Projects per year
Abstract
We show how the modular representation theory of inner forms of general linear groups over a nonArchimedean local field can be brought to bear on the complex theory in a remar kable way. Let F be a nonArchimedean locally compact field of residue characteristic p, and let G be an inner form of the general linear group GL(n,F), n > 0. We consider the problem of describing explicitly the local Jacquet–Langlands correspondence π > JL(π) between the complex discrete series representations of G and GL(n,F), in terms of type theory. We show that the congruence properties of the local Jacquet–Langlands correspondence exhibited by A. Mínguez and the first named author give information about the explicit description of this correspondence. We prove that the problem of the invariance of the endoclass by the Jacquet–Langlands correspondence can be reduced to the case where the representations π and JL(π) are both cuspidal with torsion number 1. We also give an explicit description of the Jacquet–Langlands correspondence for all essentially tame discrete series representations of G, up to an unramified twist, in terms of admissible pairs, generalizing previous results by Bushnell and Henniart. In positive depth, our results are the first beyond the case where π and JL(π) are both cuspidal.
Original language  English 

Pages (fromto)  18531887 
Number of pages  38 
Journal  Compositio Mathematica 
Volume  155 
Issue number  10 
Early online date  27 Aug 2019 
DOIs  
Publication status  Published  Oct 2019 
Profiles

Shaun Stevens
 School of Mathematics  Professor of Mathematics
 Algebra and Combinatorics  Member
Person: Research Group Member, Academic, Teaching & Research
Projects
 1 Finished

Explicit Correspondences in Number Theory.
Engineering and Physical Sciences Research Council
31/03/10 → 30/03/15
Project: Fellowship