Notes on invariant measures for loop groups

Let $K$ denote a simply connected compact Lie group and let $G=K^{\mathbb C}$, the complexification. It is known that there exists an $LK$ bi-invariant probability measure on a natural hyperfunction completion of the complex loop group $LG$. There are various generalizations, involving positive line bundle valued measures on the hyperfunction completion, replacing $K$ with a symmetric space, replacing $LK$ (the configuration space of the principal chiral model) with (the homotopy equivalent space of ) gauge equivalence classes of $K$-connections on the 2-sphere (the configuration space of $YM_3$), and so on. The purpose of these notes is to publicize a number of conjectures and questions concerning how these measures are characterized, how they are explicitly represented, and how they are potentially relevant to quantum sigma models and $YM_3$.