Universiteit Leiden Mathematisch Instituut
Geometry Seminar

Wednesday, 19 March 2003, 16:00 - 17:00, room 403

Mihai Sorin Stupariu: GIT-Quotients and symplectic reductions

Abstract: The aim of the talk is to describe links between quotients constructed using the Geometric Invariant Theory and certain quotients which arise in symplectic geometry. We will present some results of A. King concerning the case of a linear action of a complex reductive Lie group G on a finite dimensional vector space W. In this situation, any character of G yields a linearisation of the trivial line bundle over W and, phrasing Mumford's definition, one can introduce a suitable stability concept and construct a GIT-Quotient for the given action. On the other hand, considering a maximal compact subgroup K of G and endowing the vector space W with a Hermitian metric such that K acts by unitary transformations, one obtains a symplectic approach to the problem. Related to this approach are the concepts of moment map and symplectic reduction. We will briefly explain them and we will describe the (natural) bijection between the GIT-quotient and the symplectic reduction.