Gaiotto–Witten superpotential and Whittaker D-modules on monopoles

Let G   be an almost simple simply connected group over CC. For a positive element α of the coroot lattice of G   let Z∘α denote the space of maps from P1P1 to the flag variety BB of G   sending ∞∈P1∞∈P1 to a fixed point in BB of degree α. This space is known to be isomorphic to the space of framed G  -monopoles on R3R3 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].