Gelfand-Kirillov dimensions and the p-adic Langlands program

This thesis is concerned with questions in the p-adic Langlands program, in particular those involving the group GL_2K, where K is a p-adic field and p>2. The first main result is that an admissible p-adic Banach representation of GL_2K whose locally analytic vectors admit an infinitesimal character has Gelfand-Kirillov dimension at most [K:Q_p]. This is an optimal improvement of an earlier bound of Dospinescu-Paškūnas-Schraen.
The second main result is a generalization of this theorem to (admissible) p-adic Banach representations of GL_2K in families with an ``infinitesimal character in families'' in the sense of Dospinescu-Paškūnas-Schraen. In this case it is the special fiber (that is, mod p) of the family that has bounded Gelfand-Kirillov dimension, by [K:Q_p]. This allows one to generalize the bound on the Gelfand-Kirillov dimension of smooth representations found in the cohomology of Shimura varities due to Breuil-Herzig-Hu-Morra-Schraen beyond the case of K unramified.
In this context and under some favorable hypotheses, this kind of bound leads to flatness of the patched module over the patched deformation ring, as observed by Gee-Newton. Another result of this thesis uses this flatness to compute the derived pro-p Iwahori invariants of the mod p representation of Breuil-Herzig-Hu-Morra-Schraen as a complex over the (non-derived) pro-p Iwahori Hecke algebra, up to quasi-isomorphism. This result shows in particular that its quasi-isomorphism class depends only on the local mod p Galois representation and not on the global choices made in the construction of the smooth mod p representation.
Finally, the methods used to get the Gelfand-Kirillov bounds point to an annihilation condition that was already studied by Breuil-Herzig-Hu-Morra-Schraen in the unramified case. The first result of this thesis concerns the observation that one can check whether a smooth mod p representation satisfies this condition after restriction to an arbitrary open subgroup.