# An Extension of the New Einstein Geometry by Michal A.D., Botsford J.L. PDF By Michal A.D., Botsford J.L.

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LX G ! 4 a). Such a construction works in general10 and is functorial. Now observe that Ld Ge is MumLG (OQGe (1)). 3 c), which proves the lemma. Now our question of the surjectivity of f has a positive answer in view of the following. 3. Theorem. 3 c) to LX G splits. Proof. Consider the inclusion i : LX Ge ,! LGe. The map Lie(i) : LX g ,! Lg sends X f to X fb=0 where fb=0 is the Laurent development of f at x0 . 2 a) is trivial over LX g, hence LX g may be seen as a subalgebra of Lcg. Consider the basic highest weight representation H1 (0) of level one of Lcg and take coinvariants: B = [H1 (0)]LX g = H1(0)=L g:H (0): X 1 The I will use is that B 6= 0.

Math. , 164(2):385{419, 1994. A. Beauville and Y. Laszlo. Un lemme de descente. Comptes Rendus Acad. Sci. Paris, 320:335{340, 1995. A. Beauville, Y. Laszlo, and C. Sorger. The Picard group of the Moduli of G-bundles on a Curve. , 112:183{216, 1998. A. Beilinson and V. Drinfeld. Quantization of Hitchin0 s integrable system and Hecke eigensheaves. Preliminary version.  A. -P. Grivel, B. Kaup, A. Haefliger, B. Malgrange, and F. Ehlers. Algebraic D-modules. , Boston, MA, 1987.  V. G. Drinfeld and C.

2 b)  : L[ G m; m hence a line bundle L 1 on the homogeneous space QGe . 1. A line in the in nite Grassmannian. Consider the morphism of C groups ' : SL2 ! SL2 R((z))     a b 7 ! d cz 1 c d bz a and moreover the morphism of C -groups : LSL2 ! LGe deduced from the map SL2 ! Ge associated to the highest root . 2 c) ' =   : SL2 ! LG: The Borel subgroup B2  SL2 of upper triangular matrices maps to L+ Ge by construction, hence we get a morphism ' : P1C ! QGe . An easy calculation 46 shows that the derivative Lie(') maps the standard sl2 -triplet fe; f; hg = fX ; X  ; H g to the sl2 -triplet fX  z; X z 1 ; H g of Lg. 