The symplectic groupoid for Adler-Gelfand-Dikii Poisson structure

TL;DR

This work builds a global symplectic groupoid integrating the Adler-Gelfand-Dikii Poisson structure on the space of n-th order differential operators with prescribed symmetry, for G in {\\mathrm{PSL}(n), \\mathrm{PSp}(n) (even n), \\mathrm{PSO}(n) (odd n)}. It then shows a Morita equivalence with a quasi-symplectic groupoid that integrates the Dirac structure on Y_n(\\mathbf{C}), the monodromy-quotient of nondegenerate curves; this generalizes the Alekseev-Meinrenken coadjoint correspondence to a DS-reduced, groupoid setting. The method combines a DS reduction framework, a curve-operator correspondence via monodromy and the Wronskian, and a Morita reduction by a Lie 2-group to relate infinite-dimensional Poisson geometry to finite-dimensional Dirac geometry. The results provide a canonical geometric bridge between loop-space Poisson structures and finite-dimensional Dirac-type groupoids, with potential implications for integrable systems and geometric representation theory.

Abstract

The Adler-Gelfand-Dikii Poisson structure arises naturally in the study of $n$-th order differential operators on the circle and plays a central role in Poisson geometry and integrable systems. Let $G$ be one of the Lie groups $\mathrm{PSL}(n)$, $\mathrm{PSp}(n)$ (for even $n$), or $\mathrm{PSO}(n)$ (for odd $n$). In this paper, we construct the symplectic groupoid integrating the Adler-Gelfand-Dikii Poisson structure associated to $G$ and prove that it is Morita equivalent to the quasi-symplectic groupoid integrating the Dirac structure on $Y_n(\mathbf{C})$, where $Y_n(\mathbf{C})$ denotes the quotient of the space of quasi-periodic non-degenerate curves by homotopies preserving the monodromy.

Theorems & Definitions (54)

• Theorem 1.1
• Proposition 1.1
• Remark 2.1
• Example 2.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof