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