Section 2. Levi Decompositions
If the characteristi c of k i s zero and H i s an arbitrary connected linear
algebraic group over k, then H can be written as a semidirect product H =
LRrjfH), where Ry(H) i s the unipotent radical of H and L i s said to be a Levi
factor of H [25]. In positiv e characteristic s such a decomposition does not
necessaril y exist , but certain type s of groups do have Levi decompositions in
all characteristics .
If H = P j i s a parabolic subgroup of a semisimple group G then H always
has a Levi decomposition. One way to s e e thi s i s to define a torus Z to be the
identity component of n Ker a: Z = ( n Ker a)un and then define LJT to be the
aeJ a

J
centralize r CQ(Z) of Z in G. Then Lj i s automatically reductive, and a comparison
of the Lie algebras Lie(Lj) with Lie(Pj) leads to the desired decomposition. (See
$30.2 of [20].) In fact i t i s true more generally that any connected subgroup H
of G which i s normalized by T has a Levi-decomposition (see [6]). From thi s one
can se e that the Levi factor Lj of a parabolic Pj contains T and the unipotent
groups U
a
for a e $j , where * j i s the subrootsystem of $ spanned by J. Then
Ry(Pj) contains al l U
a
for a e $
+
- $ j , i.e., RytPj) - U. j which we will
henceforth abbreviate to Uj, so we have P , = LJUJ.
Now l e t P j and PK be two parabolic subgroups of G. Since each i s
normalized by T, so i s Hjjr = (Pj) ° n PK- Thus by the above a CPS subgroup als o
has a Levi decomposition. Note that Hj£ contains T and Ua for
a e $ * U -0j£. Rewriting t h i s as
$ * U (*+ * - * + *) U (-$£ - (-* + *)) we se e i t s Levi factor contains
KRJ J KflJ K KRJ
T and U^ for a e * * and that RTTK (H£) has root groups U with
G
K n J
U
a
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