Steinberg slices and group-valued moment maps

TL;DR

This paper introduces a new class of transversal slices in quasi-Poisson spaces related to complex semisimple groups, providing a multiplicative analogue of Whittaker reduction and constructing smooth compactifications with log-symplectic structures.

Contribution

It defines transversal slices in quasi-Poisson spaces, constructs their compactifications, and shows these are smooth and log-symplectic, advancing the understanding of multiplicative structures in group actions.

Findings

Defined a class of transversal slices in quasi-Poisson spaces.

Constructed smooth compactifications with log-symplectic structures.

Connected the slices to multiplicative universal centralizers.

Abstract

We define a class of transversal slices in spaces which are quasi-Poisson for the action of a complex semisimple group G. This is a multiplicative analogue of Whittaker reduction. One example is the multiplicative universal centralizer of G, which is equipped with the usual symplectic structure in this way. We construct a smooth relative compactification by taking the closure of each centralizer fiber in the wonderful compactification of G. By realizing this partial compactification as a transversal in a larger quasi-Poisson variety, we show that it is smooth and log-symplectic.