Vế hình học của công thức vết trên SL (2,IR)

✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■
❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆
✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲

❍❖⑨◆● ❚❍➚ ❉❯◆●

❱➌ ❍➐◆❍ ❍➴❈
❈Õ❆ ❈➷◆● ❚❍Ù❈ ❱➌❚ ❚❘➊◆ SL (2, R)

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❑❍❖❆ ❍➴❈

❈❤✉②➯♥ ♥❣➔♥❤✿ ❚❖⑩◆ ●■❷■ ❚➑❈❍
▼➣ sè✿ ✻✵✹✻✵✶✵✷

◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝
●❙✳❚❙❑❍✳ ✣➱ ◆●➴❈ ❉■➏P

❍⑨ ◆❐■✲ ✷✵✶✹


▼ö❝ ❧ö❝
▲í✐ ❝↔♠ ì♥
▼ð ✤➛✉

✷

✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✸

✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✶✳✶ ❙ì ❧÷ñ❝ ✈➲ SL (2, R) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✶✳✶ ❚→❝ ✤ë♥❣ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❧➯♥ ♥û❛ tr➯♥ ❝õ❛ ♠➦t ♣❤➥♥❣ ♣❤ù❝
✶✳✶✳✷ P❤➙♥ t➼❝❤ ■✇❛s❛✇❛ ✈➔ ♣❤➙♥ t➼❝❤ ❈❛rt❛♥ ❝õ❛ G ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✳✸ ◆❤â♠ ❝♦♥ ❞ø♥❣✳ ✣ë ✤♦ tr➯♥ G ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷ ▼ët sè ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷✳✶ ❚➼❝❤ ♣❤➙♥ q✉ÿ ✤↕♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷✳✷ ▲✐➯♥ ❤ñ♣ ê♥ ✤à♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷✳✸ ◆❤â♠ ❲❡✐❧ ✈➔ ♥❤â♠ ▲❛♥❣❧❛♥❞s✱ ▲✲♥❤â♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✸ ❇✐➸✉ ❞✐➵♥ ❝õ❛ SL(2, R) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✸✳✶ ●✐↔ ❤➺ sè ❝õ❛ ❝❤✉é✐ rí✐ r↕❝✱ L ✲ ❣â✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✸✳✷ ❇✐➸✉ ❞✐➵♥ ❝õ❛ GL(2, R) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✸✳✸ ❇✐➸✉ ❞✐➵♥ ❝õ❛ SL(2, R) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✹ ❚❤❛♠ sè ▲❛♥❣❧❛♥❞s ❝❤♦ SL(2, R) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✹✳✶ ❚❤❛♠ sè ▲❛♥❣❧❛♥❞s ❝❤♦ GL(2, R) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✹✳✷ ❚❤❛♠ sè ▲❛♥❣❧❛♥❞s ❝❤♦ SL(2, R) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✺ ◆❤â♠ ❝♦♥ ♥ë✐ s♦✐ ❝õ❛ SL (2, R) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✻ ❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷ ❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t

✷✳✶ ❱➳t ❝õ❛ t♦→♥ tû ❝â ♥❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷ ❈æ♥❣ t❤ù❝ tê♥❣ P♦✐ss♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✸ ❇✐➳♥ ✤ê✐ ❝æ♥❣ t❤ù❝ ✈➳t t❤❡♦ t➼❝❤ ♣❤➙♥ q✉ÿ ✤↕♦ ✳ ✳ ✳
✷✳✸✳✶ ❚r÷í♥❣ ❤ñ♣ γ ❝â ❞↕♥❣ ✤÷í♥❣ ❝❤➨♦ ❦❤✐ γ → 1
✷✳✸✳✷ ❚r÷í♥❣ ❤ñ♣ γ = r(θ) ❦❤✐ θ → 0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✹ P❤➨♣ ❝❤✉②➸♥ ✈➳ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✺ ❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
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ớ ỡ
t ữủ sỹ ộ ỹ ừ t tổ
ữủ sỹ ú ù tứ ừ t ổ

t tổ tọ ỏ t ỡ s s tợ ữớ t
ộ ồ ữớ trỹ t tr tử tự qt ữợ
ự t t ữợ tổ t ổ t
ỡ t

ổ ụ t ỡ t ổ ỡ ồ
rữớ ồ ồ tỹ ồ ố ở ỳ ữớ
trỹ t ú ù tổ tr q tr ồ t t trữớ ỡ
t t ữớ t õ õ ỵ ú ù ở tổ tr
q tr ồ t ự t
tớ tỹ ổ tự ỏ
ổ tr ọ ỳ s sõt ữủ
ỵ õ õ ừ t ổ ỗ ữủ
ỡ
ổ t ỡ
ở t
ồ





▼ð ✤➛✉
●✐↔✐ t➼❝❤ ✤✐➲✉ ❤á❛ tr➯♥ ♥❤â♠ ▲✐❡ ♥â✐ ❝❤✉♥❣ ❞➝♥ ✤➳♥ ✈✐➺❝ ♣❤➙♥ t➼❝❤ ♠ët ❜✐➸✉
❞✐➵♥ ❜➜t ❦ý r❛ tê♥❣ ❝→❝ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉②✳ ❇✐➸✉ ❞✐➵♥ ❝❤➼♥❤ q✉② ❝õ❛ ♥❤â♠
tr➯♥ ❦❤æ♥❣ ❣✐❛♥ t❤÷ì♥❣ ❝õ❛ ♥â t❤❡♦ ♥❤â♠ ❝♦♥ rí✐ r↕❝ ✤â♥❣ ✈❛✐ trá q✉❛♥ trå♥❣✳
❚❤❡♦ ❧þ t❤✉②➳t ❜✐➸✉ ❞✐➵♥ ❤➔♠ ✈➳t ✭t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ s✉② rë♥❣✮✱ ①→❝ ✤à♥❤ ❞✉②
♥❤➜t ❧î♣ t÷ì♥❣ ✤÷ì♥❣ ❝õ❛ ❜✐➸✉ ❞✐➵♥✳
❱➳t ❝õ❛ ♣❤➛♥ rí✐ r↕❝ ❝õ❛ ❜✐➸✉ ❞✐➵♥ ❝❤➼♥❤ q✉② ✤÷ñ❝ ✈✐➳t t❤➔♥❤ ❝❤✉é✐ ❝→❝ ✈➳t
❝õ❛ ❜✐➸✉ ❞✐➵♥ ♥❤å♥ ✈➔ ❞♦ ✤â ❧➔ tê♥❣ ❝→❝ t➼❝❤ ♣❤➙♥ q✉ÿ ✤↕♦ t÷ì♥❣ ù♥❣✳ ❈æ♥❣
t❤ù❝ ✈➳t ❦❤→ ♣❤ù❝ t↕♣ ♥❤÷♥❣ ❦❤✐ ❤↕♥ ❝❤➳ ①✉è♥❣ ♥❤â♠ ❝♦♥ ♥ë✐ s♦✐ t❤➻ ❦➳t q✉↔
trð ♥➯♥ t÷ì♥❣ ✤è✐ ✤ì♥ ❣✐↔♥✳ ✣➲ t➔✐ ✤÷ñ❝ ✤➦t r❛ ❧➔✿ ❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝
✈➳t tr➯♥ SL (2, R)✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ❣ç♠ ✷ ❝❤÷ì♥❣✿
•

❈❤÷ì♥❣ ✶✿ ❚â♠ t➢t ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳
❙ì ❧÷ñ❝ ❝➜✉ tró❝ ❝õ❛ SL(2, R)✳
✕ ❇✐➸✉ ❞✐➵♥ ❝õ❛ SL(2, R)✳
✕ ❚❤❛♠ sè ▲❛♥❣❧❛♥❞s ❝❤♦ SL(2, R)✳
✕ ◆❤â♠ ❝♦♥ ♥ë✐ s♦✐ ❝õ❛ SL(2, R)✳

✕

•

❈❤÷ì♥❣ ✷✿ ❚r➻♥❤ ❜➔② ✈➲ ✈➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t ♣❤➛♥ rí✐ r↕❝ ❝õ❛
❜✐➸✉ ❞✐➵♥ ❝❤➼♥❤ q✉② tr➯♥ SL(2, R) ✈➔ t❤✉ ❣å♥ ❝õ❛ ♥â tr➯♥ ♥❤â♠ ❝♦♥ ♥ë✐ s♦✐
❝õ❛ SL(2, R)✳
❱➳t ❝õ❛ t♦→♥ tû ❝â ♥❤➙♥✳
✕ ❈æ♥❣ t❤ù❝ tê♥❣ P♦✐ss♦♥✳
✕ ❇✐➳♥ ✤ê✐ ❝æ♥❣ t❤ù❝ ✈➳t t❤❡♦ t➼❝❤ ♣❤➙♥ q✉ÿ ✤↕♦✳

✕


❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R)

❍♦➔♥❣ ❚❤à ❉✉♥❣

❉♦ t❤í✐ ❣✐❛♥ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ ♥❤✐➲✉✱ ❦✐➳♥ t❤ù❝ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❦❤✐
❧➔♠ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❤↕♥ ❝❤➳ ✈➔ s❛✐ sât✳ ❚→❝ ❣✐↔ ♠♦♥❣ ♥❤➟♥
✤÷ñ❝ sü ❣â♣ þ ✈➔ ♥❤ú♥❣ þ ❦✐➳♥ ♣❤↔♥ ❜✐➺♥ ❝õ❛ q✉þ t❤➛② ❝æ ✈➔ ❜↕♥ ✤å❝✳
❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦
❍➔ ◆ë✐✱ ♥❣➔② ✷✵ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✹

❍å❝ ✈✐➯♥
❍♦➔♥❣ ❚❤à ❉✉♥❣

✹


❈❤÷ì♥❣ ✶

❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
✶✳✶ ❙ì ❧÷ñ❝ ✈➲ SL (2, R)
SL (2, R)

❜➡♥❣ ✶✿

❧➔ ♥❤â♠ ❝→❝ ♠❛ tr➟♥ ❝➜♣ 2 × 2 tr➯♥ tr÷í♥❣ sè t❤ü❝ R ✈î✐ ✤à♥❤ t❤ù❝
SL (2, R) =

a b
|a, b, c, d ∈ R; ad − bc = 1 .
c d

❚❛ ❦➼ ❤✐➺✉ G = SL (2, R)✱ ✤↕✐ sè ▲✐❡ ❝õ❛ G ❧➔ g0 = sl (2, R) ❣ç♠ ❝→❝ ♠❛ tr➟♥
t❤ü❝ ❝➜♣ 2 × 2 ❝â ✈➳t ❜➡♥❣ ✵ ✈➔ ❝â ❝ì sð ❣ç♠ ❝→❝ ♠❛ tr➟♥✿
H=

1 0
;X =
0 −1

0 1

;Y =
0 0

0 0
.
1 0

✶✳✶✳✶ ❚→❝ ✤ë♥❣ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❧➯♥ ♥û❛ tr➯♥ ❝õ❛ ♠➦t ♣❤➥♥❣ ♣❤ù❝
❑➼ ❤✐➺✉ H = {z = x + iy|x, y ∈ R, y > 0} ❧➔ ♥û❛ tr➯♥ ❝õ❛ ♠➦t ♣❤➥♥❣ ♣❤ù❝✳ ❚→❝
✤ë♥❣ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝õ❛ G tr➯♥ H ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿
❱î✐ ♠é✐ g = ac db ∈ G, z ∈ H t❛ ❝â✿
gz =

❉➵ t❤➜②✿
gz =

az + b
a b
.
z=
c d
cz + d

(az + b) (cz + d)
ac|z|2 + bd + adz + bcz
=
.
|cz + d|2
|cz + d|2

❉♦ ad − bc = 1 ♥➯♥ s✉② r❛✿

Im (gz) =

✺

Im (z)
.
|cz + d|2


❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R)

❍♦➔♥❣ ❚❤à ❉✉♥❣

❱➻ ✈➟② ♥➳✉ z ∈ H t❤➻ gz ∈ H✳
✣➦t✿ K = {g ∈ G|gi = i}✳ ❑❤✐ ✤â a2 + b2 = 1, c2 + d2 = 1 ✈➔ ad − bc = 1✳ ❍❛② K
❧➔ ♥❤â♠ ❝→❝ ♠❛ tr➟♥
r(θ) =

✈➔

cosθ sin θ
− sin θ cosθ

θ ∈ [0, 2π) .

P❤➙♥ ❧♦↕✐ ❝→❝ ♣❤➛♥ tû ❝õ❛ G
●å✐ λ ❧➔ ❣✐→ trà r✐➯♥❣ ❝õ❛ ♣❤➛♥ tû g ∈ G✱ ①➨t ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝õ❛ g✿
tr (g)2 − 4

tr (g) ±

2

λ − tr (g) λ + 1 = 0 ⇔ λ =

2

◆➳✉ |tr (g) | < 2 t❤➻ g ✤÷ñ❝ ❣å✐ ❧➔ ❡❧❧✐♣t✐❝✳
− ◆➳✉ |tr (g) | = 2 t❤➻ g ✤÷ñ❝ ❣å✐ ❧➔ ♣❛r❛❜♦❧✐❝✳
− ◆➳✉ |tr (g) | > 2 t❤➻ g ✤÷ñ❝ ❣å✐ ❧➔ ❤②♣❡r❜♦❧✐❝✳
−

✶✳✶✳✷ P❤➙♥ t➼❝❤ ■✇❛s❛✇❛ ✈➔ ♣❤➙♥ t➼❝❤ ❈❛rt❛♥ ❝õ❛ G
P❤➙♥ t➼❝❤ ■✇❛s❛✇❛ ❝õ❛ G ❧➔ ♣❤➙♥ t➼❝❤ ❝â ❞↕♥❣ G = KAN ✈î✐
cosθ sin θ
− sin θ cosθ

K=

uθ = exp θ(X − Y ) =

A=

at = exp tH =

et 0
0 e−t

N=

ns = exp sX =

1 s
0 1

|
|

|

t∈R ,
s∈R .

❚❛ ❝â K ∼= S 1, A ∼= R ✈➔ N ∼= R✳ ❈ö t❤➸ ✈î✐ ♠é✐ g =
t➼❝❤ ■✇❛s❛✇❛ ❝õ❛ ♥â ❧➔ g = uθ atns✱ tr♦♥❣ ✤â
a − ic
, et =
eiθ = √
2
2
a +c

θ ∈ [0, 2π) ,

a b
c d

∈G

t❤➻ ♣❤➙♥

ab + cd
.
a2 + c 2 , s = √
a2 + c 2

❍♦➔♥ t♦➔♥ t÷ì♥❣ tü✱ G ❝ô♥❣ ✤÷ñ❝ ♣❤➙♥ t➼❝❤ ❞÷î✐ ❞↕♥❣ G = AN K ✈➔ ❞↕♥❣
♥➔② ❝ô♥❣ ✤÷ñ❝ ❣å✐ ❧➔ ♣❤➙♥ t➼❝❤ ■✇❛s❛✇❛ ❝õ❛ G✳ ◆❣♦➔✐ r❛✱ t❛ ❝á♥ ❝â ♣❤➙♥ t➼❝❤
❈❛rt❛♥ ❝õ❛ G ❧➔ G = KAK ✳

✻


❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R)

❍♦➔♥❣ ❚❤à ❉✉♥❣

✶✳✶✳✸ ◆❤â♠ ❝♦♥ ❞ø♥❣✳ ✣ë ✤♦ tr➯♥ G
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❈❤♦ γ ∈ G✱ ♥❤â♠ ❝♦♥ ❞ø♥❣ ❝õ❛ ♣❤➛♥ tû γ tr♦♥❣ ●✱ ❦➼ ❤✐➺✉
Gγ ✱
Gγ = g ∈ G| g −1 γg = γ .

P❤➛♥ tû γ ∈ G ❧➔ ♣❤➛♥ tû ♥û❛ ✤ì♥ ❝❤➼♥❤ q✉② ♠↕♥❤ ♥➳✉ ♥❤â♠ ❝♦♥ ❞ø♥❣ Gγ
❝õ❛ ♥â ❧➔ ♠ët ①✉②➳♥ ❝ü❝ ✤↕✐ tù❝ ❧➔ Gγ = T = SO(2, R)✱ ❦❤✐ ✤â t❛ ❝ô♥❣ ❝â ♥❤â♠
t❤÷ì♥❣
Gγ \G = {Gγ x | x ∈ G}.

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ▼ët ✤ë ✤♦ µ tr➯♥ Gγ \G ✤÷ñ❝ ❣å✐ ❧➔ G ✲ ❜➜t ❜✐➳♥ ♣❤↔✐ ♥➳✉

µ(Ax) = µ(A) ✈î✐ ♠å✐ t➟♣ ❇♦r❡❧ ❆ tr♦♥❣ Gγ \G ✈➔ ♠å✐ x ∈ G✳

✣ë ✤♦ G ✲ ❜➜t ❜✐➳♥ tr→✐ ❝ô♥❣ ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ❤♦➔♥ t♦➔♥ t÷ì♥❣ tü✳ ▼ët ✤ë
✤♦ µ tr➯♥ G ❣å✐ ❧➔ ✤ë ✤♦ ❍❛❛r ♥➳✉ ♥â ❜➜t ❜✐➳♥ ❞÷î✐ t→❝ ✤ë♥❣ ❝õ❛ G✳
✣è✐ ✈î✐ ♣❤➙♥ t➼❝❤ ■✇❛s❛✇❛ G = AN K ✱ ♣❤➛♥ tû x ∈ G t❛ ❝â ♣❤➙♥ t➼❝❤ x = ank
✭✈î✐ a ∈ A, n ∈ N, k ∈ K ✮✱ ❦➼ ❤✐➺✉ da, dn, dk t÷ì♥❣ ù♥❣ ❧➔ ✤ë ✤♦ ❍❛❛r tr➯♥ A, N, K ✳
❑❤✐ ✤â ✤ë ✤♦ tr➯♥ G✱ ❦➼ ❤✐➺✉ dx✱ ✈➔ t❛ ❝â dx = da dn dk✳
❱î✐ ❤➔♠ f ①→❝ ✤à♥❤ ✈➔ ❦❤↔ t➼❝❤ tr➯♥ G✱ t❛ ❝â
f (x)dx =
G

dk

da

K

A

f (ank)dn.
N

✣è✐ ✈î✐ ♣❤➙♥ t➼❝❤ ❈❛rt❛♥ G = KAK ✱ ✈î✐ ♠å✐ x ∈ G t❛ ❝â ♣❤➙♥ t➼❝❤ x = k1ak2✱
f (x)dx =
G

K×K

|t2 − t−2 |f (k1 at k2 )da,

dk1 dk2
A

tr♦♥❣ ✤â k1, k2 ∈ K ✈➔ a ∈ A✳
✶✳✷ ▼ët sè ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥

✶✳✷✳✶ ❚➼❝❤ ♣❤➙♥ q✉ÿ ✤↕♦
❈❤♦ G = SL(2, R)✱ γ ∈ G ❧➔ ♣❤➛♥ tû ♥û❛ ✤ì♥ ❝❤➼♥❤ q✉② ♠↕♥❤✱ Gγ = T ❧➔
♥❤â♠ ❝♦♥ ❞ø♥❣ ❝õ❛ γ ✱ ❤➔♠ f ∈ Cc∞(G)✳ ❚➼❝❤ ♣❤➙♥ q✉ÿ ✤↕♦ ❝õ❛ ❤➔♠ f tr➯♥ q✉ÿ
✼


ồ ừ ổ tự t tr SL(2, R)



ừ ữủ ổ tự
f (x1 x)dx,


O (f ) =
G \G

tr õ dx ở Gt tr tữỡ G \G

ủ ờ
G = SL(2, R) , G ữủ ồ ủ tỗ t x G s
= xx1
ố ợ tỷ q ỷ ỡ t õ r , G ủ ờ
tỗ t x SL(2, C) = ac db | a, b, c, d C ; ad bc = 1 s

= xx1
f Cc(G) G tỷ q õ t q
ờ ừ f ố ợ tỷ ữủ
O (f ).

SO (f ) =
S()

r õ S() t ủ tỷ ừ ợ ủ tr ợ
ủ ờ ừ

õ õ s õ
WR õ ừ R ữ s
õ ừ C WC = Cì
õ ừ R õ tr tr SU (2) ữủ s
z 0
0 z

w =

, z Cì

0 1
1 0

.

õ SU (2) ởt õ t ợ số 22 tr
tr ợ tỷ õ tự ữủ ồ õ tr t
Gal(C/R) õ s ừ rở C/R ỗ tỷ ởt

tỷ tỹ ỗ ỗ t tỷ ỏ tỹ ỗ ủ ự
P tỷ w t ở ủ ữ tỷ ổ t tữớ tr õ
(C/R) tr Cì WR Gal(C/R) ữủ w ú ỵ



ồ ừ ổ tự t tr SL(2, R)



r w2 = 1 õ rở ừ WC = Cì Gal(C/R) rở ổ t
tữớ
õ s LF LF = WR trữớ ỡ s
C R LF = WR ì SL(2, C)

G õ ự t ồ ừ G = SL(2, R) õ G = P GL(2, C)
õ s Gal(C/R) t ở tr G q tỹ ỗ ữủ
tt ỳ t õ t t ở õ t tữớ WR t
ở tợ Gal(C/R) q tỹ ừ õ
õ ừ L G = G

WR

ừ SL(2, R)
ởt õ GL(2, R) SL(2, R) ổ
rt ởt ừ tr ởt ỗ tứ õ tỹ
t t tử GL(E) ừ
: G GL(E),

s ợ ồ tỡ v E t tứ x (x)v

tử
ữủ ồ t (x) t ợ ồ x G

ừ õ tr ổ rt
ởt ổ ừ õ t (x)W W ợ ồ
x G

ởt : G GL(E) ồ t q ổ
õ ổ t {0}

ừ tr ổ rt sỷ r
E=

En ,




❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R)

❍♦➔♥❣ ❚❤à ❉✉♥❣

sin θ
tr♦♥❣ ✤â En ❧➔ ❦❤æ♥❣ ❣✐❛♥ r✐➯♥❣ t❤ù ♥ ❝õ❛ K = −cosθ
| θ ∈ [0, 2π) .
sin θ cosθ
P❤➛♥ tû v ∈ E ❧➔ ❑✲❤ú✉ ❤↕♥ ♥➳✉ π(K)v s✐♥❤ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✈➨❝ tì ❤ú✉ ❤↕♥
❝❤✐➲✉✳

✣à♥❤ ♥❣❤➽❛ ✶✳✾✳ ❇✐➸✉ ❞✐➵♥ π ❝õ❛ ● tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❊ ✤÷ñ❝ ❣å✐ ❧➔

❝❤➜♣ ♥❤➟♥ ✤÷ñ❝ ♥➳✉ dimEn ❤ú✉ ❤↕♥ ✈î✐ ♠å✐ ♥✳

❳➨t ♣❤➙♥ t➼❝❤ ■✇❛s❛✇❛ ❝õ❛ ♥❤â♠ G = SL(2, R)✿ ● ❂ P❑ ✭✈î✐ P ❂ ❆◆✮✱ σ ❧➔
❜✐➸✉ ❞✐➵♥ ❝õ❛ P tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❱✳ ●å✐ H(σ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ →♥❤ ①↕
f : G → V s❛♦ ❝❤♦
f |K ∈ L2 (K)

✈➔ f (py) = ∆(p)

1
2

σ(p)f (y),

tr♦♥❣ ✤â ∆(p) = α(a) ❧➔ ❤➔♠ ♠♦❞✉❧❛r tr➯♥ P✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✵✳ ❇✐➸✉ ❞✐➵♥ π ❝õ❛ ● tr➯♥ H(σ) ❝❤♦ ❜ð✐ tà♥❤ t✐➳♥ ♣❤➼❛ ♣❤↔✐ tr➯♥
❜✐➳♥✱ tù❝ ❧➔ π(y)f (x) = f (xy)✱ ❣å✐ ❧➔ ❜✐➸✉ ❞✐➵♥ ❝↔♠ s✐♥❤ ❝õ❛ σ ❧➯♥ ● ✳

✣➦t ρ(a) = α(a)1/2✱ ✈î✐ ♠é✐ sè ♣❤ù❝ s ✈➔ x = ank ∈ G ①→❝ ✤à♥❤
ρs (x) = ρs (ank) = ρ(a)s+1 .

❑❤✐ ✤â
ρs (k) = ρs (n) = 1.

❉➵ t❤➜② ❤➔♠ µs : P → C∗ ❝❤♦ ❜ð✐ µs = ρ(a)s = as ❧➔ ♠ët ✤➦❝ tr÷♥❣ ✭tù❝ ❧➔ ✤ç♥❣
❝➜✉ ❧✐➯♥ tö❝ ✈➔♦ C∗✮✳ ◆➳✉ ♥â ❝â ❣✐→ trà t✉②➺t ✤è✐ ❜➡♥❣ ✶ t❤➻ µs ❧➔ ♠ët ✤➦❝ tr÷♥❣
✉♥✐t❛✳
❑➼ ❤✐➺✉ Hs ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝õ❛ ❜✐➸✉ ❞✐➵♥ πs ❝↔♠ s✐♥❤ ❜ð✐ µs✱ ♥â ❧➔ ❦❤æ♥❣ ❣✐❛♥
❍✐❧❜❡rt ❝→❝ ❤➔♠ ①→❝ ✤à♥❤ tr➯♥ ● s❛♦ ❝❤♦
✐✮ f (any) = ρs+1f (y)❀

✐✐✮ f |K ∈ L2(K)✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✶✳ ❍å ❝→❝ ❜✐➸✉ ❞✐➵♥ {πs } ①→❝ ✤à♥❤ ♥❤÷ tr➯♥ ❣å✐ ❧➔ ❜✐➸✉ ❞✐➵♥
❝❤✉é✐ ❝❤➼♥❤ ❝õ❛ SL(2, R)✳

✶✵


❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R)

❍♦➔♥❣ ❚❤à ❉✉♥❣

✶✳✸✳✶ ●✐↔ ❤➺ sè ❝õ❛ ❝❤✉é✐ rí✐ r↕❝✱ L ✲ ❣â✐
❈❤♦ G = SL(2, R)✱ t➙♠ ❝õ❛ ● ❧➔ Z(G) = {g ∈ G| ∀x ∈ G, gx = xg}✱ π ❧➔ ❜✐➸✉
❞✐➵♥ ❝❤✉é✐ rí✐ r↕❝ ❝õ❛ G✳ ❚❛ ♥â✐ ❤➔♠ f ∈ Cc∞(G) ❧➔ ♠ët ❣✐↔ ❤➺ sè ✭❝❤✉➞♥ t➢❝✮
✤è✐ ✈î✐ π ♥➳✉ ✈î✐ ❜➜t ❦➻ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② t➠♥❣ ✈ø❛ ♣❤↔✐ π t❛ ❝â
1
♥➳✉ π π,
trace π (f ) =
0 tr÷í♥❣ ❤ñ♣ ❝á♥ ❧↕✐✳
❚❛ ❦➼ ❤✐➺✉ fπ ❧➔ ❣✐↔ ❤➺ sè ✤è✐ ✈î✐ π ✭♥â ❧➔ ❦❤æ♥❣ ❞✉② ♥❤➜t✮✳ ❚➼❝❤ ♣❤➙♥ q✉ÿ ✤↕♦
❝õ❛ fπ ✤è✐ ✈î✐ ♣❤➛♥ tû ❝❤➼♥❤ q✉② ♥û❛ ✤ì♥ γ ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐✳
Θπ (γ −1 ) ♥➳✉ γ ❧➔ ❡❧❧✐♣t✐❝,
Oγ (fπ ) =
0 tr÷í♥❣ ❤ñ♣ ❝á♥ ❧↕✐✳
tr♦♥❣ ✤â Θπ ❧➔ ✤➦❝ tr÷♥❣ ❝õ❛ π✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✷✳ ❳➨t ♠ët ❜✐➸✉ ❞✐➵♥ ❝❤✉é✐ rí✐ r↕❝ π ✈➔ ❦➼ ❤✐➺✉ fπ ❧➔ ❣✐↔ ❤➺ sè
t÷ì♥❣ ù♥❣✳ ❍❛✐ ❜✐➸✉ ❞✐➵♥ ❝❤✉é✐ rí✐ r↕❝ π ✈➔ π ❝õ❛ G ✤÷ñ❝ ❣å✐ ❧➔ t❤✉ë❝ ❝ò♥❣ ♠ët
▲✲❣â✐ ♥➳✉ ✈î✐ ❜➜t ❦➻ ♣❤➛♥ tû ♥û❛ ✤ì♥ ❝❤➼♥❤ q✉② ♠↕♥❤ γ t❛ ❝â
SOγ (fπ ) = c(π, π )SOγ (fπ ),

tr♦♥❣ ✤â c(π, π ) ❧➔ ❤➡♥❣ sè ❦❤→❝ ❦❤æ♥❣✳

✶✳✸✳✷ ❇✐➸✉ ❞✐➵♥ ❝õ❛ GL(2, R)
❚➜t ❝↔ ❝→❝ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ❝❤➜♣ ♥❤➟♥ ✤÷ñ❝ ❝õ❛ GL(2, R) ✤➲✉ ❧➔ t❤÷ì♥❣
❝♦♥ ❝õ❛ ❝❤✉é✐ ❝❤➼♥❤ ρ(µ1, µ2)✱ tr♦♥❣ ✤â µi ❧➔ ✤➦❝ tr÷♥❣ ❝õ❛ R×✳ ❈→❝ ❜✐➸✉ ❞✐➵♥
❝❤✉é✐ ❝❤➼♥❤ ❧➔ ✤÷ñ❝ ❝↔♠ s✐♥❤ ❜ð✐ ❝→❝ ✤➦❝ tr÷♥❣ tø ♥❤â♠ ❝♦♥ ❇♦r❡❧✿ρ(µ1, µ2) ❧➔
❜✐➸✉ ❞✐➵♥ ❝❤➼♥❤ q✉② ♣❤↔✐ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ trì♥ s❛♦ ❝❤♦
f

α x
0 β

g

α
= µ1 (α)µ2 (β)
β

1
2

f (g).

●✐↔ sû r➡♥❣ t➼❝❤ µ1µ2 ❧➔ ✉♥✐t❛✱ t❛ ❝â ❜❛ ❧♦↕✐ t❤÷ì♥❣ ❝♦♥ t❤❡♦ ❣✐→ trà ❝õ❛ µ = µ1µ−1
2
✲ ❇✐➸✉ ❞✐➵♥ ❝❤✉é✐ ❝❤➼♥❤ ❜➜t ❦❤↔ q✉② π(µ1, µ2) ❦❤✐ µ = xn.sign(x) ✈î✐ n ∈ Z \ {0}✳
◆❤ú♥❣ ❜✐➸✉ ❞✐➵♥ ♥➔② ❧➔ ✉♥✐t❛ ❤â❛ ♥➳✉ µ ❧➔ ✉♥✐t❛ ❤♦➦❝ ♥➳✉ µ = |x|s ✈î✐ s ❧➔ sè
t❤ü❝ ✈➔ −1 < s < 1✳
✲ ❇✐➸✉ ❞✐➵♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉ π(µ1, µ2) ❦❤✐ µ = xn.sign(x)✳ ❇✐➸✉ ❞✐➵♥ ♥➔② ❧➔ ✉♥✐t❛
✶✶

❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R)

❍♦➔♥❣ ❚❤à ❉✉♥❣

❤â❛ ♥➳✉ n = ±1✳
✲ ❇✐➸✉ ❞✐➵♥ ❝❤✉é✐ rí✐ r↕❝ σ(µ1, µ2) ❦❤✐ µ = xn.sign(x) ✈î✐ n ∈ Z \ {0}✳ ◆❤ú♥❣ ❜✐➸✉
❞✐➵♥ ♥➔② ❧➔ ✉♥✐t❛ ❤â❛✳
◆❤ú♥❣ ❜✐➸✉ ❞✐➵♥ ❦❤→❝ ♥❤❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣ ❦❤✐ ❤♦→♥ ✈à µi✿ π(µ1, µ2) π(µ2, µ1)✳

✶✳✸✳✸ ❇✐➸✉ ❞✐➵♥ ❝õ❛ SL(2, R)
❇➜t ❦➻ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ❝õ❛ SL(2, R) ✤➲✉ ❧➔ ❤↕♥ ❝❤➳ ❝õ❛ ❜✐➸✉ ❞✐➵♥ ❜➜t
❦❤↔ q✉② ❝õ❛ GL(2, R)✳ ❍↕♥ ❝❤➳ ♥➔② ❤♦➦❝ ❝â ♣❤➛♥ ❝á♥ ❧↕✐ ❜➜t ❦❤↔ q✉② ✭❧➔ tr÷í♥❣
❤ñ♣ ❜✐➸✉ ❞✐➵♥ ❝❤✉é✐ ❝❤➼♥❤ ❝â ❣✐→ trà t❤❛♠ sè ❝ò♥❣ ❧♦↕✐✮ ❤♦➦❝ ❜à t→❝❤ ❧➔♠ ❤❛✐
t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ♠➔ ❤ñ♣ ❝õ❛ ♥â ❧➔ ♠ët ▲✲❣â✐ ❝❤♦ SL(2, R)✳
❍❛✐ ❜✐➸✉ ❞✐➵♥ π ✈➔ π ❧➔ ❝ò♥❣ t❤✉ë❝ ♠ët ▲✲❣â✐ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ tr➯♥ q✉❛♥ ❤➺ t÷ì♥❣
✤÷ì♥❣ ❝❤ó♥❣ ✤÷ñ❝ ❧✐➯♥ ❤ñ♣ ❜ð✐ α✿
π

π ◦ Ad(α)

tr♦♥❣ ✤â

α=

−1 0
0 1

.

❚❛ ❝â sü ♣❤➙♥ ❧♦↕✐ s❛✉ ✤➙②✿
✲ ❇✐➸✉ ❞✐➵♥ ❝❤✉é✐ ❝❤➼♥❤ ❜➜t ❦❤↔ q✉② π(µ) t❤✉ ✤÷ñ❝ ❜ð✐ ❤↕♥ ❝❤➳ ❝õ❛ π(µ1, µ2) tr➯♥
SL(2, R) ✈î✐ µ = xn .sign(x), n ∈ Z✳
✲ ❇✐➸✉ ❞✐➵♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉ π(µ) t❤✉ ✤÷ñ❝ ❜ð✐ ❤↕♥ ❝❤➳ ❝õ❛ π(µ1, µ2) tr➯♥ SL(2, R)
✈î✐ µ = xn.sign(x), n = 0✳
+
−
✲ ❇✐➸✉ ❞✐➵♥ ❝❤✉é✐ rí✐ r↕❝ ▲✲❣â✐ σ(D|n|
, D|n|
) t❤✉ ✤÷ñ❝ ❜ð✐ ❤↕♥ ❝❤➳ ❝õ❛ σ(µ1 , µ2 )
tr➯♥ SL(2, R) ✈î✐ µ = xn.sign(x), n ∈ Z \ {0}✳
✲ ●✐î✐ ❤↕♥ ❝õ❛ ❜✐➸✉ ❞✐➵♥ ❝❤✉é✐ rí✐ r↕❝ ▲✲❣â✐ σ(D0+, D0−) t❤✉ ✤÷ñ❝ ❜ð✐ ❤↕♥ ❝❤➳ ❝õ❛
π(µ1 , µ2 ) tr➯♥ SL(2, R) ✈î✐ µ = sign(x)✳
❈→❝ ▲✲❣â✐ ❝õ❛ ❜✐➸✉ ❞✐➵♥ ✤÷ñ❝ ❝❤➾ rã ❜ð✐ ❝→❝ ✤➦❝ tr÷♥❣ µ ✈➔ µ−1 ❧➔ t÷ì♥❣ ✤÷ì♥❣✳
✶✳✹ ❚❤❛♠ sè ▲❛♥❣❧❛♥❞s ❝❤♦ SL(2, R)
ˇ ❧✐➯♥ ❤ñ♣ ❝õ❛ ✤ç♥❣ ❝➜✉ ❝❤➾♥❤ ❤➻♥❤
❚❤❛♠ sè ▲❛♥❣❧❛♥❞s ❧➔ ❧î♣ G−
ϕ : LR → L G,

✶✷


ồ ừ ổ tự t tr SL(2, R)



s ủ ợ tỹ ừ LG WR t
LR L G WR ,

tỹ ừ LR tr WR s ừ tỷ ừ WR
ỷ ỡ số ữủ ồ t ủ ợ ừ tr G ổ
tr õ r trứ õ G

số s GL(2, R)
ởt t số s GL(2, R) ợ ủ ỗ ừ WR tr
GL(2, C) ợ ỷ ỡ
ợ z = .ei t s,n(z) = sein õ tr ủ ữủ
õ ữ s
ợ si C mi Z2
s1 ,m1 ,s2 ,m2 (z) =

ợ s ,m ,s ,m (w ) =

s1 ,0 (z)
0
0
s2 ,0 (z)

tr ủ t õ s ,m ,s ,m
ợ s C n Z
1

s,n (z) =

1

2

2

1

s2 ,m2 ,s1 ,m1

1

2

2



ợ s,n(w ) =

s,n (z)
0
0
s,n (z)

0
(1)m1
0
(1)m2

0 (1)n
1
0

tr ủ t õ s,n s,n

ừ t ủ ợ ủ ừ ợ ỳ t số õ

s,0

s,1,s,0

s,0,s,1

ỗ tứ WR Cì (z) = 1 (w ) = 1
ởt t số s t




tở ợ s,n ợ s t
ữỡ ự ỳ t q t số s GL(2, R) t
ữủ ữ ữợ õ ởt s tỹ ỳ ợ tữỡ ữỡ ừ



❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R)

❍♦➔♥❣ ❚❤à ❉✉♥❣

❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ❝❤➜♣ ♥❤➟♥ ✤÷ñ❝ ❝õ❛ GL(2, R) ✈➔ ❝→❝ ❧î♣ ❧✐➯♥ ❤ñ♣ ❝õ❛ ✤ç♥❣
❝➜✉ ❝❤➜♣ ♥❤➟♥ ✤÷ñ❝ ❝õ❛ WR tr♦♥❣ GL(2, C) ♥❤÷ s❛✉✿
π(µ1 , µ2 ) −→ ϕs1 ,m1 ,s2 ,m2

✈î✐ µi = |x|s sign(x)m
i

i

✈➔

✈î✐ µ1µ2(x) = |x|2ssign(x)n+1
n
tr♦♥❣ ✤â µ1µ−1
2 (x) = x sign(x)✳ ❚❤❛♠ sè ▲❛♥❣❧❛♥❞s t÷ì♥❣ ù♥❣ ✈î✐ ❜✐➸✉ ❞✐➵♥
t➠♥❣ ✈ø❛ ♣❤↔✐ ♥➳✉ ↔♥❤ ❝õ❛ →♥❤ ①↕ ❜à ❝❤➦♥ tù❝ ❧➔ si t❤✉➛♥ ↔♦✳
σ(µ1 , µ2 ) −→ ϕs,n

✶✳✹✳✷ ❚❤❛♠ sè ▲❛♥❣❧❛♥❞s ❝❤♦ SL(2, R)
❚ø s♦♥❣ →♥❤ ❣✐ú❛ ❝→❝ ❧î♣ t÷ì♥❣ ✤÷ì♥❣ ❝õ❛ ❜✐➸✉ ❞✐➵♥ ✈➔ ❧î♣ ❧✐➯♥ ❤ñ♣ ❝õ❛ t❤❛♠
sè ▲❛♥❣❧❛♥❞s ❝❤♦ GL(2, R) s✉② r❛ s♦♥❣ →♥❤ ❣✐ú❛ ❝→❝ ❧î♣ t÷ì♥❣ ✤÷ì♥❣ ▲✲❣â✐ ❝õ❛
❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ❝❤➜♣ ♥❤➟♥ ✤÷ñ❝ ❝õ❛ SL(2, R) ✈➔ ❝→❝ ❧î♣ ❧✐➯♥ ❤ñ♣ ❝õ❛ ❝→❝
✤ç♥❣ ❝➜✉ ❝❤➜♣ ♥❤➟♥ ✤÷ñ❝ ❝õ❛ WR tr♦♥❣ P GL(2, C)✳
✲ ❚❤❛♠ sè ❤â❛ ❝❤♦ π(µ) ❧➔ ❧î♣ ❧✐➯♥ ❤ñ♣ ❝õ❛ t❤❛♠ sè ❤â❛ ♣❤➨♣ ❝❤✐➳✉ ϕs,m ✤÷ñ❝
①→❝ ✤à♥❤ ❜ð✐ ϕs,m,0,0 ✈î✐ µ(x) = |x|ssign(x)m✳
✲ ❚❤❛♠ sè ❤â❛ ❝❤♦ Dn± ❧➔ ❧î♣ ❧✐➯♥ ❤ñ♣ ❝õ❛ t❤❛♠ sè ❤â❛ ♣❤➨♣ ❝❤✐➳✉ ϕn ①→❝ ✤à♥❤
❜ð✐ ϕ0,n✳
❚❛ t❤➜② r➡♥❣
ϕ0,n ⊗ ε = αϕ0,n α−1

tr♦♥❣ ✤â α =

−1 0
0 1

.

◆❤÷♥❣ ε ❝â ♠ët t➙♠ ↔♥❤ ❞♦ ✤â t❤❛♠ sè ❤â❛ ♣❤➨♣ ❝❤✐➳✉ ①→❝ ✤à♥❤ ❜ð✐ ϕ0,n ✈➔
ϕ0,n ⊗ ε ❧➔ ❜➡♥❣ ♥❤❛✉✳ ✣✐➲✉ ♥➔② ❝❤➾ r❛ r➡♥❣ ↔♥❤ ♣❤➨♣ ❝❤✐➳✉ ❝õ❛ α t❤✉ë❝ t➙♠ ❤â❛
❝õ❛ ↔♥❤ ♣❤➨♣ ❝❤✐➳✉ ❝õ❛ ϕ0,n✳
❈❤♦ ϕn ❧➔ t❤❛♠ sè ❤â❛ ♣❤➨♣ ❝❤✐➳✉ ①→❝ ✤à♥❤ ❜ð✐ ϕ0,n ✈➔ Sϕ ❧➔ t➙♠ ❤â❛ ↔♥❤ ❝õ❛
ϕn ✈➔ Sϕ ❧➔ t❤÷ì♥❣ ❝õ❛ Sϕ ❜ð✐ t❤➔♥❤ ♣❤➛♥ ❧✐➯♥ t❤æ♥❣ Sϕ0 ❝õ❛ ♥â ♥❤➙♥ ✈î✐
t➙♠ ZGˇ ❝õ❛ Gˇ ✿
✰ ❑❤✐ n = 0 t❛ ❝â Sϕ = Sϕ {1, α}✳
✰ ❑❤✐ n = 0 ♥❤â♠ Sϕ0 ❧➔ ♠ët ①✉②➳♥ ♥❤÷♥❣ Sϕ ❧↕✐ ✤÷ñ❝ s✐♥❤ ❜ð✐ ↔♥❤ ❝õ❛ α✳
n

n

n

n

n

n

0

0

✶✹


ồ ừ ổ tự t tr SL(2, R)

õ ở s ừ SL (2, R)
õ ở s ừ õ õ tỹ r
õ L H t tổ ừ t õ ừ ởt tỷ ỷ ỡ ừ
õ L G

r tt ử tr ỳ ố tữủ tr tứ ữủ t t
ủ ữợ tỷ = i tr õ ừ SO(2) tr SL(2, C)
ữ ỵ r tỷ ổ t tữớ ừ õ s t tỷ
a = w(w)1 =

1 0
0 1

s r ởt õ õ t ỗ t õ ợ H 1(C/R, SO(2))
trữ ừ õ ữủ ồ trữ ở s õ õ ở s ừ
SL(2, R) tữỡ ự ợ trữ õ ở s tữỡ ự ợ
trữ t tữớ SL(2, R) tr õ õ ở s tữỡ ự
ợ trữ ổ t tữớ t T (R) = SO(2, R)
t

ữỡ tr tự ỡ õ q ở
ừ ữỡ ữ t ở t t
t s õ ứ ở ú t ỡ trú ừ
SL (2, R) t tự ừ SL (2, R) t số õ s
õ ở s s õ trỏ ừ ốt tr ỹ ờ
ổ tự t tr SL (2, R)

❈❤÷ì♥❣ ✷

❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t
❈❤÷ì♥❣ ♥➔② s➩ tr➻♥❤ ❜➔② ✈➲ ✈➳t ❝õ❛ t♦→♥ tû ❝â ♥❤➙♥✱ ❝æ♥❣ t❤ù❝ tê♥❣ P♦✐ss♦♥✱
tø ✤â t❛ ❜✐➳♥ ✤ê✐ ❝æ♥❣ t❤ù❝ ✈➳t t❤❡♦ t➼❝❤ ♣❤➙♥ q✉ÿ ✤↕♦✳
✷✳✶ ❱➳t ❝õ❛ t♦→♥ tû ❝â ♥❤➙♥

❈❤♦ ● ❧➔ ♥❤â♠ ❝♦♠♣❛❝t ✤à❛ ♣❤÷ì♥❣✱ Γ ❧➔ ♥❤â♠ ❝♦♥ rí✐ r↕❝ ❝õ❛ ● ✈➔ R ❧➔
❜✐➸✉ ❞✐➵♥ ❝❤➼♥❤ q✉② ❝õ❛ ● tr➯♥ L2(Γ\G)
[R(g)φ](x) = φ(xg)

✈î✐ g ∈ G, x ∈ Γ\G.

Ù♥❣ ✈î✐ ❜✐➸✉ ❞✐➵♥ ✉♥✐t❛ ❝õ❛ ♥❤â♠ ● t❛ ❝â ❜✐➸✉ ❞✐➵♥ t÷ì♥❣ ù♥❣ ❝õ❛ ✤↕✐ sè ❍❛❛r
L1 (G) ✭✤è✐ ✈î✐ t➼❝❤ ❝❤➟♣✮ ❝❤♦ ❜ð✐
R(f )φ(x) =

f (x−1 g)φ(g)dg.

f (g)φ(xg)dg =
G

G

●✐↔ sû f ∈ Cc∞(G)✳ ❇➡♥❣ ❝→❝❤ t→❝❤ t➼❝❤ ♣❤➙♥✱ t❛ ❝â t❤➸ ✈✐➳t
f (x−1 γg)φ(g)dg =

R(f )φ(x) =

Γ\G γ∈Γ

Kf (x, g)φ(g)dg.
Γ\G

❉♦ ✤â R(f ) ❧➔ ♠ët t♦→♥ tû t➼❝❤ ♣❤➙♥ ✈î✐ ❤↕t ♥❤➙♥ trì♥
f (x−1 γg).

Kf (x, g) =
γ∈Γ

R(f )

❧➔ ❧î♣ ✈➳t ✈➔ ❝â t❤➸ t➼♥❤ ✈➳t ❝õ❛ ♥â t❤❡♦ ❤❛✐ ❝→❝❤✳ ✣➛✉ t✐➯♥✱ t❛ ❝â t❤➸ ✈✐➳t
trace R(f ) =

f (x−1 γx)dx.

Kf (x, x)dx =
Γ\G

Γ\G γ∈Γ

✶✻


❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R)

❍♦➔♥❣ ❚❤à ❉✉♥❣

❑➼ ❤✐➺✉ [γ] = { δ−1γδ | δ ∈ Γγ \Γ }✱ tr♦♥❣ ✤â Γγ ❧➔ t➙♠ ❤â❛ ❝õ❛ γ tr♦♥❣ Γ✳ ❑❤✐ ✤â✱
t❛ ❝â
f (x−1 δ −1 γδx)dx =

f (x−1 γx)dx = vol(Γγ \Gγ )
Γγ \G

Γ\G δ∈Γ \Γ
γ

f (x−1 γx)dx.
Gγ \G

❉♦ ✤â✱ t❛ ❝â
vol(Γγ \Gγ ) Oγ (f ).

trace R(f ) =
[γ]

✣➙② ❧➔ ❝æ♥❣ t❤ù❝ ✈➳t ❝õ❛ t♦→♥ tû ❝â ♥❤➙♥✱ ♥â ❝á♥ ✤÷ñ❝ ❣å✐ ❧➔ ❝æ♥❣ t❤ù❝ ✈➳t
❆rt❤✉r✲❙❡❧❜❡r❣✳
❚❛ ❝ô♥❣ ❝â t❤➸ t➼♥❤ trace R(f ) ❜➡♥❣ ❝→❝❤ t❤ù ❤❛✐ t❤❡♦ ❦➳t q✉↔ ❝õ❛ ●❡❧❢❛♥❞✱
●r❛❡✈ ✈➔ P✐❛t❡ts❦✐✲❙❤❛♣✐r♦✱ L2(Γ\G) ♣❤➙♥ t➼❝❤ rí✐ r↕❝ t❤➔♥❤ tê♥❣ trü❝ t✐➳♣ ❝õ❛
❝→❝ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ❝õ❛ ●✱ ①✉➜t ❤✐➺♥ ✈î✐ ♠é✐ ❜ë✐ sè ❤ú✉ ❤↕♥✳ ❱➻ ✈➟②
trace R(f ) =

m(π)trace π(f ),
ˆ
π∈G

tr♦♥❣ ✤â Gˆ ❧➔ ✤è✐ ♥❣➝✉ ✉♥✐t❛ ❝õ❛ ●✱ m(π) ❧➔ ❜ë✐ sè ❝õ❛ π ✈➔ trace π(f ) ❧➔ ✈➳t ❝õ❛
t♦→♥ tû π(f ) = G f (x)π(x)dx. ❱➻ ✈➟②✱ t❛ ❝â ❝æ♥❣ t❤ù❝
vol(Γγ \Gγ ) Oγ (f ) =

m(π)trace π(f ).
ˆ
π∈G

[γ]

▲÷✉ þ r➡♥❣ tr♦♥❣ ✈➳ tr→✐ ✭✈➳ ❤➻♥❤ ❤å❝✮ t❤ø❛ sè ✤➛✉ t✐➯♥ ♣❤ö t❤✉ë❝ ✈➔♦ Γ ♥❤÷♥❣
❦❤æ♥❣ ♣❤ö t❤✉ë❝ ✈➔♦ f tr♦♥❣ ❦❤✐ ✤â t❤ø❛ sè t❤ù ❤❛✐ ❧↕✐ ♣❤ö t❤✉ë❝ ✈➔♦ f ♠➔
❦❤æ♥❣ ♣❤ö t❤✉ë❝ ✈➔♦ Γ✳ ❚÷ì♥❣ tü ❝❤♦ ✈➳ ♣❤↔✐ ✭✈➳ ♣❤ê✮ ❝õ❛ ❝æ♥❣ t❤ù❝✳ P❤➙♥
♣❤è✐ Oγ (f ) ✈➔ trace π(f ) ❧➔ ❜➜t ❜✐➳♥ t❤❡♦ ♥❣❤➽❛ ❜➜t ❜✐➳♥ ❞÷î✐ ❧✐➯♥ ❤ñ♣ ❝õ❛ f ❜ð✐
♠ët ♣❤➛♥ tû ❝õ❛ ●✳
✷✳✷ ❈æ♥❣ t❤ù❝ tê♥❣ P♦✐ss♦♥

❳➨t tr÷í♥❣ ❤ñ♣ q✉❡♥ t❤✉ë❝ G = R, Γ = Z✱ ❣✐↔ sû r➡♥❣ f ∈ Cc∞(R)✱ ❝❤♦ t♦→♥
tû t➼❝❤ ❝❤➟♣ R(f ) tr➯♥ L2(T ) = L2(Z\R)
R(f )φ(x) =

f (y − x)φ(y)dy

f (y)φ(x + y)dy =
R

R

✶✼

❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R)

❍♦➔♥❣ ❚❤à ❉✉♥❣

f (y + n − x)φ(y)dy =

=

Kf (x, y)φ(y)dy.

T n∈Z

T

tr♦♥❣ ✤â Kf (x, y) = f (y + n − x) ∈ C ∞(T × T )✱ t❛ ❝â t❤➸ t➼♥❤ ✈➳t ❝õ❛ R(f )
n∈Z
❜➡♥❣ ❤❛✐ ❝→❝❤✳
trace R(f ) =

Kf (x, x)dx =
T

f (n).
n∈Z

▼➦t ❦❤→❝✱ t❛ ❝â t❤➸ ❝❤➨♦ ❤â❛ R(f ) sû ❞ö♥❣ ❝ì sð trü❝ ❝❤✉➞♥ en = e2πin, n ∈ Z✱
R(f ) = fˆ(n)en ✭fˆ ❧➔ ❜✐➳♥ ✤ê✐ ❋✉r✐❡r ❝õ❛ f ✮✳ ❉♦ ✤â
fˆ(n).

trace R(f ) =
n∈Z

❱➻ ✈➟②✱ t❛ ❝â ❝æ♥❣ t❤ù❝ tê♥❣ P♦✐ss♦♥
fˆ(n).

f (n) =
n∈Z

n∈Z

✷✳✸ ❇✐➳♥ ✤ê✐ ❝æ♥❣ t❤ù❝ ✈➳t t❤❡♦ t➼❝❤ ♣❤➙♥ q✉ÿ ✤↕♦

❑➼ ❤✐➺✉ G = SL(2, R), Γ = SL(2, Z) ✈➔ ❍ ❧➔ ♥❤â♠ ❝♦♥ ♥ë✐ s♦✐ ❝õ❛ ♥â ✭tù❝ ❧➔
H = SL(2, R) tr♦♥❣ tr÷í♥❣ ❤ñ♣ ✤➦❝ tr÷♥❣ t➛♠ t❤÷í♥❣ ❤♦➦❝ H = SO(2, R) tr♦♥❣
tr÷í♥❣ ❤ñ♣ ❦❤æ♥❣ t➛♠ t❤÷í♥❣✮✳ ❳➨t ①✉②➳♥ ❡❧❧✐♣t✐❝ T = SO(2, R) ✈➔ κ ❧➔ ♠ët
✤➦❝ tr÷♥❣ ♥ë✐ s♦✐ t÷ì♥❣ ù♥❣ ✈î✐ ♥❤â♠ ❝♦♥ ♥ë✐ s♦✐ ❍ ❝õ❛ ●✳ ❚❛ ❝â κ = 1 ♥➳✉
H = SL(2, R) ✈➔ κ = −1 ♥➳✉ H = SO(2, R)✳
●å✐ B ❧➔ ♥❤â♠ ❝♦♥ ❇♦r❡❧ ❝õ❛ ● ❝❤ù❛ ❚✱ B ❣ç♠ t➜t ❝↔ ❝→❝ ♠❛ tr➟♥ t❛♠ ❣✐→❝
tr➯♥ tr♦♥❣ SL(2, R) ❝â ❞↕♥❣
a b
0 a−1

.

❑➼ ❤✐➺✉
(1 − γ −α ),

∆B (γ) =
α>0

tr♦♥❣ ✤â t➼❝❤ ✤÷ñ❝ ❧➜② tr➯♥ ❝→❝ ♥❣❤✐➺♠ ❞÷ì♥❣ ①→❝ ✤à♥❤ ❜ð✐ ❇✳ ❈❤å♥ ♥❤â♠ ❝♦♥
❇♦r❡❧ BH = B tr♦♥❣ ❍ ❝❤ù❛ TH = T t÷ì♥❣ t❤➼❝❤ ✈î✐ ✤➥♥❣ ❝➜✉ ❥✿ TH T ✳
▼ët κ ✲ t➼❝❤ ♣❤➙♥ q✉ÿ ✤↕♦ ✤è✐ ✈î✐ ♣❤➛♥ tû ❝❤➼♥❤ q✉② γ ∈ T ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐✿
κ(x)f (x−1 γx)dx˙

Oγκ (f ) =
T \G

✶✽


ồ ừ ổ tự t tr SL(2, R)



= 1 t q t q ờ ữủ
SO (f )
õ ừ LG LG = G WR = P GL(2, C) WR
ữỡ tỹ õ LH = H WR ừ t tổ ừ t õ
ừ ởt tỷ ỷ ỡ tr LG
P ú ữủ ừ L H tr L G ởt
G
s ừ õ
ỗ : L H L G rở tỹ ừ H
ỗ t tr WR
tr H

sỷ õ ởt ú ữủ : L H L G õ
t ợ ở (G, H, ) ởt trữ G,H ừ T ợ t t s

ởt số ố ợ ộ rớ r tr G õ tỗ t ởt
f H tờ ủ t t ừ số ố ợ ộ rớ r tr H s
= j(H ) q tr T

SOH (f H ) = G
H (H , )O (f )

ợ G
H (H , ) tứ số ổ tự
1 1
q(G)+q(H)
G
G,H B ( 1 ).BH (H
) .
H (H , ) = (1)

P ờ f f H ừ số õ t ữủ rở tt
tr Cc(G) ữớ t rở tữỡ ự H ồ
õ ố ợ tt tỷ ỷ ỡ q tứ
số ố ợ
ỵ sỷ õ ởt ú ữủ : L H L G õ

t tứ số G
H (H , ) s ợ t f Cc (G) tỗ t ởt

f H Cc (H) ợ

SOH (f H ) = G
H (H , )O (f )

H ừ q ỷ ỡ
SOH (f H ) = 0




❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R)

❍♦➔♥❣ ❚❤à ❉✉♥❣

❦❤✐ γH ❦❤æ♥❣ ❧➔ ❞↕♥❣ ❝❤✉➞♥✳

❑❤✐ κ = 1 t❤➻ ♥❤â♠ ❝♦♥ ♥ë✐ s♦✐ ❝õ❛ ● ❧➔ ❝❤➼♥❤ ♥â tù❝ ❧➔ H = SL(2, R) ✈➔ κ✲
t➼❝❤ ♣❤➙♥ q✉ÿ ✤↕♦ ❝❤➼♥❤ ❧➔ t➼❝❤ ♣❤➙♥ q✉ÿ ✤↕♦ ê♥ ✤à♥❤✳ ✣➙② ❧➔ tr÷í♥❣ ❤ñ♣ t➛♠
t❤÷í♥❣✳
❙❛✉ ✤➙② t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❝ö t❤➸ ✤à♥❤ ❧þ ♥➔② ✤è✐ ✈î✐ SL(2, R) ❦❤✐ κ = −1 ✭♥❤â♠
❝♦♥ ♥ë✐ s♦✐ ❝õ❛ ● ❧➔ H = SO(2, R)✮ t❤æ♥❣ q✉❛ ✈✐➺❝ sû ❞ö♥❣ ❞→♥❣ ✤✐➺✉ t✐➺♠ ❝➟♥
❝õ❛ t➼❝❤ ♣❤➙♥ q✉ÿ ✤↕♦✳ ●✐↔ sû f ❧➔ ❤➔♠ trì♥ ❝â ❣✐→ ❝♦♠♣❛❝t tr➯♥ ●✱ ❦❤✐ ✤â t➼❝❤
♣❤➙♥ q✉ÿ ✤↕♦ ❝õ❛ f tr➯♥ q✉ÿ ✤↕♦ ❝õ❛ γ ∈ G ❧➔
f (x−1 γx)dx,
˙

Oγ (f ) =
Gγ \G

❈❤ó þ r➡♥❣ t➼❝❤ ♣❤➙♥ ♥➔② ♣❤ö t❤✉ë❝ ✈➔♦ sü ❧ü❛ ❝❤å♥ ✤ë ✤♦ ❍❛❛r tr➯♥ ● ✈➔ Gγ ✳
●✐↔ sû t❤➯♠ r➡♥❣ f ❧➔ ❤➔♠ ❑✲t➙♠ tù❝ ❧➔ f (xk) = f (kx)✱ ✈î✐ ♠å✐ k ∈ K ✈➔ x ∈ G
❈❤ó♥❣ t❛ s➩ ♥❣❤✐➯♥ ❝ù✉ ❞→♥❣ ✤✐➺✉ t✐➺♠ ❝➟♥ ❝õ❛ Oγ (f ) tr♦♥❣ ❤❛✐ tr÷í♥❣ ❤ñ♣✿

✷✳✸✳✶ ❚r÷í♥❣ ❤ñ♣ γ ❝â ❞↕♥❣ ✤÷í♥❣ ❝❤➨♦ ❦❤✐ γ → 1

γ=

✈î✐ ab = 1.

a 0
0 b

❱î✐ ♠é✐ x ∈ G t❛ ❝â ♣❤➙♥ t➼❝❤ ■✇❛s❛✇❛ ❝õ❛ x ❧➔ x = ank✱ tr♦♥❣ ✤â
a=

y 0
0 y −1

;n=

1 t
0 1

; k = r(θ)

✈î✐ y ∈ R+, t ∈ R, θ ∈ [0, 2π).

❑❤✐ ✤â x−1γx = (ank)−1γ(ank) = k−1n−1a−1γ ank✳ ❱➻ f ❧➔ ❤➔♠ ❑✲t➙♠ ♥➯♥
f (k −1 n−1 a−1 γ ank) = f (n−1 a−1 γ an).

▼➦t ❦❤→❝ a, γ ✤➲✉ ❧➔ ❝→❝ ♠❛ tr➟♥ ❞↕♥❣ ✤÷í♥❣ ❝❤➨♦ ♥➯♥ ❝❤ó♥❣ ❣✐❛♦ ❤♦→♥✱ tù❝ ❧➔
n−1 a−1 γ an = n−1 γa−1 an = n−1 γ n.

❉♦ ✤â ✤è✐ ✈î✐ ❝→❝❤ ❝❤å♥ ✤ë ✤♦ ❍❛❛r ❝❤✉➞♥✱ t❛ ❝â
f (n−1 γn)dn.

Oγ (f ) =
N

✷✵


❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R)

❍♦➔♥❣ ❚❤à ❉✉♥❣

❍ì♥ ♥ú❛✱
n−1 γn =

1 t
0 1

1 −t
0 1

a 0
0 b

a (b − a)t
0
b

=

.

❱➻ ✈➟②
Oγ (f ) =

f

a (b − a)t
0
b

dt.

R

✣➦t ∆(γ) = |a − b| t❤➻ ❤➔♠

h(γ) = ∆(γ)Oγ (f ) = |a − b|

a (b − a)t
0
b

f

dt

R

t❤→❝ tr✐➸♥ tî✐ ♠ët ❤➔♠ trì♥ tr➯♥ A ✭♥❤â♠ ❝→❝ ♠❛ tr➟♥ ❝â ❞↕♥❣ ✤÷í♥❣ ❝❤➨♦✮✳

✷✳✸✳✷ ❚r÷í♥❣ ❤ñ♣ γ = r(θ) ❦❤✐ θ → 0
❱î✐ x ∈ G✱ t❛ ①➨t ♣❤➙♥ t➼❝❤ ❈❛rt❛♥ ❝õ❛ x ❧➔ x = k1ak2✱ tr♦♥❣ ✤â
k1 = r(α); a =

y 0
0 y −1

; k2 = r(β)

✈î✐ y ∈ R+, α, β ∈ [0, 2π).

❑❤✐ ✤â
x−1 γx = (k1 ak2 )−1 γ(k1 ak2 ) = k2−1 a−1 k1−1 γ k1 ak2 .

❱➻ f ❧➔ ❤➔♠ ❑✲t➙♠ ♥➯♥
f (k2−1 a−1 k1−1 γ k1 ak2 ) = f (a−1 k1−1 γ k1 a).

▼➦t ❦❤→❝ k1−1γ k1 = r(α)−1r(θ)r(α) = r(−α + θ + α) = r(θ)✱ ♥➯♥
f (a−1 k1−1 γ k1 a) = f (a−1 γ a).

❍ì♥ ♥ú❛✱
cosθ sin θ
y −1 0
− sin θ cosθ
0 y
cosθ
y −2 sin θ
.
2
−y sin θ

cosθ

y 0
0 y −1

a−1 γa =
=

❉♦ ✤â
Or(θ) (f ) = c.F (sinθ),

✷✶

=

y −1 cosθ y −1 sin θ
−y sin θ y cosθ

y 0
0 y −1


❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R)

❍♦➔♥❣ ❚❤à ❉✉♥❣

✈î✐ ❤➡♥❣ sè ❝ ♣❤ö t❤✉ë❝ ✈➔♦ ✈✐➺❝ ❝❤å♥ ✤ë ✤♦ ❍❛❛r ✈➔
∞

tr♦♥❣ ✤â a(λ) =

❚❛ ❝â

√

1 − λ2

1
∞

∞

1
∞

=

a(λ)
tλ
dt −
−t−1 λ a(λ)

f
1
∞

a(λ)
tλ
dt −
−1
−t λ a(λ)

f
1

∞

=

f
1
∞

=
1

1

dt
a(λ)
tλ
(−1) 2
−t−1 λ a(λ)
t

f

a(λ)
tλ
d(t−1 )
−t−1 λ a(λ)

∞
1
∞

dt
t2

a(λ) (t−1 )−1 λ
d(t−1 )
−t−1 λ
a(λ)

f
1
0

a(λ)
tλ
dt +
−t−1 λ a(λ)

a(λ)
tλ
)(t2 − 1)
−1
−t λ a(λ)

f
1

a(λ)
tλ
dt +
−t−1 λ a(λ)

f

f

∞

a(λ)
tλ
1dt −
−t−1 λ a(λ)

f

=

✳

dt
a(λ)
tλ
|t − t−1 | =
−1
−t λ a(λ)
t

f
=

dt
,
t

1

∞

F (λ) =

a(λ)
tλ
−t−1 λ a(λ)

|t − t−1 |.f

F (λ) =

f
1

a(λ) −t−1 λ
dt
tλ
a(λ)

1

a(λ) −t−1 λ
dt.
tλ
a(λ)

(−1)f
0

❈❤ó þ r➡♥❣ f ❧➔ ❤➔♠ ❑✲t➙♠ ♥➯♥ ✈î✐ ❛✱❜✱❝ ❜➜t ❦➻✱ t❛ ❝â
f

a b
c a

b −a
a −c

=f

❉♦ ✤â

0 1
−1 0

∞

a(λ)
tλ

−1
−t λ a(λ)

ε(t − 1)f

F (λ) =

b −a
a −c

0 1
−1 0

=f

dt,

✈î✐ ε(x) = sign(x).

0

✣➸ ♥❣❤✐➯♥ ❝ù✉ t✐➺♠ ❝➟♥ ❝õ❛ ♥â ❦❤✐ λ → 0 t❛ ①➨t
∞

ε(t − 1)f

A(λ) =

a(λ) tλ
0

a(λ)

dt.

0

❚❤❡♦ ❝æ♥❣ t❤ù❝ ❚❛②❧♦r✲▲❛❣r❛♥❣❡✱ t❛ ❝â
F (λ) = A(λ) + λB(λ),

tr♦♥❣ ✤â

∞

a(λ)
tλ
−1
−t λ a(λ)

ε(t − 1)g

B(λ) =
0

✷✷

dt
,
t

=f

a −c
−b a

.


❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R)

❍♦➔♥❣ ❚❤à ❉✉♥❣

✈î✐ ❤➔♠ trì♥ g ♥➔♦ ✤â ❝â ❣✐→ ❝♦♠♣❛❝t t❤❡♦ ❜✐➳♥ ♣❤➼❛ tr➯♥ ❜➯♥ ♣❤↔✐ ✈➔ ❝â O(u)−1
♣❤➙♥ r➣ t❤❡♦ ❜✐➳♥ ❞÷î✐ ❜➯♥ tr→✐ s❛♦ ❝❤♦ t➼❝❤ ♣❤➙♥ ❧➔ ❤ë✐ tö t✉②➺t ✤è✐✳
❈❤ó þ r➡♥❣
∞

A(λ) = |λ|−1

1 ε(λ)u
0
1

f

du − 2f

1 0
0 1

+ o(λ).

0

❚ø B(λ) ❝â ✤ë t➠♥❣ ❦❤æ♥❣ q✉→ ❧♦❣❛r✐t✱ t❛ t❤➜② r➡♥❣ ❝→❝ ❤➔♠ ❝❤➤♥
G(λ) = |λ|(F (λ) + F (−λ))

✈➔
H(λ) = λ(F (λ) − F (−λ))

t❤→❝ tr✐➸♥ tî✐ ❤➔♠ ❧✐➯♥ tö❝ t↕✐ λ = 0✳
✣➸ ❝❤➼♥❤ ①→❝ ❤â❛ ❞→♥❣ ✤✐➺✉ t✐➺♠ ❝➟♥✱ t❛ q✉❛♥ s→t ❦➽ ❤ì♥ sè ❤↕♥❣
∞

a(λ)
tλ
−1
−t λ a(λ)

ε(t − 1)g

B(λ) =

dt
.
t

0

❚✉② ♥❤✐➯♥✱ ❤✐➺✉ ❝õ❛ ❤❛✐ ❜✐➸✉ t❤ù❝ ♠➔ ♣❤➛♥ ❝❤➼♥❤ ❝õ❛ ❝❤ó♥❣ t÷ì♥❣ ✤÷ì♥❣ ✈î✐
1 0

ln(|λ|−1 )g
s❛✐ ❦❤→❝ ❜✐➸✉ t❤ù❝ ❧✐➯♥ tö❝✳ ❉♦ ✤â ❇ ❧➔ ❧✐➯♥ tö❝✳ ❑❤→✐ q✉→t
0 1
❤â❛ q✉→ tr➻♥❤ ♥➔② t❛ t❤✉ ✤÷ñ❝ ❦❤❛✐ tr✐➸♥ t✐➺♠ ❝➟♥ ♠ô ❞÷î✐ ❞↕♥❣ ♥❤÷ s❛✉
N

(an |λ|−1 + bn )λ2n + o(λ2N )

G(λ) =

✈➔

n=0
N

hn λ2n + o(λ2N ).

H(λ) =
n=0

❉♦ ✤â H(λ) ❧➔ ❤➔♠ trì♥✳ ❱➻ ✈➟②✱ ❝â ♠ët ❤➔♠ trì♥ ❤ tr➯♥ T = H s❛♦ ❝❤♦
h(γ) = ∆(γ)(Oγ (f ) − Oω(γ) (f )),

✈î✐ γ = r(θ) ∈ T ✈➔ ∆(r(θ)) = −2isinθ✳
✷✳✹ P❤➨♣ ❝❤✉②➸♥ ✈➳ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t

❚❛ t❤➜② r➡♥❣ t÷ì♥❣ ù♥❣ f → f H ❦❤æ♥❣ ♣❤↔✐ ❧➔ ♠ët →♥❤ ①↕ ✈➻ f H ①→❝ ✤à♥❤
❦❤æ♥❣ ❞✉② ♥❤➜t✳ P❤➨♣ ❝❤✉②➸♥ ❤➻♥❤ ❤å❝ f → f H ❧➔ ✤è✐ ♥❣➝✉ ❝õ❛ ♣❤➨♣ ❝❤✉②➸♥
✷✸

ồ ừ ổ tự t tr SL(2, R)



ố ợ t ý t q ữủ ừ H tữỡ
ự ợ ởt tỷ G tr õ ừ ừ G ữ s
ởt t số s ố ợ H t ởt t số s ố
ợ G tr õ ú : LH LG
t õ ừ t q ữủ ừ H tữỡ ự
ợ õ ừ ừ G tữỡ ự ợ õ õ t t
rộ t số ổ t ủ ợ
ỵ ỗ t ởt
1,

:

s tỷ G tr õ ừ ỷ õ
G =

(),


ởt tữỡ ự G ố ừ ờ ồ
trace G (f ) = trace (f H ).

t ởt t số s
: LR L G

S t õ tr G ừ (WR) ú ỵ r ợ t s S

ởt õ ở s õ s tr LG t õ tổ H ừ
s tr G ừ
tỗ t ởt ú : LH LG t tỗ t t số s ố
ợ õ ởt t số s
) tr õ S0 t tổ ừ S
S = S/(S0 ì Z(G)
t ừ L G sỷ t ởt t ủ ừ õ ở s ổ
Z(G)
tữỡ ữỡ ộ ởt ú : LH LG
t ởt t số : WR LG t tổ ừ s S õ H s
ủ ợ H õ ủ ừ tứ số tr (LH) ởt õ