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 collage of Utrecht, The Netherlands, in party 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 lined extensive and relevant parts of arithmetic. notwithstanding the fourteen papers during this quantity are normally unique study contributions, a few survey articles are incorporated. Centering at the Symposium topic, such various themes are lined as Discrete Subgroups of Lie teams, Invariant thought, D-modules, Lie Algebras, designated capabilities, staff activities on Varieties.

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**Example text**

3). Consider the quotient U(g)/J as a l e f t g-module M, and take the correspon- ding sheaf of modules M:= DX~U(g) M over the sheaf Dx l o c a l i z a t i o n of of rings of d i f f e r e n t i a l operators on M on X (Beilinson-Bernstein X); then the c h a r a c t e r i s t i c v a r i e t y resp. cycle of well-defined notions from the general theory of D-modules, denoted Ch(MI here. By d e f i n i t i o n , Ch(M) is a closed subvariety in M are Ch(M) resp. T'X, and Ch[MI is a formal integer l i n e a r combination Ch(M) = z mi [ V i ] , 1 in which each i r r e d u c i b l e component Vi of the c h a r a c t e r i s t i c v a r i e t y occurs with some well-defined p o s i t i v e m u l t i p l i c i t y mi .

B u r g h e l e a also c o m p u t e s t h e cyclic h o m o l o g y ~ r o u p H C~(k [G]) , 37 for k a Q-algebra. 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 .

This is done T correspond to the exponential of i t s d i f f e r e n t i a l : ~ ~I (dx) n = x. n>O PE ~(t ; ) is any power series, then we denote edx If KT(E) onto = term, which is of course a homogeneous polynomial. [p]d i t s degree d homogeneous 30 Theorem: Let K be an o r b i t a l cone bundle. Let M = Y(E,OK), the ring of the re- gular functions on i t s f i b r e over the base point, considered as a T- equivariant S(E*)-module. Then a) As a formal power series on ~, A oh(M) has i t s lowest nonzero homogeneous term in degree d:= codimT, X Kb) The e q u i v a r i a n t c h a r a c t e r i s t i c class of K (as a polynomial on ~) is given by the formula QG(K) = [A ah(M)] d.