4 I. GENERALIZE D UNITAR Y GROUP S
(3) IfV has a totally ip-isotropic subspace U of dimension r , then there exists
an isomorphism 7 of(V, (p) to (Z , £)0(i7
r
, n r) such that E/ 7 = I
r
with a subspace
Z ofV, where £ is the restriction of (p to Z . In particular, (V , p) is isomorphic to
(Hr,rjr) if2r = dim(V).
Proof. Take a if-basi s {ei , .. . , er } o f R^iJJ). The n w e ca n find r element s
pi, .. . , gr o f V suc h tha t /?(e^ , gj) = e6ij an d /?(5 , gi) = 0 . The n w e easil y se e
that Kgi , .. . , If gr an d J 7 form a direct sum. Cal l the sum X. Tak e an E K so that
^(#; 9%) = ea a + «?* ; Pu t °hi = £p(9hi 9i) fo r h i, a
hi
= 0 for ft i, an d / * =
^ Y^j=i aijej- W e ca n easil y verif y tha t ip(ei, fj) = ebij an d ip(fi, fj) = 0 fo r
every i an d j . Let I T = XX=i(^ e i+^/*) a n d ^ V ^ e ^ n e restrictio n of/ ? to H'.
Then (#' , 77' ) is isomorphic t o (H
r
, rj
r
), X = SeH', an d p(S, H') = 0. Therefor e
we obtain th e first assertio n o f (1). The n clearl y dim(U) + r = dim(X) dim(F) .
If this is an equality, the n X = V, whic h proves (2) . I f U = R^iU), the n 5 = {0 }
and (X, f ) i s isomorphi c t o (H
r
, rj
r
). Puttin g Z = { z G V \ p(z, X) = 0 } , w e
obtain (3) .
If (V , (p) is isomorphi c t o (Z , Q 0 (H
r
, n
r
) a s i n (3 ) o f th e abov e lemma , the n
the abov e proo f show s that V ha s 2 r element s ei , fi suc h tha t
r
(1.5.1) F = ^ ( X e , + # / , ) + Z ,
(1.5.2) ^(e* , Cj) = /?(/; , /j) = 0 , y(e» , /j) = e^ j for every i and j ,
(1.5.3) Z={veV\p(ei,v) = ip(fi,v) = 0 for every i} .
Conversely suppos e tha t F ha s 2 r element s e^ , /; satisfyin g (1.5.2); defin e Z
by (1.5.3); le t £ denot e th e restrictio n o f (p t o Z . The n w e obtai n (1.5.1 ) an d
(V, (f) (Z, (" ) 0 (H
r
, r)
r
) with #
r
= J2 Ti=i(Kei + ^ / i ) - l n thi s situatio n w e cal l
the expression o f (1.5.1) a weaJ c Witt decomposition (with respect to ip); we call i t
a Witt decomposition i f £ i s anisotropic .
1.6. Lemma. Wit h (V , (p) as above the following assertions hold:
(1) There exists an anisotropic subspace XofV such that (V, ip) is isomorphic
to (X, £ ) 0 (H
m
, rjm) with some ra, where £ is the restriction of (p to X.
(2) Th e isomorphism class of (X, £ ) i s uniquely determined by (V , y?).
(3) Every totally (p-isotropic subspace ofV is contained in a totally (p-isotropic
subspace of V of dimension ra.
(4) If (Z , C ) an d (H r, rj r) are as in Lemma 1.5(3), then r m and (Z , £ ) is
isomorphic to (X, f ) 0 (H m-r, 7? m-r).
Proof. Let (Z , £ ) an d (i/
r
, ?7
r
) b e a s i n Lemm a 1.5(3). The n £ i s nondegener -
ate. I f Z i s anisotropic , w e tak e Z a s X. Otherwise , Z ha s a vecto r v / 0 suc h
that tp(v, v) = 0. Applyin g Lemm a 1.5(3) t o (Z , £) , we can find (M , // ) suc h tha t
(Z, Q i s isomorphic t o (M , u) 0 (#1, 771), so that (V , (p) is isomorphic t o (M , // ) 0
(i7
r+
i, r) r+i). Repeatin g thi s procedure , w e eventually arriv e a t X a s i n assertio n
(1). A t th e sam e tim e w e obtai n (4 ) provide d (2 ) holds . T o prov e (2) , suppos e
that (V , (p) is isomorphic t o (T , r) 0 (H n, rj n) wit h a n anisotropi c (T , r) an d som e
n. I f m n, the n (V , ip) is isomorphic t o (X, f ) 0 (i?
m
_
n
, r/ m_n) 0 (ff n, r/ n). B y
Lemma 1.3, (T , r) i s isomorphic t o (X, £ ) 0 (i7
m
_
n
, 77
m
_n). Since T i s anisotropic,
we have ra = n , whic h prove s (2) . A s for (3) , given a totally (/^-isotropi c subspac e
Previous Page Next Page