An analogue of Whittaker reduction for group-valued moment maps

TL;DR

This paper develops a new reduction technique for Poisson actions of complex Poisson-Lie groups, generalizing Whittaker reduction using transversal slices related to unipotent orbits, with interpretations in Dirac geometry.

Contribution

It introduces an analogue of Whittaker reduction for group-valued moment maps, utilizing generalized Steinberg slices and Dirac geometry for Poisson-Lie groups.

Findings

Constructed a new reduction method along transversal slices

Provided a Dirac geometric interpretation of the reduction

Described symplectic leaves of the reduced spaces

Abstract

We construct an analogue of Whittaker reduction for Poisson actions of a semisimple complex Poisson-Lie group G. The reduction takes place along a class of transversal slices to unipotent orbits in G, which are generalizations of the Steinberg cross-section and are indexed by conjugacy classes in the Weyl group. We give an interpretation of these reductions in the framework of Dirac geometry, and we use this to describe their symplectic leaves.