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 was once 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 components gave lectures which coated vast and relevant parts of arithmetic. notwithstanding the fourteen papers during this quantity are more often than not unique study contributions, a few survey articles are incorporated. Centering at the Symposium topic, such different themes are lined as Discrete Subgroups of Lie teams, Invariant conception, D-modules, Lie Algebras, certain services, workforce activities on forms.

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These r e s t r i c t i o n processes give isomorphisms KG(X) ~ KG(T*X) ~ KT(E). (*) The point of these manipulations is now that equivariant K-theory of a l i n e a r torus action can be carried out very conveniently in terms of calculations with formal characters, as I shall explain a l i t t l e more precisely below, and that we can re- 29 duce our problems from the context of G- equivariant K-theory on X into t h i s more convenient s e t t i n g . 2, the l i n k to our previous geometrical considerations is made by means of the homomorphisms KG(X) • K ( X ) / c h , H*(X).

F n o n - a r c h i m e d e a n , let U c G be t h e open set of r e g u l a r s e m i - s i m p l e e l e m e n t s , Y = G- U . Let S C ~ (G) G be t h e m u l t i p l i c a t i v e s u b s e t of formed of functions w h i c h v a n i s h n o w h e r e on U . 6 The C ¢~ (G) G - l i n e a r m a p ~ H ff(G , C c (U)) d~ff , c ~ , H i (G "c (G)) b e c o m e s a n i s o m o r p h i s m a f t e r localizing a t S . Indeed, t h e r e exists a f u n c t i o n - - t h e f u n c t i o n A - in w h i c h v a n i s h e s e x a c t l y on S.

The fact t h a t J~G is n o t a n abelian c a t e g o r y c o m p l i c a t e s t h e algebraic h o m o l o g y of JDG . DG be t h e c a t e g o r y of admissible G-modules, in t h e sense of H a r i s h - C h a n d r a . An o b j e c t of JDG is a c o m p l e x v e c t o r space M , w i t h an action of G , w h i c h is s u c h t h a t t h e stabilizer of a n y v e c t o r of M is open in G. We will also talk of differentiable (instead of admissible) modules. JDG is n o w a n abelian category, 41 The definition of the differentiable homology, due to P.