Determine the socle structure of general Nakano modules

Determine the dotted socle arrow, and hence the general socle structure, of Nakano’s modules P(τ) beyond what is established by the arguments preceding Figure 5.1 in Nakano’s construction.

Background

The paper uses Nakano’s logarithmic Virasoro extension results to identify the images of indecomposable projective objects under the McRae–Yang tensor functor and under quantum Drinfeld–Sokolov reduction. The relevant modules are denoted P(τ) and arise as middle terms of distinguished nonzero extension classes.

The authors explicitly avoid assuming a general description of the socle or contragredient self-duality of P(τ), because Nakano’s preceding arguments do not determine the dotted socle arrow in the general diagram. Establishing this missing socle information would provide a more complete structural description of the modules P(τ), although it is not needed for the results proved in the paper.