One current area of research is to obtain results in the
representation theory of Lie algebras through geometric means via
varieties attached to quivers. These techniques allow one to
prove old results in unique ways as well as discover new results.
Quivers and varieties associated with them have a long history. A
quiver is simply an oriented graph.
A representation of a quiver consists of assigning a vector space to each vertex and a
linear map to each arrow. The study of the Jordan quiver
consisting of one vertex connected by an edge with itself
is simply the study of Jordan normal forms. Flag varieties and
Hilbert schemes of points on surfaces are also both examples of
quiver varieties. The field has recently flourished with the work
of Lusztig and Nakajima who defined the quiver varieties which
yield a geometric construction of universal enveloping algebras of
Lie algebras and their representations. Nakajima's quiver varieties are constructed
by taking the spaces of quiver representations, imposing various restrictions
and quotienting by a natural group action. One then defines natural
geometric operators on the homology of these varieties and this endows the homology
with the structure of a given heighest weight representation of the Kac-Moody
algebra whose Dynkin graph is the underlying graph of the quiver you started with.
In particular, we have a quiver variety for each weight space and the number of
irreducible components of this variety is equal to the dimension of the weight space.
For example, for the zero weight space of the adjoint representation of sln,
the corresponding quiver variety is a union of n-1 copies of the complex projective line
CP1. This corresponds to the fact that the zero weight space of the adjoint
representation (i.e. the Cartan subalgebra) has dimension n-1.
By considering the equivariant K-theory (instead of homology) of quiver varieties corresponding to Dynkin graphs of finite type, one can construct finite dimensional representations of affine Lie algebras. One can also obtain geometric constructions of quantum groups and their representations as well as crystal bases which are the q=0 limit of quantum groups.
The quiver varieties of Nakajima are an extension to more general type of the varieties of affine type which were introduced by Kronheimer and Nakajima in their description of Yang-Mills instantons on gravitational instantons. Gravitational instantons, or ALE spaces, are resolutions of singularities of the quotient of C2 by a finite subgroup of SL2(C). Nakajima quiver varieties provide a large class of examples of hyper-Kahler manifolds and thus the theory of quiver varieties also has connections to the field of symplectic geometry. One of the important results from the theory of quiver varieties is the definition of the canonical and semicanonical bases in universal enveloping algebras and their representations which have remarkable properties.
There are also various other geometric constructions in representation theory. For instance, Mirkovic and Vilonen have shown that certain subvarieties of the affine Grassmannian, called MV cycles, yield bases for representations of complex semisimple groups. This construction yields a proof of the geometric Satake correspondence and is related to the geometric Langlands program.
For more details on quiver varieties and representation theory, see an overview article I have written.
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