Let F be a nonarchimedean
locally compact field of residual characteristic p, let G be a connected reductive
F-group, and let K be a special parahoric subgroup of G(F). We choose a parabolic
F-subgroup P of G with Levi decomposition P = M@N in good position with
respect to K. Let C be an algebraically closed field of characteristic p, and V an
irreducible smooth C-representation of K. We investigate the natural intertwiner
from the compact induced representation c-IndKG(F)V to the parabolic induced
representation IndP(F)G(F)c-IndM(F)∩KM(F)VN(F)∩K. Under a regularity
condition on V , we show that the intertwiner becomes an isomorphism after
localization at a specific Hecke operator. When F has characteristic 0, G is F-split
and K is hyperspecial, the result was essentially proved by Herzig. We define the
notion of K-supersingularity for an irreducible smooth C-representation of G(F)
which extends Herzig’s definition for admissible irreducible representations and we
give a list of irreducible representations which are neither supercuspidal nor
K-supersingular.
Keywords
representations modulo p of
reductive p-adic groups,
compact induction, parabolic induction, Satake isomorphism