Abstract

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 $n=mr$.

pdf

Additional Information

ISSN
1080-6377
Print ISSN
0002-9327
Pages
pp. 749-772
Launched on MUSE
2012-05-31
Open Access
No
Back To Top

This website uses cookies to ensure you get the best experience on our website. Without cookies your experience may not be seamless.