✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆ ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ❍❖⑨◆● ❚❍➚ ❉❯◆● ❱➌ ❍➐◆❍ ❍➴❈ ❈Õ❆ ❈➷◆● ❚❍Ù❈ ❱➌❚ ❚❘➊◆ SL (2, R) ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❑❍❖❆ ❍➴❈ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚❖⑩◆ ●■❷■ ❚➑❈❍ ▼➣ số ữớ ữợ ❉■➏P ❍⑨ ◆❐■✲ ✷✵✶✹ ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ▼ð ✤➛✉ ✷ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✸ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ❙ì ❧÷đ❝ ✈➲ 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♥❣ ❤đ♣ γ ❝â ❞↕♥❣ ✤÷í♥❣ ❝❤➨♦ ❦❤✐ γ → ✷✳✸✳✷ ❚r÷í♥❣ ❤đ♣ γ = r(θ) ❦❤✐ θ → ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹ P❤➨♣ ❝❤✉②➸♥ ✈➳ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✺ ❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✺ ✺ ✻ ✼ ✼ ✼ ✽ ✽ ✾ ✶✶ ✶✶ ✶✷ ✶✷ ✶✸ ✶✹ ✶✺ ✶✺ ✶✻ ✶✻ ✶✼ ✶✽ ✷✵ ✷✶ ✷✸ ✷✻ ✷✼ ✷✽ ▲í✐ ❝↔♠ ì♥ ❍♦➔♥ t❤➔♥❤ ✤÷đ❝ ❧✉➟♥ ✈➠♥ ♥➔②✱ ♥❣♦➔✐ sü ♥é ❧ü❝ ❝õ❛ ❜↔♥ t❤➙♥✱ tỉ✐ ✤➣ ♥❤➟♥ ✤÷đ❝ sü ❝❤➾ ❜↔♦✱ ❣✐ó♣ ✤ï tø ♥❤✐➲✉ ♣❤➼❛ ❝õ❛ ❝→❝ t❤➛② ❣✐→♦✱ ❝æ ❣✐→♦✱ ❣✐❛ ✤➻♥❤ ✈➔ ❜↕♥ ❜➧✳ ✣➦❝ ❜✐➺t tæ✐ ①✐♥ ❜➔② tä ỏ t ỡ s s tợ ữớ t ●❙✳❚❙❑❍✳ ✣é ◆❣å❝ ❉✐➺♣✱ ♥❣÷í✐ ✤➣ trü❝ t✐➳♣ tr✉②➲♥ t❤ư tự qt ữợ ự t t ữợ tổ t ổ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ t❤➛②✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ❦❤♦❛ ❚♦→♥ ✲ ❈ì ✲ ❚✐♥ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ tü ♥❤✐➯♥ ✲ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✱ ♥❤ú♥❣ ♥❣÷í✐ ✤➣ trü❝ t✐➳♣ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ tr÷í♥❣✳ ❈↔♠ ì♥ t♦➔♥ t❤➸ ữớ t õ õ ỵ ❣✐ó♣ ✤ï✱ ✤ë♥❣ ✈✐➯♥ tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❉♦ t❤í✐ ❣✐❛♥ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ ♥❤✐➲✉✱ ❦✐➳♥ t❤ù❝ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❦❤✐ ❧➔♠ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❤↕♥ ❝❤➳ ✈➔ s❛✐ sât✳ ❑➼♥❤ ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ þ ❦✐➳♥ ✤â♥❣ ❣â♣ ❝õ❛ ❝→❝ t❤➛② ❝æ ✈➔ ❜↕♥ ỗ ữủ ❤ì♥✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✳ ❍➔ ◆ë✐✱ ♥❣➔② ✷✵ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✹ ❍å❝ ✈✐➯♥ ❍♦➔♥❣ ❚❤à ❉✉♥❣ ✷ ▼ð ✤➛✉ ●✐↔✐ t➼❝❤ ✤✐➲✉ ❤á❛ tr➯♥ ♥❤â♠ ▲✐❡ ♥â✐ ❝❤✉♥❣ ❞➝♥ ✤➳♥ ✈✐➺❝ ♣❤➙♥ t➼❝❤ ♠ët ❜✐➸✉ ❞✐➵♥ ❜➜t ❦ý r❛ tê♥❣ ❝→❝ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉②✳ ❇✐➸✉ ❞✐➵♥ ❝❤➼♥❤ q✉② ❝õ❛ ♥❤â♠ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ t❤÷ì♥❣ ❝õ❛ ♥â t❤❡♦ ♥❤â♠ ❝♦♥ rí✐ r↕❝ ✤â♥❣ ✈❛✐ trá q trồ ỵ tt t t ✤à♥❤ ♥❣❤➽❛ ❤➔♠ s✉② rë♥❣✮✱ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❧ỵ♣ t÷ì♥❣ ✤÷ì♥❣ ❝õ❛ ❜✐➸✉ ❞✐➵♥✳ ❱➳t ❝õ❛ ♣❤➛♥ rí✐ r↕❝ ❝õ❛ ❜✐➸✉ ❞✐➵♥ ❝❤➼♥❤ q✉② ✤÷đ❝ ✈✐➳t t❤➔♥❤ ❝❤✉é✐ ❝→❝ ✈➳t ❝õ❛ ❜✐➸✉ ❞✐➵♥ ♥❤å♥ ✈➔ ❞♦ ✤â ❧➔ tê♥❣ ❝→❝ t➼❝❤ ♣❤➙♥ q✉ÿ ✤↕♦ t÷ì♥❣ ù♥❣✳ ❈ỉ♥❣ t❤ù❝ ✈➳t ❦❤→ ♣❤ù❝ t↕♣ ♥❤÷♥❣ ❦❤✐ ❤↕♥ ❝❤➳ ①✉è♥❣ ♥❤â♠ ❝♦♥ ♥ë✐ s♦✐ t❤➻ ❦➳t q✉↔ trð ♥➯♥ t÷ì♥❣ ✤è✐ ✤ì♥ ❣✐↔♥✳ ✣➲ t➔✐ ✤÷đ❝ ✤➦t r❛ ❧➔✿ ❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL (2, R)✳ ◆ë✐ ❞✉♥❣ ỗ ữỡ ã ữỡ õ t➢t ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❙ì ❧÷đ❝ ❝➜✉ tró❝ ❝õ❛ SL(2, R)✳ ✕ ❇✐➸✉ ❞✐➵♥ ❝õ❛ SL(2, R)✳ ✕ ❚❤❛♠ sè ▲❛♥❣❧❛♥❞s ❝❤♦ SL(2, R)✳ ✕ ◆❤â♠ ❝♦♥ s SL(2, R) ã ữỡ 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❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ❦❤ỉ♥❣ ♥❤✐➲✉✱ ❦✐➳♥ t❤ù❝ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❦❤✐ ❧➔♠ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❤↕♥ ❝❤➳ ✈➔ s❛✐ sât✳ ❚→❝ ❣✐↔ ♠♦♥❣ ữủ sỹ õ ỵ ỳ ỵ qỵ t ổ t❤➔♥❤ ❝↔♠ ì♥✦ ❍➔ ◆ë✐✱ ♥❣➔② ✷✵ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✹ ❍å❝ ✈✐➯♥ ❍♦➔♥❣ ❚❤à ❉✉♥❣ ✹ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ❙ì ❧÷đ❝ ✈➲ SL (2, R) SL (2, R) ❜➡♥❣ ✶✿ ❧➔ ♥❤â♠ ❝→❝ ♠❛ tr➟♥ ì tr trữớ số tỹ R ợ ✤à♥❤ t❤ù❝ SL (2, R) = a b |a, b, c, d ∈ R; ad − bc = c d ❚❛ ❦➼ ❤✐➺✉ G = SL (2, R)✱ ✤↕✐ sè ▲✐❡ ❝õ❛ G ❧➔ g0 = sl (2, R) ỗ tr tỹ ì õ t õ ỡ s ỗ ♠❛ tr➟♥✿ H= ;X = −1 ;Y = 0 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➼♥❤ G tr H ữủ ữ s ợ ♠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 = ♥➯♥ 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 = ✈➔ 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 − tr (g) ± λ − tr (g) λ + = ⇔ λ = ◆➳✉ |tr (g) | < t❤➻ g ✤÷đ❝ ❣å✐ ❧➔ ❡❧❧✐♣t✐❝✳ − ◆➳✉ |tr (g) | = t❤➻ g ✤÷đ❝ ❣å✐ ❧➔ ♣❛r❛❜♦❧✐❝✳ − ◆➳✉ |tr (g) | > t❤➻ g ✤÷đ❝ ❣å✐ ❧➔ ❤②♣❡r❜♦❧✐❝✳ − ✶✳✶✳✷ P❤➙♥ t➼❝❤ ■✇❛s❛✇❛ ✈➔ ♣❤➙♥ t➼❝❤ ❈❛rt❛♥ ❝õ❛ G P❤➙♥ t➼❝❤ ■✇❛s❛✇❛ ❝õ❛ G ❧➔ ♣❤➙♥ t➼❝❤ ❝â ❞↕♥❣ G = KAN ✈ỵ✐ cosθ sin θ − sin θ cosθ K= uθ = exp θ(X − Y ) = A= at = exp tH = et 0 e−t N= ns = exp sX = s | | | t∈R , s∈R ❚❛ ❝â K ∼= S 1, A ∼= R ✈➔ N ∼= R✳ ❈ư t❤➸ ✈ỵ✐ ♠é✐ g = t➼❝❤ ■✇❛s❛✇❛ ❝õ❛ ♥â ❧➔ g = uθ atns✱ tr♦♥❣ ✤â a − ic , et = eiθ = √ 2 a +c θ ∈ [0, 2π) , a b c d ∈G t❤➻ ♣❤➙♥ ab + cd a2 + c , s = √ a2 + c ❍♦➔♥ t♦➔♥ t÷ì♥❣ tü✱ G ụ ữủ t ữợ G = AN K ✈➔ ❞↕♥❣ ♥➔② ❝ơ♥❣ ✤÷đ❝ ❣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❛ ❝ơ♥❣ ❝â ♥❤â♠ t❤÷ì♥❣ Gγ \G = {Gγ x | x ∈ G} ✣à♥❤ ởt tr G \G ữủ ❣å✐ ❧➔ G ✲ ❜➜t ❜✐➳♥ ♣❤↔✐ ♥➳✉ µ(Ax) = µ(A) ✈ỵ✐ ♠å✐ t➟♣ ❇♦r❡❧ ❆ tr♦♥❣ Gγ \G ✈➔ ♠å✐ x ∈ G✳ ✣ë ✤♦ G ✲ ❜➜t ❜✐➳♥ tr→✐ ❝ơ♥❣ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❤♦➔♥ t♦➔♥ t÷ì♥❣ tü✳ ▼ët ✤ë ✤♦ µ tr➯♥ G ❣å✐ ❧➔ ✤ë ✤♦ ❍❛❛r õ t ữợ t G ố ✈ỵ✐ ♣❤➙♥ t➼❝❤ ■✇❛s❛✇❛ G = AN K ✱ ♣❤➛♥ tû x ∈ G t❛ ❝â ♣❤➙♥ t➼❝❤ x = ank ✭✈ỵ✐ 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✳ ❱ỵ✐ ❤➔♠ f ①→❝ ✤à♥❤ ✈➔ ❦❤↔ t➼❝❤ tr➯♥ G✱ t❛ ❝â f (x)dx = G dk da K A f (ank)dn N ✣è✐ ✈ỵ✐ ♣❤➙♥ t➼❝❤ ❈❛rt❛♥ G = KAK ✱ ✈ỵ✐ ♠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 ✼ ❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R) ❍♦➔♥❣ ❚❤à ❉✉♥❣ ✤↕♦ ❝õ❛ γ ✤÷đ❝ ❝❤♦ ❜ð✐ ❝ỉ♥❣ t❤ù❝✿ f (x−1 γx)dx, ˙ Oγ (f ) = Gγ \G tr♦♥❣ ✤â dx˙ ❧➔ ✤ë ✤♦ G✲❜➜t ❜✐➳♥ ♣❤↔✐ tr➯♥ t❤÷ì♥❣ Gγ \G✳ ✶✳✷✳✷ ▲✐➯♥ ❤ñ♣ ê♥ ✤à♥❤ ❈❤♦ G = SL(2, R)✱ γ, γ ∈ G ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ ❤đ♣ tỗ t x G s = xγx−1 ✳ ✣è✐ ✈ỵ✐ ♣❤➛♥ tû ❝❤➼♥❤ q✉② ♥û❛ ✤ì♥ ♠↕♥❤✱ t❛ ♥â✐ r➡♥❣ γ, γ ∈ G ❧➔ ❧✐➯♥ ủ tỗ t x SL(2, C) = ac db | a, b, c, d ∈ C ; ad − bc = s❛♦ ❝❤♦ γ = xγx−1 ✳ ❈❤♦ f ∈ Cc∞(G)✱ γ ∈ G ❧➔ ♣❤➛♥ tû ❝❤➼♥❤ q✉② ♠↕♥❤✱ ❦❤✐ ✤â t➼❝❤ ♣❤➙♥ q✉ÿ ✤↕♦ ê♥ ✤✐♥❤ ❝õ❛ ❤➔♠ f ✤è✐ ✈ỵ✐ ♣❤➛♥ tû γ ✤÷đ❝ ❝❤♦ ❜ð✐ Oγ (f ) SOγ (f ) = γ ∈S(γ) ❚r♦♥❣ ✤â S(γ) ❧➔ t➟♣ ❤ñ♣ ❝→❝ ♣❤➛♥ tû ✤↕✐ ❞✐➺♥ ❝õ❛ ❝→❝ ❧ỵ♣ ❧✐➯♥ ❤đ♣ tr♦♥❣ ❧ỵ♣ ❧✐➯♥ ❤đ♣ ê♥ ✤à♥❤ ❝õ❛ γ ✳ ✶✳✷✳✸ ◆❤â♠ ❲❡✐❧ ✈➔ ♥❤â♠ ▲❛♥❣❧❛♥❞s✱ ▲✲♥❤â♠ ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ❚❛ ❦➼ ❤✐➺✉ WR ❧➔ ♥❤â♠ ❲❡✐❧ ❝õ❛ R ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿ ✲ ◆❤â♠ ❲❡✐❧ ❝õ❛ C ❧➔ WC = C× ✳ ✲ ◆❤â♠ ❲❡✐❧ ❝õ❛ R ❧➔ ♥❤â♠ ❝♦♥ ❝→❝ ♠❛ tr➟♥ tr♦♥❣ SU (2) ✤÷đ❝ s✐♥❤ ❜ð✐ z 0 z¯ ✈➔ wσ = , z ∈ C× −1 ◆❤â♠ SU (2) ❧➔ ♠ët ♥❤â♠ ❝♦♠♣❛❝t ✈ỵ✐ sè ❝❤✐➲✉ 22 ❜✐➸✉ ❞✐➵♥ ❜ð✐ ❝→❝ ♠❛ tr➟♥ ✉♥✐t❛r② ✈ỵ✐ ❝→❝ ♣❤➛♥ tû ❝â ✤à♥❤ t❤ù❝ ❜➡♥❣ ✶✱ ✤÷đ❝ ❣å✐ ❧➔ ♥❤â♠ ✉♥✐t❛r② ✤➦❝ ❜✐➺t✳ ❑➼ ❤✐➺✉ Gal(C/R) õ s rở C/R ỗ tỷ ởt tỷ tỹ ỗ ỗ t tỷ ỏ tỹ ỗ ❤đ♣ ♣❤ù❝✳ P❤➛♥ tû wσ t→❝ ✤ë♥❣ ❧✐➯♥ ❤đ♣ ♥❤÷ ❧➔ ♣❤➛♥ tû ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ tr♦♥❣ ♥❤â♠ ●❛❧(C/R) tr➯♥ Cì WR Gal(C/R) ữủ w ú ỵ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R) ❍♦➔♥❣ ❚❤à ❉✉♥❣ r➡♥❣ wσ2 = −1 ❞♦ ✤â ♠ð rë♥❣ ❝õ❛ WC = Cì Gal(C/R) rở ổ t tữớ ✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ◆❤â♠ ▲❛♥❣❧❛♥❞s✱ ❦➼ ❤✐➺✉ LF ✱ LF = WR ✱ ♥➳✉ tr÷í♥❣ ❝ì sð ❋ ❧➔ C ❤♦➦❝ R ✈➔ LF = WR × SL(2, C)✱ ♥➳✉ ❋ ♣✲❛❞✐❝✳ ❑➼ ❤✐➺✉ Gˇ ❧➔ ♥❤â♠ ▲✐❡ ♣❤ù❝ t❤✉ ❣å♥ ❝õ❛ G = SL(2, R)✱ ❦❤✐ ✤â Gˇ = P GL(2, C)✳ ◆❤â♠ ●❛❧♦✐s Gal(C/R) t→❝ ✤ë♥❣ tr➯♥ Gˇ q tỹ ỗ ữủ tt ỳ ♥❣✉②➯♥ t→❝❤✳ ◆❤â♠ ● ❧➔ t→❝❤ ♥➯♥ t→❝ ✤ë♥❣ ✤â t tữớ WR t tợ Gal(C/R) q ①↕ tü ♥❤✐➯♥ ❝õ❛ ♥â✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ ▲✲♥❤â♠ ❝õ❛ ●✱ ❦➼ ❤✐➺✉ L G = Gˇ WR ✳ ✶✳✸ ❇✐➸✉ ❞✐➵♥ ❝õ❛ SL(2, R) ✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ❈❤♦ ● ❧➔ ♠ët ♥❤â♠ ✭GL(2, R) ❤♦➦❝ SL(2, R)✮✱ ❊ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ▼ët ❜✐➸✉ ❞✐➵♥ ❝õ❛ ● tr♦♥❣ ❊ ởt ỗ tứ õ tỹ ❝➜✉ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ GL(E) ❝õ❛ ❊✳ π : G → GL(E), s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ ✈➨❝ tì v ∈ E t❤➻ →♥❤ ①↕ tø ● ✈➔♦ ❊ ①→❝ ✤à♥❤ ❜ð✐ x → π(x)v ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝✳ ❇✐➸✉ ❞✐➵♥ π ✤÷đ❝ ❣å✐ ❧➔ ❜✐➸✉ ❞✐➵♥ ✉♥✐t❛ ♥➳✉ π(x) ❧➔ ✉♥✐t❛ ✈ỵ✐ ♠å✐ x ∈ G✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✼✳ ❈❤♦ π ❜✐➸✉ ❞✐➵♥ ❝õ❛ ♥❤â♠ ● tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❊✱ ❲ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ ❊✳ ❚❛ ♥â✐ ❲ ❧➔ ●✲❜➜t ❜✐➳♥ ♥➳✉ π(x)W ⊂ W ✈ỵ✐ ♠å✐ x ∈ G✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✽✳ ▼ët ❜✐➸✉ ❞✐➵♥ π : G → GL(E) ❣å✐ ❧➔ ❜➜t ❦❤↔ q✉② ♥➳✉ ❊ ❦❤æ♥❣ ❝â ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❜➜t ❜✐➳♥ ♥➔♦ ❦❤→❝ ♥❣♦➔✐ {0} ✈➔ ❊✳ ❈❤♦ π ❧➔ ❜✐➸✉ ❞✐➵♥ ❝õ❛ ● tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❊✱ ❣✐↔ sû r➡♥❣ E= En , ✾ ❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R) ❍♦➔♥❣ ❚❤à ❉✉♥❣ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ GL(2, R) ợ ủ ỗ ữủ WR tr GL(2, C) ữ s (à1 , µ2 ) −→ ϕs1 ,m1 ,s2 ,m2 ✈ỵ✐ µi = |x|s sign(x)m i i ợ à1à2(x) = |x|2ssign(x)n+1 n tr♦♥❣ ✤â µ1µ−1 (x) = x sign(x)✳ ❚❤❛♠ sè s tữỡ ự ợ t ứ ↔♥❤ ❝õ❛ →♥❤ ①↕ ❜à ❝❤➦♥ tù❝ ❧➔ si t❤✉➛♥ ↔♦✳ σ(µ1 , µ2 ) −→ ϕs,n ✶✳✹✳✷ ❚❤❛♠ sè ▲❛♥❣❧❛♥❞s ❝❤♦ SL(2, R) ❚ø s♦♥❣ →♥❤ ❣✐ú❛ ❝→❝ ❧ỵ♣ tữỡ ữỡ ợ ủ t❤❛♠ sè ▲❛♥❣❧❛♥❞s ❝❤♦ GL(2, R) s✉② r❛ s♦♥❣ →♥❤ ỳ ợ tữỡ ữỡ õ t ❦❤↔ q✉② ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ SL(2, R) ✈➔ ❝→❝ ợ ủ ỗ ữủ ❝õ❛ WR tr♦♥❣ P GL(2, C)✳ ✲ ❚❤❛♠ sè ❤â❛ (à) ợ ủ t số õ s,m ữủ s,m,0,0 ợ à(x) = |x|ssign(x)m✳ ✲ ❚❤❛♠ sè ❤â❛ ❝❤♦ Dn± ❧➔ ❧ỵ♣ ❧✐➯♥ ❤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 ◆❤÷♥❣ 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❤ỉ♥❣ Sϕ0 ❝õ❛ ♥â ♥❤➙♥ ✈ỵ✐ t➙♠ ZGˇ ❝õ❛ Gˇ ✿ ✰ ❑❤✐ n = t❛ ❝â Sϕ = Sϕ {1, α}✳ ✰ ❑❤✐ n = ♥❤â♠ Sϕ0 ❧➔ ♠ët ①✉②➳♥ ♥❤÷♥❣ Sϕ ❧↕✐ ✤÷đ❝ s✐♥❤ ❜ð✐ ↔♥❤ ❝õ❛ α✳ n n n n n n 0 ✶✹ ❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R) ❍♦➔♥❣ ❚❤à ❉✉♥❣ ✶✳✺ ◆❤â♠ ❝♦♥ ♥ë✐ s♦✐ ❝õ❛ SL (2, R) ✣à♥❤ ♥❣❤➽❛ ✶✳✶✸✳ ◆❤â♠ ❝♦♥ ♥ë✐ s♦✐ ❍ ❝õ❛ ♥❤â♠ ● ❧➔ ♥❤â♠ tü❛ ❝❤➫ r❛ ♠➔ ▲✲♥❤â♠ L H ❧➔ t❤➔♥❤ ♣❤➛♥ ❧✐➯♥ t❤æ♥❣ ❝õ❛ t➙♠ ❤â❛ ❝õ❛ ♠ët ♣❤➛♥ tû ♥û❛ ✤ì♥ ❝õ❛ ▲✲♥❤â♠ L G✳ ❚r♦♥❣ t➜t ❝↔ ❝→❝ ✈➼ ❞ư ð tr➯♥ ♥❤ú♥❣ ✤è✐ t÷đ♥❣ tr♦♥❣ tø♥❣ ❝➦♣ ✤÷đ❝ t t ủ ữợ tỷ = iα tr♦♥❣ ❝❤✉➞♥ ❤â❛ ❝õ❛ SO(2) tr♦♥❣ SL(2, C)✳ ▲÷✉ þ r➡♥❣ ♥➳✉ σ ❧➔ ♣❤➛♥ tû ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ ❝õ❛ ♥❤â♠ ●❛❧♦✐s t❤➻ ♣❤➛♥ tû aσ = wσ(w)−1 = −1 0 −1 s✐♥❤ r❛ ♠ët ♥❤â♠ ❝♦♥ ❝➜♣ õ t ỗ t õ ợ H 1(C/R, SO(2))✳ ✣➦❝ tr÷♥❣ ❝õ❛ ✷✲♥❤â♠ ✤÷đ❝ ❣å✐ ❧➔ ✤➦❝ tr÷♥❣ ♥ë✐ s♦✐✱ ❝â ❤❛✐ ♥❤â♠ ❝♦♥ ♥ë✐ s♦✐ ❝õ❛ SL(2, R) tữỡ ự ợ trữ õ s tữỡ ự ợ trữ t tữớ ❝❤➼♥❤ SL(2, R)✱ tr♦♥❣ ❦❤✐ ✤â ♥❤â♠ ❝♦♥ ♥ë✐ s♦✐ tữỡ ự ợ trữ ổ t tữớ ❝♦♠♣❛❝t T (R) = SO(2, R)✳ ✶✳✻ ❑➳t ❧✉➟♥ ❈❤÷ì♥❣ ♥➔② ✤➣ tr➻♥❤ ❜➔② ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ð ❝❤÷ì♥❣ ✷✳ ❈→❝ ❦❤→✐ ♥✐➺♠ ♥❤÷ t→❝ ✤ë♥❣ ♣❤➙♥ t✉②➳♥ t➼♥❤✱ ♣❤➙♥ t➼❝❤ ■✇❛s❛✇❛✱ ♥❤â♠ ❝♦♥ ❞ø♥❣✱ ✤ë ✤♦ ❣✐ó♣ t❛ ❤✐➸✉ ❤ì♥ ✈➲ ❝➜✉ tró❝ ❝õ❛ SL (2, R)✳ ✣➦❝ ❜✐➺t ❦✐➳♥ t❤ù❝ ✈➲ ❜✐➸✉ ❞✐➵♥ ❝õ❛ SL (2, R)✱ t❤❛♠ sè ❤â❛ ▲❛♥❣❧❛♥❞s ✈➔ ♥❤â♠ ❝♦♥ ♥ë✐ s♦✐ s➩ ✤â♥❣ ✈❛✐ trá ❝❤õ ❝❤èt tr♦♥❣ ❝→❝❤ ①➙② ❞ü♥❣ ✈➔ ❜✐➳♥ ✤ê✐ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL (2, R)✳ ✶✺ ❈❤÷ì♥❣ ✷ ❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝ỉ♥❣ 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) ✈ỵ✐ g ∈ G, x ∈ Γ\G Ù♥❣ ✈ỵ✐ ❜✐➸✉ ❞✐➵♥ ✉♥✐t❛ ❝õ❛ ♥❤â♠ ● t❛ ❝â ❜✐➸✉ ❞✐➵♥ t÷ì♥❣ ù♥❣ ❝õ❛ ✤↕✐ sè ❍❛❛r L1 (G) ✭✤è✐ ✈ỵ✐ 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➼❝❤ ♣❤➙♥ ✈ỵ✐ ❤↕t ♥❤➙♥ trì♥ f (x−1 γg) Kf (x, g) = γ∈Γ R(f ) ❧➔ ❧ỵ♣ ✈➳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♥ ✤÷đ❝ ❣å✐ ❧➔ ❝ỉ♥❣ 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 ❤✐➺♥ ✈ỵ✐ ♠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()trace (f ) G [] ữ ỵ r tr♦♥❣ ✈➳ tr→✐ ✭✈➳ ❤➻♥❤ ❤å❝✮ t❤ø❛ sè ✤➛✉ t✐➯♥ ♣❤ư t❤✉ë❝ ✈➔♦ Γ ♥❤÷♥❣ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ f tr♦♥❣ ❦❤✐ ✤â t❤ø❛ sè t❤ù ❤❛✐ ❧↕✐ ♣❤ö t❤✉ë❝ ✈➔♦ f ♠➔ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ Γ✳ ❚÷ì♥❣ tü ❝❤♦ ✈➳ ♣❤↔✐ ✭✈➳ ♣❤ê✮ ❝õ❛ ❝æ♥❣ t❤ù❝✳ P❤➙♥ ♣❤è✐ Oγ (f ) ✈➔ trace π(f ) ❧➔ ❜➜t ❜✐➳♥ t t ữợ ủ f ♠ët ♣❤➛♥ tû ❝õ❛ ●✳ ✷✳✷ ❈æ♥❣ t❤ù❝ tê♥❣ P♦✐ss♦♥ ❳➨t tr÷í♥❣ ❤đ♣ 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 ❞ư♥❣ ❝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♥❣ ❤đ♣ ✤➦❝ tr÷♥❣ t➛♠ t❤÷í♥❣ ❤♦➦❝ H = SO(2, R) tr♦♥❣ tr÷í♥❣ ❤đ♣ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣✮✳ ❳➨t ①✉②➳♥ ❡❧❧✐♣t✐❝ T = SO(2, R) ✈➔ ởt trữ s tữỡ ự ợ ♥❤â♠ ❝♦♥ ♥ë✐ s♦✐ ❍ ❝õ❛ ●✳ ❚❛ ❝â κ = ♥➳✉ H = SL(2, R) ✈➔ κ = −1 ♥➳✉ H = SO(2, R)✳ ●å✐ B ❧➔ ♥❤â♠ r ự B ỗ tt ❝→❝ ♠❛ tr➟♥ t❛♠ ❣✐→❝ tr➯♥ tr♦♥❣ SL(2, R) ❝â ❞↕♥❣ a b a−1 ❑➼ ❤✐➺✉ (1 − γ −α ), ∆B (γ) = α>0 tr♦♥❣ ✤â t➼❝❤ ✤÷đ❝ ❧➜② tr➯♥ ❝→❝ ♥❣❤✐➺♠ ❞÷ì♥❣ ①→❝ ✤à♥❤ ❜ð✐ ❇✳ ❈❤å♥ ♥❤â♠ ❝♦♥ ❇♦r❡❧ BH = B tr♦♥❣ ❍ ❝❤ù❛ TH = T tữỡ t ợ TH T ✳ ▼ët κ ✲ t➼❝❤ ♣❤➙♥ q✉ÿ ✤↕♦ ✤è✐ ợ tỷ q T ữủ ✤à♥❤ ❜ð✐✿ κ(x)f (x−1 γx)dx˙ Oγκ (f ) = T \G ✶✽ ❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R) ❍♦➔♥❣ ❚❤à ❉✉♥❣ ❑❤✐ κ = 1✱ κ ✲ t➼❝❤ ♣❤➙♥ q✉ÿ ✤↕♦ ❧➔ t➼❝❤ ♣❤➙♥ q✉ÿ ✤↕♦ ê♥ ✤à♥❤ ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ SOγ (f )✳ ❚❛ ♥❤➢❝ ❧↕✐ ▲✲♥❤â♠ ❝õ❛ ●✱ ❦➼ ❤✐➺✉ LG ✈➔ LG = Gˇ WR = P GL(2, C) WR✳ ❚÷ì♥❣ tü✱ ▲✲♥❤â♠ LH = Hˇ WR ❝õ❛ ❍✱ ❧➔ t❤➔♥❤ ♣❤➛♥ ❧✐➯♥ t❤æ♥❣ ❝õ❛ t➙♠ ❤â❛ ❝õ❛ ♠ët ♣❤➛♥ tû ♥û❛ ✤ì♥ tr♦♥❣ LG✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳ P❤➨♣ ♥❤ó♥❣ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ L H ✈➔♦ tr♦♥❣ L G ❧➔ ♠ët ˇ →G ˇ s❛♦ ❝❤♦ ❤↕♥ ❝❤➳ ❝õ❛ ♥â ỗ : L H L G rë♥❣ tü ♥❤✐➯♥ ❝õ❛ H ˇ ❧➔ ❝❤➾♥❤ ❤➻♥❤ ✈➔ ỗ t tr WR tr H ✷✳✶✳ ●✐↔ sû ❝â ♠ët ♣❤➨♣ ♥❤ó♥❣ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ η : L H → L G✳ ❚❛ ❝â t❤➸ ợ (G, H, ) ởt trữ χG,H ❝õ❛ T ✈ỵ✐ t➼♥❤ ❝❤➜t s❛✉✳ ❈❤♦ ❢ ❧➔ ♠ët ❣✐↔ ❤➺ sè ✤è✐ ✈ỵ✐ ❝❤✉é✐ rí✐ r↕❝ tr➯♥ G õ tỗ t ởt f H tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❝õ❛ ❝→❝ ❣✐↔ ❤➺ sè ✤è✐ ✈ỵ✐ ❝❤✉é✐ rí✐ r↕❝ tr➯♥ H s❛♦ ❝❤♦ γ = j(γH ) ❝❤➼♥❤ q✉② tr♦♥❣ T ✈➔ κ SOγH (f H ) = ∆G H (γH , γ)Oγ (f ) ✈ỵ✐ ∆G H (γH , γ) ❧➔ t❤ø❛ sè ❝❤✉②➸♥ ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝ −1 −1 q(G)+q(H) ∆G χG,H ∆B (γ −1 ).∆BH (γH ) H (γH , γ) = (−1) P❤➨♣ ❜✐➳♥ ✤ê✐ f → f H ❝õ❛ ❣✐↔ ❤➺ sè ❝â t❤➸ ✤÷đ❝ ♠ð rë♥❣ ❝❤♦ t➜t ❝↔ ❝→❝ ❤➔♠ tr♦♥❣ Cc∞(G)❀ ✤➸ ❧➔♠ ✤✐➲✉ ♥➔② ♥❣÷í✐ t❛ ♣❤↔✐ ♠ð rë♥❣ t÷ì♥❣ ù♥❣ γ → γH ✭❣å✐ ❧➔ ❝❤✉➞♥ ❤â❛✮✱ ✤è✐ ✈ỵ✐ t➜t ❝↔ ❝→❝ ♣❤➛♥ tû ♥û❛ ✤ì♥ ❝❤➼♥❤ q✉② ✈➔ ①→❝ ✤à♥❤ ❝→❝ t❤ø❛ sè ố ợ ỵ sỷ õ ♠ët ♣❤➨♣ ♥❤ó♥❣ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ η : L H → L G✳ ❚❛ ❝â ∞ t❤➸ ①→❝ ✤à♥❤ t❤ø❛ sè ❝❤✉②➸♥ ∆G H (γH , γ) s❛♦ ❝❤♦ ✈ỵ✐ t f Cc (G) tỗ t ởt f H ∈ Cc∞ (H) ✈ỵ✐ κ SOγH (f H ) = ∆G H (γH , γ)Oγ (f ) ❦❤✐ γH ❧➔ ❞↕♥❣ ❝❤✉➞♥ ❝õ❛ γ ❝❤➼♥❤ q✉② ♥û❛ ✤ì♥ ✈➔ SOγH (f H ) = ✶✾ ❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R) ❍♦➔♥❣ ❚❤à ❉✉♥❣ ❦❤✐ γH ❦❤æ♥❣ ❧➔ ❞↕♥❣ ❝❤✉➞♥✳ ❑❤✐ κ = t❤➻ ♥❤â♠ ❝♦♥ ♥ë✐ s♦✐ ❝õ❛ ● ❧➔ ❝❤➼♥❤ ♥â tù❝ ❧➔ H = SL(2, R) ✈➔ κ✲ t➼❝❤ ♣❤➙♥ q✉ÿ ✤↕♦ ❝❤➼♥❤ ❧➔ t➼❝❤ ♣❤➙♥ q✉ÿ ✤↕♦ ê♥ ✤à♥❤✳ ✣➙② ❧➔ tr÷í♥❣ ❤đ♣ t➛♠ t❤÷í♥❣✳ ❙❛✉ ✤➙② t❛ s ự t ỵ ố ợ 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 x)dx, O (f ) = G \G ú ỵ r➡♥❣ t➼❝❤ ♣❤➙♥ ♥➔② ♣❤ö t❤✉ë❝ ✈➔♦ sü ❧ü❛ ❝❤å♥ ✤ë ✤♦ ❍❛❛r tr➯♥ ● ✈➔ Gγ ✳ ●✐↔ sû t❤➯♠ r➡♥❣ f ❧➔ ❤➔♠ ❑✲t➙♠ tù❝ ❧➔ f (xk) = f (kx)✱ ✈ỵ✐ ♠å✐ k ∈ K ✈➔ x ∈ G ❈❤ó♥❣ t❛ s➩ ♥❣❤✐➯♥ ❝ù✉ ❞→♥❣ ✤✐➺✉ t✐➺♠ ❝➟♥ ❝õ❛ Oγ (f ) tr♦♥❣ ❤❛✐ tr÷í♥❣ ❤đ♣✿ ✷✳✸✳✶ ❚r÷í♥❣ ❤đ♣ γ ❝â ❞↕♥❣ ✤÷í♥❣ ❝❤➨♦ ❦❤✐ γ → γ= ✈ỵ✐ ab = a 0 b ❱ỵ✐ ♠é✐ x ∈ G t❛ ❝â ♣❤➙♥ t➼❝❤ ■✇❛s❛✇❛ ❝õ❛ x ❧➔ x = ank✱ tr♦♥❣ ✤â a= y 0 y −1 ;n= t ; k = r(θ) ✈ỵ✐ 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✐ ✈ỵ✐ ❝→❝❤ ❝❤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 = t 1 −t a 0 b a (b − a)t b = ❱➻ ✈➟② Oγ (f ) = f a (b − a)t b dt R ✣➦t ∆(γ) = |a − b| t❤➻ ❤➔♠ h(γ) = ∆(γ)Oγ (f ) = |a − b| a (b − a)t b f dt R t❤→❝ tr✐➸♥ tỵ✐ ♠ët ❤➔♠ trì♥ tr➯♥ A ✭♥❤â♠ ❝→❝ ♠❛ tr➟♥ ❝â ❞↕♥❣ ✤÷í♥❣ ❝❤➨♦✮✳ ✷✳✸✳✷ ❚r÷í♥❣ ❤đ♣ γ = r(θ) ❦❤✐ θ → ❱ỵ✐ x ∈ G✱ t❛ ①➨t ♣❤➙♥ t➼❝❤ ❈❛rt❛♥ ❝õ❛ x ❧➔ x = k1ak2✱ tr♦♥❣ ✤â k1 = r(α); a = y 0 y −1 ; k2 = r(β) ✈ỵ✐ 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 − sin θ cosθ y cosθ y −2 sin θ −y sin θ cosθ y 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 y −1 ❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R) ❍♦➔♥❣ ❚❤à ❉✉♥❣ ✈ỵ✐ ❤➡♥❣ sè ❝ ♣❤ö t❤✉ë❝ ✈➔♦ ✈✐➺❝ ❝❤å♥ ✤ë ✤♦ ❍❛❛r ✈➔ ∞ tr♦♥❣ ✤â a(λ) = ❚❛ ❝â √ − λ2 ∞ ∞ ∞ = a(λ) tλ dt − −t−1 λ a(λ) f ∞ a(λ) tλ dt − −1 −t λ a(λ) f ∞ = f ∞ = 1 dt a(λ) tλ (−1) −t−1 λ a(λ) t f a(λ) tλ d(t−1 ) −t−1 λ a(λ) ∞ ∞ dt t2 a(λ) (t−1 )−1 λ d(t−1 ) −t−1 λ a(λ) f a(λ) tλ dt + −t−1 λ a(λ) a(λ) tλ )(t2 − 1) −1 −t λ a(λ) f 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 ∞ F (λ) = a(λ) tλ −t−1 λ a(λ) |t − t−1 |.f F (λ) = f a(λ) −t−1 λ dt tλ a(λ) a(λ) −t−1 λ dt tλ a(λ) (−1)f ❈❤ó þ r➡♥❣ f ❧➔ ❤➔♠ ❑✲t➙♠ ♥➯♥ ✈ỵ✐ ❛✱❜✱❝ ❜➜t ❦➻✱ t❛ ❝â f a b c a b −a a −c =f ❉♦ ✤â −1 ∞ a(λ) tλ −1 −t λ a(λ) ε(t − 1)f F (λ) = b −a a −c −1 =f dt, ✈ỵ✐ ε(x) = sign(x) ✣➸ ♥❣❤✐➯♥ ❝ù✉ t✐➺♠ ❝➟♥ ❝õ❛ ♥â ❦❤✐ λ → t❛ ①➨t ∞ ε(t − 1)f A(λ) = a(λ) tλ a(λ) dt ❚❤❡♦ ❝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) = ✷✷ dt , t =f a −c −b a ❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R) ❍♦➔♥❣ ❚❤à ❉✉♥❣ ✈ỵ✐ ❤➔♠ trì♥ g ♥➔♦ ✤â ❝â ❣✐→ ❝♦♠♣❛❝t t❤❡♦ ❜✐➳♥ ♣❤➼❛ tr➯♥ ❜➯♥ ♣❤↔✐ ✈➔ ❝â O(u)−1 r t ữợ tr s t tử tt ố ú ỵ r A(λ) = |λ|−1 ε(λ)u f du − 2f 0 + o(λ) ❚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✐➸♥ tỵ✐ ❤➔♠ ❧✐➯♥ 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 ❚✉② ♥❤✐➯♥✱ ❤✐➺✉ ❝õ❛ ❤❛✐ ❜✐➸✉ t❤ù❝ ♠➔ ♣❤➛♥ ❝❤➼♥❤ ❝õ❛ ❝❤ó♥❣ tữỡ ữỡ ợ ln(||1 )g s t❤ù❝ ❧✐➯♥ tö❝✳ ❉♦ ✤â ❇ ❧➔ ❧✐➯♥ tö❝✳ ❑❤→✐ q✉→t ❤â❛ q✉→ tr➻♥❤ ♥➔② t❛ t❤✉ ✤÷đ❝ tr t ụ ữợ ữ 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 )), ✈ỵ✐ γ = r(θ) ∈ T ✈➔ ∆(r(θ)) = −2isinθ✳ ✷✳✹ P❤➨♣ ❝❤✉②➸♥ ✈➳ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t ❚❛ t❤➜② r➡♥❣ t÷ì♥❣ ù♥❣ f → f H ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ♠ët →♥❤ ①↕ ✈➻ f H ①→❝ ✤à♥❤ ❦❤æ♥❣ ❞✉② ♥❤➜t✳ P❤➨♣ ❝❤✉②➸♥ ❤➻♥❤ ❤å❝ f → f H ❧➔ ✤è✐ ♥❣➝✉ ❝õ❛ ♣❤➨♣ ❝❤✉②➸♥ ✷✸ ❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R) ❍♦➔♥❣ ❚❤à ❉✉♥❣ ✤è✐ ✈ỵ✐ ❜✐➸✉ ❞✐➵♥✳ ❇➜t ❦ý ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ❝❤➜♣ ữủ H tữỡ ự ợ ởt tû σG tr♦♥❣ ❜❛♦ ♥❤â♠ ❝õ❛ ❜✐➸✉ ❞✐➵♥ ↔♦ ❝õ❛ G ♥❤÷ s❛✉✳ ❈❤♦ ϕ ❧➔ ♠ët t❤❛♠ sè ▲❛♥❣❧❛♥❞s ✤è✐ ✈ỵ✐ H t❤➻ η ◦ ϕ ❧➔ ♠ët t❤❛♠ sè ▲❛♥❣❧❛♥❞s ✤è✐ ✈ỵ✐ G tr♦♥❣ ✤â η ❧➔ ♣❤➨♣ ♥❤ó♥❣ η : LH → LG✳ ❚❛ ①➨t ❧➔ ▲✲❣â✐ ❝õ❛ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ H tữỡ ự ợ õ G tữỡ ự ợ õ ❝â t❤➸ ❧➔ t➟♣ ré♥❣ ♥➳✉ t❤❛♠ sè ♥➔② ❦❤æ♥❣ t ủ ợ ỵ ỗ t ởt ❤➔♠ → ±1, ε: s❛♦ ❝❤♦ ♣❤➛♥ tû σG tr♦♥❣ ❜❛♦ ♥❤â♠ ❝õ❛ ♥û❛ ♥❤â♠ ❝→❝ ❜✐➸✉ ❞✐➵♥ ❝❤♦ ❜ð✐ σG = ε(π)π, π∈ ①→❝ ✤à♥❤ ♠ët t÷ì♥❣ ù♥❣ σ → σG ❧➔ ✤è✐ ♥❣➝✉ ❝õ❛ ❜✐➳♥ ✤ê✐ ❤➻♥❤ ❤å❝ trace σG (f ) = trace σ(f H ) ❳➨t ♠ët t❤❛♠ sè ▲❛♥❣❧❛♥❞s ϕ : LR → L G ❦➼ ❤✐➺✉ Sϕ ❧➔ t➙♠ ❤â❛ tr♦♥❣ Gˇ ❝õ❛ ϕ(WR)✳ ú ỵ r ợ t s S ①→❝ ✤à♥❤ ♠ët ♥❤â♠ ♥ë✐ s♦✐ ❍✳ ◆❤â♠ ❍ s✐♥❤ tr♦♥❣ LG ❜ð✐ t➙♠ ❤â❛ ❧✐➯♥ t❤æ♥❣ Hˇ ❝õ❛ s tr G tỗ t ởt ú : LH LG t tỗ t t❤❛♠ sè ▲❛♥❣❧❛♥❞s ϕ ✤è✐ ✈ỵ✐ ● ✈➔ ❞♦ ✤â ①→❝ ✤à♥❤ ♠ët t❤❛♠ sè ▲❛♥❣❧❛♥❞s η ◦ ϕ ❝❤♦ ❍✳ ˇ Γ )✱ tr♦♥❣ ✤â Sϕ0 ❧➔ t❤➔♥❤ ♣❤➛♥ ❧✐➯♥ t❤æ♥❣ ❝õ❛ Sϕ ✈➔ ❑➼ ❤✐➺✉ Sϕ = Sϕ/(Sϕ0 × Z(G) ˇ Γ ❧➔ t➙♠ ❝õ❛ L G✳ ●✐↔ sû t❛ ❝❤♦ ♠ët t➟♣ ❤ñ♣ ✤➛② ✤õ ❝→❝ ♥❤â♠ ♥ë✐ s♦✐ ❦❤ỉ♥❣ Z(G) t÷ì♥❣ ✤÷ì♥❣ ❍ ✈➔ ❝❤♦ ♠é✐ ❍ ♠ët ♣❤➨♣ ♥❤ó♥❣ η : LH → LG✳ ❚❛ ①➨t ♠ët t❤❛♠ sè ϕ : WR → LG✱ t➙♠ ❤♦→ ❧✐➯♥ t❤æ♥❣ ❝õ❛ s ∈ Sϕ ❧➔ ♥❤â♠ Hˇ s ❧✐➯♥ ❤đ♣ ✈ỵ✐ Hˇ ✈➔ ❞♦ ✤â ❧✐➯♥ ❤đ♣ ❝õ❛ t❤ø❛ sè ϕ tr♦♥❣ η(LH) ①→❝ ✤✐♥❤ ♠ët ▲✲❣â✐ ✷✹ ❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R) ❍♦➔♥❣ ❚❤à ❉✉♥❣ ❝õ❛ ❜✐➸✉ ❞✐➵♥ ❝õ❛ ❍ ✳✭◆❤➻♥ ❝❤✉♥❣ ▲✲❣â✐ ❧➔ ❦❤ỉ♥❣ ❞✉② ♥❤➜t ♥â ♣❤ư t❤✉ë❝ ✈➔♦ sü ❧ü❛ ❝❤å♥ ❧✐➯♥ ❤đ♣ ❝â t❤➸ ❦❤ỉ♥❣ ❞✉② ♥❤➜t✮✳ ▼➦t ❦❤→❝✱ ❙❤❡❧t❛❞ ✤➣ ✤à♥❤ ♥❣❤➽❛ ♠ët ❝➦♣ s, π ỳ S () ỗ tớ r r s ε(π) = c(s) s, π ✈➔ ❞♦ ✤â trace σ(f H ) = s✉② r❛ σ∈ ε(π) trace π(f ), π∈ s ˜ s (f H ) = s, π trace π(f ), π∈ tr♦♥❣ ✤â ˜ s (f H ) = c(s)−1 trace σ(f H ) σ∈ ❚ø ✤â t õ ỵ s ỵ trace (f ) = #Sϕ s, π ˜ s∈Sϕ s (f H ) t ố ợ SL(2, R) ú ỵ r➡♥❣✱ ♥➳✉ π ❧➔ ❜✐➸✉ ❞✐➵♥ ✉♥✐t❛ ❜➜t ❦❤↔ q✉② ỳ t trữ ố ợ ♣❤➛♥ tû ❝❤➼♥❤ q✉② g ∈ G ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ Θ (π, g) = trace π(g) ❳➨t t♦→♥ tû π(f ) = G π(g)f (g)dg✱ ✈ỵ✐ f ∈ Cc∞(G) ✈➔ dg ❧➔ ✤ë ✤♦ ❍❛❛r tr➯♥ ●✱ t❤❡♦ ❍❛r✐s❤✲❈❤❛♥❞r❛ t❛ ❝â trace π(f ) = Θ(π, g)f (g)dg G ✣➦❝ tr÷♥❣ ❝õ❛ Dn+ tr➯♥ SO(2, R) ✤÷đ❝ ❝❤♦ ❜ð✐ Θ+ n (r(θ)) = −einθ ei(n+1)θ = − e2iθ eiθ − e−iθ ✣➦❝ tr÷♥❣ ❝õ❛ Dn− tr➯♥ SO(2, R) ✤÷đ❝ ❝❤♦ ❜ð✐ ❧✐➯♥ ❤đ♣ ♣❤ù❝ Θ− n (r(θ)) = e−i(n+1)θ e−inθ = − e−2iθ eiθ − e−iθ ✷✺ ❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R) ú ỵ r tờ − SΘn = Θ+ n + Θn t❤❛② Θ+n ✈➔ Θ−n t❛ ✤÷đ❝ SΘn = − einθ − e−inθ eiθ ei õ õ t ữợ ❤đ♣ ê♥ ✤à♥❤✱ t❛ ❣å✐ SΘn ❧➔ ✤➦❝ tr÷♥❣ ê♥ ✤à♥❤✳ ❚r♦♥❣ ❦❤✐ − inθ ∆(r(θ))(Θ+ + e−inθ n − Θn )(r(θ)) = e tr♦♥❣ ✤â ∆(r(θ)) = −2i.sinθ ❧➔ tê♥❣ ❤❛✐ ✤➦❝ tr÷♥❣ ❝õ❛ T (R)✳ ✣➙② ❧➔ ♣❤➨♣ ❝❤✉②➸♥ ✈➳ ♣❤ê ❝❤♦ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL(2, R)✳ ✷✳✺ ❑➳t ❧✉➟♥ ❈❤÷ì♥❣ ♥➔② ✤➣ tr➻♥❤ ❜➔② ♥ë✐ ❞✉♥❣ ỗ ổ tự t t ♣❤➛♥ rí✐ r↕❝ ❝õ❛ ❜✐➸✉ ❞✐➵♥ ❝❤➼♥❤ q✉✐✱ ❜✐➳♥ ✤ê✐ ✈➳t t❤➔♥❤ tê♥❣ ❝→❝ t➼❝❤ ♣❤➙♥ q✉ÿ ✤↕♦✱ t➼♥❤ ✈➳t ❝õ❛ ❜✐➸✉ ❞✐➵♥ t❤✉ ❣å♥ ✈➲ ♥❤â♠ ❝♦♥ ♥ë✐ s♦✐ ❝õ❛ SL (2, R)✳ ✷✻ ❑➳t ❧✉➟♥ ◆❤í ❦✐➳♥ t❤ù❝ t➼❝❤ ❧ơ② q✉❛ t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ t↕✐ ❦❤♦❛ ❚♦→♥ ✲ ❈ì ✲ ❚✐♥ ❤å❝✱ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ❚ü ♥❤✐➯♥✱ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐❀ ✤➦❝ ❜✐➺t ♥❤í sü ❝❤➾ ❜↔♦ ✈➔ ❣✐ó♣ ✤ï t➟♥ t➻♥❤ ❝õ❛ ●❙✳❚❙❑❍✳ ✣é ◆❣å❝ ❉✐➺♣✱ tæ✐ ✤➣ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ❦❤♦❛ ❤å❝ ✈ỵ✐ ✤➲ t➔✐✿ ❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ ✈➳t tr➯♥ SL (2, R)✳ ✣➙② ❧➔ ❜➔✐ t♦→♥ ♥➡♠ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ ▲❛♥❣❧❛♥❞s ❝❤♦ ♥❤â♠ SL (2, R)✳ ❈→❝ ❦➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❜❛♦ ỗ ổ tự t t t tỷ õ ♥❤➙♥ tr➯♥SL(2, R) ✈➔ ❝ỉ♥❣ t❤ù❝ tê♥❣ P♦✐ss♦♥ t÷ì♥❣ ù♥❣✳ ✷✳ ❱➳ ❤➻♥❤ ❤å❝ ❝õ❛ ❝æ♥❣ t❤ù❝ t➼♥❤ ✈➳t ♣❤➛♥ rí✐ r↕❝ ❝õ❛ ❜✐➸✉ ❞✐➵♥ ❝❤➼♥❤ q✉✐ tr➯♥ SL (2, R)✳ ✸✳ ❈æ♥❣ t❤ù❝ ✈➳t ❝õ❛ ❜✐➸✉ ❞✐➵♥ ❝❤➼♥❤ q✉② t❤✉ ❣å♥ tr➯♥ ♥❤â♠ ❝♦♥ ♥ë✐ s♦✐ ❝õ❛ SL(2, R)✳ ❚✉② ♥❤✐➯♥ ❞♦ t❤í✐ ❣✐❛♥ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ❦❤ỉ♥❣ ♥❤✐➲✉ ♥➯♥ ❝á♥ ❝â ♥❤ú♥❣ t❤✐➳✉ sât✱ tæ✐ r➜t ♠♦♥❣ ữủ sỹ õ ỵ qỵ t ổ ❜↕♥ ✤å❝✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✳ ✷✼ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬❈❪ ◆❣æ ❇↔♦ ❈❤➙✉✱ ❆✉t♦♠♦r♣❤✐❝ ❢♦r♠s ♦♥ GL2✱ Pr❡♣r✐♥t✳✷✵✶✶✱ ❈❤✐❝❛❣♦ ❯♥✐✲ ✈❡rs✐t②✳ ❬●❪ ❙✳ ●❡❧❜❛rt✱ ▲❛♥❣❧❛♥❞s ♣✐❝t✉r❡ ♦❢ ❛✉t♦♠♦r♣❤✐❝ ❢♦r♠s ❛♥❞ ▲✲❢✉♥❝t✐♦♥s✱▲❡❝t✉r❡ ◆♦t❡s✳ ▲❡❝t✉r❡ ❙❡r✐❡s ❛t ❙❤❛♥❣❞♦♥❣ ❯♥✐✈❡rs✐t②✱ ❈❤✐♥❛✱ ♣♣✳ ✶✲✺✳ ❬▲❪ ❙✳ ▲❛♥❣✱ ❙▲✭✷✱❘✮✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ②♦r❦ ✲ ❇❡r❧✐♥ ✲ ❍❡✐❞❡❧❜❡r❣ ✲ ❚♦❦②♦✳ ❬▲▲❪ ❏✳✲P✳ ▲❛❜❡ss❡✱ ❘✳ ▲❛♥❣❧❛♥❞s✱ ▲✲❞✐st✐♥❣✉✐s❤❛❜✐❧✐t② ❢♦r ❙▲✭✷✮✱ ❈❛♥✳ ❏✳ ▼❛t❤✱ ✈♦❧✳ ①①①✐✱ ◆♦ ✹✱ ✼✷✻✲✼✽✺✳ ✷✽