Associated varieties of minimal highest weight modules (Q6639761)

Let \(\mathfrak g\) be a complex simple Lie algebra, \(\mathfrak h \subset \mathfrak b\) is a Cartan (Borel) subalgebra of \(\mathfrak g\) and \(\Delta\) the simple roots for the positive roots attached to \(\mathfrak b\). For \(\lambda \in \mathfrak h^\star\), \(L(\lambda)\) denotes the irreducible module of highest weight \(\lambda -\rho\). A simple \(\mathfrak g\)-module is called \textit{minimal} if the associated variety of its annihilator ideal coincides with the closure of the minimal nilpotent coadjoint orbit. The main result of this paper is a classification of minimal highest weight modules for \(\mathfrak g\). Moreover, the authors have determined the associated varieties of these modules. For \(\lambda \in \mathfrak h^\star\), set \(I_\lambda=\{ \alpha \in \Delta : \lambda(\alpha^\wedge)\in \mathbb Z_{>0} \}\). \N\NThe authors show: \N\Na) If \(L(\lambda)\) is a minimal highest weight module, then the associated variety to \(L(\lambda)\) is \(\bigcup_{\alpha \in \Delta /I_{\lambda}} \overline{Be_\alpha}\). Here, \(e_\alpha\) is a non-zero root vector associated to \(\alpha\). Furthermore, \(I_\lambda\) contains all the simple short roots. \N\Nb) Suppose \(\lambda\) is nonintegral. Then \(L(\lambda)\) is minimal if and only if the following conditions are satisfied: \N(1) the weight \(L(\lambda)\) is dominant regular. \N(2) the root system \(\Phi(\mathfrak g, \mathfrak h)\) has type \(A_n\), \(B_n\), \(C_n\), \(F_4\),or \(G_2\). \N(3) aligning with condition (2), the integral root system attached to \(\lambda\) has type \(A_{n-1}, B_1 \times B_{n-1}, D_n, C_4\) or \(A_2\) respectively. \N\Nc) Suppose \(\lambda\) is integral. Let \(\gamma_1, \dots, \gamma_m\) its linked sequence. Then \(L(\lambda)\) is minimal if and only if the following conditions are satisfied: \N(1) \(\lambda\) is regular or semiregular. \N(2) \(S_{\gamma_m}\cdots S_{\gamma_1}\lambda\) is dominant. \N(3) \(\Phi(\mathfrak g, \mathfrak h)\) is of type \(A,D, or E\).