1. ELEMENTAR Y PROPERTIE S O F HERMITIA N FORM S 3
u £ Y U Yp. Defin e q G Hom(F + Ku, V) b y uq u an d q = p o n Y. The n
Ker(g) = {0 } an d w e easil y se e tha t q ip = (p o n7 + Ku, bu t thi s contradict s
the abov e propert y o f X. Thu s H C Y o r H C Fp . Sinc e y ^ V , we have H = Y
or H = Yp. li H = Y, w e have ip(s, 5 t) = 0 , so that y?(s , t) = p(s , 5) = /?(t , t).
Thus /?(t , 5 t) = 0 , tha t is , t £ H. Similarl y i f i J yp , changin g s fo r t , w e
find tha t s H. Therefor e i n eithe r cas e we have H = Y = Fp .
Now take any v G V, ^ Y an d put 6 = y?(v , s -1). Then V = F + ifv an d 6 ^ 0 .
Since t £ X , w e can find an element r EV suc h that p(r , t) = b an d /?(r , X) = 0 .
Put X ' = { y G y I p(r, y) = 0 } . The n X C X' C y , an d henc e X ' = X . Sinc e
s ^ X , w e hav e p(r , 5)7 ^ 0. Pu t a = y?(r , 5) an d w = v + r + c(t s) wit h an y
c G X; defin e g G Hom(V, V ) b y g = p o n Y an d v # = w. Observ e that g ip = (p
(and hence g G G^) i f tp(v—w, X) = 0, y?(v , s) = /?(u , t), an d p(i; , i) = y?(w , it;).
The first tw o condition s ar e satisfie d becaus e o f our choic e of r. No w
p(w, w) p(v, v) =(p(v + r , v + r ) p(v, v)
+ y?( v + r , c( t - s) ) + p(c( £ s), i; + r )
and ^(v+r , s t) = (p(v, s—i) + ip(r, s) (p(r, i) = a . Therefor e p(i , v) = (p(w, w)
holds i f we can tak e c s o tha t
(p(v + r, v + r) ip(v, v) = ac p + £ca p.
Condition (* ) allow s u s t o writ e th e left-han d sid e i n th e for m z + ez
p
wit h som e
z G K. Puttin g c = (a~
1 z)p,
w e ca n establis h g G G^. Sinc e g = f o n W, thi s
completes th e proof .
1.3. Lemma. Let (V, (p) = (5 , a) 0 (T , r ) - (S' , a') 0 (T' , r' ) wit h nonde -
generate (p satisfying (*) above. If (S , J ) i s isomorphic to (S f,crf), then (T , r) i s
isomorphic to (X" , T') .
Proof. Let / i b e a if-linea r isomorphis m o f S ont o 5 ' suc h tha t h - a' o. B y
Theorem 1.2, / i ca n b e extende d t o a n elemen t g G G^. The n ip(Tg, S
f)
= 0 , s o
that Tg = T
f.
Sinc e g (p = tp, we have g - r' r a s desired .
1.4. Give n a nonnegative intege r r , w e shall always denote by (H r, rj r), I' r, an d
Ir th e symbol s give n a s follows: I f r 0, then H
r
I'
r
^ I
r
, I
r
~ I'
r
= K}, an d
(1.4.1) 77
r
(x + ix , y + v) = w •V + ex lvp (x , y G i£; w , i; G Jr ).
Clearly rj
r
i s a nondegenerat e £-hermitia n for m o n H r; bot h I
r
an d i £ ar e totall y
nr-isotropic. Notic e tha t th e ma p (x , u) h- (u , ex ) fo r x , i x G K} define s a n
element o f G(rj r) tha t exchange s I'
r
an d I r. W e easily se e that (H r, rj r) 0 (i/
s
, 7/ s)
is isomorphic t o (-ff r+s, ry r+s)- We understand tha t i7
0
= {0 } and r/
0
= 0 .
1.5. Lemma. Wit h (V, p) satisfying (*) of Theorem 1.2, the following assertions
hold:
(1) Let U be a subspace of V , S a subspace of U such that [ 7 = 5 0 i?^(E/) ,
and r = di m (i^(£7)). The n ther e exis t a subspace XofV containing U and an
isomorphism 7 of (X , £ ) t o (5 , cr) 0 (i/
r
, r^ r) suc h tha t R^(U)^f = 7 r, wher e ^
(resp. a) is the restriction of (p to X (resp. S). In particular dim(C/)+ r dim(F) .
(2) If dim(C/)+ r = dim(V ) i n (1), then (V , y?) is isomorphic to (5 , cr)0(iif r, r/ r).
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