Some not-so-complicated representation theory of complex reflection groups

A complex reflection group $G$ is a (finite) subgroup of $GL_n(\mathbb{C})$ that is generated by pseudo-reflections; that is, $\mathbb{C}$-linear transformations that fix a hyperplane and have finite order. These groups generalize finite reflections groups, have been completely classified and have a very rich representation and invariant theory.

In the first lecture we'll discuss some numerical invariants associated to them (degrees and (co-)exponents), but more importantly we'll start to study their co-invariant algebra: It is defined as the quotient $S/\langle S^G_{+}\rangle$ of the polynomial algebra $S:=\mathbb{C}[x_1,\cdots ,x_n]$ by the (ideal generated by) positive degree invariant polynomials $S^G_{+}$. We will calculate its Poincare polynomial and show it carries the regular representation of the group $G$. We will even produce a/three natural combinatorial base(is) for it.

In the second lecture we will show that the exterior powers of the (natural from the inclusion $G\subset GL_n(\mathbb{C})$ ) reflection representation of $G$ are inequivalent irreducible modules and deduce more interpretations for the exponents. More importantly we will discuss some connections with Schubert calculus: In the case of a Weyl group $W$, the coinvariant algebra is $\mathbb{C}[W]$-isomorphic to the cohomology ring of the associated flag manifold $G/B$ (where $G$ is the corresponding Lie group and $B$ a Borel).