Canonical bases, toric degenerations, and collective integrable systems

Jeremy Lane, McMaster University

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Passcode: 998749

There are three important settings for studying actions of reductive Lie groups: modules, algebraic group actions, and Hamiltonian group actions. In the study of modules one encounters various constructions of nice bases which are in some sense canonical (e.g. Gelfand-Zeitlin, Lusztig).  In the study of algebraic group actions canonical bases give rise to toric degenerations; deformations of the G-variety to a toric variety (cf. Caldero, Alexeev-Brion). 

In this talk I will discuss the symplectic analogue of these constructions: integrable systems. We show how  toric degenerations  give rise to integrable systems on arbitrary symplectic manifolds equipped with Hamiltonian group actions. This generalizes a family of well-known examples called Gelfand-Zeitlin integrable systems due to Guillemin and Sternberg.  As a by-product, we  generalize results of Harada and Kaveh on construction integrable systems from toric degenerations.

This talk is based on joint work with Benjamin Hoffman.