Invariant theory of little adjoint modules (Q2905192)

Let \(G\) be a complex simple group with Lie algebra \(\mathfrak g\). Fixing a triangular decomposition and the relevant root system objects, let \(\theta_s\) be the short dominant root and \(V_{\theta_s}\) the simple \(G-\) module with highest weight \(\theta_s\), called little adjoint. In the paper under review, the author studies the invariant theoretic properties of \(V_{\theta_s}\), only some of them being presented below.NEWLINENEWLINELet \(\Pi_s\) be the set of short simple roots, \(W(\Pi_s)\) the subgroup of the Weyl group generated by the corresponding reflections and \(V^0_{\theta_s}\) the zero weight space of \(V_{\theta_s}\). It is then proved, among others, that \(\dim V^0_{\theta_s}=\#\Pi_s\) and \(\mathbb{C}[V_{\theta_s}]^G\simeq \mathbb{C}[V^0_{\theta_s}]^{W(\Pi_s)}\). The author also shows that the orbit of highest weight vectors in \(V_{\theta_s}\) is of dimension \(2\mathrm{ht}(\theta_s)\) and \(\dim V_{\theta_s}=(h+1)\cdot\#\Pi_s\), \(h\) being the Coxeter number of \(G\). Let also \(\mathfrak{l}:=\mathfrak g(\Pi_s)\subset\mathfrak g\) be the semisimple subalgebra with \(\Pi_s\) as its set of simple roots, and \(L\) the corresponding group. The author proves an isomorphism \(\mathbb{C}[V_{\theta_s}]^G\simeq \mathbb{C}[\mathfrak{l}]^L\) and finishes proving case-by-case that if \(\mathfrak{R}(\mathfrak{l})\) is the set of nilpotent elements of \(\mathfrak{l}\) and \(\mathcal{O}\) is an \(L-\) orbit, then the map \(\mathcal{O}\mapsto G\cdot\mathcal{O}\) is a bijection between \(\mathfrak{R}(\mathfrak{l})\) and \(\mathfrak{R}(V_{\theta_s})\).