Gaiotto–Witten superpotential and Whittaker D-modules on monopoles

Let G be an almost simple simply connected group over C. For a positive element α of the coroot lattice of G let Z∘^α denote the space of maps from P¹ to the flag variety B of G sending ∞∈P¹ to a fixed point in B of degree α. This space is known to be isomorphic to the space of framed G-monopoles on R³ with maximal symmetry breaking at infinity of charge α. In [6] a system of (étale, rational) coordinates on Z∘^α is introduced. In this note we compute various known structures on Z∘^α in terms of the above coordinates. As a byproduct we give a natural interpretation of the Gaiotto–Witten superpotential studied in [8] and relate it to the theory of Whittaker D-modules discussed in [9].