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

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.