Differential operators on the moduli space of G-bundles on algebraic curve and Lie algebra cohomologies

Abstract

In this paper we discuss the following problem. Let 𝔊 be a semi-simple complex Lie algebra, 𝔊̂ — the corresponding affine Kac-Moody algebra, K is the central element of 𝔊̂, U _k (𝔊̂), the universal enveloping algebra of 𝔊̂ where we suppose, that K is equal to the number k ∈ 𝐂 ; U _k (𝔊̂) contains infinite combination of the generators, which are acting in the representations of (𝔊̂) with highest weight. It was noticed some time ago that there is a remarkable value of k, which is equal to —g, where g is the dual Coxeter number of 𝔊. For such k the algebra U _-g (𝔊̂) has a large center. Algebra U _k (𝔊̂) is a quantization of the Lie algebra of currents on the circle 𝔊^S , k is the paramenter of quantization. In some sense the current algebra 𝔊^S is a sum of algebras ⊕φ𝔊(φ), where 𝔊(φ) is 𝔊 attached to a point φ ∈ S. Each 𝔊(φ) has a family of Casimir operators, which are destroyed after the quantization. For example, the famous Sugawara construction provides the Virasoro algebra from the quadratic central elements of 𝔊. But the center appears again for the special value of k. It was proved by Hayashi [5], Malikov [4], and Goodman and Wallach [9].