Titel: Complex symplectomorphisms, Kähler geodesics and representation theory
Dieser Vortrag findet statt im Rahmen des Oberseminars "Algebraische Methoden der harmonischen Analysis"
Abstract: The geodesics for the Mabuchi metric on the space H of Kähler metrics on a compact symplectic manifold M correspond to solutions of a homogeneous complex Monge-Ampere (HCMA) equation. The space H is an infinite dimensional analogue of the symmetric spaces of noncompact type G_C/G for compact Lie groups G. In H the role of G is being played by the group of Hamiltonian symplectomorphisms.
We will describe a method for reducing the relevant Cauchy problem for the HCMA equation with analytic initial data to finding a related Hamiltonian flow followed by a "complexification".
For Hamiltonian G-spaces, with G-invariant Kähler structure, the geodesic corresponding to the norm square of the moment map or its Hamiltonian flow in imaginary time (= gradient flow for the changing metric following the geodesic) leads to the convergence of the holomorphic sections to sections supported on Bohr-Sommerfeld leaves.
For M=T*G, starting from the vertical or Schrödinger polarization, one obtains the Segal-Bargmann-Hall coherent state transform.