By Louis Auslander
Complaints of the yank Mathematical Society
Vol. sixteen, No. 6 (Dec., 1965), pp. 1230-1236
Published via: American Mathematical Society
Stable URL: http://www.jstor.org/stable/2035904
Page count number: 7
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So y(E) leaves this subspace invariant, the f u l l s t a b i l i z er being a proper parabolic subgroup of L<|_*. Taking preimages under 4> we obtain a proper parabolic subgroup of Py containing E. As above, E is contained in a Levi f a c t o r of this parabolic subgroup, and this contradicts m i n i m a l i t y of Py. So the claim holds. We have now established the existence of a parabolic subgroup 34 GARYM. SEITZ Py satisfying (1), (11), and (iii). For uniqueness we use the fact that for a given Levi factor Ly of Py there are only two parabolic subgroups of Y with Levi factor equal to Ly-namely Py and its opposite.
I i ) . ® (V1 ^ 1 (Q)) r d. Proof. 6). The next several results are technical but Important as they link the group theory and the representation theory. The f i r s t result shows one can usually choose Py so that r | Z = cc|Z f o r each r € Tr(Y)-TT(l_Y). A consequence w i l l be that in most cases V|X i s a basic module, although this w i l l not be proved until §10. 6). Assume Y is a classical group w i t h natural module W and W|X is an irreducible basic or p-basic module. Then Py can be chosen so that the followin g hold: (i).
P r o o f . 4) Qy ^ contains an L y - c o m p o s i t i o n factor isomorphic to I ^ , f o r some q. 4)(iv) shows such a composition facto r exists which is covered by Q. 4)(iii) j ^ | T = -cc. Therefore, q = 1. ,Df be the L'-composition f a c t o r s of V L . ( - r ) . (-r)lL')(S)D( < for some k. Decompose D[< into a tensor product of t w i s t s of r e s t r i c t e d modules, consider high weights, and obtain the result. 10). 9) and M£ is t r i v i a l for MAXIMAL SUBGROUPS OF CLASSICAL GROUPS each £ ^ i w i t h L# adjoining r.
An Account of the Theory of Crystallographic Groups by Louis Auslander