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An analogue of Whittaker reduction for group-valued moment maps

Ana Balibanu

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.

An analogue of Whittaker reduction for group-valued moment maps

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.

Paper Structure

This paper contains 16 sections, 9 theorems, 113 equations.

Table of Contents

  1. Review of Whittaker reduction
  2. Slodowy slices
  3. Whittaker reduction
  4. Poisson actions of Poisson--Lie groups
  5. Poisson--Lie groups
  6. Poisson structures on the double
  7. Poisson actions
  8. Moment maps for the dressing action
  9. Poisson reduction
  10. Groupy Whittaker
  11. Multiplicative transversal slices
  12. Characteristic distributions in $(G,\pi_0)$
  13. Reduction along the slice $\Sigma$
  14. Symplectic leaves of the reduction
  15. Recollections on Dirac geometry
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\mathfrak{c}$ be the orthogonal complement of the fixed-point set $\mathfrak{t}^w$ in the maximal Cartan $\mathfrak{t}$ of the Lie algebra $\mathfrak{g}$. The quotient $Q$ carries a natural Poisson bracket $\{\cdot,\cdot\}_Q$ which is uniquely characterized by the property that for all functions $f,g\in \mathcal{O}_Q$ and all $\mathfrak{c}$-invariant lifts $F,G\in\mathcal{O}_M$ that satisfy

Theorems & Definitions (31)

  • Theorem
  • Theorem
  • Remark 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.8
  • Example 2.9
  • ...and 21 more