Weyl invariance

I haven’t blogged for a while due to various occupations of my time – I returned to America, moved into a new place, started the school year,… – however I plan to do so on a reasonably regular basis now that my routine is up and running.

I have my qual coming up in October and so most of my time is spent preparing for that at the moment. One of my topics is the representation theory of Lie groups and algebras. I was taught a course on Lie algebras by a finite group theorist who refused to cover any more representation theory than was necessary to classify the semisimple complex Lie algebras, and so my last week or so has been spent extending my rigorous knowledge of root decompositions to a similarly firm grasp of my intuition said had to be true for arbitrary representations of semisimple Lie algebras.

One of the aspects of the theory that I had underappreciated at the first time of learning was why one should expect Lie theoretic objects to be invariant under the action of the associated Weyl group. As my notes describe, the appearance of the Weyl group is totally unsurprising when considered from the right point of view.

The roots of a (semi)simple Lie algebra \mathfrak{g} come in +/- pairs and together produce a one dimensional subspace of the designated Cartan subalgebra as their commutator. The sum of these three 1-dimensional spaces is a subalgebra isomorphic to \mathfrak{sl}_2(\mathbb{C}). Now there is a lot of structure in the irreducible representations of \mathfrak{sl}_2(\mathbb{C})! In particular, their weights or eigenvalues are symmetric about zero and form a connected string. The symmetry manifests in the weights of a representation of \mathfrak{g} as symmetry about hyperplanes of the form \alpha^\perp for \alpha a root of \mathfrak{g} and the orthogonal taken with respect to the Killing form. In other words, reflections in these hyperplanes preserve weights of representations. What is the Weyl group? Exactly the group generated by these reflections.

This whole stream of theory has the effect of reducing many decomposition questions about representations of semisimple Lie algebras to combinatorial activities decomposing very symmetric polygons into other very symmetric polygons. I’ll describe some examples next time…