Modules universels de GL(3) sur un corps p-adique en caractéristique p

Rachel Ollivier (1, 2) , Vincent Sécherre (1)

1 LMV - Laboratoire de Mathématiques de Versailles
2 Columbia University [New York]

Let F be a p-adic field with residue class field k. We investigate the structure of certain mod p universal modules for GL(3,F) over the corresponding Hecke algebras. To this end, we first study the structure of some mod p universal modules for the finite group GL(n,k) as modules over the corresponding Hecke algebras. We then relate this finite case to the p-adic one by using homological coefficient systems on the the affine Bruhat-Tits building of GL(3). Suppose now that k has cardinality p. We prove that the mod p universal module of GL(3,F) relative to the Iwahori subroup is flat and projective over the Iwahori-Hecke algebra. When replacing the Iwahori subgroup of GL(3,F) by its pro-p-radical, we prove that the corresponding module is flat over the pro-p Iwahori-Hecke algebra if and only if p=2.

Mots clés: reductive p-adic group; building; homological coefficient system; universal module

Domaines: Théorie des représentations [math.RT]

Rachel Ollivier, Vincent Sécherre. Modules universels de GL(3) sur un corps p-adique en caractéristique p. 2011. ⟨hal-00593423⟩