On the constituents of the mod $p$ cohomology of Shimura curves

Let $p$ be a prime number and $K$ a finite unramified extension of ${\mathbb{Q}}_p$. When $p$ is large enough with respect to $[K:{\mathbb{Q}}_p]$ and under mild genericity assumptions, we proved in our previous work that the admissible smooth representations $\pi$ of ${\mathrm{G}\mathrm{L}}_2(K)$ that occur in Hecke eigenspaces of the mod $p$ cohomology are of finite length. In this paper we obtain various refined results about the structure of subquotients of $\pi$, such as their Iwahori-socle filtrations and $K_1$-invariants, where $K_1$ is the principal congruence subgroup of ${\mathrm{G}\mathrm{L}}_2({\mathcal{O}}_K)$. We also determine the Hilbert series of $\pi$ as Iwahori-representation under these conditions.