Contraction of Hamiltonian $K$-spaces

TL;DR

This paper introduces an intrinsic symplectic contraction for Hamiltonian spaces, connecting horospherical degenerations with symplectic reduction techniques, and applies it to representation theory and integrable systems.

Contribution

It formulates a new symplectic contraction map for Hamiltonian spaces, linking horospherical degenerations with symplectic geometry and representation theory.

Findings

The contraction map is surjective and a symplectomorphism on a dense open subset.

Gradient-Hamiltonian flow induces the contraction from general to special fibers.

Application to the Gel'fand-Tsetlin integrable system demonstrates its geometric origin.

Abstract

In the spirit of recent work of Harada-Kaveh and Nishinou-Nohara-Ueda, we study the symplectic geometry of Popov's horospherical degenerations of complex algebraic varieties with the action of a complex linearly reductive group. We formulate an intrinsic symplectic contraction of a Hamiltonian space, which is a surjective, continuous map onto a new Hamiltonian space that is a symplectomorphism on an explicitly defined dense open subspace. This map is given by a precise formula, using techniques from the theory of symplectic reduction and symplectic implosion. We then show, using the Vinberg monoid, that the gradient-Hamiltonian flow for a horospherical degeneration of an algebraic variety gives rise to this contraction from a general fiber to the special fiber. We apply this construction to branching problems in representation theory, and finally we show how the Gel'fand-Tsetlin integrable system can be understood to arise this way.