Représentations modulo p de GL(m,D), D algèbre à division sur un corps local

2014/09/16 by Ly, Tony
#20C08 #22E50 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1409.4686

Abstract

This Ph.D. thesis belongs to the realm of mod p representation theory of p-adic groups. The main object of study is the inner form of the general linear group GL(m,D) where D is a division algebra over a non-Archimedean local field. The first part focuses on the rank 1 case (m=2): it develops in detail the irreducibility criterion for parabolically induced representations and a classification of irreducible admissible smooth representations "à la Barthel-Livné". Then one turns to the generalized Steinberg representations: building on ideas of Grosse-Klönne and Herzig, this chapter manages to get a result that is valid for any reductive group over a non-Archimedean local field. Finally the last part exploits the two previous ones, and by overcoming some combinatorial difficulties that arise in the rank >1 case, one gets a classification "à la Herzig" for GL(3,D) and GL(m,D), m>3, with some technical assumptions on D.

Related

• On modular representations of inner forms of GL_n over a local non-archimedean field
• Distinction of the Steinberg representation for inner forms of GL(n)
• Smooth representations and Hecke algebras of p-adic GL_n(D)
• Extensions of mod p representations of division algebras over non-Archimedean local fields
• Types modulo ℓ pour les formes intérieures de GL_n sur un corps local non archimédien