γ(ℂ^×)-action on T* M and fixed-point equality

Determine whether the γ(ℂ^×)-action defined on T* M, for a Hamiltonian G-variety M, preserves the intersection T* M ×_{[\mathfrak{g}^*/G]} T* G //_{e} U_{e}, and ascertain whether the equality of fixed-point sets M^T = M^{T_e^∘ γ(ℂ^×)} holds without the assumption that all stabilizers in M are of maximal rank.

Background

To extend fixed-point computations beyond cotangent bundles of homogeneous spaces, the authors attempt to generalize their torus and ℂ^×-action framework to T* M for a general Hamiltonian G-variety M.

They define a natural γ(ℂ^×)-action but explicitly note two unresolved issues: whether this action preserves the intersection variety with T* G //{e} U{e}, and whether fixed-point sets agree for the relevant torus actions without additional assumptions on stabilizers. Resolving these would broaden the applicability of their fixed-point machinery.