Irreducibility of standard representations for Iwahori-spherical representations.
Let \(\mathbb G\) be a split connected reductive group over a local nonarchimedean field \(F\) of characteristic zero, and let \(\mathbb B = \mathbb T \mathbb U\) be a Borel subgroup of \(\mathbb G\), where \(\mathbb T\) is a maximal split torus of \(\mathbb B\) and \(\mathbb U\) is its unipotent radical. Let \(\mathbb P = \mathbb M \mathbb N\) be a Levi decomposition of a parabolic subgroup \(\mathbb P\) of \(\mathbb G\) with \(\mathbb T \subset \mathbb M\) and \(\mathbb N \subset \mathbb U\), and let \(\sigma\) be an Iwahori-spherical tempered generic representation of \(M = \mathbb M (F)\). Fix an element \(\nu\) of the complex dual \(\mathfrak a^\ast_{\mathbb C}\) of the real Lie algebra of the split component \(\mathbb A\) of \(\mathbb M\) in the positive Weyl chamber, and denote by \(I(\nu,\sigma)\) the standard representation induced from \(\nu\) and \(\sigma\). In this paper the authors prove that \(I(\nu, \sigma)\) is irreducible if and only if the Langlands quotient \(J(\nu, \sigma)\) of \(I(\nu, \sigma)\) is generic. They also show that points of reducibility can be computed explicitly.
scientific article; zbMATH DE number 1221879