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