Each extremal harmonic space is one-dimensional

By [L1] and [L3] the operator $\mathrm{\square }$ is diagonalizable on $C^q$ with eigenvalues $\frac{1}{2}(\mathrm{\parallel }\lambda +\rho {\mathrm{\parallel }}^2-\mathrm{\parallel }\mu +\rho {\mathrm{\parallel }}^2)$ on $C_{\mu }^q$, so $\ker \mathrm{\square }\cap C^q$ is the direct sum of the weight components $C_{\mu }^q$ with $\mathrm{\parallel }\mu +\rho \mathrm{\parallel }=\mathrm{\parallel }\lambda +\rho \mathrm{\parallel }$, that is, with $\mu \in W\cdot \lambda$ by the equality case of [L1].

For each $w\in W$ the component at $\mu =w\cdot \lambda$ contributes $C_{w\cdot \lambda }^q$, which by [L2] is $0$ unless $q=\mathrm{\ell }(w)$, and is the one-dimensional space $\mathbb{C}{\gamma }_w$ when $q=\mathrm{\ell }(w)$; the dot weights $w\cdot \lambda$ for distinct $w$ are distinct, so these contributions form a direct sum.

Summing the contributions of step 2.1 over all $w\in W$ gives $\ker \mathrm{\square }\cap C^q={⨁}_{\mathrm{\ell }(w)=q}\mathbb{C}{\gamma }_w$, each summand one-dimensional, so the dimension is the number of Weyl elements of length $q$; the identification with cohomology is [L4]. The case $q=0$ is included: ${\gamma }_1=v_{\lambda }$ spans the invariants, and for the zero Lie algebra all statements reduce to the single line $\mathbb{C}{\gamma }_1$ in degree zero.