Poisson Geometry of Monic Matrix Polynomials

TL;DR

This paper explores the Poisson geometric structure of a specific subgroup of the loop group for GL_m and SL_m, classifying symplectic leaves via Smith Normal Forms and relating them to affine Grassmannians and monopoles.

Contribution

It provides a classification of symplectic leaves in the Poisson structure of the first congruence subgroup, extending known descriptions to new geometric contexts.

Findings

Classified symplectic leaves using Smith Normal Forms.

Identified open charts with Poisson transition functions.

Connected results to affine Grassmannians and Zastava spaces.

Abstract

We study the Poisson geometry of the first congruence subgroup $G_1[[z^{-1}]]$ of the loop group $G[[z^{-1}]]$ endowed with the rational r-matrix Poisson structure for $G=GL_m$ and $SL_m$. We classify all the symplectic leaves on a certain ind-subvariety of $G_1[[z^{-1}]]$ in terms of Smith Normal Forms. This classification extends known descriptions of symplectic leaves on the (thin) affine Grassmannian and the space of $SL_m$-monopoles. We show that a generic leaf is covered by open charts with Poisson transition functions, the charts being birationally isomorphic to products of coadjoint $GL_m$ orbits. Finally, we discuss our results in terms of (thick) affine Grassmannians and Zastava spaces.