Bernstein eigenvarieties

We construct parabolic analogues of (global) eigenvarieties, of patched eigenvarieties and of (local) trianguline varieties, that we call respectively Bernstein eigenvarieties, patched Bernstein eigenvarieties, and Bernstein paraboline varieties. We study the geometry of these rigid analytic spaces, in particular (generalizing results of Breuil-Hellmann-Schraen) we show that their local geometry can be described by certain algebraic schemes related to the generalized Grothendieck-Springer resolution. We deduce several local-global compatibility results, including a classicality result (with no trianguline assumption at $p$), and new cases towards the locally analytic socle conjecture of Breuil in the non-trianguline case.