Fock Space Realizations of Imaginary Verma Modules

The motivation for this paper goes back to at least the work of A. Borel, R. Bott and A. Weil on their geometric construction of irreducible representations of complex simple Lie groups, and the related work of B. Kostant [10] and J. Bernšteı̆n, I. Gel′fand and S. Gel′fand [2]. Articles more closely related to ours are those of H. Jakobsen and V. Kac [9] and then presumably Wakimoto [11] and more generally B. Feı̆gin and È. Frenkel [4–6]. These latter authors in particular gave geometric constructions of certain representations of affine Kac–Moody algebras in terms of infinite sums of partial differential operators acting on a Fock space. Our work expands on the last section of the paper of D. Bernard and G. Felder (see [1]) to the case of ŝln(C) and then notes that, in the generic case, these geometric constructions are just realizations of the imaginary Verma modules studied by V. Futorny (see [8]). We’ll now describe in more detail the cited work of D. Bernard and G. Felder and its relationship to our main result. Fix a positive integer n, γ ∈ C∗ and for each 1 ! i ! n fix λi ∈ C. Set 2/ c = γ 2. We will let