Geodesic
Localization of $\mathfrak g$-modules on the affine Grassmannian
Edward Frenkel1 ; Dennis Gaitsgory2
1Department of Mathematics<br/>University of California, Berkeley<br/>970 Evans Hall #3840<br/>Berkeley, CA 94720-3840<br/>United States
2Department of Mathematics<br/>Harvard University<br/>One Oxford Street<br/>Cambridge, MA 02138<br/>United States
Annals of mathematics, Tome 170 (2009) no. 3, pp. 1339-1381
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We consider the category of modules over the affine Kac-Moody algebra $\widehat{\mathfrak g}$ of critical level with regular central character. In our previous paper we conjectured that this category is equivalent to the category of Hecke eigen-D-modules on the affine Grassmannian $G(\!(t)\!)/G[\mskip-2mu[t]\mskip-2mu]$. This conjecture was motivated by our proposal for a local geometric Langlands correspondence. In this paper we prove this conjecture for the corresponding $I^0$ equivariant categories, where $I^0$ is the radical of the Iwahori subgroup of $G(\!(t)\!)$. Our result may be viewed as an affine analogue of the equivalence of categories of ${\mathfrak g}$-modules and D-modules on the flag variety $G/B$, due to Beilinson-Bernstein and Brylinski-Kashiwara.

DOI : 10.4007/annals.2009.170.1339

Edward Frenkel  1   ; Dennis Gaitsgory  2

1 Department of Mathematics<br/>University of California, Berkeley<br/>970 Evans Hall #3840<br/>Berkeley, CA 94720-3840<br/>United States
2 Department of Mathematics<br/>Harvard University<br/>One Oxford Street<br/>Cambridge, MA 02138<br/>United States 
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     title = {Localization of $\mathfrak g$-modules on the affine {Grassmannian}},
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Edward Frenkel; Dennis Gaitsgory. Localization of $\mathfrak g$-modules on the affine Grassmannian. Annals of mathematics, Tome 170 (2009) no. 3, pp. 1339-1381. doi: 10.4007/annals.2009.170.1339

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