36

M. L. RACINE

must show s - s, = 0 and t - t. = r - r., l j , J? n or s - s = r - r

0

and t - t. = 0, 1 j , o\ n. Assume s

0

- s = s

0

- s = . . . = s, - s. .

£ j — * — 2 1 3 2 l i - l

and t

0

- t . = r

0

- r_, t - t = r - r

0

, . . . , t. - t. = r. - r. . If

2 1 2 1* 3 2 3 2 ' i

I - I I

l- l

s,^, - s. = r, ,. - r, then t. - t. = 0 (r. ,. - r. 0). Hence s . . . - s_ =

l+l l

I + I I I + I

l l+l

I

— l+l 1

r, , - r. and t, ,, - t . = r. - r, . Now one of them is zero, so either

i+n l

I

i+I l i l

s, ,, - s. = r, ,, - r. = 0 in which cas e s. ,, - s, = 0, or t. , _ - t . =

i+l 1 i+l

I

i+l

I

i+l 1

r. - r = 0 so that r. = r = . . . = r. and t

0

- t = t , - t = . . . = t. - t. _ = 0

i l 1 2

I

2 1 3 2

I

i- l

and t - t. = t. - t = 0 . Induction on i yields the desired result.

q . e . d .

PROPOSITION 5. Let * be an involution of & . If F is a maximal

n

^-stabl e order of & , there exist s a maximal order E of i9 with F = E n E*

n n

and if F C E , E a maximal order, then E = E or E*.

PROOF. We know that F C some maximal order E. Therefore

F = F* C E * and by maximality F = E n E*. If F c E E a maximal

order, then the same argument yields F = E n E.#. Clearly E* is maximal

for if not then E* C E maximal and E = E** c E *, a contradiction. By

Lemma 12 E n E* = E nE * implies E = E or E* = E

q. e. d.

Let h be a non-degenerate hermitian or skew-hermitian form on V an

n-dimensional left & -vecto r space ; & act s on V and h induces an involu-

n

tion * of & : A - A* where A* is th e adjoint of A relative to h, so

n

h(xA, y) = h(x, yA*) for all x, y

€

V. If H is the matrix of h with respect to

a fixed bas e of V, AH = HA* , so A* = HA H . Any involution of & may be

n