Contragredient representations and characterizing the local Langlands correspondence

Contragredient representations and characterizing the local Langlands correspondence

It is surprising that the following question has not been addressed in the literature: what is the contragredient in terms of Langlands parameters? Thus suppose G is a connected, reductive algebraic group defined over a local field F , and G(F ) is its F -points. According to the local Langlands conjecture, associated to an admissible homomorphism φ from the WeilDeligne group of F into the L-gr...

On the Local Langlands Correspondence

The local Langlands correspondence for GL(n) of a non-Archimedean local field F parametrizes irreducible admissible representations of GL(n, F ) in terms of representations of the Weil-Deligne group WDF of F . The correspondence, whose existence for p-adic fields was proved in joint work of the author with R. Taylor, and then more simply by G. Henniart, is characterized by its preservation of s...

An introduction to the local Langlands correspondence

In these notes, based on my lectures at the FRG workshop on “Characters, Liftings, and Types” at American University in June 2012, I give an introduction to the conjectural Local Langlands Correspondence (LLC), for split semisimple groups over a nonarchimedean local field. This conjecture has been evolving over the past 45 years (with roots going back much further) to the point that today’s sta...

Local Langlands Correspondence in Rigid Families

We show that local-global compatibility (at split primes) away from p holds at all points of the p-adic eigenvariety of a definite n-variable unitary group. The novelty is we allow non-classical points, possibly non-étale over weight space. More precisely we interpolate the local Langlands correspondence for GLn across the eigenvariety by considering the fibers of its defining coherent sheaf. W...

Local geometric Langlands correspondence and representations of affine Kac - Moody algebras