to the context of Remark 3 1 ).

Ü

I f r i s a p o la r iz a t io n o f (V ,n ) then ( r e f e r r in g to (1 .4 0 )) we denote by Mpc (V , il; r ) the f u l l preimage o f S p (V ,fl;r) c Sp(V ,n) under a:Mpc (V ,n ) -► S p (V ,n ) .

I f r « Lag+ (V ,n) then ( in view o f P ro p o sitio n 5 .6 ) the complex lin e ( E ' ) r 1s mapped to I t s e l f under the m e ta p le ctlc a c tio n o f Mpc (V ,fl;r) , d e fin in g a ch aracte r

x r : Mpc (V , n ;r ) -

i

x r (U) - X D e t^ I+ Z g Z j,)* 1 (5.1 0 )

Detj.00 .

Proposition 5.7 : I f r e Lag+(V ,n ) then

(tj, ) 2 • Dttj.00 = nr . (5.11)

Proof: Let U e Mpc (V ,i2;r) have parameters ( x , g ) . Then we have

n (U ) - {X Deti (I+ZgZr ) _1) 2{D e t(C g (I+ZgZr ) ) } . (5.12)

(5 .1 1 ) now follows from (1.53) (5.10) and (5 .1 2 ).

□

Remark 5.8 : I f r i s an a r b it r a r y p o la r iz a t io n o f (V ,n ) then Mpc (V,£J;r) a c ts on the complex lin e H ^ r ^ ( r ; E ') and the ch ara cte r tj, o f Mpc ( V , n ;r ) so defined continues to s a t i s f y (5 .1 1 ). See B la tt n e r & Rawnsley CBR] f o r d e t a ils .

//

The fo llo w in g r e s u lt sh ou ld be compared w ith P r o p o s itio n s 1.17(11) and 1.21 :

P ro p o sitio n 5.9 : I f (r^ .T g ) 1s a tra n sv e rse p a ir o f p o s it iv e p o la r iz a t io n s o f (V ,n ) then ( * . * ) o r e s t r i c t s to d e fin e a n o n sin g u la r sesqu111near p a ir in g

r

r

( E ‘ ) 1 x ( E ‘ ) 2 -► (I and

( f , f ) Q = Det*(I-Z Z f 1 . (5 .1 3 )

r l 12 u 2 1

Proof: An instance of Proposition 3.8.

□ Remark 5.10 : In order to determine the way in which the pairing

(5 .1 3 ) transforms under the action of g e S p (V ,n ) on Lag+(V ,n ) , we can e ith e r proceed as in Rawnsley [Ry51 o r apply Proposition 5.6 ; we obtain

t V , , V 2 , o ' | t e t c g | t e t , ( 1 * V s )D' * ! < I *Z9 V < f r r V 0 • <5' , 4 ) //

I f L i s a L agran gian subspace o f (V ,n) then L* « Lag+ (V,fl) (in d e e d , L® has type (m,0) ; moreover, a l l type (m,0) p o la r iz a t io n s o f (V ,n ) a r i s e 1n t h is way). The r e s u lt s o f t h is se c tio n consequently have Im p lic a tio n s f o r L . In the next se c tio n we co n sid e r the case o f an Is o t r o p ic subspace o f (V ,n ) which 1s not L agran gian .

§6. Passing to Symplectic Normals.

Let L be an isotropic subspace of (V ,n ) such that L * * 0 + LX/L .

We denote by ( E 1)^ the space of a ll vectors in E' which are annihilated by L in the representation W , thus

( E ' ) L = { f e E' | t e L » N (t « 0 ) f * 0 } . ( 6 . 1 )

This extends the concept of vacuum state from §5. However, as we shall see in Proposition 6 . 4 , ( E ' ) L is infinite-dimensional as lV l + 0 .

Proposition 6.1 : ( E ' ) L c £' is stable under W(v,0) f o r v « L 1 and W(v,0) acts t r i v i a l l y on ( E 1)*" fo r v « L . Thus W induces a representation

WL :N (La/L,Ol ) A u t ( E ' ) L . ( 6 . 2 )

• •

Proof: For convenience, w rit e W(v) ■ W(v,0) and W(v) ■ W(v ® 0) for v « V . I f v.j ,Vg e V then

which upon d if fe r e n tia tio n yields

[W(V l ),W (v2 ) ] = ^ n ( v 1, v 2 )W(v1) . (6 .3 )

I f f « ( E 1)*" and v-| e L1 then ( 6 . 3 ) implies v2 « L => W(v2 )W (v^)f * 0

whence W (v^)f e ( E ' ) L . This establishes the L1 - s t a b i l i t y of ( E 1)*" c E ‘ . I f f e ( E 1)*- then Integration of the constraint

v « L *> W (v)f * 0

a lo n g one-parameter subgroups shows th a t

v e L ■> W (v )f * f

whence L a c ts t r i v i a l l y on ( E ' ) L . We may thus d efine (6 .2 ) by

WL (v L v . t ) - W ( v , t ) | ( E * )L (6 .4 )

f o r v c L1 and t « P . I t 1s ro u tin e to check th a t W*" i s a re p re se n ta tio n .

r L L

z e V => f L ( z ) = exp < z,Z Lz> .

Since L® c r L i t is imnediate that « ( E ' ) L •

Proposition 6.2 : I f f e ( E 1)*" then

f * (foPL ) - f L .

Proof: I f g:V^ 0 is holomorphic then

W (v«0)(gfL ) ( z ) = - f L (z )d g z ( v )

fo r v € L and z e V ; consequently gfL e ( E ' ) L i f f dg for a l l z « V . Since dgz 1s J-11near f o r a l l z « V 1 that

gf^ e ( E ' ) L <■ > (z e V •> dgz |L#JL = 0) .

Now suppose f « (E ' ) L and d efine g by

f - g - f L •

From (6 .8 ), g e q u a ls hoP|_ fo r some holomorphic fu n c tio n sin c e f J VL s i » 1 t 1s c le a r th a t h ■ f|VL . (6 .5 ) (6.6) ( 6 . 7 ) follows (6.8) h:VL 8 ; □

Let E |_ c ^|_ c be the r ^99ed H ilb e r t space in the BS model fo r (LA/L .Q L ;JL ) . R e call th a t we id e n t if y ( VL ,n| VL ;J | VL ) and

, v (L /L ,tlL ;J L ) by means o f (1 .7 7 ). Proposition 6.3 : ( i ) I f f e ( E •) L then f|VL £ . ( i i ) I f h c E^ then (hoPL ) . f L £ ( E 1) L .

P ro o f; Holom orphlcity i s c le a r ; we need o n ly check the a p p ro p riate growth c o n d itio n s.

( i ) I f p < 0 , Zj £ VL , z2 £ L9JL , then

(1 ♦ |z1 |2 )p ( l +

|

2|2) P

|

1+

2|2)

A p p ly in g the F u b in l-T o n e lli theorem to (6 .9 ) we see th a t 1 f f £ Fp (V ,0 ;J ) fo r some p < 0 then f|VL £ Fp (VL ,o|VL JJ|VL ) and

I M VlI Ip s ( l ! f L M p r 1||f||p . (6.10)

(11) I f p < 0 , z, < VL , z2 c L I JL , then

Applying the Fu b in i-To n e lli theorem to (6.11) we see that i f

h e Fp (VL ,n|VL;J|VL ) fo r some p < - dim L then (hoPL ) - f L e F^piV.n-.J)

and

| I ( hopL )* f L l l 2p s ( I I f L I I p ) I I h I Ip •

□

We now have the following description of ( E ' ) L :

Proposition 6.4 : A topological li n e a r isomorphism

Rl : (E*)L - E^

is d efine d by

R|_f - f|vL f o r f « ( E ' ) L .

P ro o f: That RL i s a w e ll-d e fin e d li n e a r isomorphism 1s c le a r from P ro p o sitio n 6.2 and 6.3. Th at RL 1s a c t u a lly an Isomorphism o f to p o lo g ic a l vector spaces i s a consequence o f the norm e stim a te s (6.10)

(6 .1 2 ) 1n view o f the comments preceding P ro p o sitio n 3.6.

□