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 collage of Utrecht, The Netherlands, in occasion of the 350th birthday of the college and the sixtieth of T.A. Springer. famous leaders within the box of algebraic teams and comparable parts gave lectures which coated vast and valuable parts of arithmetic. even though the fourteen papers during this quantity are often unique examine contributions, a few survey articles are integrated. Centering at the Symposium topic, such different subject matters are lined as Discrete Subgroups of Lie teams, Invariant thought, D-modules, Lie Algebras, specific services, team activities on kinds.

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G , a n d of Even though cyclic homology is in some sense d e t e r m i n e d by Hochschild homology, it is not clear w h a t the cyclic homology of Cc (G) is. One r e m a r k a b l e f e a t u r e though, is t h a t cyclic homology does m a k e a difference b e t w e e n the c o m p a c t c o n j u g a c y classes in G and the others. In particular, for o-adic Lie Groups, P. Blanc and I prove an a b s t r a c t Selberg principle. w h i c h says t h e following: if e is an i d e m p o t e n t of C~ '~C (G) ' and if ~ is ' a regular e l e m e n t of G w h i c h is not compact, t h e n ZG/G~ e(g ~ g-l) dg = 0 , where G~ is t h e c e n t r a l i z e r of ~' Such an "abstract Selberg principle" w a s first proven, for G of split-rank [18 I, by Julg and Valette ], by v e r y different methods.

3). - Hence we may w r i t e our c h a r a c t e r i s t i c cycle as well as r Z z i [K i ] i=I Oh(M) = where we admit some of the " m u l t i p l i c i t i e s " zi 4_:_3 D e f i n i t i o n . to be zero. Now we define the c h a r a c t e r i s t i c class of J in H*(X) as the cha- r a c t e r i s t i c class of i t s c h a r a c t e r i s t i c v a r i e t y by: r P(d):= Q(Ch(M)):= z z i q(Ki). 6). 4-:4 Theorem: Let J1 . . Jr ~ Xo be the c o l l e c t i o n of p r i m i t i v e ideals correspon- ding to a n i l p o t e n t o r b i t ~ ( t h a t is with associated v a r i e t y a) The character~st__iic classes b) They span a c) This P(JI ) .

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. 2 Springer's resolution of the n i l p o t e n t cone. 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.