The Poisson geometry of Plancherel formulas for triangular groups

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-gener...

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Bibliographic Details
Published in:Physica. D 2023-11, Vol.453, p.133801, Article 133801
Main Author: Ercolani, Nicholas M.
Format: Article
Language:English
Subjects:
Dixmier–Pukanszky operator
Invariant theory
Orbit method
Polarizations
Toda lattice
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Summary: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. •New links explored between dual groups of Lie groups and their Poisson geometry.•Dixmier–Pukanszky operator of Plancherel theory related to Toda invariant theory.•Canonical coordinates established for symplectic leaves of full Toda systems.
ISSN:0167-2789
1872-8022
DOI:10.1016/j.physd.2023.133801