4
DAVID C. VELLA
suite d to our purposes. Indeed t h i s theorem provides the motivation for most of
the r e s u l t s proved below.
Let A be a base for the root system $ of G, with corresponding Borel
subgroup B =
Ua
aeA*
L e t J

A a n d
*
e t P
J
b e t
^
i e s  t a n c
*
a r d
parabolic
determined by B and J. Let wQ be the long word of the Weyl group W of $. If
H Q G i s any subgroup and V i s a rational Hmodule, l e t V denote V regarded
w
o *
a s a rational H module, with action through conjugation by wQ. Let X » X
be the opposition involution on $. Then 4.5 of [12] s a y s that if J and K are
* J
w
o
proper subset s of A such that J UK = A, and we define H = H£ = (Pj) n PK#
then for any rational Pjmodule V we have an isomorphism of PK~modules:
(«) v?jpK ,
VW°H£K.
We will refer to such a twisted intersectio n Hj£ of two standard parabolic
subgroups a s a coupled parabolic system or j u s t a "CPS"subgroup for
brevity.
We will briefly indicate how t h i s theorem i s obtained. The proof i s s e t in
the category of kgroup schemes. Let G be a reduced algebraic group over k,
and l e t H,L be two subgroup schemes of G. Suppose that L has an open orbit
G c G/H = X. Choose a point f in Q(k) and x e G(k) such that *(x) = f (where K
i s th e quotient map G  G/H). Let L be the stabilize r of x in L: L
x
= Hx XQ L,
where Hx = x _ 1 Hx. For any rational Hmodule V, denote by V x the Hxmodule V
with action vi a conjugation by x. Then Theorem 4.4 of [12] states : Let
d = codim(X  Q). Then for 0 * i d  l
/
w e have an isomorphism H^G/^KV)) a
H^L/L^UV3*)).
In particular, if d £ 2, v 
G

L
s V
X

L

L
. The proof proceeds in
severa l steps . First, one shows that L/L
x
= Q and that the bundles JUV)Q and
L(VX))
are isomorphic (in the right category). Next, l e t Z = X  Q. For z e Z,
we have depth Oy _ = Krull dim Oy^d because X i s smooth. But if V i s finit e
dimensional, L(V) i s locall y free of finit e rank, s o the above give s depth
£(V)2 £ d for al l z e Z. Then by (3.8) of [15] the local cohomology sheave s