Comparison of compact induction with parabolic induction

Let F be any non archimedean locally compact field of residual characteristic p, let G be any reductive connected F -group and let K be any special parahoric subgroup of G(F ). We choose a parabolic F -subgroup P of G with Levi decomposition P = MN in good position with respect to K. Let C be an algebraically closed field of characteristic p. We choose an irreducible smooth C-representation V of K. We investigate the natural intertwiner from the compact induced representation c-Ind G(F ) K V to the parabolically induced representation Ind G(F ) P (F )(c-Ind M(F ) M(F )∩K VN(F )∩K). Under a regularity condition on V , we show that the intertwiner becomes an isomorphism after a localisation 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-supersingular irreducible smooth C-representation of G(F ) which extends Herzig’s definition for admissible irreducible representations and we give a list of K-supersingular irreducible representations which are supercuspidal and conversely a list of supercuspidal representations which are K-supersingular.