27. PROPERTIE S O F TH E WIT T RIN G
67
(2) D o no t assum e tha t d(q) = 1, bu t choos e a n elemen t r G W(ko) suc h tha t
r-(l,—d(q)) = 0 i n W(ko). Sho w tha t th e produc t r-l(a,q) i s independen t o f
the choic e o f a ; i t i s a Wit t invarian t o f Quad
n d
^, whic h w e ma y denot e b y
er(q).
(3) Wit h th e notatio n o f (2) , sho w tha t ever y Wit t invarian t o f Quad
n s
ca n b e
written i n a uniqu e wa y a s
ra—-1
i = 0
where th e n an d r belon g t o W(ko) an d th e produc t r-(l , —S) i s 0 in W(ko).
[Hints: (1) ma y b e prove d b y th e usua l chai n device . Th e sam e applie s t o (2) , o r (2 )
may b e deduce d fro m (1). Th e proo f o f (3 ) i s similar t o tha t o f Th . 20.6. ]
E X A M P L E 27.20 . Th e Li e algebr a o f derivation s o f a n octonio n (resp . Albert )
algebra i s o f typ e G
2
(resp . F4) . I f th e characteristi c i s ^ 2,3 , it s Killin g for m i s
nondegenerate an d give s a n invarian t Oc t » W (resp . Al b W). B y 27.17, thi s
invariant ca n b e writte n i n term s o f th e invarian t q% (resp . th e invariant s q% an d
q§). Whe n on e doe s th e computation , on e finds:
G2 CASE : Killin g = ( - 1 , -3)(q3 - 1)
F
4
CASE : Killin g = (-2)(q
5
- q
3
) + ( - 1 , - 1 , - 1 , -l)(q
3
- 1)
(Here, an d i n wha t follows , i f q i s a quadrati c for m o f ran k n whic h represent s 1,
we writ e q 1 for th e quadrati c for m q f o f ran k n 1 such tha t q = (1) 0 q f. Th e
quadratic for m q$ q3 i s define d similarly , cf . Th . 22.4. )
The formul a fo r F 4 ca n b e foun d i n [LoOl , (33) ] o r ca n b e deduce d fro m [Ja71,
p. I l l , (133)].
EXERCISE 27.21. Kee p th e notatio n above , bu t assum e th e characteristi c i s 3.
(1) I n th e G2 case, prov e that th e Killin g for m ha s ran k 7 instead o f 14 and tha t i t
is isomorphi c t o (—1)(#3 1).
(2) I n th e F4 case , prov e tha t th e Killin g for m i s 0 , bu t tha t ther e exist s a non -
degenerate invarian t quadrati c for m o n th e Li e algebr a whic h i s isomorphi c t o
(#5 1) 0 (1,1, l)(j3 1). [Hint : us e th e "normalize d Killin g form" , i.e. , lif t t o
the Wit t vectors , divid e th e Killin g for m b y 36 , an d reduc e mo d 3. ]
27.22. W I T T INVARIANT S O F Herm
n
. Recal l tha t w e conside r hermitia n form s
relative t o a quadrati c extensio n fci = fco(V^) o f fco- Fi x a n i 1 an d a n elemen t
x e W(k
0
) suc h tha t x G (1, -S)W(k0) = ker[W(k
0
) - W(fci)] . I f ft G Herm
n
(/c)
is represente d b y q G Quadn(/c), w e hav e x-X l(q) G W(k). Moreover :
L E M M A. The product x-X l(q) depends only on x and ft.
P R O O F . Writ e x a s (l,—5)y, an d observ e tha t
x-\i(q)=r(l,-6):\i(q)=yN\%
where no w \ %h i s th e z-t h exterio r produc t o f ft (i n th e sens e o f hermitia n forms )
and A^ : Herm
n
Quad2
n
i s a s define d i n 21.1.
We ma y the n defin e X l
x
(h) a s bein g x-\ l(h).
T H E O R E M . Every Witt invariant o/Herm
n
can be written uniquely as
Xie{i,-S)'W(ko) ifii.
XQ + V ^ A^. . (ft) , with
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