By Arjeh M. Cohen, Wim H. Hesselink, Wilberd L.J. van der Kallen, Jan R. Strooker

From 1-4 April 1986 a Symposium on Algebraic teams used to be held on the college of Utrecht, The Netherlands, in get together of the 350th birthday of the college and the sixtieth of T.A. Springer. famous leaders within the box of algebraic teams and similar components gave lectures which coated extensive and imperative components of arithmetic. even though the fourteen papers during this quantity are often unique learn contributions, a few survey articles are incorporated. Centering at the Symposium topic, such various subject matters are lined as Discrete Subgroups of Lie teams, Invariant conception, D-modules, Lie Algebras, unique features, team activities on kinds.

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Acts on We wilt s t a t e his r e s u l t as follows. The group Gx , t h e g e n e r a t o r of Z acting b y m u l t i p l i c a t i o n by Z x ; hence, up to h o m o t o p y , S 1 = BZ acts on BG×. The r e s u l t is S1 ® H. (BG x , k ) . (BG x , k) ® H . ( B S i) . (BN x , k ) . ~ B Gx l B Nx . The Gysirl e x a c t s e q u e n c e for this fibration: H . - I ( B N x , k) , H . ( B Gx, k) * H . _2(k[G] ) In p a r t i c u l a r , B a n d S h a v e a clean g e o m e t r i c i n t e r p r e t a t i o n .

This l e m m a m a y be extracted from [20, ~4 ]. 55 We t h u s obtain a m o r p h i s m of complexes of sheaves: L "DX ® "DX -* ~ H ° m x x x ( i * f~X' i*f2X)[2n]" DX m DO Using t h e inclusion k % QX , w e get a m a p ~x E ® 0 ~X -~ I R H ° m x x x ( i . k, Qx)[2n) = Q x [ 2 n ] . 8 L ® DX Dx m o ~X --* £2X [2n] Dx is a quasi-isomorphism. 1. Proof (sketch): The question being local on t h e r e exists a n etale m o r p h i s m x X is open and closed in %o: X X , one m a y a s s u m e ~ AI~. The diagonal A x c X ~0-1 (AAn) .

Conjugacy classes in algebraic groups; Springer LNM 366 (1974). Ao: Trigonometric sums, Green functions of f i n i t e groups, and representations of Wey! groups; Invent. Math. 36 (1976), 173-207. A. The theory of algebraic groups and their representations has made important progress in the last decade; let us point out two remarkable aspects of this progress. l) the use of sophisticated (co)homology theories like ~tate cohomotogy and intersection cohornology, in the work of Deligne, Kazhdan, Lusztig, Springer, and m a n y others.