❛❦❡ V ✲ ❛ s✐♠♣❧❡ ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛✳
❚❛❦❡ ✐ts ❝♦♠♣❧❡①✐✜❝❛t✐♦♥ W = V C✳
❇✉✐❧❞ ❛ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣ H = W × R✳
❚❛❦❡ U = (❙tr(W ) ∩ U(W ))✵✱ t❤❡♥ (U ⋉ H, U) ✐s ❛ ●❡❧❢❛♥❞

♣❛✐r ✐✳❡✳ L✶(H/U) ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛✳ ❚❤✐s ✇✐❧❧ r❡s✉❧t ✐♥ ❛ ▼❛r❦♦✈ s❡♠✐❣r♦✉♣ ♦❢ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ♦♥ t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ ♦❢ ❈∗(H/U)✱ ✇❤✐❝❤ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ ❈∗(H)✳ ❋✐♥❛❧❧②✱ ✇❡ ✇✐❧❧ ✜♥❞ ❛♥ ❡♠❜❡❞❞✐♥❣ ♦❢ t❤❡ s♣❡❝tr✉♠ ✐♥t♦ ❛ s✉❜s❡t ♦❢ ❛ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ✭❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❍❡✐s❡♥❜❡r❣ ❢❛♥✮✳ ❚❤✉s ✇❡ ✇✐❧❧ ♦❜t❛✐♥ ❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss ✭t❤❡ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss✮✳

✾ ✴ ✷✺

SLIDE 35

▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❛♥❛❧♦❣✉❡ ♦❢ ❇✐❛♥❡✬s ❝♦♥str✉❝t✐♦♥

✳

❚❛❦❡ V ✲ ❛ s✐♠♣❧❡ ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛✳
❚❛❦❡ ✐ts ❝♦♠♣❧❡①✐✜❝❛t✐♦♥ W = V C✳
❇✉✐❧❞ ❛ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣ H = W × R✳
❚❛❦❡ U = (❙tr(W ) ∩ U(W ))✵✱ t❤❡♥ (U ⋉ H, U) ✐s ❛ ●❡❧❢❛♥❞

♣❛✐r ✐✳❡✳ L✶(H/U) ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛✳ ❚❤✐s ✇✐❧❧ r❡s✉❧t ✐♥ ❛ ▼❛r❦♦✈ s❡♠✐❣r♦✉♣ ♦❢ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ♦♥ t❤❡ ●❡❧❢❛♥❞ s♣❡❝tr✉♠ ♦❢ ❈∗(H/U)✱ ✇❤✐❝❤ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ ❈∗(H)✳ ❋✐♥❛❧❧②✱ ✇❡ ✇✐❧❧ ✜♥❞ ❛♥ ❡♠❜❡❞❞✐♥❣ ♦❢ t❤❡ s♣❡❝tr✉♠ ✐♥t♦ ❛ s✉❜s❡t ♦❢ ❛ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ✭❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❍❡✐s❡♥❜❡r❣ ❢❛♥✮✳ ❚❤✉s ✇❡ ✇✐❧❧ ♦❜t❛✐♥ ❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ▼❛r❦♦✈ ♣r♦❝❡ss ✭t❤❡ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss✮✳

✾ ✴ ✷✺

SLIDE 36

❏♦r❞❛♥ ❛❧❣❡❜r❛s

▲❡t V ❜❡ ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛ ✭r❡❛❧ ♦r ❝♦♠♣❧❡①✮ ✇✐t❤ ♥❡✉tr❛❧ ❡❧❡♠❡♥t e✳ ❲❡ s❛② t❤❛t V ✐s ❏♦r❞❛♥ ❛❧❣❡❜r❛ ✐❢ x✷(xy) = x(x✷y). ❆ r❡❛❧ ❏♦r❞❛♥ ❛❧❣❡❜r❛ ✐s ❝❛❧❧❡❞ ❊✉❝❧✐❞❡❛♥ ✐❢ t❤❡r❡ ✐s ❛ s❝❛❧❛r ♣r♦❞✉❝t ♦♥ V s✉❝❤ t❤❛t (xy|z) = (y|xz) ∀x, y, z ∈ V . ❇❛s✐❝ ❡①❛♠♣❧❡✿ t❤❡ s♣❛❝❡ V = ❙②♠(m, R) ♦❢ r❡❛❧ s②♠♠❡tr✐❝ ♠❛tr✐❝❡s ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐♦♥ x ◦ y = xy + yx ✷ , ❛♥❞ s❝❛❧❛r ♣r♦❞✉❝t (x|y) = tr(xy) ✐s ❛ s✐♠♣❧❡ ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛✳

✶✵ ✴ ✷✺

SLIDE 37

❏♦r❞❛♥ ❛❧❣❡❜r❛s

▲❡t V ❜❡ ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛ ✭r❡❛❧ ♦r ❝♦♠♣❧❡①✮ ✇✐t❤ ♥❡✉tr❛❧ ❡❧❡♠❡♥t e✳ ❲❡ s❛② t❤❛t V ✐s ❏♦r❞❛♥ ❛❧❣❡❜r❛ ✐❢ x✷(xy) = x(x✷y). ❆ r❡❛❧ ❏♦r❞❛♥ ❛❧❣❡❜r❛ ✐s ❝❛❧❧❡❞ ❊✉❝❧✐❞❡❛♥ ✐❢ t❤❡r❡ ✐s ❛ s❝❛❧❛r ♣r♦❞✉❝t ♦♥ V s✉❝❤ t❤❛t (xy|z) = (y|xz) ∀x, y, z ∈ V . ❇❛s✐❝ ❡①❛♠♣❧❡✿ t❤❡ s♣❛❝❡ V = ❙②♠(m, R) ♦❢ r❡❛❧ s②♠♠❡tr✐❝ ♠❛tr✐❝❡s ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐♦♥ x ◦ y = xy + yx ✷ , ❛♥❞ s❝❛❧❛r ♣r♦❞✉❝t (x|y) = tr(xy) ✐s ❛ s✐♠♣❧❡ ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛✳

✶✵ ✴ ✷✺

SLIDE 38

❏♦r❞❛♥ ❛❧❣❡❜r❛s

▲❡t V ❜❡ ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛ ✭r❡❛❧ ♦r ❝♦♠♣❧❡①✮ ✇✐t❤ ♥❡✉tr❛❧ ❡❧❡♠❡♥t e✳ ❲❡ s❛② t❤❛t V ✐s ❏♦r❞❛♥ ❛❧❣❡❜r❛ ✐❢ x✷(xy) = x(x✷y). ❆ r❡❛❧ ❏♦r❞❛♥ ❛❧❣❡❜r❛ ✐s ❝❛❧❧❡❞ ❊✉❝❧✐❞❡❛♥ ✐❢ t❤❡r❡ ✐s ❛ s❝❛❧❛r ♣r♦❞✉❝t ♦♥ V s✉❝❤ t❤❛t (xy|z) = (y|xz) ∀x, y, z ∈ V . ❇❛s✐❝ ❡①❛♠♣❧❡✿ t❤❡ s♣❛❝❡ V = ❙②♠(m, R) ♦❢ r❡❛❧ s②♠♠❡tr✐❝ ♠❛tr✐❝❡s ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐♦♥ x ◦ y = xy + yx ✷ , ❛♥❞ s❝❛❧❛r ♣r♦❞✉❝t (x|y) = tr(xy) ✐s ❛ s✐♠♣❧❡ ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛✳

✶✵ ✴ ✷✺

SLIDE 39

❏♦r❞❛♥ ❛❧❣❡❜r❛s

❋♦r ❛♥② ❏♦r❞❛♥ ❛❧❣❡❜r❛ V ✇❡ ❝❛♥ ❞❡✜♥❡ ✐ts r❛♥❦ r ❛♥❞ tr❛❝❡ tr(x) ❛♥❞ ❞❡t❡r♠✐♥❛♥t ∆(x) ♦❢ x ∈ V ✳ ■❢ V ✐s ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛ t❤❡♥ ❢♦r ❛♥② x ∈❱ t❤❡r❡ ❡①✐sts ❛ ❏♦r❞❛♥ ❢r❛♠❡ ✐✳❡ ✭♠❛①✐♠❛❧✮ s❡q✉❡♥❝❡ c✶, c✷, . . . , cr ♦❢ ♣r✐♠✐t✐✈❡ ♦rt❤♦❣♦♥❛❧ ✐❞❡♠♣♦t❡♥ts ✭cicj = δj

i cj✮✱ ❛♥❞ ❛ ✉♥✐q✉❡ ✉♣ t♦

♣❡r♠✉t❛t✐♦♥ s❡q✉❡♥❝❡ ♦❢ r❡❛❧ ♥✉♠❜❡rs λ✶, ..., λr s✉❝❤ t❤❛t x = λ✶c✶ + λ✷c✷ + . . . + λrcr. tr x = λ✶ + λ✷ + · · · + λr, ∆(x) = λ✶ · λ✷ · · · λr.

✶✶ ✴ ✷✺

SLIDE 40

❏♦r❞❛♥ ❛❧❣❡❜r❛s

❋♦r ❛♥② ❏♦r❞❛♥ ❛❧❣❡❜r❛ V ✇❡ ❝❛♥ ❞❡✜♥❡ ✐ts r❛♥❦ r ❛♥❞ tr❛❝❡ tr(x) ❛♥❞ ❞❡t❡r♠✐♥❛♥t ∆(x) ♦❢ x ∈ V ✳ ■❢ V ✐s ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛ t❤❡♥ ❢♦r ❛♥② x ∈❱ t❤❡r❡ ❡①✐sts ❛ ❏♦r❞❛♥ ❢r❛♠❡ ✐✳❡ ✭♠❛①✐♠❛❧✮ s❡q✉❡♥❝❡ c✶, c✷, . . . , cr ♦❢ ♣r✐♠✐t✐✈❡ ♦rt❤♦❣♦♥❛❧ ✐❞❡♠♣♦t❡♥ts ✭cicj = δj

i cj✮✱ ❛♥❞ ❛ ✉♥✐q✉❡ ✉♣ t♦

♣❡r♠✉t❛t✐♦♥ s❡q✉❡♥❝❡ ♦❢ r❡❛❧ ♥✉♠❜❡rs λ✶, ..., λr s✉❝❤ t❤❛t x = λ✶c✶ + λ✷c✷ + . . . + λrcr. tr x = λ✶ + λ✷ + · · · + λr, ∆(x) = λ✶ · λ✷ · · · λr.

✶✶ ✴ ✷✺

SLIDE 41

❏♦r❞❛♥ ❛❧❣❡❜r❛s

❋♦r ❛♥② ❏♦r❞❛♥ ❛❧❣❡❜r❛ V ✇❡ ❝❛♥ ❞❡✜♥❡ ✐ts r❛♥❦ r ❛♥❞ tr❛❝❡ tr(x) ❛♥❞ ❞❡t❡r♠✐♥❛♥t ∆(x) ♦❢ x ∈ V ✳ ■❢ V ✐s ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛ t❤❡♥ ❢♦r ❛♥② x ∈❱ t❤❡r❡ ❡①✐sts ❛ ❏♦r❞❛♥ ❢r❛♠❡ ✐✳❡ ✭♠❛①✐♠❛❧✮ s❡q✉❡♥❝❡ c✶, c✷, . . . , cr ♦❢ ♣r✐♠✐t✐✈❡ ♦rt❤♦❣♦♥❛❧ ✐❞❡♠♣♦t❡♥ts ✭cicj = δj

i cj✮✱ ❛♥❞ ❛ ✉♥✐q✉❡ ✉♣ t♦

♣❡r♠✉t❛t✐♦♥ s❡q✉❡♥❝❡ ♦❢ r❡❛❧ ♥✉♠❜❡rs λ✶, ..., λr s✉❝❤ t❤❛t x = λ✶c✶ + λ✷c✷ + . . . + λrcr. tr x = λ✶ + λ✷ + · · · + λr, ∆(x) = λ✶ · λ✷ · · · λr.

✶✶ ✴ ✷✺

SLIDE 42

P✐❡r❝❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ❛♥❞ ❝❧❛ss✐✜❝❛t✐♦♥

❆♥② ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛ ❤❛s ❛ P✐❡r❝❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ❛ ✜①❡❞ ❏♦r❞❛♥ ❢r❛♠❡✱ V =

✶≤i≤j≤m

Vij, ✇❤❡r❡ Vii = Rci ✐s ♦♥❡ ❞✐♠❡♥s✐♦♥❛❧✳ ■❢ V ✐s s✐♠♣❧❡ ❊✉❝❧✐❞❡❛♥ d = ❞✐♠Vij, ❢♦r i = j. ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ (i, j) ❛♥❞ ✐s ❝❛❧❧❡❞ t❤❡ P✐❡r❝❡ ❝♦♥st❛♥t ♦❢ V ✳ V n = ❞✐♠(V ) d r ❙②♠(m, R) m(m + ✶)/✷ ✶ m ❍❡r♠(m, C) m✷ ✷ m ❍❡r♠(m, H) m(✷m − ✶) ✹ m ❍❡r♠(✸, O) ✷✼ ✽ ✸ R × Rl−✶ l l − ✷ ✷

❚❛❜❧❡ ✿

✶✷ ✴ ✷✺

SLIDE 43

P✐❡r❝❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ❛♥❞ ❝❧❛ss✐✜❝❛t✐♦♥

❆♥② ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛ ❤❛s ❛ P✐❡r❝❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ❛ ✜①❡❞ ❏♦r❞❛♥ ❢r❛♠❡✱ V =

✶≤i≤j≤m

Vij, ✇❤❡r❡ Vii = Rci ✐s ♦♥❡ ❞✐♠❡♥s✐♦♥❛❧✳ ■❢ V ✐s s✐♠♣❧❡ ❊✉❝❧✐❞❡❛♥ d = ❞✐♠Vij, ❢♦r i = j. ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ (i, j) ❛♥❞ ✐s ❝❛❧❧❡❞ t❤❡ P✐❡r❝❡ ❝♦♥st❛♥t ♦❢ V ✳ V n = ❞✐♠(V ) d r ❙②♠(m, R) m(m + ✶)/✷ ✶ m ❍❡r♠(m, C) m✷ ✷ m ❍❡r♠(m, H) m(✷m − ✶) ✹ m ❍❡r♠(✸, O) ✷✼ ✽ ✸ R × Rl−✶ l l − ✷ ✷

❚❛❜❧❡ ✿

✶✷ ✴ ✷✺

SLIDE 44

❚❤❡ ❈♦♥❡ ♦❢ sq✉❛r❡s

■❢ V ✐s ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛ t❤❡♥ t❤❡ s❡t Ω = ■♥t{x✷ : x ∈ V }, ✐s ❛♥ ♦♣❡♥ ❝♦♥✈❡① s②♠♠❡tr✐❝ ❝♦♥❡✳ ❉❡✜♥❡ G(Ω) = {g ∈ GL(V ) : gΩ = Ω} K = G(Ω)✵ ∩ O(V )

r♦✉♣ K ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ t❤❡ s❡t ♦❢ ❏♦r❞❛♥ ❢r❛♠❡s✳

✶✸ ✴ ✷✺

SLIDE 45

❚❤❡ ❈♦♥❡ ♦❢ sq✉❛r❡s

■❢ V ✐s ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛ t❤❡♥ t❤❡ s❡t Ω = ■♥t{x✷ : x ∈ V }, ✐s ❛♥ ♦♣❡♥ ❝♦♥✈❡① s②♠♠❡tr✐❝ ❝♦♥❡✳ ❉❡✜♥❡ G(Ω) = {g ∈ GL(V ) : gΩ = Ω} K = G(Ω)✵ ∩ O(V )

r♦✉♣ K ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ t❤❡ s❡t ♦❢ ❏♦r❞❛♥ ❢r❛♠❡s✳

✶✸ ✴ ✷✺

SLIDE 46

❚❤❡ ❈♦♥❡ ♦❢ sq✉❛r❡s

■❢ V ✐s ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛ t❤❡♥ t❤❡ s❡t Ω = ■♥t{x✷ : x ∈ V }, ✐s ❛♥ ♦♣❡♥ ❝♦♥✈❡① s②♠♠❡tr✐❝ ❝♦♥❡✳ ❉❡✜♥❡ G(Ω) = {g ∈ GL(V ) : gΩ = Ω} K = G(Ω)✵ ∩ O(V )

r♦✉♣ K ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ t❤❡ s❡t ♦❢ ❏♦r❞❛♥ ❢r❛♠❡s✳

✶✸ ✴ ✷✺

SLIDE 47

❙♣❤❡r✐❝❛❧ ♣♦❧②♥♦♠✐❛❧s

▲❡t V ❜❡ ❛ s✐♠♣❧❡ ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛ ❛♥❞ W = V + iV ✐ts ❝♦♠♣❧❡①✐✜❝❛t✐♦♥✳ ❈♦♥s✐❞❡r t❤❡ s♣❛❝❡ P(W ) ♦❢ ❝♦♠♣❧❡① ♣♦❧②♥♦♠✐❛❧s ♦♥ W ✳ ❚❤❡r❡ ✐s ❛ ♥❛t✉r❛❧ r❡♣r❡s❡♥t❛t✐♦♥ π ♦❢ U = (❙tr(W ) ∩ U(W ))✵ ♦♥ P(W ) (π(g)p)(z) = p(g−✶z), g ∈ U, p ∈ P(W ). ❚❤❡r❡ ✐s ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥t♦ ✐♥✈❛r✐❛♥t s✉❜s♣❛❝❡s P(W ) =

♠∈P❛rt✭r✮

P♠, ✇❤❡r❡ P❛rt(r) = {(mk)k ∈ Zr : m✶ ≥ m✷ ≥ . . . ≥ mr ≥ ✵} ❛♥❞ (π, P♠) ❛r❡ ✐rr❡❞✉❝✐❜❧❡✱ ♣❛✐r✇✐s❡ ✐♥❡q✉✐✈❛❧❡♥t✱ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ U✳ ▲❡t dm ❞❡♥♦t❡ d♠ = ❞✐♠P♠

✶✹ ✴ ✷✺

SLIDE 48

❙♣❤❡r✐❝❛❧ ♣♦❧②♥♦♠✐❛❧s

▲❡t V ❜❡ ❛ s✐♠♣❧❡ ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛ ❛♥❞ W = V + iV ✐ts ❝♦♠♣❧❡①✐✜❝❛t✐♦♥✳ ❈♦♥s✐❞❡r t❤❡ s♣❛❝❡ P(W ) ♦❢ ❝♦♠♣❧❡① ♣♦❧②♥♦♠✐❛❧s ♦♥ W ✳ ❚❤❡r❡ ✐s ❛ ♥❛t✉r❛❧ r❡♣r❡s❡♥t❛t✐♦♥ π ♦❢ U = (❙tr(W ) ∩ U(W ))✵ ♦♥ P(W ) (π(g)p)(z) = p(g−✶z), g ∈ U, p ∈ P(W ). ❚❤❡r❡ ✐s ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥t♦ ✐♥✈❛r✐❛♥t s✉❜s♣❛❝❡s P(W ) =

♠∈P❛rt✭r✮

P♠, ✇❤❡r❡ P❛rt(r) = {(mk)k ∈ Zr : m✶ ≥ m✷ ≥ . . . ≥ mr ≥ ✵} ❛♥❞ (π, P♠) ❛r❡ ✐rr❡❞✉❝✐❜❧❡✱ ♣❛✐r✇✐s❡ ✐♥❡q✉✐✈❛❧❡♥t✱ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ U✳ ▲❡t dm ❞❡♥♦t❡ d♠ = ❞✐♠P♠

✶✹ ✴ ✷✺

SLIDE 49

❙♣❤❡r✐❝❛❧ ♣♦❧②♥♦♠✐❛❧s

▲❡t V ❜❡ ❛ s✐♠♣❧❡ ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛ ❛♥❞ W = V + iV ✐ts ❝♦♠♣❧❡①✐✜❝❛t✐♦♥✳ ❈♦♥s✐❞❡r t❤❡ s♣❛❝❡ P(W ) ♦❢ ❝♦♠♣❧❡① ♣♦❧②♥♦♠✐❛❧s ♦♥ W ✳ ❚❤❡r❡ ✐s ❛ ♥❛t✉r❛❧ r❡♣r❡s❡♥t❛t✐♦♥ π ♦❢ U = (❙tr(W ) ∩ U(W ))✵ ♦♥ P(W ) (π(g)p)(z) = p(g−✶z), g ∈ U, p ∈ P(W ). ❚❤❡r❡ ✐s ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥t♦ ✐♥✈❛r✐❛♥t s✉❜s♣❛❝❡s P(W ) =

♠∈P❛rt✭r✮

P♠, ✇❤❡r❡ P❛rt(r) = {(mk)k ∈ Zr : m✶ ≥ m✷ ≥ . . . ≥ mr ≥ ✵} ❛♥❞ (π, P♠) ❛r❡ ✐rr❡❞✉❝✐❜❧❡✱ ♣❛✐r✇✐s❡ ✐♥❡q✉✐✈❛❧❡♥t✱ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ U✳ ▲❡t dm ❞❡♥♦t❡ d♠ = ❞✐♠P♠

✶✹ ✴ ✷✺

SLIDE 50

❙♣❤❡r✐❝❛❧ ♣♦❧②♥♦♠✐❛❧s

❊❛❝❤ s✉❜s♣❛❝❡ P♠ ❝♦♥t❛✐♥s ❛ ♦♥❡ ❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ K✲✐♥✈❛r✐❛♥t ♣♦❧②♥♦♠✐❛❧s✳ ❚❤❡ ✉♥✐q✉❡ K ✐♥✈❛r✐❛♥t ♣♦❧②♥♦♠✐❛❧ p ∈ P♠ s✉❝❤ t❤❛t p(e) = ✶ ✐s ❝❛❧❧❡❞ s♣❤❡r✐❝❛❧ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ✐t ✐s ❞❡♥♦t❡❞ Φ♠(x)✳ ❯♣ t♦ ♥♦r♠❛❧✐③❛t✐♦♥✱

■❢ V = ❙②♠(m, R) t❤❡♥ Φ♠ = ③♦♥❛❧ ♣♦❧②♥♦♠✐❛❧✳
✐❢ V = ❍❡r♠(m, C)✱ t❤❡♥ Φ♠ = ❙❝❤✉r ♣♦❧②♥♦♠✐❛❧✱
❢♦r ❛♥② ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛✱ Φ♠ ❛r❡ s♣❡❝✐❛❧ ❝❛s❡s ♦❢ t❤❡

❏❛❝❦ ♣♦❧②♥♦♠✐❛❧s✳

✶✺ ✴ ✷✺

SLIDE 51

❙♣❤❡r✐❝❛❧ ♣♦❧②♥♦♠✐❛❧s

❊❛❝❤ s✉❜s♣❛❝❡ P♠ ❝♦♥t❛✐♥s ❛ ♦♥❡ ❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ K✲✐♥✈❛r✐❛♥t ♣♦❧②♥♦♠✐❛❧s✳ ❚❤❡ ✉♥✐q✉❡ K ✐♥✈❛r✐❛♥t ♣♦❧②♥♦♠✐❛❧ p ∈ P♠ s✉❝❤ t❤❛t p(e) = ✶ ✐s ❝❛❧❧❡❞ s♣❤❡r✐❝❛❧ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ✐t ✐s ❞❡♥♦t❡❞ Φ♠(x)✳ ❯♣ t♦ ♥♦r♠❛❧✐③❛t✐♦♥✱

■❢ V = ❙②♠(m, R) t❤❡♥ Φ♠ = ③♦♥❛❧ ♣♦❧②♥♦♠✐❛❧✳
✐❢ V = ❍❡r♠(m, C)✱ t❤❡♥ Φ♠ = ❙❝❤✉r ♣♦❧②♥♦♠✐❛❧✱
❢♦r ❛♥② ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛✱ Φ♠ ❛r❡ s♣❡❝✐❛❧ ❝❛s❡s ♦❢ t❤❡

❏❛❝❦ ♣♦❧②♥♦♠✐❛❧s✳

✶✺ ✴ ✷✺

SLIDE 52

❙♣❤❡r✐❝❛❧ ♣♦❧②♥♦♠✐❛❧s

❊❛❝❤ s✉❜s♣❛❝❡ P♠ ❝♦♥t❛✐♥s ❛ ♦♥❡ ❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ K✲✐♥✈❛r✐❛♥t ♣♦❧②♥♦♠✐❛❧s✳ ❚❤❡ ✉♥✐q✉❡ K ✐♥✈❛r✐❛♥t ♣♦❧②♥♦♠✐❛❧ p ∈ P♠ s✉❝❤ t❤❛t p(e) = ✶ ✐s ❝❛❧❧❡❞ s♣❤❡r✐❝❛❧ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ✐t ✐s ❞❡♥♦t❡❞ Φ♠(x)✳ ❯♣ t♦ ♥♦r♠❛❧✐③❛t✐♦♥✱

■❢ V = ❙②♠(m, R) t❤❡♥ Φ♠ = ③♦♥❛❧ ♣♦❧②♥♦♠✐❛❧✳
✐❢ V = ❍❡r♠(m, C)✱ t❤❡♥ Φ♠ = ❙❝❤✉r ♣♦❧②♥♦♠✐❛❧✱
❢♦r ❛♥② ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛✱ Φ♠ ❛r❡ s♣❡❝✐❛❧ ❝❛s❡s ♦❢ t❤❡

❏❛❝❦ ♣♦❧②♥♦♠✐❛❧s✳

✶✺ ✴ ✷✺

SLIDE 53

❙♣❤❡r✐❝❛❧ ♣♦❧②♥♦♠✐❛❧s

❊❛❝❤ s✉❜s♣❛❝❡ P♠ ❝♦♥t❛✐♥s ❛ ♦♥❡ ❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ K✲✐♥✈❛r✐❛♥t ♣♦❧②♥♦♠✐❛❧s✳ ❚❤❡ ✉♥✐q✉❡ K ✐♥✈❛r✐❛♥t ♣♦❧②♥♦♠✐❛❧ p ∈ P♠ s✉❝❤ t❤❛t p(e) = ✶ ✐s ❝❛❧❧❡❞ s♣❤❡r✐❝❛❧ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ✐t ✐s ❞❡♥♦t❡❞ Φ♠(x)✳ ❯♣ t♦ ♥♦r♠❛❧✐③❛t✐♦♥✱

■❢ V = ❙②♠(m, R) t❤❡♥ Φ♠ = ③♦♥❛❧ ♣♦❧②♥♦♠✐❛❧✳
✐❢ V = ❍❡r♠(m, C)✱ t❤❡♥ Φ♠ = ❙❝❤✉r ♣♦❧②♥♦♠✐❛❧✱
❢♦r ❛♥② ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛✱ Φ♠ ❛r❡ s♣❡❝✐❛❧ ❝❛s❡s ♦❢ t❤❡

❏❛❝❦ ♣♦❧②♥♦♠✐❛❧s✳

✶✺ ✴ ✷✺

SLIDE 54

❙♣❤❡r✐❝❛❧ ♣♦❧②♥♦♠✐❛❧s

❊❛❝❤ s✉❜s♣❛❝❡ P♠ ❝♦♥t❛✐♥s ❛ ♦♥❡ ❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ K✲✐♥✈❛r✐❛♥t ♣♦❧②♥♦♠✐❛❧s✳ ❚❤❡ ✉♥✐q✉❡ K ✐♥✈❛r✐❛♥t ♣♦❧②♥♦♠✐❛❧ p ∈ P♠ s✉❝❤ t❤❛t p(e) = ✶ ✐s ❝❛❧❧❡❞ s♣❤❡r✐❝❛❧ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ✐t ✐s ❞❡♥♦t❡❞ Φ♠(x)✳ ❯♣ t♦ ♥♦r♠❛❧✐③❛t✐♦♥✱

■❢ V = ❙②♠(m, R) t❤❡♥ Φ♠ = ③♦♥❛❧ ♣♦❧②♥♦♠✐❛❧✳
✐❢ V = ❍❡r♠(m, C)✱ t❤❡♥ Φ♠ = ❙❝❤✉r ♣♦❧②♥♦♠✐❛❧✱
❢♦r ❛♥② ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛✱ Φ♠ ❛r❡ s♣❡❝✐❛❧ ❝❛s❡s ♦❢ t❤❡

❏❛❝❦ ♣♦❧②♥♦♠✐❛❧s✳

✶✺ ✴ ✷✺

SLIDE 55

❛♠♠❛ ❋✉♥❝t✐♦♥
❡♥❡r❛❧✐③❡❞ ●❛♠♠❛ ❢✉♥❝t✐♦♥✿

ΓΩ(s) =

e− tr x∆s(x)∆(x)−n/r❞x, s ∈ Cr ❋♦r ❘❡(sj) > (j − ✶) d

✷ t❤❡ ✐♥t❡❣r❛❧ ✐s ❝♦♥✈❡r❣❡♥t ❛♥❞ ✐s ❡q✉❛❧ t♦

ΓΩ(s) = (✷π)(n−r)/✷

r

j=✶

Γ

sj − (j − ✶)d

✷

P♦❝❤❤❛♠♠❡r s②♠❜♦❧ (s)♠ = ΓΩ(s + ♠) ΓΩ(s) =

r

i=✶
si − (i − ✶)d

✷

✶✻ ✴ ✷✺

SLIDE 56

❛♠♠❛ ❋✉♥❝t✐♦♥
❡♥❡r❛❧✐③❡❞ ●❛♠♠❛ ❢✉♥❝t✐♦♥✿

ΓΩ(s) =

e− tr x∆s(x)∆(x)−n/r❞x, s ∈ Cr ❋♦r ❘❡(sj) > (j − ✶) d

✷ t❤❡ ✐♥t❡❣r❛❧ ✐s ❝♦♥✈❡r❣❡♥t ❛♥❞ ✐s ❡q✉❛❧ t♦

ΓΩ(s) = (✷π)(n−r)/✷

r

j=✶

Γ

sj − (j − ✶)d

✷

P♦❝❤❤❛♠♠❡r s②♠❜♦❧ (s)♠ = ΓΩ(s + ♠) ΓΩ(s) =

r

i=✶
si − (i − ✶)d

✷

✶✻ ✴ ✷✺

SLIDE 57

❛♠♠❛ ❋✉♥❝t✐♦♥
❡♥❡r❛❧✐③❡❞ ●❛♠♠❛ ❢✉♥❝t✐♦♥✿

ΓΩ(s) =

e− tr x∆s(x)∆(x)−n/r❞x, s ∈ Cr ❋♦r ❘❡(sj) > (j − ✶) d

✷ t❤❡ ✐♥t❡❣r❛❧ ✐s ❝♦♥✈❡r❣❡♥t ❛♥❞ ✐s ❡q✉❛❧ t♦

ΓΩ(s) = (✷π)(n−r)/✷

r

j=✶

Γ

sj − (j − ✶)d

✷

P♦❝❤❤❛♠♠❡r s②♠❜♦❧ (s)♠ = ΓΩ(s + ♠) ΓΩ(s) =

r

i=✶
si − (i − ✶)d

✷

✶✻ ✴ ✷✺

SLIDE 58

❇✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts ❛♥❞ ▲❛❣✉❡rr❡ ♣♦❧②♥♦♠✐❛❧s

❋♦r ❛♥② t✇♦ ♣❛rt✐t✐♦♥s ♥✱ ♠ ♦❢ ❧❡♥❣t❤ ≤ r t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❜✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥t ✐s ❞❡✜♥❡❞ ❜② ❡q✉❛❧✐t②✿ Φ♠(e + x) =

♠ ♥

Φ♥(x).
❡♥❡r❛❧✐③❡❞ ▲❛❣✉❡rr❡ ♣♦❧②♥♦♠✐❛❧s✿

Lν

♠(x) = (ν)♠

|❦|≤|♠|

♠ ❦ ✶ (ν)❦ Φ❦(−x).

✶✼ ✴ ✷✺

SLIDE 59

❇✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts ❛♥❞ ▲❛❣✉❡rr❡ ♣♦❧②♥♦♠✐❛❧s

❋♦r ❛♥② t✇♦ ♣❛rt✐t✐♦♥s ♥✱ ♠ ♦❢ ❧❡♥❣t❤ ≤ r t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❜✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥t ✐s ❞❡✜♥❡❞ ❜② ❡q✉❛❧✐t②✿ Φ♠(e + x) =

♠ ♥

Φ♥(x).
❡♥❡r❛❧✐③❡❞ ▲❛❣✉❡rr❡ ♣♦❧②♥♦♠✐❛❧s✿

Lν

♠(x) = (ν)♠

|❦|≤|♠|

♠ ❦ ✶ (ν)❦ Φ❦(−x).

✶✼ ✴ ✷✺

SLIDE 60

▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss

▲❡t H = W × R ❜❡ ❛ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣ ❜✉✐❧❞ ❢r♦♠ t❤❡ ❝♦♠♣❧❡①✐✜❝❛t✐♦♥ ♦❢ ❛ s✐♠♣❧❡ ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛ ❛♥❞ U = (❙tr(W ) ∩ U(W ))✵✱ t❤❡♥ (U ⋉ H, U) ✐s ❛ ●❡❧❢❛♥❞ ♣❛✐r ✐✳❡✳ L✶(H/U) ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛✳ ❍❡♥❝❡ (Qt)t s✉❝❤ t❤❛t Qt(f )(z, w) = etψ(z,w)f (z, w) ✐s ❛ s❡♠✐❣r♦✉♣ ♦❢ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥s ♦♥ ❝♦♠♠✉t❛t✐✈❡ ❈∗ ❛❧❣❡❜r❛ ❈∗(H/U)✳ V Ω W = V C U ❙②♠(m, R) Πm(R) ❙②♠(m, C) U(m) ❍❡r♠(m, C) Πm(C) M(m, C) U(m) × U(m) ❍❡r♠(m, H) Πm(H) ❙❦❡✇(✷m, C) U(✷m) ❍❡r♠(✸, O) Π✸(O) ❍❡r♠(✸, O) ⊗ C E✻ × T Rl Λl Cl SO(l) × T

✶✽ ✴ ✷✺

SLIDE 61

▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss

▲❡t H = W × R ❜❡ ❛ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣ ❜✉✐❧❞ ❢r♦♠ t❤❡ ❝♦♠♣❧❡①✐✜❝❛t✐♦♥ ♦❢ ❛ s✐♠♣❧❡ ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛ ❛♥❞ U = (❙tr(W ) ∩ U(W ))✵✱ t❤❡♥ (U ⋉ H, U) ✐s ❛ ●❡❧❢❛♥❞ ♣❛✐r ✐✳❡✳ L✶(H/U) ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛✳ ❍❡♥❝❡ (Qt)t s✉❝❤ t❤❛t Qt(f )(z, w) = etψ(z,w)f (z, w) ✐s ❛ s❡♠✐❣r♦✉♣ ♦❢ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥s ♦♥ ❝♦♠♠✉t❛t✐✈❡ ❈∗ ❛❧❣❡❜r❛ ❈∗(H/U)✳ V Ω W = V C U ❙②♠(m, R) Πm(R) ❙②♠(m, C) U(m) ❍❡r♠(m, C) Πm(C) M(m, C) U(m) × U(m) ❍❡r♠(m, H) Πm(H) ❙❦❡✇(✷m, C) U(✷m) ❍❡r♠(✸, O) Π✸(O) ❍❡r♠(✸, O) ⊗ C E✻ × T Rl Λl Cl SO(l) × T

✶✽ ✴ ✷✺

SLIDE 62

❇♦✉♥❞❡❞ s♣❤❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✿

❙♣❡❝tr✉♠ ♦❢ ❈∗(H/U) ❝❛♥ ❜❡ ✐❞❡♥t✐✜❡❞ ✇✐t❤ t❤❡ s❡t Σ = Σ✶ ∪ Σ✷ ♦❢ ❜♦✉♥❞❡❞ U s♣❤❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ✇❤❡r❡ Σ✶ = {ϕ(µ, ♠; ·, ·) : µ ∈ R \ {✵}, ♠ ✲ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❧❡♥❣t❤ ≤ r} , ❛♥❞ Σ✷ = {ϕ(τ; ·, ·) : τ = (τ✶, . . . , τr) ∈ Rr, τ✶ ≥ . . . ≥ τr ≥ ✵} . ϕ(µ, ♠; z, w) = ✶ (n/r)♠ ❡①♣

iµw − |µ|z✷

✷

Ln/r

♠ (|µ|v✷),

✇❤❡r❡ v ∈ V ✐s s✉❝❤ t❤❛t z = u(v) ❢♦r u ∈ U✳ ϕ(τ; z, w) =

❦≥✵

d❦ ((n/r)❦)✷ (−✶)|❦|Φ❦(τ)Φ❦

v✷

;

✶✾ ✴ ✷✺

SLIDE 63

❇♦✉♥❞❡❞ s♣❤❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✿

❙♣❡❝tr✉♠ ♦❢ ❈∗(H/U) ❝❛♥ ❜❡ ✐❞❡♥t✐✜❡❞ ✇✐t❤ t❤❡ s❡t Σ = Σ✶ ∪ Σ✷ ♦❢ ❜♦✉♥❞❡❞ U s♣❤❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ✇❤❡r❡ Σ✶ = {ϕ(µ, ♠; ·, ·) : µ ∈ R \ {✵}, ♠ ✲ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❧❡♥❣t❤ ≤ r} , ❛♥❞ Σ✷ = {ϕ(τ; ·, ·) : τ = (τ✶, . . . , τr) ∈ Rr, τ✶ ≥ . . . ≥ τr ≥ ✵} . ϕ(µ, ♠; z, w) = ✶ (n/r)♠ ❡①♣

iµw − |µ|z✷

✷

Ln/r

♠ (|µ|v✷),

✇❤❡r❡ v ∈ V ✐s s✉❝❤ t❤❛t z = u(v) ❢♦r u ∈ U✳ ϕ(τ; z, w) =

❦≥✵

d❦ ((n/r)❦)✷ (−✶)|❦|Φ❦(τ)Φ❦

v✷

;

✶✾ ✴ ✷✺

SLIDE 64

❖♥❡ ❝❛♥ s❤♦✇ t❤❛t t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ Σ ✐♥t♦ Rr+✶ ✈✐❛ Σ✶ ∋ ϕ(µ, ♠; ·, ·) → (µ, µm✶, . . . , µmr), Σ✷ ∋ ϕ(τ; ·, ·) → (✵, τ✶, . . . , τr) ✐s ❛ ❤♦♠❡♦♠♦r♣❤✐s♠ ♦♥t♦ ✐ts ✐♠❛❣❡✳ ❚❤❡ ✐♠❛❣❡ Σ′ = Σ′(r) ♦❢ Σ ✉♥❞❡r t❤✐s ♠❛♣✱ ❣✐✈❡♥ ❡①♣❧✐❝✐t❧② ❛s Σ = {(s, |s|k✶, . . . , |s|kr) : s < ✵, κ = (k✶, . . . , kr) ∈ P❛rt(r)} ∪ {(✵, τ✶, . . . , τr) : τ✶ ≥ τ✷ ≥ . . . ≥ τr ≥ ✵} ∪ {(s, |s|k✶, . . . , |s|kr) : s > ✵, κ ∈ P❛rt(r)}. ✭t❤❡ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❍❡✐s❡♥❜❡r❣ ❢❛♥✮

✷✵ ✴ ✷✺

SLIDE 65

❚r❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ❢♦r ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ◗✉❛♥t✉♠ ❇❡ss❡❧ Pr♦❝❡ss

■❢ ① = (κ, −κ❦) ✇✐t❤ κ < ✵✱ t ≥ ✵ ❛♥❞ µ = κ + t < ✵ t❤❡♥ qt(①, ❞②) =

|♠|≥|❦|

♠ ❦ d♠ d❦

✶ − µ

κ |♠|−|❦| µ κ |❦|+n δ(µ,−µ♠)(❞②). ■❢ ① = (κ, −κ❦) ✇✐t❤ κ < ✵ ❛♥❞ t = −κ t❤❡♥ qt(①, ❞②) = t−nr! (Γ(✶ + d/✷))r (n/r)❦ r

j=✶ Γ (✶ + (j − ✶)d/✷))Γ(✶ + jd/✷) ❡①♣ (− tr(a/t))

× Φ❦(a/t)

✶≤i<j≤r

(aj − ai)d✶ ✶R++(a)

δ✵ ⊗ λ⊗r

(❞②) ✇❤❡r❡ a ✐s ❛♥ ❡❧❡♠❡♥t ♦❢ R++ = {a ∈ Rr : a✶ > . . . > ar > ✵}, ② = (a✵, a✶, . . . , ar)✱ ❛♥❞ λ ❞❡♥♦t❡s t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✳

✷✶ ✴ ✷✺

SLIDE 66

❚r❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ❢♦r ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ◗✉❛♥t✉♠ ❇❡ss❡❧ Pr♦❝❡ss

■❢ ① = (κ, −κ❦) ✇✐t❤ κ < ✵✱ t ≥ ✵ ❛♥❞ µ = κ + t < ✵ t❤❡♥ qt(①, ❞②) =

|♠|≥|❦|

♠ ❦ d♠ d❦

✶ − µ

κ |♠|−|❦| µ κ |❦|+n δ(µ,−µ♠)(❞②). ■❢ ① = (κ, −κ❦) ✇✐t❤ κ < ✵ ❛♥❞ t = −κ t❤❡♥ qt(①, ❞②) = t−nr! (Γ(✶ + d/✷))r (n/r)❦ r

j=✶ Γ (✶ + (j − ✶)d/✷))Γ(✶ + jd/✷) ❡①♣ (− tr(a/t))

× Φ❦(a/t)

✶≤i<j≤r

(aj − ai)d✶ ✶R++(a)

δ✵ ⊗ λ⊗r

(❞②) ✇❤❡r❡ a ✐s ❛♥ ❡❧❡♠❡♥t ♦❢ R++ = {a ∈ Rr : a✶ > . . . > ar > ✵}, ② = (a✵, a✶, . . . , ar)✱ ❛♥❞ λ ❞❡♥♦t❡s t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✳

✷✶ ✴ ✷✺

SLIDE 67

▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ◗✉❛♥t✉♠ ❇❡ss❡❧ Pr♦❝❡ss ✲ ❝♦♥t✬❞

■❢ ① = (κ, −κ❦) ✇✐t❤ κ < ✵ ❛♥❞ µ = κ + t > ✵ t❤❡♥ qt(①, ❞②) =

♠≥✵

d♠ µ t |❦|+n ✶ − µ t |♠| ❦

♥

♠

♥

d♥

δ(µ,µ♠)(❞②), ✇❤❡r❡ t❤❡ ❧❛st s✉♠♠❛t✐♦♥ ✐s ♦✈❡r ♣❛rt✐t✐♦♥s ♥ ♦❢ t❤❡ ❝♦♥s❡❝✉t✐✈❡ ✐♥t❡❣❡rs r❛♥❣✐♥❣ ❢r♦♠ ✵ t♦ ♠✐♥(|❦|, |♠|)✳ ■❢ ① = (✵, τ✶, . . . , τr) t❤❡♥ qt(①, ❞②) =

♠≥✵

d♠ (n/r)♠ Φ♠(τ/t)e− tr(τ/t)δ(t,t♠)(❞②) ■❢ ① = (κ, κ❦) ✐s s✉❝❤ t❤❛t κ > ✵✱ t ≥ ✵ ❛♥❞ µ = κ + t > ✵ t❤❡♥ qt(①, ❞②) =

|♠|≤|❦|

❦ ♠ κ µ |♠| ✶ − κ µ |❦|−|♠| δ(µ,µ♠)(❞②).

✷✷ ✴ ✷✺

SLIDE 68

▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ◗✉❛♥t✉♠ ❇❡ss❡❧ Pr♦❝❡ss ✲ ❝♦♥t✬❞

■❢ ① = (κ, −κ❦) ✇✐t❤ κ < ✵ ❛♥❞ µ = κ + t > ✵ t❤❡♥ qt(①, ❞②) =

♠≥✵

d♠ µ t |❦|+n ✶ − µ t |♠| ❦

♥

♠

♥

d♥

δ(µ,µ♠)(❞②), ✇❤❡r❡ t❤❡ ❧❛st s✉♠♠❛t✐♦♥ ✐s ♦✈❡r ♣❛rt✐t✐♦♥s ♥ ♦❢ t❤❡ ❝♦♥s❡❝✉t✐✈❡ ✐♥t❡❣❡rs r❛♥❣✐♥❣ ❢r♦♠ ✵ t♦ ♠✐♥(|❦|, |♠|)✳ ■❢ ① = (✵, τ✶, . . . , τr) t❤❡♥ qt(①, ❞②) =

♠≥✵

d♠ (n/r)♠ Φ♠(τ/t)e− tr(τ/t)δ(t,t♠)(❞②) ■❢ ① = (κ, κ❦) ✐s s✉❝❤ t❤❛t κ > ✵✱ t ≥ ✵ ❛♥❞ µ = κ + t > ✵ t❤❡♥ qt(①, ❞②) =

|♠|≤|❦|

❦ ♠ κ µ |♠| ✶ − κ µ |❦|−|♠| δ(µ,µ♠)(❞②).

✷✷ ✴ ✷✺

SLIDE 69

▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ◗✉❛♥t✉♠ ❇❡ss❡❧ Pr♦❝❡ss ✲ ❝♦♥t✬❞

■❢ ① = (κ, −κ❦) ✇✐t❤ κ < ✵ ❛♥❞ µ = κ + t > ✵ t❤❡♥ qt(①, ❞②) =

♠≥✵

d♠ µ t |❦|+n ✶ − µ t |♠| ❦

♥

♠

♥

d♥

δ(µ,µ♠)(❞②), ✇❤❡r❡ t❤❡ ❧❛st s✉♠♠❛t✐♦♥ ✐s ♦✈❡r ♣❛rt✐t✐♦♥s ♥ ♦❢ t❤❡ ❝♦♥s❡❝✉t✐✈❡ ✐♥t❡❣❡rs r❛♥❣✐♥❣ ❢r♦♠ ✵ t♦ ♠✐♥(|❦|, |♠|)✳ ■❢ ① = (✵, τ✶, . . . , τr) t❤❡♥ qt(①, ❞②) =

♠≥✵

d♠ (n/r)♠ Φ♠(τ/t)e− tr(τ/t)δ(t,t♠)(❞②) ■❢ ① = (κ, κ❦) ✐s s✉❝❤ t❤❛t κ > ✵✱ t ≥ ✵ ❛♥❞ µ = κ + t > ✵ t❤❡♥ qt(①, ❞②) =

|♠|≤|❦|

❦ ♠ κ µ |♠| ✶ − κ µ |❦|−|♠| δ(µ,µ♠)(❞②).

✷✷ ✴ ✷✺

SLIDE 70

❆ ♣r♦❝❡ss ♦♥ ♣❛rt✐t✐♦♥s

▲❡t ❳t = (t, Xt)t ❜❡ ❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ◗✉❛♥t✉♠ ❇❡ss❡❧ Pr♦❝❡ss✳ ❉❡✜♥❡ Yt = Xt |t| ❚❤❡♥ ❛ t✐♠❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♣r♦❝❡ss Yt ❣✐✈❡s ❛ ❤♦♠♦❣❡♥♦✉s ▼❛r❦♦✈ ♣r♦❝❡ss Rt ♦♥ t❤❡ s❡t ♦❢ ♣❛rt✐t✐♦♥s P❛rt(r)✳ ■ts s❡♠✐❣r♦✉♣ ❝❛♥ ❜❡ ❝♦♥str✉❝t❡❞ ❢r♦♠ ❛ s❡♠✐❣r♦✉♣ ♦❢ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥ ♦♥ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ K(L✷(R✷n))

✷✸ ✴ ✷✺

SLIDE 71

❆ ♣r♦❝❡ss ♦♥ ♣❛rt✐t✐♦♥s

▲❡t ❳t = (t, Xt)t ❜❡ ❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ◗✉❛♥t✉♠ ❇❡ss❡❧ Pr♦❝❡ss✳ ❉❡✜♥❡ Yt = Xt |t| ❚❤❡♥ ❛ t✐♠❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♣r♦❝❡ss Yt ❣✐✈❡s ❛ ❤♦♠♦❣❡♥♦✉s ▼❛r❦♦✈ ♣r♦❝❡ss Rt ♦♥ t❤❡ s❡t ♦❢ ♣❛rt✐t✐♦♥s P❛rt(r)✳ ■ts s❡♠✐❣r♦✉♣ ❝❛♥ ❜❡ ❝♦♥str✉❝t❡❞ ❢r♦♠ ❛ s❡♠✐❣r♦✉♣ ♦❢ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥ ♦♥ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ K(L✷(R✷n))

✷✸ ✴ ✷✺

SLIDE 72

❆ ♣r♦❝❡ss ♦♥ ♣❛rt✐t✐♦♥s

▲❡t ❳t = (t, Xt)t ❜❡ ❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ◗✉❛♥t✉♠ ❇❡ss❡❧ Pr♦❝❡ss✳ ❉❡✜♥❡ Yt = Xt |t| ❚❤❡♥ ❛ t✐♠❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♣r♦❝❡ss Yt ❣✐✈❡s ❛ ❤♦♠♦❣❡♥♦✉s ▼❛r❦♦✈ ♣r♦❝❡ss Rt ♦♥ t❤❡ s❡t ♦❢ ♣❛rt✐t✐♦♥s P❛rt(r)✳ ■ts s❡♠✐❣r♦✉♣ ❝❛♥ ❜❡ ❝♦♥str✉❝t❡❞ ❢r♦♠ ❛ s❡♠✐❣r♦✉♣ ♦❢ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥ ♦♥ ❝♦♠♠✉t❛t✐✈❡ s✉❜✲❈∗✲❛❧❣❡❜r❛ ♦❢ K(L✷(R✷n))

✷✸ ✴ ✷✺

SLIDE 73

■♥t❡r♣r❡t❛t✐♦♥s

❖♥❡ ❝❛♥ s❤♦✇ t❤❛t |Rt| ✐s ❛ ♣✉r❡ ❜✐rt❤ ♣r♦❝❡ss ✭❨✉❧❡ ♣r♦❝❡ss✮✳ ❲❡ ❝❛♥ t❤✐♥❦ ♦❢ Rt ❛s ❛ ♣✉r❡ ❜✐rt❤ ♣r♦❝❡ss ✇❤✐❝❤ ❞❡s❝r✐❜❡s ❛ ♣♦♣✉❧❛t✐♦♥ ♦❢ |m| ♣❛rt✐❝❧❡s ❞✐✈✐❞❡❞ ✐♥t♦ r s✉❜♣♦♣✉❧❛t✐♦♥s✳ ❚❤❡ ❧❛r❣❡st ♦♥❡ ❤❛s m✶ ♣❛rt✐❝❧❡s✱ t❤❡ s❡❝♦♥❞ ❧❛r❣❡st m✷✱ ❡t❝✳✳✳ ❚❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥ ❝❤❛♥❣❡s ✐s ❡♥❝♦❞❡❞ ✐♥ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❜✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ s✐♠♣❧❡ ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛s✳ ❚❤❡ ❜✐rt❤ ✭❛♥❞ ❞❡❛t❤✮ ♣r♦❝❡ss❡s ♦r✐❣✐♥❛t✐♥❣ ✐♥ t❤✐s ✇❛② ❛r❡ ❝✉rr❡♥t❧② ✉♥❞❡r st✉❞②✳

✷✹ ✴ ✷✺

SLIDE 74

■♥t❡r♣r❡t❛t✐♦♥s

❖♥❡ ❝❛♥ s❤♦✇ t❤❛t |Rt| ✐s ❛ ♣✉r❡ ❜✐rt❤ ♣r♦❝❡ss ✭❨✉❧❡ ♣r♦❝❡ss✮✳ ❲❡ ❝❛♥ t❤✐♥❦ ♦❢ Rt ❛s ❛ ♣✉r❡ ❜✐rt❤ ♣r♦❝❡ss ✇❤✐❝❤ ❞❡s❝r✐❜❡s ❛ ♣♦♣✉❧❛t✐♦♥ ♦❢ |m| ♣❛rt✐❝❧❡s ❞✐✈✐❞❡❞ ✐♥t♦ r s✉❜♣♦♣✉❧❛t✐♦♥s✳ ❚❤❡ ❧❛r❣❡st ♦♥❡ ❤❛s m✶ ♣❛rt✐❝❧❡s✱ t❤❡ s❡❝♦♥❞ ❧❛r❣❡st m✷✱ ❡t❝✳✳✳ ❚❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥ ❝❤❛♥❣❡s ✐s ❡♥❝♦❞❡❞ ✐♥ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❜✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ s✐♠♣❧❡ ❊✉❝❧✐❞❡❛♥ ❏♦r❞❛♥ ❛❧❣❡❜r❛s✳ ❚❤❡ ❜✐rt❤ ✭❛♥❞ ❞❡❛t❤✮ ♣r♦❝❡ss❡s ♦r✐❣✐♥❛t✐♥❣ ✐♥ t❤✐s ✇❛② ❛r❡ ❝✉rr❡♥t❧② ✉♥❞❡r st✉❞②✳

✷✹ ✴ ✷✺

SLIDE 75

❚❤❡ ❡♥❞ ❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

✷✺ ✴ ✷✺