  Selfdual representations of any group fall into two classes when they are
irreducible: 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. Our results treat more generally the case of
GL_m(B), for B a division algebra over k of index r, and n=mr.
