- REDUCTIVE GROUPS (REVIEW) 13
4.1.2. Parabolic subgroups. The general definition of parabolic subgroups will be
recalled in ( 4.1.2). For GLn or SLn they are the stability groups of flags, and we
t a ke this as a definition. A flag is an increasing sequence F
(19) {O} =Vo C V1 C · · · C Vs-l C Vs =!Rn of subspaces of !Rn.
It is conjugate to a standard flag in which the Vi are coordinate subspaces. Let
ni =dim Vi/Vi-1. Then n = n1 + n2 + · · · + ns, and dim Vi = n1 + · · · + ni. The
stabilizer P of F is the group of matrices which are "block triangular"(20)
*
*
0
:),
As
Ai E GLn; · Pis a semi-direct product P = Lp.Np where Np is the "unipotent
radical" and consists of upper triangular matrices with I in the blocks and Lp is
reductive, and equal to
GLn, (JR) x GLn2 (JR) x ... x GLn. (JR).Let Ap b e the intersection of A with the center of Lp. It consists of diagonal
ma trices which are scalar multiples cJn; of the identity in the ith block. We have
(21) Lp =Mp x Ap
where Mp consists of matrices (g 1 , ... ,gs) with 9i E GLn;(IR) of determinant ±1.
Given J C ,0,., we let A J = na:EJ ker a. Then we h ave
(22) Ap = AJ where J = ,0,. - { O:n 1 , O:n 1 +n 2 , • • ·, O'.n 1 +···+n._ 1 }.
We shall also write PJ for the present P. Thus P0 is the group of upper triangular
matrices and Pf:}. = G. We shall also write NJ and MJ for Np and Mp. Note
that LJ = Z(AJ). Thus we h ave PJ = MJAJNJ where MJAJ = Z(AJ). The Lie
algebra of MJ (resp. NJ) is spanned by the g,e (/3 linear combination of elements
of J) (resp. f3 > 0, not in the span of J).
This was all for GLn(IR). One gets the similar objects for SLn(IR) by taking
subgroups of elements of determinant l.
4.2. In this section, we review some general facts about linear algebraic groups
and in the next one we sp ecialize to the main case of interest in this course. See
l for a more extended survey and [5], [17] for a systematic exposition. F is
a field (commutative as usual) and F an algebraically closed extension of infinite
transcendence degree over its prime subfield.
4.2.l. The group G c GLn( F) is algebraic if there exists a set of polynomials
Pi (i E J) in n^2 varia bles with coefficients in F, such that
(23) G = {g = (9iJ) E GLn(F), Pi(g11,912, ... , 9nn) = 0, (i E J)}.
It is defined over F if the ideal of polynomials in F[X11, X12, ... , Xnn] vanishing
on G is generated by elements with coefficiens in F. Its coordinate ring, or ring of
regular functions (resp. defined over F) is the ring generated over F (resp. F) by
the 9iJ and ( det g )-^1. We also say that G is an F-group if it defined over F. For
any extension F' of Fin F, we let G(F') be the subgroup of elements in GLn( F').
If G' is another F-group, a morphism q : G --> G' is a group homomorphism