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
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