Geometry and Topology Seminar: "Equivariant differential operators and contact geometry"

Michael Deutsch, Universidade Federal do Rio de Janeiro

Abstract: Given a group G of automorphisms of a complex manifold M, effective methods to construct meromorphic functions on M which are invariant under the action of G are classical, dating back to the 19th century in the case of complex curves. In this talk we describe a method to produce a more general class of functions, those which are *equivariant* with respect to a projective representation of G. In the 1-dimensional case, we use hyperbolic geometry to extend an idea of Doyle and McMullen and obtain an equivariant analogue of the classical Schwarzian derivative. This construction leads via contact geometry to a kind of Cayley transform parametrizing the full monoid of equivariant differential operators, the action of which generates arbitrary equivariant functions from a given one. This in turn generalizes to algebra-valued functions in arbitrary dimension, yielding what appear to be the first examples of such functions in the multi-variable case when the image of G is non-compact.

Host: Michael Landry