Representation Theory and Complex Geometry, By

Representation theory. Let us try to study a given object X through vector spaces V associated naturally to it. The symmetries S of X can form a group or more generally some kind of an algebra usually related to a group (Lie algebra, quantum group,...). The Vector space V inherits the symmetries of X, and they act on V by linear operators. This is what one means by “V is a representation of the structure S”. A representation theorist produces the list of all irreducible representations of S (those that do not admit a non-trivial S-invariant subspace, i.e., the simplest ones), studies irreducible representations in detail and reconstructs interesting representations V by gluing together the irreducible ones. Two sources of interest in representation theory are particle physics and the Langlands conjectures in number theory. While the traditional representation theorist is primarily an algebraist, most of the successes in the Langlands program use some geometric methods.