Selfdual representations of division algebras and Weil groups: A contrast

TL;DR

This paper investigates how the Langlands correspondence preserves the selfdual classification of representations, showing that for even n, symplectic and orthogonal types are exchanged between Weil groups and division algebra representations.

Contribution

It establishes a global method to analyze the behavior of the Langlands correspondence concerning selfdual representation types, especially for even dimensions.

Findings

For even n, symplectic Weil group representations correspond to orthogonal division algebra representations.

The results extend to GL_m(B) where B is a division algebra, generalizing previous cases.

The study clarifies the relationship between selfdual representation types under the Langlands correspondence.

Abstract

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.