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There has long been interest in Beilinson-Bernstein localization for the affine Grassmannian (or affine flag variety). First, Kashiwara-Tanisaki treated the so-called negative level case in the 90's. Some ten years later, Frenkel-Gaitsgory (following work of Feigin-Frenkel and Beilinson-Drinfeld) formulated a conjecture at the critical level and made some progress on it. Their conjecture is more subtle than its negative level counterpart, but also more satisfying. We will review the necessary background from representation theory of Kac-Moody algebras at critical level, formulate the Frenkel-Gaitsgory conjecture, and outline the proof for GL_2.

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