REDUCTIVE GROUPS

15

3.9.

EXAMPLES,

(i) The preceding results apply when k = Q, the absolute case.

In particular, we then have the properties of parabolics of 3.6 and the Bruhat

decomposition of 3.7.

(ii) G = GL(w) (k arbitrary). This is indeed a reductive k-gvoup. Its Lie algebra

g is the Lie algebra of all n x /7-matrices.

Let 5 be the subgroup of diagonal matrices. This is a maximal /split torus which

is also a maximal torus of G (in the absolute sense). Let e{ e X = X*(S) map

s e S onto its /th diagonal element. The et- form a basis of X. The root system

0 = 0(G, 5), which coincides with the relative root system

k

0, consists of the

et - e}- e X with / ^ j . One checks that the root datum of G is given by X = Z",

Xy

=

ZM,

0 = {?,- - £/},-/, 0

7

= {*/- *}},•=,-, where (e)) is the basis of X dual

to (e,).

The subgroup B of all upper triangular matrices is a minimal parabolic k-

subgroup. It is a Borel subgroup. Its unipotent radical U is the group of all upper

triangular matrices with ones in the diagonal. The basis J of 0 defined by B is

(e{ — e,-+i)i£i:»-i- The Weyl group W (which coincides with the relative Weyl

group

k

W) is isomorphic to the symmetric group 3„, viewed as the group of per-

mutations of the basis (et).

The parabolic subgroups P ^ B are the groups of block matrices

iAn Al7i ••• A1SK

0 A22 \

where A(j is an wf- x /iy-matrix with «! + •••+ ns = n, the AH being nonsingular. Its

unipotent radical consists of these matrices where Au = 1 (1 g / ^ s). The sub-

group of P of matrices with A{j = 0 fory / is a Levi subgroup of P. The center

of L consists of those elements of S at which the elements of J different from one of

the en. — en.+i (1 g i ^ s) are trivial. Hence with the notations of 3.6, we have

P = Pe, where d e J is the complement of the set of these roots.

A more geometric description of parabolic subgroups is as follows.

Let V —

Qn.

A flag in V is a sequence 0 = F0 c Kx c ••• c K5 = K of distinct

subspaces of K A Ar-^g is one where all V{ are defined over k, i.e., have a basis

consisting of vectors in

kn.

G operates on the set of all flags. The parabolic subgroups of G are then the

isotropy groups of flags. One sees that there is a bijection of the set of all parabol-

ic subgroups of G onto the set of all flags, under which /:-subgroups correspond

to /:-flags.

If P is a parabolic subgroup, then the points of G/P can be viewed as the flags

of the same type as P (i.e., such that the subspaces of the flags have a constant

dimension).

The Tits building of (G, k) can then also be described in terms of flags: The sim-

plices correspond to the nontrivial Aflags (i.e., those with s 1). If af is the simplex

defined by the flag/, then af is a face of af, if and only if/' refines/(in the obvious

sense). The chambers correspond to the maximal flags (s = n, dim V{ = i) and the

vertices of 38 are described by the nontrivial /:-subspaces of V. We see that the com-

binatorial structure of ^ pictures the incidences in the projective space Pn^x(k).