Highest weight modules associated with classical irreducible regular prehomogeneous vector spaces of commutative parabolic type (Q1191040)

The purpose of this paper is to give realization of highest weight modules in terms of irreducible relative invariants of prehomogeneous vector spaces. Let \({\mathfrak g}\) be a classical complex simple Lie algebra. We assume that \({\mathfrak g}\) has a \(\mathbb{Z}\)-gradation of the form \({\mathfrak g}={\mathfrak g}(-1)+{\mathfrak g}(0)+{\mathfrak g}(1)\). Then the natural action of \({\mathfrak g}(0)\) on \({\mathfrak g}(\pm 1)\) makes an irreducible prehomogeneous vector space. Let \(d\lambda\) be a 1-dimensional representation of the parabolic subalgebra \({\mathfrak p}={\mathfrak g}(0)+{\mathfrak g}(1)\) and let \(\mathbb{C}_{d\lambda}\) be its representation space. Let \(U({\mathfrak g})\) and \(U({\mathfrak p})\) be the universal enveloping algebras of \({\mathfrak g}\) and \({\mathfrak p}\), respectively. We denote by \(V(d\lambda)\) the generalized Verma module induced from \(d\lambda:V(d\lambda)=U({\mathfrak g})\otimes_{U({\mathfrak p})}\mathbb{C}_{d\lambda}\) and by \(L(d\lambda)\) its irreducible quotient. Then the \(U({\mathfrak g})\)-module \(L(d\lambda)\) is realized using the irreducible relative invariant polynomial of the prehomogeneous vector space \(({\mathfrak g}(0),{\mathfrak g}(-1))\). As an application, we recover the reducibility criterion of \(V(d\lambda)\) and show that it has a natural interpretation in terms of the zeros of the \(b\)-function of the irreducible relative invariant.