Algebraic Groups by Arjeh M. Cohen, Wim H. Hesselink, Wilberd L.J. van der

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 get together of the 350th birthday of the collage and the sixtieth of T.A. Springer. famous leaders within the box of algebraic teams and comparable parts gave lectures which coated large and primary components of arithmetic. although the fourteen papers during this quantity are as a rule unique examine contributions, a few survey articles are incorporated. Centering at the Symposium topic, such varied subject matters are lined as Discrete Subgroups of Lie teams, Invariant conception, D-modules, Lie Algebras, targeted features, staff activities on forms.

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As a subring of the ring by making a character × of A S(t*) of formal power series on We consider t . 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.

5 For zero", H iff(G , Cc (G)) is c o n j e c t u r e d t h e case F by iff all orbital P. Blanc non archimedean for F is d u e to ]. 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 .

5) combine with Springer's and Joseph's correspondences to a commutative t r i a n g l e ? Or in other words: Is to p~, i f J corresponds to ~? In case o(J) equivalent G = SLn, the e x p l i c i t combinatorial des- c r i p t i o n s given above prove that t h i s is true. Barbasch and Vogan have v e r i f i e d t h i s 24 as a matter of f a c t f o r a l l cases, by an enormous amount of e x p l i c i t c a l c u l a t i o n s in [BVI,2]. More conceptual reasons are offered by Hotta and Kashiwara [HK] and below.

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