The cotangent bundle of $G/U_P$ and Kostant-Whittaker descent
Tom Gannon
TL;DR
The paper addresses the problem of describing the algebra of functions on the cotangent bundle $T^*(G/U_P)$ for a reductive group $G$ and a parabolic $P$ in a way that is compatible with parabolic and modular settings. It develops a Kostant-Whittaker descent framework and a Whittaker reduction to produce a canonical invariant description: $\mathcal{O}(T^*(G/U_P))$ is identified with $\mathcal{O}(G \times \mathfrak{c}_L \times L)^{J_G}$, where $J_G$ is the group scheme of universal centralizers. The construction extends to the partial Whittaker cotangent bundle, proving a conjecture of Devalapurkar and yielding corollaries that relate to dual Hamiltonian spaces in relative Langlands duality. The methods connect to geometric Langlands, representation theory of Lie algebras, and Coulomb branches, providing explicit invariant-algebra realizations that bridge classical and modular settings. This enables new avenues for understanding dualities and reductions in the parabolic base affine context.
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.