2 I. GENERALIZE D UNITAR Y GROUP S
and cal l U totally ip-isotropic i f U RffJJ). W e also call (p degenerate o r nonde-
generate accordin g a s R^CV) ^ {0 } or R^CV) = {0} . Clearly th e restrictio n o f tp
to a n anisotropi c subspac e i s nondegenerate. Whe n ip is nondegenerate, w e pu t
(1.1.6) G* = G(ip) = G(V, p) = { 7 e GL(V) | 7 V = V ? }
This i s calle d th e unitar y group of (p. If p = id^ , the n i f i s commutative . I n
this cas e (p is calle d symmetric o r alternating accordin g a s £ = 1 or e 1, an d
G^ i s calle d a n orthogonal group o r a symplectic group accordingly . B y abus e o f
language w e cal l a symmetri c for m als o a quadratic form . I f p ^ id# , the n tp i s
called hermitian o r skew-hermitian accordin g a s £ = 1 or e = 1.
Given p on V a s abov e an d a n ^-hermitia n for m I/J o n a if-modul e V T we say
that (V , ip) is isomorphic t o (W , ip) if p = f -ip wit h a if-linear isomorphis m / o f
V ont o W ! We can als o define a n e-hermitia n for m (p(Bif o n V 0 W b y
(1.1.7) {ip 0 ^)( x + y, x' + y
7)
- (p(x, x
l)
+ ^(y , y
7)
(x , a; ' G 7; y , 2 / G W).
We then write (V®W , ^0V O = (V , /?)0 (W, V 7)- Clearly (^0 ^ i s nondegenerate if
and only if bothy ? and ^ ar e nondegenerate. I n such a case, we can view G^ x G^
as a subgrou p o f G^®^ . Th e elemen t (a , /? ) o f G ^ x G ^ viewe d a s a n elemen t o f
G ^ wil l be denote d b y a x (3 or b y (a , /?) .
Hereafter w e fix V an d a nondegenerat e / ? o n V unti l th e en d o f §1.6. W e
now stat e a generalize d versio n o f Witt' s Theorem , wit h a modificatio n whic h i s
inessential bu t convenien t fo r ou r late r applications .
1.2. Theorem . Let W be a K-module and let / , / ' G Hom(W, V) ; suppose
that Ker(f) = Ker(f) and f (p = f (p. Then there exists an element g of G^
such that f f'g, provided that the following condition is satisfied:
(*) For every x G V there exists an element z G K such that p(x, x) z + ez p.
Proof. Assuming thi s t o b e tru e whe n W C V an d / ' i s the identit y injection ,
let u s prove the genera l case. Give n / an d /' , w e can defin e a n isomorphis m j o f
W'f ont o Wf s o that fj = f. The n f (j-ip) = f -p = f -p, an d henc e j tp = ip
on Wf '. B y ou r assumption , j g o n Wf wit h som e g G G^. The n fg = / .
Thus ou r tas k i s to sho w that i f W C V, / G Inj(W, V), an d f (p = ip on W , the n
f = g o n W wit h som e # G G^. W e prov e thi s b y inductio n o n dim(VT) . Clearl y
we may assum e tha t dim(VF ) 0 . Tak e a subspac e U o f W o f codimension 1. B y
the inductio n assumption , w e find h G G^ suc h tha t h = f o n U. Changin g /
for / / i
- 1,
w e may assum e tha t uf u fo r ever y u G t/. W e consider subspace s X
of V containin g U suc h tha t / ca n b e extende d t o a n elemen t p G Inj(W + X , V )
such tha t xp = x fo r x G X an d p - (p (p on W + X. Choos e X t o b e o f th e
largest dimensio n amon g such subspaces. Pu t Y = W-\-X. HY = X, the n xf x
for x G W, an d s o we hav e th e desire d conclusio n wit h g = 1. Therefor e w e ma y
assume Y ^ X . The n X ha s codimension 1 in Y. Ou r choice of X mean s that i f Z i s
a subspac e of V containin g Y , g G Inj(Z, V) , g = p o n Y , and q-p = (p on Z, the n
X = { x G Z | x g = x } . No w w e ca n pu t Y = X 0 i f 5 with s G W. Pu t sp = t ;
then Yp = X + i f t an d s ^ t ^ X. W e ma y assum e tha t 7 / 7 , sinc e i f Y = V ,
then we can take p t o be the desired g. Pu t i f = { x G F | (p(s t, x) = 0 } . Then
^(2/P

2/5 -^0
=
0
f°r
ever y 1/6 7 .
Suppose H. (jLY an d ii " ^ Yp . The n H n Y an d i f D Yp ar e prope r subspace s
of if , an d henc e i f ha s a n elemen t u no t containe d i n thes e subspaces . The n
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