The cotangent bundle of $G/U_P$ and Kostant-Whittaker descent

We prove that the algebra of functions on the cotangent bundle $T^*(G/U_P)$ of the parabolic base affine space for a reductive group $G$ and a parabolic subgroup $P$ is isomorphic to the subalgebra of the functions on $G \times L \times \mathfrak{l}//L$ which are invariant under a certain action of the group scheme of universal centralizers on $G$, where $L$ is a Levi subgroup of $P$ and $\mathfrak{l}$ is its Lie algebra, upgrading an isomorphism of Ginzburg and Kazhdan simultaneously to the parabolic and the modular setting. We also derive a related isomorphism for the partial Whittaker cotangent bundle of $G$, which proves a conjecture of Devalapurkar.