Self-dual representations of division algebras and Weil groups: A contrast

Irreducible selfdual representations of any group fall into two classes: those which carry a symmetric bilinear form, and the others which carry an alternating bilinear form. The Langlands correspondence, which matches the irreducible representations $\sigma$ of the Weil group of a local field $k$ of dimension $n$ with the irreducible representations $\pi$ of the invertible elements of a division algebra $D$ over $k$ of index $n$, takes selfdual representations to selfdual representations. In this paper we use global methods to study how the Langlands correspondence behaves relative to this distinction among selfdual representations. We prove in particular that for $n$ even, $\sigma$ is symplectic if and only if $\pi$ is orthogonal. More generally, we treat the case of ${\rm GL}_m(B)$, for $B$ a division algebra over $k$ of index $r$, and $n=mr$.