Morita Equivalences, Moduli Spaces, and Flag varieties

Abstract Double Bruhat cells in a connected complex semisimple Lie group G emerged as a crucial concept in the work of S. Fomin and A. Zelevinsky on total positivity and cluster algebras. These cells are special instances of a broader class of cluster varieties known as generalized double Bruhat cells, which can be studied collectively as Poisson subvarieties of F₂₍ = B^2n-1 G, where B is the flag variety of G. The spaces F₂₍ are Poisson groupoids over B^n and were introduced by J. -H. Lu, V. Mouquin, and S. Yu in the study of configuration Poisson groupoids of flags. In this work, we describe the spaces F₂₍ as decorated moduli spaces of flat G-bundles over a disc. This perspective yields the following results: (1) We explicitly integrate the Poisson groupoids F₂₍ to symplectic double groupoids, which are complex algebraic varieties. Furthermore, we show that these integrations are symplectically Morita equivalent for all n. (2) Using this construction, we integrate the Poisson subgroupoids of F₂₍ formed by unions of generalized double Bruhat cells to explicit symplectic double groupoids. As a corollary, we obtain integrations for the top-dimensional generalized double Bruhat cells contained therein. (3) Finally, we relate our integration to the work of P. Boalch on meromorphic connections. We lift the torus actions on F₂₍ to the double groupoid level and show that they correspond to the quasi-Hamiltonian actions on the fission spaces of irregular singularities.