Hecke eigenvalues in 𝑝-modular representations of unramified 𝑈(2,1)

Abstract

Let G G be the unramified unitary group U ( 2 , 1 ) ( E / F ) U(2, 1)(E/F) defined over a non-archimedean local field F F of odd residue characteristic p p , and let K K be a maximal compact open subgroup of G G . For an irreducible smooth F ¯ p \overline {\mathbf {F}}_p -representation π \pi of G G , and a weight σ \sigma of K K contained in π \pi , we prove that π \pi admits eigenvectors for the spherical Hecke algebra H ( K , σ ) \mathcal {H}(K, \sigma ) . Our approach is close to that of Barthel–Livné [Duke Math. J. 75 (1994), pp. 261–292] for G L 2 GL_2 . For G L n ( n ≥ 3 ) GL_n (n\geq 3) , we give a sketch of an essential case where the analogue of a crucial ingredient in loc.cit fails.