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Symplectic fibrations and multiplicity diagrams. (English) Zbl 0870.58023

Cambridge: Cambridge Univ. Press. xiv, 222 p. (1996).
In this research monograph, the authors use techniques from symplectic group actions in order to study multiplicities of weights in representations of compact Lie groups.
To each irreducible representation \(\rho\) of a compact connected Lie group \(H\) one can assign a certain orbit \(\mathcal O\) of the coadjoint representation of \(H\) on \(\mathfrak h^*\). This orbit carries a natural symplectic structure, and the \(H\)-action on \(\mathcal O\) is Hamiltonian. Let \(G\) be a maximal torus of \(H\) and let \(\Phi\colon\mathcal O\to\mathfrak g^*\) be the moment map of the \(G\)-action on \(\mathcal O\). The image of \(\Phi\) is a convex polytope \(\Delta=\bigcup\limits^.\Delta_i\subset\mathfrak g^*\). The pushforward of the Liouville measure on \(\mathcal O\) to \(\Delta\) has a density with respect to the Lebesgue measure which is given by the so-called “Duistermaat-Heckman polynomials” \(f_i\colon\Delta_i\to\mathbb R\) of degree at most \(\dim(\mathcal O)/2-\dim(G)\). For “large” representations \(\rho\) these polynomials are asymptotic to the polynomials that describe the “multiplicity diagram” of \(\rho\), i.e., the multiplicities of weights of \(G\) in \(\rho\).
Many coadjoint orbits \(\mathcal O\) fibre over coadjoint orbits of lower dimension, and these fibrations are symplectic. In this situation one can compute the Duistermaat-Heckman polynomials of \(\mathcal O\) in terms of the corresponding polynomials of the fibre and the base. The authors show how this method can be used to inductively compute Duistermaat-Heckman measures of all coadjoint orbits of certain Lie groups \(H\). By a weak coupling limit construction the authors also explain why some of the \(f_i\) may have lower than maximal degree. These \(f_i\) correspond to so-called “lacunary regions” in the multiplicity diagram of \(\rho\).
The book requires some familiarity with symplectic geometry and the theory of Hamiltonian group actions (as it is developped e.g.in “Symplectic techniques in physics” (1984; Zbl 0576.58012) and (1990; Zbl 0734.58005) by V. Guillemin and S. Sternberg), as well as with representation theory. In addition to its main subject, it contains a lot of interesting references to related topics as well as appendices on multiplicity formulas and equivariant cohomology. The authors also give a survey of recent developments.

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
53D50 Geometric quantization
81R40 Symmetry breaking in quantum theory
57S15 Compact Lie groups of differentiable transformations
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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