Maximal Subgroups of Classical Algebraic Groups by Gary M. Seitz

By Gary M. Seitz

285 pages

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Certain d i f f i c u l t i e s arise when V^ is r e s t r i c t e d but not basic and we w i l l use the followin g notation. For V^ r e s t r i c t e d w r i t e V i = V i s <3 V ^ , where V^ = V^ s unless X has type B n ; C n , F 4 ; or G2, w i t h p = 2 , 2 ; 2 ; 3 ; respectively. 6) ( so V^ s is basic and V^ is p-basic ). We w i l l sometimes w r i t e V i ~ to indicate one of V^; V^ s ; or V^1. For each r € Tt(Y)-Tt(Ly) we define a certain normal subgroup K y ^ of P Y ; which in most cases is just the largest normal subgroup of Py which is contained in Qy and does not contain the T y - r o o t corresponding to - r .

Then the weights of V are X a , X^-a, X a - a : - 2 £ ; \ - 2 a - 2 J 3 . If rf(L) = { a } , then we see there are no weights of Q-level 1. 4). ,Qk and let Q* = Q^ X ... X Qk. Then f o r d > 0, [Q* d , V i S L . ® [Q k d k, V k ] , the sum ranging over sets of nonnegative integers d ^ , . . +d k = d. Proof. It w i l l suffice to establish the result f o r k = 2. Clearly, 30 GARYM. SEITZ [ Q l d l , y±l ® [ Q 2 d 2 , V 2 ] < [ Q d l + d 2 , Vj_ ® V 2 ] . For just use d] elements f r o m Qj_, followed by d 2 elements f r o m Q 2 , and take commutators.

V k ] = (**) 2 ( v i ® . . ®V| < ) Mow (mod M). [ Q ; [ Q * d " 1 , V 0 ] ] = S is T - i n v a r i a n t , so S+M is a sum of T - w e i g h t spaces and we consider the sum of t e r m s in the r i g h t side of (**) corresponding to a fixed weight. ®V k ), where ] r qj = j s q^ for each pair of term s in the sum. and so j r ^ js. (***) For r ^ s, q* ^ qi Letting t vary we conclude each summand in ( * * * ) is in S+M. It f o l l o w s that r + M € S + M. This shows [ Q , ^ 1 ^ ] ] = [Q* d ,V 0 ] (mod M).

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