Fock Space and Kazhdan-lusztig Polynomials

1. Lecture 1: Affine Lie algebras and the Fock representation of ĝln. 1.1. The loop algebra construction. Let g be a complex reductive Lie algebra and let L denote the algebra of Laurent polynomials in one variable L = C[t, t−1]. The loop algebra over g is L(g) = L ⊗ g, which is a Lie algebra with the bracket [t ⊗ x, t ⊗ y]0 = t[x, y]. (1.1) The elements of the loop algebra may be regarded as regular rational functions on C× with values in g. If V is a g-module, then L(V ) = L ⊗ V is an L(g)-module with the action (t ⊗ x)(t ⊗ v) = txv. (1.2) Define a 2-cocycle on g using a non-degenerate symmetric associative bilinear form (·, ·) on g (namely, the Killing form on the derived algebra of g, direct sum any nondegenerate symmetric form on the center of g.) Define α : g× g −→ C by α(x(t), y(t)) = Res0((x ′(t), y(t))). (1.3) In particular, α(t ⊗ x, t ⊗ y) = δr+s,0 r (x, y). (1.4) Then α is a 2-cocycle, namely it is antisymmetric and satisfies α([x(t), y(t)], z(t)) + α([y(t), z(t)], x(t)) +α([z(t), x(t)], y(t)) = 0. (1.5)