On the constituents of the mod  p cohomology of Shimura curves

Let p be a prime number and K a finite unramified extension of ℚ p . When p is large enough with respect to [ K : ℚ p ] , and under mild genericity assumptions, we proved in our previous work that the admissible smooth representations π of GL 2 ( K ) which occur in Hecke eigenspaces of the mod p cohomology of Shimura curves are of finite length. In this paper, we obtain various refined results about the structure of subquotients of π , such as their Iwahori-socle filtrations and K 1 -invariants, where K 1 is the principal congruence subgroup of GL 2 ( 𝒪 K ) . We also determine the Hilbert series of π as a Iwahori-representation under these conditions.