Hecke modules and supersingular representations of U(2,1)

Abstract

Let F F be a nonarchimedean local field of odd residual characteristic p p . We classify finite-dimensional simple right modules for the pro- p p -Iwahori-Hecke algebra H C ( G , I ( 1 ) ) \mathcal {H}_C(G,I(1)) , where G G is the unramified unitary group U ( 2 , 1 ) ( E / F ) \textrm {U}(2,1)(E/F) in three variables. Using this description when C = F ¯ p C = \overline {\mathbb {F}}_p , we define supersingular Hecke modules and show that the functor of I ( 1 ) I(1) -invariants induces a bijection between irreducible nonsupersingular mod- p p representations of G G and nonsupersingular simple right H C ( G , I ( 1 ) ) \mathcal {H}_C(G,I(1)) -modules. We then use an argument of Paškūnas to construct supersingular representations of G G .