The simplest geometric invariant of a differential equation $Df=0$ is its characteristic variety: the collection of zeros (in the cotangent bundle) of the principal symbol of $D$. Elliptic equations, which have few solutions, are those for which the characteristic variety is zero. In general the size of the characteristic variety exercises some control over the number of solutions.

For an infinite-dimensional representation of a reductive Lie group, there is a similar invariant, still called the characteristic variety, which offers a nice geometric picture of the representation. In the case of $GL(n,{\mathbb R})$, the characteristic variety is just a conjugacy class of nilpotent matrices: a partition of $n$.

Surprisingly, decades of powerful results about these representations have left us still unable to compute the characteristic variety. I'll explain what the characteristic variety is, why it is such a natural and powerful invariant of a representation, and how we're finally learning to calculate it in general. The little bits of new math here are joint work with Jeff Adams of University of Maryland.