  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.
