Reductive Dual Pair Weil compact Weil Howe duality non-compact pair Sp(2n; R) O(k) U(p; q) U(k) popular K-type Sp O x7 [Kas
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1 Reductive Dual Pair Weil compact Weil Howe duality non-compact pair Sp(n; R) O(k) U(p; q) U(k) popular K-type Sp O x7 [Kashiwara-Vergne] [Howe4, Howe6] 3 (?) x7 Gelfand-Kirillov Bernstein Bernstein Selberg 1 G : Lie ( ) G (Cartan-Weyl ) G GC G GC 1 1 Weyl Flensted-Jensen Flensted-Jensen duality 3 1
2 Young [Fulton-Harris], [Knapp], [ ] bg G b G V G ( ) (ρ; H) H ' X Φ XΦ X Φ Hom G (V ;H) Ω V ' (H Ω V Λ ) G Ω V b G b G b G (H Ω V Λ ) G Ω V 3 X h Ω v Λ Ω v 7! v Λ (v)h H (H Ω V Λ )G V H X G G-( ) H = L (X) G (H Ω V Λ ) G = (L (X) Ω V Λ) G ' ff : X! V Λ j f(xg) = Λ (g) 1 f(x) (g G)g =: L (X; V Λ) L (X) L (X) ' X Φ ' b G L (X; V Λ) Ω V X Φ b G HomC V Λ;L (X; V Λ) P : L (X)! HomC Z V Λ ;L (X; V Λ ) ; (P f)(v Λ )(x) = f(xg) Λ (g)v Λ dg G V fe i j 1» i» dim V g fe Λ i j 1» i» dim V gρ V Λ = V Λ X dim XV Z L (X) 3 f(x) $ f(xg) Λ (g)e Λ i dg X Ω e i L (X; V Λ) Ω V b G b G i=1 G () Peter-Weyl
3 Exercise 1.1 (1) () X = G L (G; V Λ) G L (G; V Λ) ' V Λ Peter-Weyl () Peter-Weyl L (G) G G dual pair Weil tensor 1 Sp(n; R) Weil (L; L (R n )) k tensor tensor 1. unitary highest weight Weil K-weight ( ) tensor () " 3. (pluriharmonic functions) tensor ( lowest K-type) tensor Ω k L (R n ) ' L (R n Φ ΦR n ) ' L (R n Ω R k ) ' L (M(n; k; R)) (L; L (R n )) Sp(n; R) Φ Ψ» 0 Sp(n; R) = g SL(n; R) j t 1n gjg = J ; J = 1 n 0 4 L Ωk» a t a 1 f(x) = (det a) k= f( t ax)(a GL(n; R)) (.1) L Ωk» 1 b 1 L Ωk» f(x) = exp( i Tr t xbx=)f(x)( t b = b) (.) f(x) = i ß nk= Z exp(i Tr t xy)f(y)dy (.3) M(n;k;R) metaplectic theta (Gauss ) Sp(n; R) 3 [EHW], [Jakobsen], [Parthasarathy] 4 Sp(n; R) Sp(n; R) 3
4 k k O(k) L (M(n; k; R)) Sp(n; R) Fourier Tr ( ) O(k) Exercise.1 O(k) Sp(n; R) Example. k = 1 Weil O(1) = Z intertwiner Weil L = L + Φ L O(1) R n f±1g 5 O(k) tensor Sp(n; R) O(k) ρ Sp(nk; R) dual pair Definition.3 ([Howe], [Howe4]) (reductive) dual pair Sp(n; R) reductive (G; G 0 ) G Sp(n; R) commutant subgroup G 0 G 0 commutant subgroup G Remark. dual pair Sp(n; R) Weil dual pair commutant algebra dual pair Howe correspondence Theorem.4 ([Howe4]) (G; G 0 ) ρ Sp(n; R) dual pair G 0 X Φ L ' Hom G 0(V ;L) Ω V c G 0 Hom G 0(V ;L)=L (R n ; Λ ) G (G 0 )^ 3 7! L (R n ; Λ ) G^ L (R n ; Λ ) 6= 0 (Howe ) 5 O(1) = Z metaplectic 4 Z 4 intertwiner 4 " f( x) =if(x) L 0 dual pair 4
5 Remark. G Hermitian type L (R n ; Λ ) 6= 0 non-compact ([Howe5]) [Howe1] primitive O(p; q) SL(; R) Example.5 U(p; q) U(1) ρ Sp(n; R) (n = p + q; p q) dual pair (U(1) U(p; q) ) U(1) U(1) 3 e i 7! e ik (k Z) k Sp(n; R) Weil (L; L (R n )) dual pair X (L; L (R n Φ )) ' L (R n ; k ) Ω k kz Z ladder Sp(n; R) Weil k tensor dual pair Sp(n; R) O(k) ρ Sp(nk; R) Sp Weil pair L (M(n; k; R)) ' X Φ O(k)^ L (M(n; k; R); V ) Ω V Λ L (M(n; k; R); V )= ff : M(n; k; R)! V j f(xh) = (h) 1 f(x) Sp(n; R) dim V Λ (x M(n; k; R);h O(k))g 3 : L (M(n; k; R); V ) Siegel G = Sp(n; R) Siegel H n H n ' G=K (K ' U(n)) K ρ» ff A B K = B A fi A + ib U(n) G Lie g gc k; kc g = k Φ p Cartan pc K adjoint pc K p ± gc = p Φ kc Φ p + (Exercise 7.4 ) 5
6 Exercise 3.1 G=K ' H n = fz Sym(n; C ) j Im z > 0g (1) G = Sp(n; R) H n» a b G 3 g = H c d n 3 z 7! g z =(az + b)(cz + d) 1 H n well-defined (Hint) () p 1 1 n H n K G=K ' H n (1) G=K 3 gk $ g p 1 1 n H n Exercise 3. (1) K G K K = fg G j gkg 1 = Kg [Hint] Cartan 6 G = KAK ([Knapp, Theorem 5.0]) A G split Cartan () well-defined G=K 3 gk 7! gkg 1 fk G- g G G Siegel (3) p- Siegel H n Sp(n; R) cohomological induction 7 (fi;u fi ) KC = GL(n; C ) O(H n ; fi) = ff : G! U fi : C 1 j f(gk) =fi(k 1 )f(g);r(x)f =0(X p )g ' ff : H n! U fi : holomorphic g R(X) X gc X g R(X)f(g) = d dt f(g exp tx) fi t=0 gc G O(H n ; fi) G Siegel a b (T (fi)(g)f)(z) =fi( t (cz + d))f((az + b)(cz + d) 1 ) z H n ;g 1 = (3.1) c d 6 Lie Cartan A Lie p ( Cartan ) 7 cohomological induction [Knapp-Vogan, Theorem 8.], [Wallach, Theorem 6.7.6] lowest K-type 1 cohomological induction ( ) lowest K-type 1 [Adams] 6
7 Exercise 3.3 (1) G = Sp(; R) = SL(; R) P : C 1 (G; fi) 3 f 7! F O(H;fi)» a x F (z) =fi(a)f (z = ax + a i (a >0)) 0 a 1 fi(e i )=e im fi(a) =a m (a C ) C 1 (G; fi) =ff : G! U fi : C 1 j f(gk) =fi(k) 1 f(g)g kc = ρ» 0 0 fi C ff ; p ± = ρ» ' ±i' ±i' ' fi ' C ff R(X)f(g) =0(X p ) F (z) Cauchy-Riemann () P G- O(H;fi) G (3.1) (3) F (z) =fi((z + i)=i) F (z) X p + O(H n ; fi) fi 8 Weil intertwining F P : L (M(n; k; R); V )!O(H n ; fi Ω det k= ) P : M(n; k; R)! HomC (U fi ;V ) ' U fi Λ Ω V : (F P f)(z) = Z M(n;k;R) e (i=) Tr(t xzx) P (x) Λ f(x)dx (f L (M(n; k; R); V )) (3.) F P Sp(n; R) intertwining P (x) (1) P (xh) = (h) 1 P (x) (h O(k)) () P (ax) =P (x)fi(a) 1 (a GL(n; C )) (3) P (x) O(k)-harmonic P (x) Λ U fi Ω V Λ ;U fi ) F P f O(H n ; fi) (1) V L f(x) (h)f(xh) 8 O(H n ; fi) Verma paring hf;d Ω vi = h(r(d)f)(1);vi fi (f O(H n ; fi);dω v U(g C ) Ω U(k CΦp ) U fi ) 7
8 f(x) (h O(k)) F P L (M(n; k; R); V ) (h)f(xh) =f(x) () g(a) F P t(b) F P g(a) = a t a 1 (a GL(n; R)); t(b) = 1 b 1 ( t b = b) (3) ff (Fourier ; (.3) ) Exercise ff = 1 0 Theorem 3.4 ([Kashiwara-Vergne]) P (x) (1) (3) F P : L (M(n; k; R); V )! O(H n ; fi Ω det k= ) intertwining O(H n ; fi Ω det k= ) L (M(n; k; R); V ) 9 Remark. L (M(n; k; R); V ) k» n 1 k n +1 =) L (M(n; k; R); V ) k =n =) L (M(n; k; R); V ) k» n 1 =) L (M(n; k; R); V ) Exercise 3.5 F P ff (1) Parseval P Fourier e (i=)tr txzx P (x) Λ ^ (y) = det z k= e (i=)tr t y( z 1)y P ( z 1 y) Λ i (x; y M(n; k; R);z H n ) F ^(y) R nk ' M(n; k; R) Fourier () z = iff ( t ff = ff) z = iff expf (1=) Tr t xxgp (x) Λ ^ (y) = expf (1=) Tr t yygp ( iy) Λ 9 L (M(n; k; R); V ) dual pair (.4) [Kashiwara-Vergne] 8
9 (3) y y = ifi P (fi) Λ = (nk=) P (r! + fi) ß ZΩ Λ d! (r >0; d! (nk 1)- Ω ) nk= () Exercise 3.6 P (x) (1) C = fο M(n; n; R) j t ο = ο;ο 0g C k = fο C j rank ο» kg Q : M(n; k; R)=O(k)! C k [j [j x 7! ο(x) =x t x GL(n; R)- () f(x) L (M(n; k; R); V ) P (x) Λ f(x) O(k)- '(x t x)=p (x) Λ f(x) C k (3) C k d k ο C k ' Z C k '(ο)d k ο = Z M(n;k;R) '(x t x)dx F P f(z) = Z C k e (i=)tr ο z '(ο)d k ο F P '(ο)d k ο Fourier-Laplace 4 F P P (x) O(H n ; fi)? P (x) H =(O(k)- ) K V K- K- ( [Helgason] ; [ 1] ) f(v) C [V ] def =0(8h S(V ) K + ) 9
10 O(k) H = ( f : M(n; k; C )! C fi i;jf = j;ν f =0(1» i» j» n) (1) (3) P (x) H GL(n; C ) O(k;C ) joint action Theorem 4.1 ([Howe6, Proposition 3.6.3]) H GL(n; C ) O(k;C ) 1 X Φ H = U fi (D) Ω V (D) D D Young 10 (= `(D)) minfk; ng D =(μ 1 ; ;μ k ) `(D) > k= μ j = 1 (`(D) j > k `(D)) ρ (μ1 ; ;μ fi(d) =(μ 1 ; ;μ n ); (D) = k ) GL(n; C ) O(k;C ) 11 (`(D)» k=) (μ 1 ; ;μ k `(D) ) (`(D) >k=) Theorem 4. (1) M(n; k; C ) C [M(n; k; C )] O(k;C )- : C [M(n; k; C )] = H C [M(n; k; C )] O(k;C ) O(k;C )^ O(k;C ) C [M(n; k; C )]( ) = H( ) C [M(n; k; C )] O(k;C ) H( ) GL(n; C ) O(k;C ) H( ) =(C [M(n; k; C )]( ) ) 10 Young D μ =(μ 1 ; ;μ k ) μ 1 μ μ l > 0=μ l+1 = = μ k 0 `(D) =`(μ) =l (0 μ i ) 11 SO(k) O(k) [Howe6, x3.6.] SO(k) ffl `(D) =l<k= V (D) SO(k) D =(μ 1 ; ;μ l ) SO(k) ffl `(D) =k= V (D) SO(k) (μ 1 ; ; ±μ k= ) ffl `(D) =l>k= V (D) SO(k) D 0 =(μ 1 ; ;μ k l ) SO(k) O(k) V (D) = V (D 0 ) Ω det ) 10
11 () k>n tensor C [M(n; k; C )] = HΩC [M(n; k; C )] O(k;C ) O(k;C ) eο i;j (x) = kx ν=1 x i;ν x j;ν (x M(n; k; C ); 1» i; j» n) e οi;j C [M(n; k; C )] O(k;C ) : C [M(n; k; C )] O(k;C ) = C [e οi;j j 1» i; j» n] n k Sym(n) Q Q : M(n; k; C )! Sym(n) [j [j x 7! x t x (Exercise 3.6 ) Q(ax) =aq(x) t a (a GL(n; C )) Q(xh) =Q(x) (h O(k;C )) Q Λ : C [Sym(n)]! C [M(n; k; C )] O(k;C ) : surjective GL(n; C )-homomorphism Theorem 4.3 (1) GL(n; C ) C [Sym(n)] C [M(n; k; C )] O(k;C ) C [Sym(n)] ' () GL(n; C ) X Φ D;`(D)»n U fi (D) ; C [M(n; k; C )] O(k;C ) ' k n C [Sym(n)] ' C [M(n; k; C )] O(k;C ), k<n C [Sym(n)] 6' C [M(n; k; C )] O(k;C ) X Φ E;`(E)»minfn;kg U fi (E) fe οi;j j 1» i» j» ng k n k<n P : M(n; k; C )! V Λ Ω U fi 3.4 GL(n; C ) O(k; C ) P (HΩ(V X GL(n;C ) O(k;C ) Λ Ω U fi )) ) O(k;C ) = (V (D) Ω U fi (D) ) Ω (V Λ Ω U fi ) GL(n;C D = X D V (D) Ω V Λ O(k;C ) Ω U fi (D) Ω U fi GL(n;C ) ' ρ C if = (D) andfi Λ = fi(d) for 9D 0 otherwise 11
12 P 6= 0 ( Λ ;fi)=( (D) Λ ;fi(d) Λ ) P 5 Weil tensor Weil infinitesimal Lie tensor lowest K-type S(R n ) ρ L (R n ) Schwartz G = Sp(n; R) Lie gc = sp(n; C )» Ei;j A i;j = 0 t 7! x E i + 1 j ffi i;j» 0 Ei;j + E B i;j = j;i 7! p 1 x 0 0 i x j C i;j =» 0 0 E i;j + E j;i 0 7! j E i;j (i; j) 1 Exercise 5.1 A i;j ;B i;j (.1) (.) C i;j» 0 1 Adff(B i;j )= C i;j ; ff = 1 0 ff K ( [Howe-Tan, xiii..1] ) Fock type L Schrödinger type a i =(x i ) ; a Λ i =(x i i ) v = exp( jxj =) S(R n ) Φ Φ:C [a i j 1» i» n] 3 p(a 1 ; ;a n ) 7! p(a 1 ; ;a n )v S(R n ) a Λ i S(Rn ) Φ a Λ i v =0; [a Λ i ;a j ]=ffi i;j a Λ i C [a i j 1» i» i gc C [a i j 1» i» n] 1
13 kc (1) A i;j A j;i (i 6= j) () B i;j C i;j (1) C [a i j 1» i» n] x x i = a i + a Λ i aλ j a j x j = i (a ia Λ j a i a j ) $ a j () B i;j C i;j p 1 x i x $ p j a j (1), () kc a + j i i a j gl(n; C ) 1= "(renormalized) k ( ) pc (3) A i;j + A j;i (4) B i;j + C i;j (3) : x i p 1 ψ n X j=1 a + + x j $ i a ia j j (4) : p 1 x i x j $ p j 1 ia j j pc a i a j 13
14 kc ff p,fa i a j g; p j (5.1) k tensor L (M(n; k; R)) ffs(m(n; k; R)) ff C [M(n; k; R)]e Tr t xx= =Φ(C [a i;j j 1» i» n; 1» j» k]) (5.) C [M(n; k; R)]e Tr t xx= (gc ;K)- C [a i;j j 1» i» n; 1» j» k] (gc ;K)- Lemma 5. Sp(nk; R) L (M(n; k; R)) Harish-Chandra (gc ;K)- 1 C [a i;j j 1» i» n; 1» j» k] Φ (lowest K-type), p + C [a i;j j 1»i»n;1»j»k] f p +, f O(k)- i.e., f H Theorem 4. C [a i;j j 1»i»n;1»j»k] = H C [a i;j j 1»i»n;1»j»k] O(k) = X Φ D;`(D)»minfn;kg H( (D)) C [a i;j j 1»i»n;1»j»k] O(k) (5.3) H( (D)) ' V (D) Ω U fi (D)+ k 1 13 U fi (D)+ k 1 lowest K-type sp(n; C ) L(fi(D)+ k 1) H( ) C [a i;j j 1»i»n;1»j»k] O(k) = V Ω L(fi(D)+ k 1) 1 =(1; 1; ; 1) (gc ;K)- L(fi(D)+ k 1)=L (M(n; k; R) : Λ ) K 14 G V V K V K- 1 Harish-Chandra 13 K C ' GL(n; C ) renormalize k 1 1 =(1; 1; ; 1) 14 L (M(n; k; R); ) 'O(H n ; fi Ω det k= ) det k= K C -weight k 1 k 1 14
15 k>n Theorem 4. tensor X C [a i;j ]=HΩC[a i;j ] O(k) Φ = V (D) Ω L(fi(D)+ k 1) = U fi (D)+ k 1 Ω C [a i;j ] O(k) = 8 >< >: D U fi (D)+ k 1 Ω X Φ E;`(E)»n U fi (E) U fi (D)+ k 1 Ω C [a i;j ] O(k) (KC = GL(n; C ) ) U(gC ) Ω U(p+ ΦkC) U fi (D)+ k 1 ((gc ;K) ) L(fi(D)+ k 1) [ ], [Schmid], [Varadarajan] K-type K- type Blattner [Knapp, p. 736], [Knapp-Vogan, (5.108b)], [Hecht-Schmid] K- type U fi (D)+ k 1 Ω U fi (E) K-type 15 : Fock type lowest K-type 6 (cf. [, x., x.3]) ( Fourier ) x4 P (x) M(n; k; R) O(k) GL(n; C ) ( ; fi)- P 6= 0 Young D ( ; fi) =( (D);fi(D) Λ ) (cf. Theorem 4.1) P (x) Fourier F P (3.) Definition 6.1 H n H n U Λ fi Ω U fi - K D (z; w) (1) K D (z; w) Λ = K D (w; z) () 8u U fi K D (z; w)u z H n U fi - K D (z; w)u O(H n ; fi) (3) L fi = L(fi + k 1) ρ O(H n; fi) 8f(z) L fi 8u U fi hf(w);ui Ufi = hf(z);k D (z; w)ui Lfi 15 Steinberg [Humphreys, x4.4] Littlewood-Richardson tensor [Macdonald, xi.9] 15
16 L fi h; i Lfi L (M(n; k; R); V ) L h; i L hf(z);g(z)i Lfi = hf 1 1 P f(x); F P g(x)i L Remark. [Kashiwara-Vergne] K (z; w) fi Young D K D (z; w) Exercise 6. unique (cf. [, p.51]) Theorem 6.3 K D (z; w) = Z M(n;k;R) e (i=)tr t x(z w)x P (x) Λ P (x)dx Proof. (1), () (3) u U fi f(z) L fi ρo(h n ; fi) 9'(x) L (M(n; k; R); V ) F P ' = f hf(z);k D (z; w)ui Lfi = h(f P ')(z);k D (z; w)ui Lfi = h'(x); (F P ) 1 K D (z; w)ui L K D (z; w)u = Z M(n;k;R) e (i=)tr txzx P (x) Λ e (i=)tr txwx P (x)u dx = F P e (i=)tr t xwx P (x)u (z) ( ) = h'(x);e (i=)tr txwx P (x)ui L = = = Z Z M(n;k;R) fiz M(n;k;R) M(n;k;R) e (i=)tr t xwx h'(x);p(x)ui V dx e (i=)tr t xwx hp (x) Λ '(x);ui Ufi dx e (i=)tr t xwx P (x) Λ '(x)dx; u = hf P '(w);ui Ufi = hf(w);ui Ufi () 16 fl U fi
17 K D (z; w)u = F P e (i=)tr t xwx P (x)u w = i, u = u fi U fi K D (z; i)u fi = F P e (1=)Tr txx P (x)u fi (z) e (1=)Tr t xx P (x)u fi x5 (5.) (5.3) L (M(n; k; R); V ) K D (z; i)u fi Fourier Exercise 6.4 w 6= i K D (z; w)u fi (cf. Exercise 3.) (z) z w K D (z; w) =ff fi i ff R ff = ke (1=)Tr t xx P (x)u fi k L ku fi k fi 7 Gelfand-Kirillov Bernstein Howe n (n =1 SL(; R) ) 7.1 ([EHW], [Jakobsen], [Parthasarathy]) Gelfand-Kirillov Bernstein non-compact Howe G Lie K G = Sp(n; R) V (gc ;K) Gelfand-Kirillov V 17
18 ( base ) V 0 V K- V U(gC ) U l (gc )=fx U(gC ) j X = X t»l X i1 X it (X ij gc )g V l = U l (gc )V 0 (l 0) V K- gr V = 1M l=0 V l =V l 1 (V 1 = (0)) gr V S(gC )=gru(gc )- h(t) = tx l=0 dim gr l V (t Z 0 ) h(t) t t Hilbert h(t) = b d d! td +(lower terms) (7.1) V 0 Definition 7.1 ([Vogan1]) (7.1) d V Gelfand-Kirillov d = Dim V b d V Bernstein Deg V Remark. Gelfand-Kirillov Dim V Bernstein Deg V Bernstein Gelfand-Kirillov Bernstein ( [, 7.1] ) gr V S(gC )- Ann(gr V ) Ann(gr V )=fx S(gC ) j X gr V = (0)g Definition 7. ([Vogan], [Joseph]) S(gC ) g Λ C g Λ C Ann(gr V ) (V 0 ) Ass V = ff g Λ C j X(f) =0(8X Ann(gr V ))g V 18
19 g Λ C g C Killing Ass V ρ g C g C s N s s gr V Theorem 7.3 ([Vogan, Th. 8.4]) V (g C ;K)- V I I Ass I (1) g C = k C Φ p C Cartan Ass V N K p C C - () Ass I N p C ff Ass V (3) Ass I G C O O N = p C ra i=1 O i K C Ass V K C fo i (1» i» r)g (4) 1»8i» r dim O = dim O i = Dim V V 9O V ρ N p C : K C - Ass V = O V ( ) O V V K C - [Vogan], [Ohta], [ ] Gelfand-Kirillov [Yamashita] 7. Weil Gelfand-Kirillov Bernstein (?) Weil Weil (g C ;K)- Fock type C [a i j 1» i» n] (L + ) K =( ); (L ) K =( ) (L ± ) K K- L ± Harish-Chandra (g C ;K)- K- (L + ) K;0 = C =( ); (L ) K;0 = V = (L + ) K V 0 p + () V l U(g C ) U(p )=S(p ) p L K nm i=1 C a i V l = U l (g C )V 0 = U l (p )V 0 =(l ) V 19
20 gr l V =(l ) tx tx n +l 1 =) h(t) = gr l V = n 1 l=0 = n 1 t n + O(t n 1 ) n! Dim L + = n; Deg L + = n 1 Dim L =Dim(L + ΦL )= n; Deg(L + Φ L )= n Dim L = n; Deg L = n 1 Ass(L ± ) V = (L + ) K V 0 = ( ) gr V S(g C ) p + k C gr V l V l p + k C g C V l V l+1 gr V mod V l p + k C Ass V U(p )=S(p ) p x4 S(p ) ' C [Sym(n)] Q Λ (k =1) Ann(gr V )=S(g C )p + + S(g C )k C l=0 +ker(q : C n 3 x 7! x t x Sym(n)) Λ ker Q Λ 1 1 K C i 1 0 n 1 0 n 1 1 i 0 n 1 0 n p+ K C (Killing (p ) Λ = p + ) L + Exercise 7.4 (1) (5.1) ρ» ia A p ± = A ±ia fi t A = Aff () c Sp(n; C ) (Cayley )» c = p 1 1n i1 n i1 n 1 n p + =Adc ρ» ff 0n A 0 n 0 n fi t A = A ; p =Adc 0 ρ» ff 0n 0 n A 0 n fi t A = A
21 L Ann(gr (L ) K )=S(g C )p + + S(g C )k C + Ann(gr V ) U(p ) Ann(gr V ) U(p ) p Ann(gr (L ) K ) = Ann(gr (L + ) K ) k1 L( k 1) Theorem 4. L( k 1) lowest K-type Theorem 4.3 L( k1) ' U ) fi ( k 1) Ω C [M(n; k; C )]O(k;C = M `(μ)»minfn;kg U fi ( k 1+μ) K- k n Gelfand-Kirillov Bernstein k = n k» n V = L( k1) K V V 0 = U fi ( k 1) =( ) V t = U t (g C )V 0 = M `(μ)»minfn;kg;jμj»t U fi ( k 1+μ) (jμj = μ μ k ) dim U fi ( k 1+μ) = dim U fi (μ) dim V t = X `(μ)»minfn;kg;jμj»t dim U fi (μ) Weyl dim U fi (μ) = Q Q 1»i<j»n (μ 1»i<j»n i μ j + j i) (j i) = Q nk k(k+1)= k l=1 (n l)! Y 1»i<j»k (μ i μ j ) ψ Y 1»i»k μ i! n k +(μ ) 16 Ann(L )K = Ann(L + )K 1
22 μ i =0(k <i» n) Q k l=1 dim V t = nk k(k+1)= (n l)! Z 1 x 1 x x k 0; 1 x 1 + +x k 0 Y (x i x j ) ψ Y 1»i<j»k 1»i»k! n k x i dx 1 dx k t k(k 1)=+k(n k)+k +(t ) k(k 1) Dim L( k 1)=k(k 1)=+k(n k)+k = nk Deg L( k nk k(k+1)= (nk k(k 1) +1) 1) = (k +1) (n l +1) ZΩk Y 1»i<j»k Q k l=1 jx i x j j (x 1 x x k ) n k dx 1 dx k (7.) Ω k = f(x 1 ; ;x k ) [0; 1] k j x x k» 1g (7.3) Z[0;1]k Y 1»i<j»k jx i x j j (x 1 x x k ) n k dx 1 dx k Selberg [ -, ] Exercise 7.5 k = Deg L(1) = 1 n n 1? Bernstein Gelfand- Kirillov fi = fi(d) Theorem 4.1 Dim L( k1 + fi) = Dim L( k(k 1) k 1)=nk
23 Theorem 7.6 (1) 1» k» n fi = fi(d) Theorem 4.1 k(k 1) Dim L( k 1 + fi) =nk fi =0 Deg L( k 1) (7.) () k n Gelfand-Kirillov Bernstein Dim L( k +1) 1 + fi) =n(n ; Deg L( k 1 + fi) =dimu fi Deg L( k 1 + fi) n<k» n fi =017 Gelfand-Kirillov Bernstein Proof. () Deg L () K L( k1 + fi) ' U Ω k 1+fi U(p ) U(p ) n(n +1)= Bernstein dim U =dimu k 1+fi fi Bernstein Theorem 7.6 () Corollary 7.7 Ω n (7.3) n ZΩn Y 1»i<j»n Q n l=0 (l +1) jx i x j jdx 1 dx n = n(n 1)= n(n+1) +1 L( k 1 + fi) k» n L( k1) V = L( k1) K Weil Ann(gr V )=S(g C )p + + S(g C )k C +ker(q : M(n; k; C ) 3 x 7! x t x Sym(n)) Λ Im Q = fx Sym(n) j rank X» kg Ass V = fx p + j rank X» kg = O k ; O k = AdK C 6 4 i1 k 1 k 0 n k 0 n k 1 k i1 k 0 n k 0 n k ρ p+ (7.4) V = L( k1 + fi) K Weil p + ; k C ρ Ann(gr V ) U(p ) Ann(gr V ) 17 n<k» n 3
24 p U(p ) Ann(gr V ) ( gr V ) Ann(gr L( k 1 + fi) K) = Ann(gr L( k 1) K); =) Ass L( k 1 + fi) =AssL( k 1) K = O k Ann L( k 1 + fi) K = Ann L( k 1) K ([, 1.5.5] ) Theorem 7.8 (1) 1» k» n fi = fi(d) Theorem 4.1 Ass L( k 1 + fi) =O k K C - O k (7.4) () k n Ass L( k 1 + fi) =O n = p + Corollary 7.9 (1) 1» k» n fi = fi(d) Theorem 4.1 L( k 1 + fi) AdG C 6 4 i1 k 1 k 0 n k 0 n k 1 k i1 k 0 n k 0 n k =AdG C 4 0 n 1 k 0 n k 0 n 0 n () k n L( k 1 + fi) AdG C (p + ) Theorem 7.3 ([Yamashita, Th. 3.] ) [Kobayashi, Theorems 3.1 & 3.7] dual pair Gelfand- Kirillov dim Ass V = Dim V Ass V ρ p + p + K C - p + = `n O k=0 k Ass V Exercise 7.10 dim O k 3 5 4
25 [Howe3] (rank) Ass V = O k Rank V = k () Weil tensor Ass V (wave front set) G- (cf. Exercise 3.6) ([Sekiguchi]) Ass V References [Adams] J. Adamas, Unitary highest weight modules, Adv. in Math., 63(1987), [ - ] /,,,, [EHW] T. Enright, R. Howe and N. Wallach, A classification of unitary highest weight modules, in Representation theory of reductive groups, Progress in Math. 40, Birkhäuser, 1983, pp [Fulton-Harris] W. Fulton and J. Harris, Representation Theory A First Course, GTM (RIM) 19, Springer-Verlag, [Hecht-Schmid] H. Hecht and W. Schmid, A proof of Blattner's conjecture, Invent. Math., 31(1975), [Helgason] S. Helgason, Groups and Geometric Analysis, Academic Press, [ ],,, [Howe1] R. Howe, On some results of Strichartz and of Rallis and Schiffman ( ), J. Funct. Anal., 3(1979), [Howe] R. Howe, Dual pairs in physics: Harmonic oscillators, photons, electrons, and singletons, in Applications of Group Theory in Physics and Mathematical Physics, Lectures in Applied Mathematics vol. 1, AMS, 1985, pp [Howe3] R. Howe, Small unitary representations of classical groups, in Group representations, ergodic theory, operator algebras, and mathematical physics, MSRI Publications vol. 6, Springer-Verlag, 1987, pp [Howe4] R. Howe, Remarks on classical invariant theory, Trans. AMS, 313(1989), [Howe5] R. Howe, Transcending classical invariant theory, Journ. AMS, (1989),
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量子力学 問題
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第5章 偏微分方程式の境界値問題
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(Gen KUROKI) 1 1 : Riemann Spec Z 2? 3 : 4 2 Riemann Riemann Riemann 1 C 5 Riemann Riemann R compact R K C ( C(x) ) K C(R) Riemann R 6 (E-mail address: kuroki@math.tohoku.ac.jp) 1 1 ( 5 ) 2 ( Q ) Spec