Calculus of Finite Differences - eDisk

– p. 11/?Verma modulesLet λ ∈ C. The Verma module V (λ) has basis {v i } i≥0whereuv i = v i+1 , dv 0 = 0, hv 0 = λv 0 ;other relations are derived from the defining relations.Thus, for i ≥ 0,hv i = (λ + iγ)v i ,dv i+1 = [f(λ) + f(λ + γ) + · · · + f(λ + iγ)]v i .(You can see why u is the “up” operator, d is the“down” operator, and h is the “horizontal” operator.)