Self-dual representations of division algebras and Weil groups: a contrast (Q2892854)

To a self-dual representation \((\rho, V)\) of a locally compact group, the authors define an invariant \(c(\rho)\) by setting \(c(\rho)=1\) (resp. \(c(\rho)=-1\)) if \(\rho\) fixes a non-degenerate symmetric (resp. alternating) bilinear form on \(V\). When the group is compact, this is also known as the Schur-Frobenius indicator of \(\rho\).NEWLINENEWLINEThe authors consider the case when \(k\) is a local field and \(G = GL_m(D)\), where \(D\) is a finite-dimensional division \(k\)-algebra with center \(k\). It is an inner form of \(G' = GL_n(k)\). Using the local Langlands correspondence for \(G'\) together with the Jacquet-Langlands correspondence, they obtain a bijection between discrete series representations of \(G\) and certain \(n\)-dimensional irreducible representations of \(W'_k\), the Weil-Deligne group of \(k\). Write this bijection as \(\pi \mapsto \sigma\); one would like to compare \(c(\pi)\) and \(c(\sigma)\).NEWLINENEWLINEThe main result (Theorem A) in this article asserts that \((-1)^m c(\pi) = (-1)^n c(\sigma)^m\). The idea of the proof is based on a local-global argument, together with a product formula (Theorem C) for the local invariants \(c(\cdots)\) of a self-dual automorphic representation. Roughly speaking, the globalization argument is based on works of Jiang and Soudry for generic representations [\textit{D. Jiang} and \textit{D. Soudry}, in: Contributions to automorphic forms, geometry, and number theory. Baltimore, MD: Johns Hopkins University Press. 457--519 (2004; Zbl 1062.11077)], while the ``local input'' comes from an explicit calculation of level-one representations of \(D^\times\). The authors remark that the main difficulty is the globalization of self-dual representations using Poincaré series; however, this obstacle may be removed by invoking Arthur's endoscopic classification for classical groups.NEWLINENEWLINEFinally, recall that the Schur-Frobenius indicator is related to the field of definition of a unitary representation. In the last section, the authors state interesting questions and conjectures on this aspect.