1.2. Radiation s

For an y nonzer o a i n F defin e th e linear transformatio n r

a

b y

rax = ax V x E V.

Thus r

a

i s in GL n(V). An y a i n GL

n

whic h has the for m o = ra fo r som e suc h a

will be called a radiation o f K Th e set o f radiation s o f V i s a normal subgroup of GL n(V

which will be written RL n(V). Th e isomorphism /?Z,

W

* F i s obvious.

1.2.1. Le r o be any element of GL n(V). Then o is in RL n(V) if and only if

oL = L for all lines L in V. In particular

ker (P\GL

n

) = RLn, ke r (P\SL

n

) = SLn n /tt „

«?!„ ^ CI„//?L„ , iSZ, „ - SLjSL

n

n *£„ .

PROOF. Fi x z i n K . Ther e i s then a | 3 i n F suc h tha t oz = |3z. W e have t o prov

that ox = j3x fo r a typical x i n K B y hypothesis, ox = ax fo r som e a i n F . I f x

is in Fz , the n x ha s the for m x = Xz, s o

ax = o(\z) = X(az) = Xj3z = |3x.

If x i s not i n Fz

t

the n

ax 4 - 0z = a(x + z ) = 7(x + z) ,

so a = 7 = |3 b y th e independenc e o f x an d z . Q . E. D.

1.2.2. PSL

n

(V) is a normal subgroup of PGL

n

(V) with PGL

n

/PSLn ^ F/F n.

PROOF. Normalit y i s clear. Nex t verif y tha t th e kerne l o f th e composit e homomor -

phism

GLn - * PGL

n

-^PGLJPSL

n

n

p

n

n a t

nl n

is the grou p G = {o E GL

n

| det a E F"} . The n verify tha t th e kernel of th e composit e

homomorphism

GLn-*F-+ F/F

n

n de t na t '

is also G. Finally ,

PGLjPSLn = GLJG =

F/Fn.

Q.E.D .

1.3. Residue s

Consider o i n GL n(V). W e define th e residua l spac e R, th e fixe d spac e P, an d th

residue re s a, o f a b y th e equation s

R = (o- \ V)V, P=ker(a - \

v

\

res a = dim R.

The subspace s / ? an d P o f F ar e called th e space s of a . W e have