Twisted generating functions incorporating singular vectors in verma modules and their localizations, I

In this first paper in a series, we construct certain “twisted” generating functions, ranging over the highest weights as well as the weight spaces, incorporating the well-known classical singular vectors in the Verma modules for sl(2), and we give a characterization of singular vectors by deriving simple partial differential equations for the generating functions. We also give a second interpretation of singular vectors using twisted generating functions incorporating singular vectors in certain localizations of the Verma modules. For this, we introduce certain natural “negative counterparts” of Verma modules for a certain natural “negative counterpart” of U(sl(2)). Our twisted generating functions for the second interpretation are based on a certain divergent formal series which, as it turns out, appeared in Euler’s study of the Euler-Gompertz constant.