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).
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