CONSTRUCTION OF CANONICAL COORDINATES FOR EXPONENTIAL LIE GROUPS

• Published in: Transactions of the American Mathematical Society 2009-12, Vol.361 (12), p.6283-6348
• Main Authors: ARNAL, DIDIER, CURREY, BRADLEY, DALI, BECHIR
• Format: Article
• Language: English
• Subjects: Algebra; Analytic functions; Coordinate systems; Exact sciences and technology; Functions of a complex variable; Group theory; Lie groups; Manifolds and cell complexes; Mathematical analysis; Mathematical functions; Mathematical theorems; Mathematics; Rational functions; Real functions; Sciences and techniques of general use; Sine function; Subalgebras; Topological groups, lie groups; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Vector fields
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-09-04936-8

Given an exponential Lie group G, we show that the constructions of B. Currey, 1992, go through for a less restrictive choice of the Jordan-Holder basis. Thus we obtain a stratification of g* into G-invariant algebraic subsets, and for each such subset Ω, an explicit cross-section Σ ⊂ Ω for coadjoint orbits in Ω, so that each pair (Ω Σ) behaves predictably under the associated restriction maps on g*. The cross-section mapping σ→ Σ is explicitly shown to be real analytic. The associated Vergne polarizations are not necessarily real even in the nilpotent case, and vary rationally with l ∈ Σ. For each Ω, algebras µ⁰ (Ω) and µ¹ (Ω) of polarized and quantizable functions, respectively, are defined in a natural and intrinsic way. Now let 2d > 0 be the dimension of coadjoint orbits in Ω. An explicit algorithm is given for the construction of complex-valued real analytic functions {q1, q2, · · · , q d } and {p1, P2, · · · , P d } such that on each coadjoint orbit O in Ω, the canonical 2-form is given by $\sum {dp_k \wedge dq_k } $ The functions {q1,q2, · · . , q d } belong to µ (Ω), and the functions {p1, P2, · · · , P d } belong to µ¹ (Ω). The associated geometric polarization on each orbit coincides with the complex Vergne polarization, and a global Darboux chart on O is obtained in a simple way from the coordinate functions (p1,...,pd,q1,..., qd) (restricted to O). Finally, the liner evaluation functions l → l(X) are be quantizanle as well.