REDUCTIVE GROUPS

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3.5. Properties of reductive k-groups. We next review the properties of reductive

groups. The reference for these is [5].

Let G be a connected reductive fc-group. Let S be a maximal £split torus of G,

i.e., a A:-subtorus of G which is Asplit and maximal for these properties. Any two

such tori are conjugate over k, i.e., by an element of G(k). Their dimension is called

the k-rank ofG.

The root system 0(G9 S) of G with respect to S (see 2.1) is called the relative root

system of G (notation k0 or k0(G)). This is indeed a root system in the sense of

[7], lying in the subspace V of X*(S) ® Q spanned by k0. Its Weyl group is the

relative Weyl group of G (notation kW or kW(G)). Let N(S) and Z(S) denote nor-

malizer and centralizer of S in G; these are ^-subgroups. Then N(S)/Z(S) operates

on

h

0 and in V. In fact it can be identified with

k

W. Any coset of N{S)\Z(S) can be

represented by an element in N(S) (k).

Z(S) is a connected reductive Agroup. Its derived group Z{S)' is a semisimple

£-group which is anisotropic. To a certain extent G can be recovered from Z(S)'

and the relative root system

k

0 (for details see [19]). There is a decomposition of the

Lie algebra g of G:

9 = 9 o + S 9«

where for cceX*(S) we have defined ga = {Xe$\Ad(s)X = saX, seS}. Then

g0 is the Lie algebra of Z(S). If a e

k

0 there is a unique unipotent ^-subgroup C/a

of G normalized by S, such that its Lie algebra is ga.

In the absolute case (k = £) S is a maximal torus, 0 is the ordinary root system

and the Ua are as in 2.3. If (7 is split over k then 5 is a maximal torus of G and k0

coincides with the absolute root system 0.

In the general case k0 need not be reduced, nor is dim ga = dim Ua always 1.

3.6. Parabolic subgroups. Recall that a parabolic subgroup P of an algebraic

group G is a closed subgroup such that G/P is a projective variety. Equivalently, P

is parabolic if P contains a Borel subgroup of G.

Now let G be as in 3.5. Then the minimal parabolic Asubgroups of G are con-

jugate over k. If P is one, there is a maximal Asplit torus of G such that P is the

semidirect product of /:-groups P = Z(S) • i?«(/ ) (RU(P) denotes the unipotent

radical). There is an ordering of

k

0 such that P is generated by Z(S) and the Ua

of 3.5 with a 0. The minimal parabolic Ar-subgroups containing a given 5 cor-

respond to the Weyl chambers of k0. They are permuted simply transitively by the

relative Weyl group.

Fix an ordering of k0 and let kA be the basis of k0 defined by it. For any subset

0 a kA denote by P0 the subgroup of G generated by Z(S) and the U where

a e k0 is a linear combination of the roots of

k

J in which all roots not in 0 occur

with a coefficient ^ 0. Then PkJ = G, P^ = P and P6 ZD P.

The Pe are the standard parabolic subgroups of G containing P. Any parabolic

A:-subgroup Q of G is Ar-conjugate to a unique iV If 5^ is the identity component of

Qae0(Ker 0) then Se is a /:-split torus of G and we have PQ = Z(55) • Ru(Pe),

a semidirect product of £groups. The unipotent radical Ru(Pe) is generated by the

Ua where a is a positive root which is not a linear combination of elements of 6.

Let Q be any parabolic Ar-subgroup of G, with unipotent radical V (which is

defined over k). A Levi subgroup of Q is a fc-subgroup L such that Q is the semi-