The Poisson geometry of Plancherel formulas for triangular groups

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In this paper we establish the existence of canonical coordinates for generic co-adjoint orbits on triangular groups. These orbits correspond to a set of full Plancherel measure on the associated dual groups. This generalizes a well-known coordinatization of co-adjoint orbits of a minimal (non-generic) type originally discovered by Flaschka. The latter had strong connections to the classical Toda lattice and its associated Poisson geometry. Our results develop connections with the Full Kostant–Toda lattice and its Poisson geometry. This leads to novel insights relating the details of Plancherel theorems for Borel Lie groups to the invariant theory for Borels and their subgroups. We also discuss some implications for the quantum integrability of the Full Kostant Toda lattice.

Original languageEnglish (US)
Article number133801
JournalPhysica D: Nonlinear Phenomena
StatePublished - Nov 2023


  • Dixmier–Pukanszky operator
  • Invariant theory
  • Orbit method
  • Polarizations
  • Toda lattice

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics


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