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

TL;DR

This paper establishes an isomorphism between the algebra of functions on the cotangent bundle of a parabolic base affine space and a subalgebra of functions on a product involving the Levi subgroup, extending previous work to the parabolic and modular contexts.

Contribution

It generalizes Ginzburg and Kazhdan's isomorphism to the parabolic and modular setting, and proves a conjecture of Devalapurkar regarding the partial Whittaker cotangent bundle.

Findings

Isomorphism between function algebras on cotangent bundles and invariant subalgebras.

Extension of Ginzburg and Kazhdan's isomorphism to parabolic and modular cases.

Proof of Devalapurkar's conjecture on the partial Whittaker cotangent bundle.

Abstract

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.