A quasi-Poisson structure on the multiplicative Grothendieck-Springer resolution

TL;DR

This paper demonstrates that the multiplicative Grothendieck-Springer space naturally admits a quasi-Poisson structure, extending classical Poisson geometry to a multiplicative setting with explicit geometric and algebraic implications.

Contribution

It introduces a quasi-Poisson structure on the multiplicative Grothendieck-Springer space and relates it to group-valued moment maps and Steinberg fibers, building on recent reduction techniques.

Findings

The multiplicative Grothendieck-Springer space has a natural quasi-Poisson structure.

The associated moment map is the resolution morphism.

Quasi-Hamiltonian leaves correspond to preimages of Steinberg fibers.

Abstract

In this note we show that the multiplicative Grothendieck-Springer space has a natural quasi-Poisson structure. The associated group-valued moment map is the resolution morphism, and the quasi-Hamiltonian leaves are the connected components of the preimages of Steinberg fibers. This is a multiplicative analogue of the standard Poisson structure on the additive Grothendieck-Springer resolution, and an explicit illustration of a more general procedure of reduction along Dirac realizations which is developed in recent work of the author and Mayrand.