16 ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPS
4.4.3. The parabolic F-subgroups of G are, by definition, the groups of real points
of the parabolic F-subgroups of G. There exist proper ones if and only if DG is
isotropic. Fix an ordering on p
the set of simple F-roots. Given J C pD.., let as in Section 4.1.2.
(29)Then the parabolic subgroup PJ or pPJ is generated by Z(AJ) and the subgroup
N with Lie algebra EB.e>og,e. It admits the semidirect decomposition
(30)where LJ = Z(AJ). Let
elements in J. The group NJ is the unipotent radical of PJ. It has Lie algebra
(31) nJ = EB,e>O,MJg,e.
The Lie algebra [J of L J is sum of the g,e ((3 E J) and of m EB a. This describes the
identity component L°.J of LJ, but LJ is not necessarily connected. It is generated
by L°.J and Z(A). By applying (27) we get
(32) LJ = MJ x AJ, where MJ =^0 LJ.
If P is defined over F, we denote Lp, Ap, Mp the corresponding data. We have
therefore the decompositions
(33)(34)P = N p.Ap.Mp (semidirect)G = Np.Ap.Mp.K
If we write g = n.a.m.k (n E Np, a E Ap, m E Mp, k E K), then n and a
are uniquely determined and will be denoted n(g), a(g), whereas m and k are
determined up to an element of Mp n K, but their product is unique.4.4.4. Langlands decomposition. In (33), (34), Np is uniquely determined, but Mp
and Ap are determined only up to conjugacy. The group K being fixed once and for
all , it is customary to normalize t he choice of Lp, Ap, Mp be requesting them to be
invariant under the Cartan involution B =BK having K as its fixed point set. This
determines t hem uniquely. Note that by doing so, one usually drops t he requirement
that they be defined over F. However , t he projection P __, P /Np = Lp is defined
over F and induces an isomorphism of any Levi subgroup of P onto Lp, so that
the notions defined over F in Lp can be transported to any Levi subgroup. The
decomposition (33), (34) so normalized are called La nglands decompositions. Note
that if P C P', then Ap1 C Ap , Mp C Mp1 (besides N p1 C Np which is true
regardless of normalization).4.4.5. Two parabolic F-subgroups are associate if a conjugate of one h as a common
Levi subgroup with the other and are opposite if their intersection reduces to a Levi
subgroup. The transform B(P) is a parabolic subgroup P' opposite to P, the only
one such that P n P' is B-stable. It is not necessarily defined over F (unless K is
defined over F), but it is conjugate to an F -subgroup. Indeed P n P' is conjugate
under N p to a Levi F -subgroup and this conjugation brings P' to an F-subgroup
P". (If P = PJ is standard, then the Lie algebra of the unipotent radical of
P" = Pi is spanned by the spaces g,e ((3 < 0 is not a linear combination of
elements in J).)