Local geometric Langlands correspondence and affine Kac-Moody algebras
Let \( \mathfrak{g} \) be a simple Lie algebra over ℂ and G a connected algebraic group with Lie algebra \( \mathfrak{g} \). The affine Kac-Moody algebra \( \hat {\mathfrak{g}} \) is the universal central extension of the formal loop agebra \( \mathfrak{g} \)((t)). Representations of \( \hat {\mathfrak{g}} \) have a parameter, an invariant bilinear form on \( \mathfrak{g} \), which is called the level. Representations corresponding to the bilinear form which is equal to minus one half of the Killing form are called representations of critical level. Such representations can be realized in spaces of global sections of twisted D-modules on the quotient of the loop group G((t)) by its “open compact” subgroup K, such as G[[t]] or the Iwahori subgroup I.