ix

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,...

aK

— coupled parabolic system, Hji = (Pj) D PK where

(Pj) ° = w0PjW0. (see IIand S2)

(J,K) — overlap index of J and K (se e S6)

A+(A)

— ( 2 naa n

Q

£ 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

w

have Ry(B ) =

Ry(Bo

) = 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

n

j(__). The symbol D

denotes th e end (or absence) of a proof.