i(V) length of any composition s e r i e s of V
U J J unipotent group with root s i n $ j - $ p if I £ J
R^H) unipotent radical of H
X(H) character group of H
(__) I H induction functor from Rat(H) t o Rat(G)
LjJ Q(_) derived functors of (_j|jj# for n = 0,1,...
coupled parabolic system, Hji = (Pj) D PK where
(Pj) ° = w0PjW0. (see IIand S2)
(J,K) overlap index of J and K (se e S6)
( 2 naa n
£ 0 and n
f l
an integer} for
any finit e subse t A c E (se e %7)
A(V) s e t of T-weights of th e module V
Note that i n our notation UA ^ represent s R^B). We will abbreviate t h i s t o
U. We als o us e Uj j t o denote a unipotent group with roots i n - * j - (-*t).
As an example, take R^B"), where B~ i s th e opposite Borel subgroup of B. We
have Ry(B ) =
) = U A 0' which we abbreviate t o U . Similar abbreviations
are made for Ry(Pj) i n S2.
The bracket notation [S:V] above will be used for other filtration s beside s
composition series . For example, [u:V] will denote th e multiplicity dim V of M
a s a weight of V. Finally, when speaking of th e derived functors of th e functor
(_) J p for Pj C p
J #
we will shorten th e above notation t o L
j(__). The symbol D
denotes th e end (or absence) of a proof.
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