Algebraic Groups Utrecht 1986: Proceedings of a Symposium in by Armand Borel (auth.), Arjeh M. Cohen, Wim H. Hesselink,

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 get together 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 huge and critical components of arithmetic. notwithstanding the fourteen papers during this quantity are normally unique study contributions, a few survey articles are integrated. Centering at the Symposium topic, such assorted themes are lined as Discrete Subgroups of Lie teams, Invariant conception, D-modules, Lie Algebras, certain services, team activities on Varieties.

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1, t h e proposition follows f r o m a n i s o m o r p h i s m of complexes given by: c p ' M @ Cc (G i+1) given by ~(F) (S0 . . . 1 (~ > M • C c (G1*1) gi) = g O .. • g i F ( g 0 . . . , t h e s a m e is a m a p f r o m t h e Hochschild c o m p l e x to t h e s t a n d a r d complex of g r o u p homology). Now w e c o n c e n t r a t e on t h e birnoduie M = C c (G) . The definition of B in [13, II, ~3], [24 ] is not applicable, since o u r algebra h a s no unit, a n d B involves a h o m o t o p y s for t h e cyclic c o m p l e x (C~ (G i+1) , b / ) , w h i c h uses a unit.

Hence ' Z x is non-zero in Hidiff(Gx, (t]) . tt follows t h a t J'G/Gx e(g x g-l) d g m u s t be 0 . 5 Let (abstract Selberg principle) G be a connected reductive Lie Group over F , e be an i d e m p o t e n t in Cc (G) , x a regular semi-simple e l e m e n t of G w h i c h is not compact. Then ~GIG x e(gxg-t)dg = O. The t e r m "abstract Selberg principle" was coined by Jutg and Valette, who established it for groups of F-split r a n k one [18 ]. They use analysis on t h e Bruhat-Tits tree, and Fredholm modules (in t h e sense of Connes), The "concrete" Selberg principle is deduced by taking e(g) = X< ~(g) v , v * > where (~,V) e to be i s a cuspidal representation of G (here F i s n o n - a r c h i m e d e a n ) , ~ ¢ V , v* E V* w i t h ( v , v*) = O, and X is a suitable constant.

N') be a bounded complex of finitely-generated right (resp. left) A-modules. 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).

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