Richard Kirwan, F.R.S., 1733–1812

The Kirwan map , equivariant Kirwan maps , and their kernels Lisa

Consider a Hamiltonian action of a compact Lie group K on a compact symplectic manifold. We find descriptions of the kernel of the Kirwan map corresponding to a regular value of the moment map κK . We start with the case when K is a torus T : we determine the kernel of the equivariant Kirwan map (defined by Goldin in [Go]) corresponding to a generic circle S ⊂ T , and show how to recover from t...

The Kirwan map , equivariant Kirwan maps , and their kernels

Consider a Hamiltonian action of a compact Lie group K on a compact symplectic manifold. We find descriptions of the kernel of the Kirwan map corresponding to a regular value of the moment map κK . We start with the case when K is a torus T : we determine the kernel of the equivariant Kirwan map (defined by Goldin in [Go]) corresponding to a generic circle S ⊂ T , and show how to recover from t...

Elected FRS 1976

A Kirwan-ness Stratification for Loop Groups

We study a stratification of Kirwan-Ness type for the action of the loop group on the moduli space of framed bundles on a surface cut along a circle. The genus zero case is equivalent to Birkhoff decomposition of the loop group. The stratification is used to prove versions of Kähler quantization commutes with reduction and Kirwan surjectivity.

A Right Inverse of the Kirwan Map

We construct a right inverse of the Kirwan map for any Hamiltonian action of a compact torus T on a compact symplectic manifold. When T = S 1 this right inverse is canonical. For general T , it depends on a choice of a generic subgroup S 1 ⊂ T .