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 was once held on the collage of Utrecht, The Netherlands, in party of the 350th birthday of the college and the sixtieth of T.A. Springer. well-known leaders within the box of algebraic teams and comparable components gave lectures which lined broad and principal parts of arithmetic. although the fourteen papers during this quantity are in general unique learn contributions, a few survey articles are integrated. Centering at the Symposium topic, such assorted themes are coated as Discrete Subgroups of Lie teams, Invariant thought, D-modules, Lie Algebras, specific features, staff activities on Varieties.

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Recent advances in enveloping algebras of semisimple Lie algebras; S~minaire Bourbaki 1976, Springer LNM 67___77(1978),exDos#489. J . L . : Equivariant K-theory approach to nilpotent orbits; IHES preprint M/86/13, March 1986. Einh~llende Algebren halbeinfacher Lie-Algebren; Springer: Berlin-Heidelberg-New York-Tokio 1983. : Go]die rank in the enveloping algebra of a semisimp]e Lie algebra I , I I ; J. : Kostant's problem, Goldie rank, and the Gelfand-Kirillov conjecture; Invent. Math. 56 (1980), 191-213.

There is a natural isomorphism L N. ~ ~HOmA((M). * , N ' ) . M" ® A These considerations also hold for A a n o e t h e r i a n sheaf of rings (in t h e sense of [21 ]), of finite global homological dimension. One t h e n considers derived categories of bounded complexes of coherent (sheaves of) A-modules. Now in t h e case of the sheaf ~Dx, t h e r e is an equivalence of 52 categories between left 4~x-modules and right 4Jx-modules (see [20, $1] n, for details). Here cox = 9X is t h e canonical sheaf, w h e r e n = d i m (X) .

Dx x x - m o d u l e ttRX_(Ox x X) constant, equal to k* . The l e m m a t h e n follows from is Hl ( x , k*) = 0; recall a constant sheaf is flasque, for the Zariski topology. N o w use the Japanese notation ~BAx/XxX = HRx(Oxx x). W e have obtained so far: IL ~x _(1) = ® o ~Dx a ~Ux ® ~X [] ~Dx = RHomh xx ( , )[2n] . Recall the following t e m m a , w h e r e for Y a smooth v a r i e t y , ~ a left ~Dy-module, D R (Tll) denotes the de R h a m complex: ~l d Q~, ~y ~]t ~ "'' d,QdimY O Oy ! 7 If Z ~ Y over k , D R ( ~ Z / Y ) is a closed immersion of smooth varieties is quasi-isomorphic to i .