Affine Beilinson-Bernstein at the critical level for GL_2

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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 Beilinson-Drinfeld and Feigin-Frenkel) 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 a proof for GL_2. Time permitting, we will discuss how our result provides a test of the Frenkel-Gaitsgory proposal for local geometric Langlands.