By Armand Borel (auth.), Arjeh M. Cohen, Wim H. Hesselink, Wilberd L. J. van der Kallen, Jan R. Strooker (eds.)

From 1-4 April 1986 a Symposium on Algebraic teams used to be held on the college of Utrecht, The Netherlands, in party of the 350th birthday of the collage and the sixtieth of T.A. Springer. well-known leaders within the box of algebraic teams and comparable parts gave lectures which coated vast and primary parts of arithmetic. notwithstanding the fourteen papers during this quantity are generally unique study contributions, a few survey articles are integrated. Centering at the Symposium topic, such various issues are lined as Discrete Subgroups of Lie teams, Invariant idea, D-modules, Lie Algebras, targeted features, crew activities on Varieties.

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**Example text**

M ® (go ' "''' j=O (gi m) ® (go ' '" + ( -1)i+1 gj+l gj ' "''' gi ' gi-1) • The second group is the homology of the s t a n d a r d complex , M®k[Gi+l]-k d ~ Mek[G i]~-" k d ( m (~ (go'''''gi)) = (go1 m g o ) ® ( g l ' . . ,gi) where i-I ' i + E (-i)J+ m ® ( g O ' " ' g j g j , l ' " ' g i j=O + (-1) i+1 An isomorphism m ® (go ' .... gi-1) • ~0 f r o m the first complex to the second is c o n s t r u c t e d as follows: ~ ( m ® (go . . . gi)) = (go " gi m) ® (go . . . gi) • Next, for M = k[G], one m a y c o m p u t e the operator B/= ~ o B o ~ - l .

Construction is the famous Springer map ~:T*X Another key ingredient f o r our . ~ N, which w i l l allow us to pass from n i l p o t e n t o r b i t s to the geometry of the f l a g v a r i e t y . obtained by transposition and pro- j e c t i o n . The remarkable point about potent cone g g is the n i l - N. ,,l,potent orbit. Starting from produce a c o l l e c t i o n of cone bundles on X by under Springer's map ~ and then decomposing the closure into i r r e d u c i b l e components KI . . .

For the elaboration of more d e t a i l s , I r e f e r to our o r i g i n a l papers [BBMI-3]. 3~ Characteristic classes of cone bundles. The concept of a cone bundle generalizes that of a vector bundle: The bundle map K locally trivial f i b r a t i o n of of Chern classes of s(K) ( I ) f o r a vector-bundle s(K) is assumed to be a K by cones (in vector spaces). To extend the theory of a cone bundle. I t may be characterized by two axioms K, s(K) = c(K) -I is the inverse of the t o t a l Chern class is f u n c t o r i a l under proper push-forwards.