| Abstract: 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 just
the zero section of the cotangent bundle. In general the size of the
characteristic variety exercises some control over the number of
solutions of the equation. 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,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: Leticia Barchini and Roger Zierau have recent deep papers
about special cases. 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. |