14
T. A. SPRINGER
direct product of Agroups Q = L V. It follows from the above that such L exist.
Two Levi subgroups of Q are ^-conjugate. If A is a maximal /:-split torus in the cen-
tre of L, then L = Z(A). If A is any fc-split subtorus of G then there is a parabolic
Asubgroup Q of G with Levi subgroup L. Two such Q are not necessarily k-
conjugate (as they are when A is a maximal fc-split torus). Two parabolic &-sub-
grdups Qi and Q2 are associated if they have Levi subgroups which are Aconjugate.
This defines an equivalence relation on the set of parabolic Asubgroups.
If Qi and Q2 are two parabolic Asubgroups, then {Qx f] Q2) RU(Q\) iS also a
parabolic Asubgroup, contained in Qx. It is equal to Qi if and only if there is a Levi
subgroup of Qi containing a Levi subgroup of Q2. Q\ and Q2 are called opposite if
Gi fl (?2 is a Levi subgroup of Qx and Q2-
3.7. Bruhat decomposition ofG(k). Let P and £ be as in 3.5 and put U = RU(P).
If we
k
W denote by nw a representation in N(S)(k). The Bruhat decomposition
of G(k) asserts that G(k) is the disjoint union of the double cosets U(k)nwP(k)
(wzkW).
One can phrase this in a more precise way. If w e kWthere exist two £-sub-
groups U'w9 U^ of U such that U = U'w x £/* (product of /c-varieties) and that
the map U'w x P -+ UnwP sending (x, y) onto X/I^J is an isomorphism. We then
have
(G/P)(k) = G(k)/P(k) = U *{U'JLk%
wzkW
where TT is the projection G - G/P.
If fe = fl this gives a cellular decomposition of the projective variety G/P.
If 0 e kA let W0 be the subgroup of kW generated by the reflections defined by
the a e kA. IfO, 0' e kA there is a bijection of double cosets
Pe(k)\G(k)/Pd,(k) * W{d)\kWIW{0').
Let 2 be the set of generators of kW defined by kA. The above assertions (except for
the algebro-geometric ones) then all follow from the fact that (G(k), P(k\ Z(S)(k), 2)
is a Tits system in the sense of [7].
3.8. The Tits building. Let G be the connected reductive Agroup. We define a
simplicial complex ^ , the (simplicial) Tits building of(G, k), as follows.
The vertices of % are the maximal nontrivial parabolic /^-subgroups of G. A set
(P1?---, Pn) of distinct vertices determines a simplex of & if and only if P =
^i fl *' fl ?n
is
parabolic. In that case, the P{ are uniquely determined by P. It fol-
lows that the simplices of & correspond to the nontrivial parabolic fc-subgroups
of G. Let Gp be the simplex defined by P. Then aP is a face of aP. if and only if
P' = P. The maximal simplices correspond to minimal &-parabolics. These sim-
plices are called chambers. A codimension 1 face of a chamber is a wall. Two
chambers are adjacent if they are distinct and have a wall in common. One shows
that any two chambers a, & can be joined by a gallery, i.e., a set of chambers a =
0"o 0"i ' •" 3*
=
0"'*
s u c
h
t r i a t
0 V
anc*
ov+i are adjacent (0 ^ / s).
It is clear that G{k) operates on %.
One can show (using a concrete geometric realization of the abstract simplicial
complex %) that % has the homotopy type of a bouquet of spheres.
For more details about buildings see [20].
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