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By Bak A. (ed.)

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Extra resources for Algebraic K-Theory, Number Theory, Geometry and Analysis: Proceedings of July 26-30, 1982

Sample text

Proof. 1 that the kernel of the forgetful map f : PicLX Ge (QGe ) ! Pic(QGe ) identi es to the character group 12 This is not really a restriction: actually the result is the same for the other components. 13 This is related to the fact that the center of Spin is Z2  Z2 which has a non trivial 4m central extension. 52 X (LX G) of LX G. 3. I will not prove (ii) here. 3). Once we know that we can0 t do better than `b , we have to show that OMG;X e (`b ) actually descends. This may be easy, as for G = PGLr , where a pfaan of DAd is convenient (just look at the numbers of Tables B and C) or more complicated, as for SLr =s with s j r (see [13]).

0 X Once we know the lemma, we do the following. Let g 2 LX Ge(C ) and let K be the eld of rational functions on X . Using the fact (cf. [21]) that Ge(K ) is generated by the standard unipotent subgroups U (K ), 2 , we may Q suppose that g is of the form j 2J exp(fj nj ) where the nj are nilpotent 44 elements of g and fj 2 K . Let fp1 ; : : : ; pi g be the poles of the functions fj ; j 2 J . LiX Ge Y t 7! exp(tfj nj ) j 2J is a path from g to 1 in LiX Ge. By the above, the morphism 0 (LX Ge) !

Each connected component of QG is isomorphic to QGe . Proof. By [21], there exists a nite family of homomorphisms x : G a ! Ge such that for any extension K of C , the subgroups x (K ) generate Ge(K ). Since the ind-group G a (C ((z ))) is connected, it follows that LGe is connected. In the general case, consider the exact sequence 1 ! 1 (G) ! Ge ! G ! 1 as an exact sequence of etale sheaves on D := Spec C((z )). Since H 1 (D ; Ge) Moduli of G-bundles 43 is trivial [22], it gives rise to an exact sequence of C -groups e 1 (G) !

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