Quadratic unipotent representations: \(p\)-adic classical groups (Q1922125)

In [Orbites unipotentes et représentations. II: Groupes \(p\)-adiques et réels. Astérisque 171-172, 13-72 (1989; Zbl 0728.22014)] \textit{J. Arthur} gave a conjectural description of the discrete spectrum attached to the automorphic forms on a general reductive group. The main qualitative feature was a Jordan decomposition into semisimple and unipotent constituents based on an extended Langlands correspondence. In the present paper this correspondence is given for unipotent representations of an orthogonal or symplectic group \(G\) over a \(p\)-adic field. The result is expressed in terms of the wave front set, an idea going back to \textit{N. Kawanaka} [Representation of finite groups, Proc. Conf., Arcata/CA 1986, Pt. 1. Proc. Symp. Pure Math. 47, 147-163 (1987; Zbl 0654.20046)]. The wave front set of a representation is a set of pairs consisting of a unipotent conjugacy class \(U\) in \(G\) and a \(G\)-equivariant system of irreducible representations of the groups \(\text{Cent}_G (u)\), \(u\in U\). The main theorem of the present paper states a bijection between the set of unipotent quadratic Arthur parameters and the set of \(G\)-representations with wave front set satisfying certain restrictions on the size of the Jordan blocks.