Reeder's conjecture on graded multiplicities in the exterior algebra

Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$, fix a Cartan subalgebra $\mathfrak{h}$, and let $\Phi$ be the associated root system with Weyl group $W$. Choose positive roots $\Phi^+$ and let $\rho$ be the corresponding Weyl vector. For a dominant weight $\lambda$, write $V_\lambda$ for the corresponding irreducible representation and $V_\lambda^0$ for its zero weight space. Let $H^h$ denote the degree-$h$ part of the space of $W$-harmonic polynomials on $\mathfrak{h}$. Define

$$P(V_\lambda,\bigwedge \mathfrak{g},u)=\sum_{n\geq 0}\dim\operatorname{Hom}_{\mathfrak{g}}(V_\lambda,\bigwedge^n\mathfrak{g})u^n,$$

and

$$P_W(V_\lambda^0,q,y)=\sum_{h,k\geq 0}\dim\operatorname{Hom}_W(V_\lambda^0,\bigwedge^k\mathfrak{h}\otimes H^h)q^hy^k.$$

A dominant weight $\lambda$ is small if it lies in the root lattice and $2\alpha$ is not smaller than $\lambda$ in the dominant order for every positive root $\alpha$. Reeder's conjecture. If $V_\lambda$ is a small representation, then

$$P(V_\lambda,\bigwedge \mathfrak{g},q)=P_W(V_\lambda^0,q^2,q).$$

Equivalently, the graded multiplicities of small representations in $\bigwedge\mathfrak{g}$ are determined by the Weyl-group representation on their zero weight spaces. The ungraded multiplicity is $2^{\operatorname{rk}\mathfrak{g}}\dim V_\lambda^0$. The conjecture is resolved in the paper for simple Lie algebras of type $D$ and for the exceptional cases, completing the case-by-case proof; the displayed equality is therefore solved in those cases, while the source does not assert a broader unresolved status.