Linearization of Poisson actions and singular values of matrix products

Abstract

We prove that the linearization functor from the category of Hamiltonian K-actions with group-valued moment maps in the sense of Lu, to the category of ordinary Hamiltonian K-actions, preserves products up to symplectic isomorphism. As an application, we give a new proof of the Thompson conjecture on singular values of matrix products and extend this result to the case of real matrices. We give a formula for the Liouville volume of these spaces and obtain from it a hyperbolic version of the Duflo isomorphism.

Decomposable representations and Lagrangian submanifolds of moduli spaces associated to surface groups

“…It was first proved by Klyachko in [19] and then proved using symplectic geometry in [3]. Statements (iii) and (iv) are real versions of (i) and (ii), respectively.…”

“…As we have seen in section 5, this ultimately relies on the real convexity theorem for group-valued momentum maps 5.1, just as the proof of theorem 7.1 given in [3] ultimately relies on a real convexity theorem for Lie-algebra-valued momentum maps. Finally, as for Thompson's problem, precise conditions on the λ j above for statement (i) to be true are already known (we refer for instance to [1]): these conditions are linear inequalities in the λ j .…”

“…Statements (iii) and (iv) are real versions of (i) and (ii), respectively. We refer to [3] to see how the equivalence of (ii) of (iv) relies on a real convexity theorem for Lie-algebravalued momentum maps. For a thorough treatment of the Lie-theoretic approach to Thompson's conjecture, we refer to the work of Evens and Lu in [10].…”

“…The result we will obtain there (theorem 7.2) can be thought of as an analogue of Thompson's problem in the case of a compact connected Lie group. We will underline the analogy with the symplectic geometry of Thompson's problem as it is treated in [3]. Other examples of Lagrangian submanifolds of representation spaces are given in [14] and [16].…”

Lie theory and the Chern–Weil homomorphism

“…Another example is obtained by adjoining d as an odd element, defining a semi-direct product (4) Fd ⋉ g where the action of d on g is as the differential, dξ = ξ, dξ = 0. (Note that a g − ds can be defined equivalently as a module for the super Lie algebra (4); this is the point of view taken in the book [18].)…”