Let \(G\) be a quasisplit connected reductive group defined over a non-archimedean local field \(F\) with characteristic 0. Let \(B:= TU\) be a Borel subgroup of \(G\) with a maximal torus \(T\) and the unipotent part \(U\) and let \(P= MN\) be a maximal parabolic subgroup with the Langlands decomposition satisfying \(M\supset T\) and \(N\supset U\). Let \(\sigma\) be an irreducible unitary supercuspidal representation of \(M\) and let \(\nu\in {\mathfrak n}_{\mathbb{C}}^*\) be the complex dual of the real Lie algebra of the split component \(A\) of the center of \(M\). Let \(I(\nu, \sigma)\) be an induced representation form \((\nu, \sigma)\). We consider the standard intertwining operator \(A(\nu, \sigma, w_0)\) from \(I(\nu, \sigma)\) to \(I(w_0(\nu), w_0(\sigma))\) where \(w_0\) is the longest element of the Weyl group in the maximal split torus \(A_0\) of \(T\) in \(G\). Then we see that \(I(0,\sigma)\) is irreducible if and only if \(A(\nu, \sigma, w_0)\) has a pole at \(\nu=0\) provided that \(w_0(\sigma) \cong \sigma\).
The main result of this paper is to give a condition for the irreducibility of \(I(0, \sigma)\) by utilizing this property in the case that \(N\) is Abelian and the action of \(M\) to \({\mathfrak n}\) is prehomogeneous, i.e., \({\mathfrak n}\) has a finite number of open dense \(M\)-orbits whose union is dense in \({\mathfrak n}\).