Cohomologie non abelienne by Jean Giraud

By Jean Giraud

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We shall therefore concentrate on the determination of the G-orbits of nilpotent elements in L( G). 5 The lacobson-Morozov Theorem The Lie algebra of the group SL 2 (K) is the algebra sI2(K) of all 2 x 2 matrices of trace 0 under Lie multiplication. This algebra has a basis (e, h,f) where h=G 0) /=(0 0) = =-2/ = e=(~ ~) -1 1 0 These basis elements satisfy the relations [he] 2e [hf] [ef] h The elements e and / are nilpotent. The Jacobson-Morozov theorem asserts that, under certain circumstances, every non-zero nilpotent element e E L(G) lies in a three-dimensional subalgebra (e, h,f) isomorphic to sI2(K).

Y. -stable points by considering a geometrical construct called the Brauer complex. This is defined as follows. Let F- 1 (y) = {y E Y®

Then End (1gJ) is the Hecke algebra H(GF , BF ). 3 that this algebra has dimension IWFI and basis Tw, WE WF. WF is a Coxeter group with Coxeter generators SJ corresponding to the F-orbits J on the Dynkin diagram of G. The multiplication of the basis elements is determined by the relations T. T. Jw+(PJ-1)Tw ifl(sJw)=I(w)-1 where W E WF, i is the length function on WF, and PJ = IUF n (UF)WOSJI. We now turn to the general case. It was shown by Howlett and Lehrer that End (,pff:) has dimension IWol where Wo is the subgroup of WF given by J Wo = {w E WF; w(J) = J, W,p = ,p}.

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