42 M. L. RACINE
with at leas t one of (3,7 non zero. Let b = [ £] = P
0
; we have
\^ W 2
_
/7^e
+
P*n
Yy2
+ P* \
bab =
2
. If (3 ^ 0, letting A = 1, p. = £ = TI = 0 one
obtains bab = f LJ , and letting r\ = 1, ^ = £ = A = 0, bab = f
Similarly if 7 = 0 the subspac e of P spanned by elements of the form
bab, where b is allowed to vary in P , is M(P - ) . Therefore M(P -)e, ,
is spanned by b . ^ . b . , * and hence contained in $, ® P. But
JJ JJ JJ 1
e..H(P
0
,-)e..e
iij
. * = tf(P0,-)e..11 C £_ ® P. Therefore &. ® P = tt(& ® P, *) and
ij 2 JJ 2' 1 1
arguing as above $ = H($, *).
q . e . d .
In this section, unles s specified otherwise, K will be a complete
discret e valuation field. Let {&, - ) be a central associativ e division algebra
with involution over K. Let | | , 0, ^ and p be as in the las t section .
Let & , n_2, act on v an n-dimensional left ^-vector spac e and let * be
an involution of $ induced by a hermitian form h. Let &^ be a subspac e
n 0
of W(&, -) such that 1 & and x& x C & V x

$. If H is the matrix of
h with respect to a basi s of V compatible with the above action of & on V,
n
then tf(JB , #) = {HA|A
e
& ,A = A}. If the diagonal entries of H belong to
&_, {HA|A = (a..) e & , A = A, a,,

£ } is a subalgebra of H(& , *) which we
0 ij n 11 0 n
denote H(& , & , *). It is eas y to se e that the diagonal entries of H belong to
$ if and only if h(v, v) $
n
, V v e V and that therefore this condition is
independent of the choice of b a s i s . If H is the identity matrix we drop the
* and write M(fl , JB ) ([17], p. 2.14). In fact H(fl , & , *), * arbitrary is an
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