Patching and multiplicities of p-adic eigenforms

We prove the existence of non-classical p-adic automorphic eigenforms associated to a classical system of eigenvalues on definite unitary groups in 3 variables. These eigenforms are associated to Galois representations which are crystalline but very critical at p. We use patching techniques related to the trianguline variety of local Galois representations and its local model. The new input is a comparison of the coherent sheaves appearing in the patching process with coherent sheaves on the Grothendieck--Springer version of the Steinberg variety given by a functor constructed by Bezrukavnikov.