重複度１の表現と複素多様体上の
可視的な作用
M u lt ip li c it y -f re e re p re s e n t a t io n s a n d v is ib le a c t io n s o n c o m p le x m a n if o ld s T o s h iy u k i K o b a y a s h i ( R IM S , K y o t o U n iv e rs it y ) S y m p o s iu m o n R e p re s e n t a t io n T h e o ry 2 0 0 5 K a k e g a w a , S h iz u o k a N o v e m b e r 1 5 -1 8 , 2 0 0 5 アブストラクト： 正則 ベク トル 束に 群作 用が 与え られ てい ると きに 、そ の 正 則 な 大 域 切 断 の 空 間 に 実 現 さ れ た 表 現 を 考 え る 。 こ の 表 現 は 、 1) 底 空 間 が 非 コ ン パ ク ト な ら ば 一 般 に 無 限 次 元 表 現 で あ り 、 2) 底 空 間 に お け る 群 軌 道 が 無 限 個 な ら ば 一 般 に 既 約 とは程遠い表現になる。 しか し、 ある 種の 幾何 的な 条件 （ 『可 視的 な作 用』 ） が 満 た さ れ て い れ ば 、 重 複 度 が １ と い う 性 質 が フ ァ イ バ ー へ の 表現⇒大域切断における表現に伝播することが証明できる。 この 概説 講演 では 、有限 次元 およ び無 限次 元の 表現 に お け る 重 複 度 １ の 表 現 の た く さ ん の 例 を 紹 介 し 、 そ れ が 『 可 視 的 な 作 用 』 と い う 幾 何 的 な 条 件 か ら ど の よ う に 理 解 で き る かを説明する予定です 。 [1 ] M u lt ip lic it y -f re e th eo re m in b ra n ch in g p ro b le m s of u n i-ta ry h ig h es t w eig h t m o d u le s, P ro c. S y m p os iu m on R ep -re se n ta tio n T h eo ry h eld at S ag a, K y u sh u 19 97 (e d . K . M im ac h i) , (1 99 7) , 9– 17 . [2 ] M u lt ip lic it y on e th eo re m in th e or b it m et h o d (w it h Na s-rin ), in m em or y of K ar p ele v ic “L ie G ro u p s an d S y m m et -ric S p ac es ”, (e d s. S . G in d ik in ), 16 1– 16 9 Am er . M at h . S o c. 20 03 [3 ] G eo m et ry of M u lt ip lic it y -f re e re p re se n ta tio n s of G L (n ), v is ib le ac tio n s on fl ag va rie tie s, an d tr iu n it y, Ac ta Ap p l. M at h . 81 (2 00 4) , 12 9-14 6. [4 ] M u lt ip lic it y -f re e re p re se n ta tio n s an d v is ib le ac tio n s on co m p le x m an if old s, P u b l. R IM S 41 (2 00 5) , 49 7-54 9. 表現論シンポジウム講演集，2005 pp.33-63E g . 1 ( E ig e n s p a ce d e comp os it ion ) H : V e ct o r s p ./ C , di m < ∞ A ∈ E nd C (H ) 1
°
s .t . A is d ia g on a liz a b le , a ll e ig e n v a lu e s a re d is t in ct . ⇒ H = C e 1 ⊕ · · · ⊕ C en ≃ C n ( ca n on ica l) 1°
& de t A 6= 0 ⇓ π A : Z − → G L C (H ) is M F ∈ ∈ n 7− → A n E g . 2 ( F ou rie r s e rie s e x p a n s ion ) L 2 (S 1 ) ≃ X ⊕ n ∈ Z C e inx f (x ) = X n∈ Z an e inx T ra n s la t ion ( ⇒ re p . of t h e g rou p S 1 ) f ( ·) 7→ f ( · − c ) (c ∈ S 1 ≃ R / 2 π Z ) S 1 y L 2 (S 1 ) is M Fπ : G → G L C ( H ) g rou p D e f. ( n a iv e ) ( π , H ) is M F mu lt ip licit y -f re e if di m Hom G ( τ , π ) ≦ 1 ( ∀ τ : ir re d . re p . of G ) . L 2 ( S 1 ) ∃ 1 ← ֓ e inx ⇔ di m Hom S 1 ( τ , L 2 ( S 1 )) = 1 ( ∀ τ : ir re d . re p . of S 1 ) ⇒ S 1 y L 2 ( S 1 ) is M F
E g . 3 ( T a y lo r e x p a n s ion , L a u re n t e x p a n s ion ) f ( z 1 , . . . , zn ) = X α =( α1 ,. .. ,α n ) aα z α1 1 ·· · z αn n P oin t ( t o o ob v iou s ) ∃ 1 aα ∈ C fo r e a ch α ⇑ di m Hom ( S 1 ) n ( τ ,O ({ 0 } ) ≦ 1 ( ∀ τ : ir re d . re p . of ( S 1 ) n ) i.e . MF E g . 4 ( P e t e r-W e y l) ¡ ¡ ¡ ir re d . re p . of G × G G : comp a ct ( L ie ) g rou p L 2 ( G ) ≃ X ⊕ τ ∈ c G τ ⊠ τ ∗ T ra n s la t ion ( ⇒ re p . of G × G ) f ( ·) 7→ f ( g − 1 1 · g 2 ) ⇒ G × G y L 2 ( G ) is M F
E g . 5 ( Sp h e rica l h a rmon ics ) H l := ( f ∈ C ∞ ( S n − 1 ) : ∆ S n − 1 f = − l( l + n − 2) f ) L 2 ( S n − 1 ) ≃ ∞ X ⊕ l=0 H l O ( n ) ir re d . ⇓ O ( n ) y L 2 ( S n − 1 ) is M F ⊗ -p ro d u ct re p . S L2 ( C ) πk y ir re d . S k ( C 2 ) ( k = 0 , 1 , 2 , . . .) E g . 6 ( Cle b s ch -G o rd a n ) π k ⊗ π l ≃ π k + l ⊕ π k + l− 2 ⊕ ·· · ⊕ π |k − l| ↑ MF
Nota tion Hig h te s t w e ig h t λ = ( λ1 . . . , λn ) ∈ Z n , λ1 ≥ λ2 ≥ · · · ≥ λn ⇓ π GL n λ ≡ π λ : ir r e d . r e p . of GL n ( C ) E g . λ = ( k , 0 , . . . , 0) ↔ GL n ( C ) y S k ( C n ) λ = (1 , . . . , 1 | {z } k , 0 , . . . , 0) ↔ GL n ( C ) y Λ k ( C n ) ⊗ -p ro d u ct re p . ( GL n -ca s e ) E g .7 (Pie ri’ s la w ) π (λ 1 ,. .. ,λ n ) ⊗ π (k ,0 ,. .. ,0) ≃ M µ1 ≥ λ1 ≥· ·· ≥ µn ≥ λn P (µ i − λi )= k π (µ 1 ,. .. ,µ n ) ↑ MF a s a GL n -mo d u le .
⊗ -p ro d u ct re p . (con tin u e d ) E g . (cou n te re x a mp le ) π (2 ,1 ,0) ⊗ π (2 ,1 ,0) is NOT MF a s a GL 3 ( C )-mo d u le . ⊗ -p ro d u ct re p . ( con t in u e d ) λ = ( a , · · · , a | {z } p , b , · · · , b | {z } q ) ∈ Z n , a ≥ b E g . 8. (Ste mb rid g e 2001, K–) πλ ⊗ πν is MF a s a G Ln ( C ) -mo d u le if 1) mi n( a − b , p , q ) = 1 (a nd ν is a ny ), o r 2) mi n( a − b , p , q ) = 2 a nd ν is of the fo rm ν = ( x · · · x | {z } n1 y · · · y | {z } n2 z · · · z | {z } n3 ) ( x ≥ y ≥ z ), o r 3) mi n( a − b , p , q ) ≥ 3, & mi n( x − y , y − z , n1 , n2 , n3 ) = 1. ⋆ ⋆
E g .9 (GL n ↓ GL n − 1 ) π GL n (λ 1 ,. .. ,λ n ) |GL n − 1 ≃ ⊕ λ1 ≥ µ1 ≥· ·· ≥ µn − 1 ≥ λn π GL n − 1 (µ 1 ,. .. ,µ n − 1 ) ↑ MF a s a GL n − 1 ( C ) -mo d u le ⇓ Ap p lica t ion G e lf a n d -T s e tlin b a s is GL n πλ y V F in d a ‘g o o d ’ b a s is of V R e ma r k . V is n ot n e ce s s a r ily MF a s a ( C × ) n -mo d u le . Id e a of G e lf a n d -T s e tlin b a s is GL n ( C ) ∪ y MF C × × GL n − 1 ( C ) ∪ y MF ( C × ) 2 × GL n − 2 ( C ) ∪ y MF . . . . . . ∪ y MF ( C × ) n
In fin ite d ime n s ion a l ca s e (cf . E g .9) E g .10. ( U ( p , q ) ↓ U ( p − 1 , q )) ∀ π : ir re d . u n ita ry re p . of U ( p , q ) w ith h ig e s t w e ig h t ⇒ re s tr iction π | U ( p − 1 ,q ) is MF a s a U ( p − 1 , q )-mo d u le E g .11. ( GL − GL d u a lit y ) N = m n ⇒ GL m × GL n y S ( C N ) ≃ S ( M ( m , n ; C )) T h is re p . is MF • E x p licit fo rmu la · · · GL m − GL n d u a lit y • G e n e ra liz a tion (B ra n ch in g la w of h olo. d is c. w .r .t. s y mme tr ic p a ir ) Hu a -Kos ta n t-Sch mid - K-¢ K-¢ ¢ ¢ £ £ £ £ e ach comp on e n t · · · fin ite d im ∞ − dim
An oth e r g e n e ra liz a tion E g .12 (Ka c’ s MF s p a ce ) S ( C N ) is s till MF a s a GL m − 1 × GL n mo d u le W e re ca ll: π : G gr oup → GL C ( H ) D e f. (n a iv e ) ( π , H ) is MF mu lt ip licit y -f re e if di m Hom G ( τ , π ) ≤ 1 ( ∀ τ : ir re d . re p . of G ).
Ob s e rv a t ion n ≤ 1 ⇔ E n d ( C n ) is commu t a t iv e . ( π , H ): u n it a ry re p . of G D e f. ( π , H ) is M F if E n d G ( H ) is commu t a t iv e . Mo re g e n e ra lly , ( ̟ , W ): t op . re p . D e f. ( ̟ , W ) is M F if a n y u n it a ry s u b re p . ( π , H ) is MF ( ∃ G -in j. con t . h om. H ֒→ W ) E g . 2 ( F ou rie r s e rie s e x p a n s ion ) L 2 (S 1 ) ≃ X ⊕ n ∈ Z C e inx f (x ) = X n∈ Z an e inx e inx (cf . ce n tr if u g a l se p a ra to r) T ra n s la t ion ( ⇒ re p . of t h e g rou p S 1 ) f ( ·) 7→ f ( · − c ) (c ∈ S 1 ≃ R / 2 π Z ) S 1 y L 2 (S 1 ) is M F
E g .13 (F ou rie r tr a n s fo rm) L 2 ( R ) ≃ Z ⊕ R C e iζ x dζ (d ir e ct in te g ra l of Hilb e rt sp .) f ( x )= Z R ˜ f(ζ ) e iζ x dζ T ra n s la tion ( ⇒ re g u la r re p re se n ta tion on L 2( R )) f ( ·) 7→ f ( · − c ) R y L 2 ( R ) is “MF ” . . . con t in u ou s s p e ctr u m Sy mme t ric Sp a ce E g .14. G / K : R ie ma n n ia n Sy mm. Sp a ce ⇒ G y L 2 (G / K ) is M F . E g .15. ( cou n t e r e x a mp le ) G / H : Se mis imp le Sy mm. Sp a ce ⇒ G y L 2 (G / H ) is NOT a lw a y s M F .
E g .14-A. G / K = SL (n , R )/ SO (n ) L 2 (G / K ) ≃ Z ⊕ λ1 ≥· ·· ≥ λn Σ λi =0 H λ dλ con t. sp e c. MF H λ : ∞ -d im, ir r e d . r e p . of G E g .15-A. G / H = SL (n , R )/ SO (p ,q ) Mu lt ip licit y of mos t con t . s p e c. in L 2 (G / H ) = n ! p !q ! > 1 if p ,q > 0 . ⇒ NOT M F V e ct o r b u n d le ca s e E g .15-B . G / K = G L (n , R )/ O (n ) Vτ := G × K τ → G / K τ : u n it a r y r e p . of K ⇒ G y L 2 ( Vτ ) u n it a r y G y L 2 ( Vτ ) is NOT a lw a y s MF . b u t it is MF if τ ≃ Λ k ( C n ) (0 ≤ k ≤ n )
E x a mp le s of M u lt ip licit y -F re e re p s • P e te r-W e y l th e o re m • Ca rta n -He lg a s on th e o re m • B ra n ch in g la w s : GL n ↓ GL n − 1 , On ↓ On − 1 • Cle b s ch -G o rd a n fo rmu la • Pie ri’ s la w • GL m -GL n d u a lit y • Pla n ch e re l fo rmu la fo r L 2 (G/ K ) (G/ K : R ie ma nni a n sy mme tr ic spa c e s) • (G e lf a n d -G ra e v -V e rs h ik ) ca n on ica l re p re s e n ta tion s • Hu a -Kos ta n t-Sch mid K -t y p e fo rmu la • (Ka c) lin e a r mu ltip licit y -f re e s p a ce s • (P a n y u s h e v ) s p h e rica l n ilp ote n t o rb it s • (Ste mb rid g e ) mu lt ip licit y -f re e t e n s o r p ro d u ct re p re s e n ta tion s of GL n e tc. Acco rd in g ly , v a riou s t e ch n iq u e s ca n b e a p p lie d in e a ch M F ca s e . F o r e x a mp le , on e ca n 1) lo ok fo r a n op e n o rb it of a B o re l s u b g rou p . 2) a p p ly L it t le w o o d -R ich a rd s on ru le s a n d v a ria n t s . 3) u s e comp u t a t ’l comb in a t o rics . 4) e mp lo y t h e commu t a t iv it y of t h e He ck e a lg e b ra . 5) a p p ly Sch u r-W e y l d u a lit y a n d Ho w e d u a lit y .
Aim · · · T o g iv e a s imp le p r in cip le t h a t e x p la in s t h e p r op e r t y MF of a ll t h e s e e x a mp le s , a n d mo r e . § 2 MF t h e o r e m mu ltip licit y fr e e H , K : L ie g r ou p s D : comp le x mf d . P → D : H -e q u iv . p r in cip a l K-b ’d le µ : K → G L C (V ) ⇓ Se t t in g 1 H -e q u iv . h olo. v e cto r b ’d le : V := P × K V → D ⇓ H y O (D ,V ) = { h olo. s e ct ion s}
Se t t in g 2 σ1 y P d iffe o. σ σ2 y K a u t o. σ 3 y H a u t o. s .t . σ (h p k ) = σ (h )σ (p )σ (k ) ∋ ∋ ∋ H P K σ y D ≃ P / K a n ti-h olomo r p h ic R e ca ll H y P x K ↓ D B B B B B B B B B B B M ∩ ∩ H y P x K µ − → G L C (V ) ↓ D B ⊂ P σ := { p ∈ P : σ (p ) = p } su b se t ⇓ M ≡ M B := { k ∈ K : bk ∈ H b ( ∀ b ∈ B )} As s u mp t ion ( a ) H B K = P ( b ) µ | M is MF s a y , µ | M ≃ l ⊕ i=1 νi ( c) µ ◦ σ ≃ µ ∗ a s K -mo d u le s νi ◦ σ ≃ ν ∗ i a s M -mo d u le s ( ∀ i)
As s u mp t ion ( a ) H B K con t a in s a n in t e rio r p oin t of P ( b ) µ | M is MF s a y , µ | M ≃ l ⊕ i=1 νi ( c) µ ◦ σ ≃ µ ∗ a s K -mo d u le s νi ◦ σ ≃ ν ∗ i a s M -mo d u le s ( ∀ i) P oin t of As s u mp t ion s ( a ) · · · b a s e s p . ( b ) · · · fi b e r ( c) · · · of t e n a u t oma t ic T h e o re m ( MF t h e o re m) As s u me ∃ σ a n d ∃ B ⊂ P σ s a t is fy in g ( a ) ∼ ( c) . ⇒ H y O ( D , V ) is M F .
P oin t • p r op a g a t ion of M F p r op e r t y fi b e r MF ⇒ s e ct ion s M F • g e ome t r y of b a s e s p a ce · · · ‘ v is ib le a ct ion ’ y V H ↓ y D ⇒ H y O ( D , V ) As s u mp tion (c) µ ◦ σ ≃ µ ∗ a s K -mo d u le s h old s if σ is a W e y l in v olu tion i. e . σ∈ A ut( K ), σ 2 = id s .t. σ (g ) = g − 1 on s ome ma x to ru s of K e .g . K = U (n ), σ (g ) = ¯g
§ 3 Vis ib le a ct ion h olomo rp h ic H y D comp le x mf d , con n e ct e d D e f. T h e a ct ion is (s t ron g ly ) v is ib le ∃ D ′ ⊂ D op e n s u b s e t if ∃ σ y D a n t i-h olomo rp h ic ∃ N ⊂ D t ot a lly re a l s .t . σ | N = id N me e t s e v e ry H -o rb it in D ′ σ s t a b iliz e s e v e ry H -o rb it . As s u mp t ion ( a ) + ( s t ron g ly ) v is ib le a ct ion As s u mp t ion ( a ) : H B K = P fo r s ome B ⊂ P σ ⇒ N := P σ K / K me e t s e v e ry H -o rb it on D := P / K ⇒ H y D v is ib le ( σ : in v olu s ion ⇒ N : tota lly re a l)
E x a mp le of Vis ib le a ct ion s T = { a ∈ C : | a| = 1 } (≃ S 1 ) E g . T y C ⊃ R ∈ a z 7→ a z R me e t s e v e r y T -o r b it R ⇒ T -a ct ion on C is v is ib le . h olomo r p h ic H y ( D , J ) comp le x mf d , con n e ct e d D e f. Act ion is v is ib le if ∃ D ′ ⊂ D , op e n ∃ N ⊂ D s .t . totally r e al N me e t s e v e r y H -o r b it Jx ( Tx N ) ⊂ Tx ( H · x ) ( x ∈ N )
sy mp le ctic H y ( D , ω ) s y mp le ct ic mf d D e f. ( G u ille min -St e rn b e rg , Hu ck le b e rr y -W u rz b a ch e r) Act ion is cois ot rop ic ( o r mu lt ip licit y -f re e ) if T(x H · x ) ⊥ ω ⊂ Tx ( H · x ) ( x ∈ D ) isome tr ic H y ( D , g ) R ie ma n n ia n mf d D e f. ( P o d e s t` a -T h o rb e rg s s on ) Act ion is p ola r if ∃ N ⊂ D s .t . N me e t s e v e ry H -o rb it . Tx N ⊥ Tx ( H · x ) ( x ∈ N )
H y D comp a ct comp a ct , K ¨a h le r R ie ma n n ia n P ola r Visib le & dim S = dim M Coisotr op ic (mu ltip licit y -fr e e ) Comp le x Sy mp le ctic Str on g ly Visib le Comp le x E x a mp le s of Vis ib le a ct ion s E g . T y C is v is ib le . ∪ R ⇓ E g . T n y C n is v is ib le . ∪ n R ⇓ E g . T n y P n − 1 C is v is ib le . ∪ P n − 1 R ⇓ E g . U (1) × U ( n − 1) y Bn ( fu ll fl a g v a rie ty ) is v is ib le . E g . U ( n ) y P n − 1 C × Bn is v is ib le .
Unde rs ta nding of v is ible ac tions H L ⊃ ⊃ G ∪ σ G := T n U (1) × U (n − 1) ⊃ ⊃ U (n ) ∪ O (n ) G e ome tr y ² ± ¯ ° P n − 1 R me e ts e v e ry T n -o rbit on P n − 1 C k G σ / G σ ∩ L k H k G/ L (v isible ac tion) ← → G roup ® © ª G = H G σ L ⇒ Eg.3 l G roup ® © ª G = LG σ H ⇒ Eg.9 l G roup ² ± ¯ ° (G × G ) = diag( G )( G σ × G σ )( H × L ) ⇒ Eg.7 ⇓ T h e o re m V a riou s k in d s of MF re s u lt in clu d in g • (F ou rie r s e rie s ) T y L 2 ( T ) E g .2 • (T a y lo r s e rie s ) T n y O ( C n ) E g .3 • ( GL n ↓ GL n − 1 ) R e s tr iction π |GL n − 1 E g .9 • (Pie ri) π ⊗ S k ( C n ) E g .7 • (Ka c) GL m − 1 × GL n y S ( C m n ) E g .12
E g . G / K comp a ct s y mm. s p . ⇒ G y G C / K C is v is ib le . ⇓ T h e o re m E g .16. G y L 2 (G / K ) is M F . E g .17. ( v e ct o r b u n d le ca s e ) V k = U (n ) × O (n ) Λ k ( C n ) → U (n )/ O (n ) U (n ) y L 2 ( V k ) is M F . E g . G / K n on -comp a ct s y mm. s p . ⇒ G y Ω ⊂ cr o w n G C / K C is v is ib le . ⇓ T h e o re m E g .16 ′ . G y L 2 (G / K ) is M F . E g .17 ′ . ( v e ct o r b u n d le ca s e ) V k = G L (n , R ) × O (n ) Λ k ( C n ) → U (n )/ O (n ) G L (n , R ) y L 2 ( V k ) is M F .
G / K He r mit ia n s y mm. s p a ce E g . G = S L (2 , R ) K = S O (2) H = a 0 0 a − 1 : a > 0 G / K ≃ { z ∈ C : |z | < 1 } K y G / K v is ib le H y G / K v is ib le K -o r b it s H -o r b it s T h e o r e m H ⊂ G ⊃ K As s u me G / K He r mitia n s y mm. s p . (G ,H ) s y mme tr ic p a ir ⇒ H y G / K is v is ib le ⇓ T h e o r e m E g . 18 π λ ,π µ : h ig h e s t w t . mo d u le s of s ca la r t y p e ⇒ π λ ⊗ πµ is M F E g . 19 π λ : h ig h e s t w t . mo d u le of s ca la r t y p e (G ,H ) : s y mme tr ic p a ir ⇒ π λ | H is M F
Als o, fo r fi n it e d ime n s ion a l ca s e ⇓ T h e o re m E g .20 ( Ok a d a , 1998) g C = gl ( n , C ) λ = ( a ,· ·· , a | {z } p , b ,· ·· , b | {z } n − p ) ∈ Z n , a ≥ b π λ | h C is MF if h C = gl ( k , C ) + gl ( n − k , C ) (1 ≤ k ≤ n ) o ( n , C ) sp ( n , 2 C ) ( n : e v e n ) G : n on -comp a ct, s imp le L ie g p ., G/ K He r mitia n E g . S U (p ,q ), S O (n, 2) ,S p (n, R ) , S O ∗(2 n ), E6( − 14) ,E 7( − 25) gC = kC + p + + p − D e f. (π ,V ) ∈ c G u n ita r y h ig h e st w t r e p ⇔ { υ ∈ V ∞ : dπ (X )υ = 0 ( ∀ X ∈ p + )} 6= 0 µ K W r ite π = π G (µ ) (µ ∈ c K ) D e f. π : h olomo r p h ic d is cr e te s e r ie s ⇔ Hom G (π ,L 2 (G )) 6= 0 π : s ca la r t y p e ⇔ di m µ = 1
D e f. (G ,H ) s y mme t r ic p a ir , h olomo r p h ic t y p e ⇔ ∃ τ ∈ A ut( G ), τ 2 = id s .t . (G τ ) 0 ⊂ H ⊂ G τ τ y G / K h olomo r p h ic g C = k C + p + + p− ∪ ∪ ∪ ∪ h C = g τ C = k τ + C p τ ++ p τ − t τ ⊂ k τ Ca r ta n ∩ ∩ Ca r ta n t ⊂ k { ν 1 ,. .. ,ν k } ma x ima l s e t of s t r on g ly o r t h . r o ot s in ∆ ( p − τ + , t τ ) Not e : k = R -r a n k G / H D e f. (G ,H ) s y mme t r ic p a ir , h olomo r p h ic t y p e ⇔ ∃ τ ∈ A ut( G ), τ 2 = id s .t . (G τ ) 0 ⊂ H ⊂ G τ τ y G / K h olomo r p h ic E g . τ = θ , H = K E g . G = S p (n , R ) (n = p + q ) H = S p (p , R ) × S p (q , R ) H = U (p ,q ) H = U (n ) (= K ) cf . H = G L (n , R ) n ot h olo. t y p e
T h e o re m π G (µ ) ∈ ˆ G: h olo. d is c. s e rie s , s ca la r ty p e (G , H ) : s y mme t ric p a ir , h olomo rp h ic ty p e ⇒ π G (µ ) H ≃ X ⊕ a1 ≥· ·· ≥ ak ≥ 0 aj ∈ N π H µ ¯ ¯ ¯ t τ − k X j=1 aj νj H = K · · · Sch mid ( 1969) G L ( n , C ) y N ⊂ n ilp ote n t o rb it M ( n , C ) E g .21 ( P a n y u s h e v 1994) O ( N ) is MF a s G L ( n , C )-mo d u le ⇔ N = N (2 p ,1 n − 2 p ) (0 ≤ ∃ p ≤ n ) 2 N (2 p ,1 n − 2 p ) ∋ 0 1 0 0 p ·· · z }| { 0 1 0 0 0 ·· · n − 2 p 0 z }| {
E g . 0 ≤ 2 p ≤ n 1) U ( n ) y N (2 p ,1 n − 2 p ) is v is ib le . 2) (P a n y u s h e v) O ( N 2 p ,1 n − 2 p ) is MF N (2 p ,1 n − 2 p ) ∋ 0 1 0 0 p ·· · z }| { 0 1 0 0 0 ·· · n − 2 p 0 z }| { Sk e t ch of p ro of G = U ( n ) H = U ( p ) × U ( q ) ( p + q = n ) G × H M ( p , q ; C ) → N (2 p ,1 n − 2 p ) H y M ( p , q ; C ) v is ib le ⇒ G y N (2 p ,1 n − 2 p ) v is ib le Mo re e x a mp le s of v is ib le a ct ion s E g . n1 + n2 + n3 = p + q = n Gr p ( R n ) me e t s e v e ry o rb it of U ( n 1 ) × U ( n 2 ) × U ( n 3 ) y Gr p ( C n ) ( ⇒ v is ib le a ct ion ) ⇔ ( U ( n1 ) × U ( n2 ) × U ( n3 )) × O ( n ) × ( U ( p ) × U ( q )) ։ U ( n ) ⇔ m in( n 1 + 1 , n 2 + 1 , n 3 + 1 , p , q ) ≦ 2
⇓ T h e o r e m MF p r op e r t y of t h e follo w in g • GL m × GL n y S ( C m n ) E g .11 • GL m − 1 × GL n y S ( C m n ) E g .12 • th e Ste mb r id g e lis t of πλ ⊗ πν E g .8 • GL n ↓ ( GL p × GL q ) E g .22 • GL n ↓ ( GL n1 × GL n2 × GL n3 ) E g .23 • ∞ -d ime n s ion a l v e r s ion s . . . ⊗ -p ro d u ct re p . ( con t in u e d ) λ = ( a , · · · , a | {z } p , b , · · · , b | {z } q ) ∈ Z n , a ≥ b E g . 8. (Ste mb rid g e 2001, K–) πλ ⊗ πν is MF a s a G Ln ( C ) -mo d u le if 1) mi n( a − b , p , q ) = 1 (a nd ν is a ny ), o r 2) mi n( a − b , p , q ) = 2 a nd ν is of the fo rm ν = ( x · · · x | {z } n1 y · · · y | {z } n2 z · · · z | {z } n3 ) ( x ≥ y ≥ z ), o r 3) mi n( a − b , p , q ) ≥ 3, & mi n( x − y , y − z , n1 , n2 , n3 ) = 1. ⋆ ⋆
E g .22. ( GL n ↓ ( GL p × GL q )) n = p + q π GL n ( x, .. ., x, y ,. .. ,y ,z ,. .. ,z ) |GL p × GL q is MF µ n1 µ n2 µ n3 if mi n( p , q ) ≤ 2 o r if mi n( n1 , n2 , n3 , x − y , y − z ) ≤ 1 (Kos ta n t n3 = 0 ; Kr a tte n th a le r 1998) E g .23. ( GL n ↓ ( GL n1 × GL n2 × GL n3 )) n = n1 + n2 + n3 π GL n ( a ,. .. ,a ,b, .. ., b) |GL n1 × GL n2 × GL n3 is MF µ p µ q if mi n( n1 , n2 , n3 ) ≤ 1 o r if mi n( p , q , a − b) ≤ 2 Or b it me th o d G A d y g ∗ ⊃ O G ∪ ↓ p r H A d ∗ y h ∗ ⊃ O H H ⊂ G ⊃ K sy mm. p air He rmitia n sy mm. p air g ⊃ k ⊃ ce n te r ⇔ R · z ⊂ g ∗ 6= 0 T h e o re m (Na s rin & K-) If O G ∩ R · z 6= ∅ , th e n # (O G ∩ p r − 1 (O H )) / H ≦ 1 fo r a n y O H .
